SPECTROSCOPY OF ASSOCIATING SYSTEMS: LEVERAGING MOLECULAR INSIGHT
TO IMPROVE THERMODYNAMIC MODELING
By
William George Killian
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Chemical Engineering – Doctor of Philosophy
Chemistry – Dual Major
2022
ABSTRACT
Revitalized interest in biorenewable materials has revealed some accompanying
challenges. For example, many compounds of interest, such as alcohols, are polar and readily self-
associate, causing them to behave in a non-ideal manner. Equations of state (EOSs) such as the
statistical associating fluid theory (SAFT), cubic plus association (CPA), and Elliot-Suresh-
Donohue (ESD) are attractive options for modeling because they explicitly account for hydrogen
bonding. However, these EOSs are typically parameterized by fitting macroscopic pressure-
volume-temperature data, a practice that ignores molecular measurements of the bonding.
Advancing the predictive power of thermodynamic models for polar systems requires molecular-
level awareness, which can be provided by spectroscopy.
This work implements variable-temperature infrared spectroscopy guided by insight from
computational quantum mechanics to quantify the extent of hydrogen bonding in alcohol +
cyclohexane systems based on the alcohol’s hydroxyl stretching vibration. A new scaling
technique is developed that provides for the first time a temperature-independent integrated area
for the hydroxyl stretching region. For further validation of the new scaling method, the scaled
infrared spectra are correlated to the nuclear magnetic resonance spectra for 1-butanol +
cyclohexane and 2-propanol + cyclohexane using quantum calculations with minor empirical
adjustments. The infrared measurements are used to parameterize two association constants for
each binary system, which are implemented in a new activity coefficient model based on the
resummed form of Wertheim’s perturbation theory (RTPT). The widely used implementation of
one association parameter for each binary (TPT-1) in PC-SAFT, CPA, and ESD is shown to be
inadequate for fitting the spectroscopic data.
The RTPT model succeeds in recovering the hydroxyl bond type distributions from the
infrared measurements. When the association constants from spectroscopy are applied to the
modeling of phase equilibria, association is demonstrated to be the dominant contribution to
solution non-ideality. When combined with combinatorial and residual models, RTPT provides an
improved representation of experimental phase equilibria and excess enthalpies when compared
to the TPT-1 model.
TABLE OF CONTENTS
CHAPTER 1: Introduction.............................................................................................................. 1
1.1 Significance and Overarching Objective ............................................................................ 1
1.2 Association: A Molecular Approach .................................................................................. 2
1.3 Measuring Association: Infrared Spectroscopy .................................................................. 5
1.4 Modeling Association ......................................................................................................... 9
1.5 Overview of this work ...................................................................................................... 13
CHAPTER 2: A MATLAB® Application for Calculation of Cell Pathlength in
Absorption/Transmission Infrared Spectroscopy ......................................................................... 15
2.1 Preface............................................................................................................................... 15
2.2 Publisher Permission......................................................................................................... 15
2.3 Introduction ....................................................................................................................... 15
2.4 Pathlength Measurement Using the Interference Pattern.................................................. 17
2.5 The Pathlength Application .............................................................................................. 18
2.6 Summary and Conclusions ............................................................................................... 20
CHAPTER 3: Quantitative Analysis of Infrared Spectra of Binary Alcohol + Cyclohexane
Solutions with Quantum Chemical Calculations .......................................................................... 21
3.1 Preface............................................................................................................................... 21
3.2 Publisher Permission......................................................................................................... 21
3.3 Introduction ....................................................................................................................... 22
3.4 Analysis of the O-H Stretching Band ............................................................................... 25
3.5 Motivation ......................................................................................................................... 25
3.6 Computational Methods .................................................................................................... 28
3.7 Experimental Methods ...................................................................................................... 33
3.8 Results and Discussion ..................................................................................................... 35
3.9 Summary and Conclusions ............................................................................................... 51
3.10 Acknowledgements ........................................................................................................... 52
CHAPTER 4: Parameterization of a RTPT Association Activity Coefficient Model using
Spectroscopic Data........................................................................................................................ 53
4.1 Preface............................................................................................................................... 53
4.2 Publisher Permission......................................................................................................... 53
4.3 Introduction ....................................................................................................................... 54
4.4 Background ....................................................................................................................... 55
4.5 Methods and Derivations .................................................................................................. 61
4.6 Results and Discussion ..................................................................................................... 71
4.7 Summary and Conclusions ............................................................................................... 87
4.8 Acknowledgements ........................................................................................................... 88
CHAPTER 5: Infrared Quantification of Ethanol and 1-Butanol Hydrogen Bonded Hydroxyl
Distributions in Cyclohexane ........................................................................................................ 89
5.1 Preface............................................................................................................................... 89
5.2 Publisher Permission......................................................................................................... 89
5.3 Introduction ....................................................................................................................... 90
iv
5.4 Background ....................................................................................................................... 91
5.5 Methods............................................................................................................................. 94
5.6 Results and Discussion ................................................................................................... 100
5.7 Summary and Conclusions ............................................................................................. 119
5.8 Acknowledgments........................................................................................................... 120
CHAPTER 6: Modeling Phase Equilibria Using Infrared Spectroscopy ................................... 122
6.1 Introduction ..................................................................................................................... 122
6.2 Background ..................................................................................................................... 124
6.3 Methods and Modeling ................................................................................................... 125
6.4 Results and Discussion ................................................................................................... 133
6.5 Summary and Conclusions ............................................................................................. 154
CHAPTER 7: Relation of Hydroxyl NMR Chemical Shift to Infrared Vibrational Frequency . 157
7.1 Introduction ..................................................................................................................... 157
7.2 Background ..................................................................................................................... 158
7.3 Methods........................................................................................................................... 159
7.4 Results ............................................................................................................................. 162
7.5 Discussion ....................................................................................................................... 162
7.6 Conclusions ..................................................................................................................... 165
CHAPTER 8: Densities of Selected Deuterated Solvents .......................................................... 166
8.1 Preface............................................................................................................................. 166
8.2 Publisher Permission....................................................................................................... 166
8.3 Introduction ..................................................................................................................... 166
8.4 Experimental Methods .................................................................................................... 167
8.5 Results ............................................................................................................................. 171
8.6 Discussion ....................................................................................................................... 174
8.7 Summary and Conclusions ............................................................................................. 185
8.8 Acknowledgements ......................................................................................................... 185
CHAPTER 9: Conclusions and Future Directions ...................................................................... 187
9.1 Conclusions ..................................................................................................................... 187
9.2 Future work ..................................................................................................................... 188
BIBLIOGRAPHY ....................................................................................................................... 191
APPENDIX A: Detailed Summary of Attenuation Coefficient Function .................................. 210
APPENDIX B: Processing of Spectra ........................................................................................ 213
APPENDIX C: Relation of RTPT to Kretschmer-Wiebe ........................................................... 215
APPENDIX D: Conversion of Extensive Helmholtz Energy to Molar ...................................... 219
APPENDIX E: Key Material Balance Equations ....................................................................... 223
APPENDIX F: Excess Helmholtz Energy .................................................................................. 226
v
APPENDIX G: Activity Coefficients ......................................................................................... 227
APPENDIX H: Regression Flow Diagram ................................................................................. 232
APPENDIX I: Individual Isotherm Regression (Stage-1) .......................................................... 233
APPENDIX J: Scaling Parameters for Ethanol and 1-Butanol .................................................. 239
APPENDIX K: Parity Plots ........................................................................................................ 240
APPENDIX L: Hydroxyl Populations ........................................................................................ 241
APPENDIX M: Hydroxyl Fractions ........................................................................................... 242
APPENDIX N: Limiting Activity Coefficient Regressions ....................................................... 243
APPENDIX O: Mapping RTPT onto TPT-1 .............................................................................. 245
APPENDIX P: Phase Equilibria and Excess Enthalpy ............................................................... 247
APPENDIX Q: Contributions to the Excess Enthalpy at 318.15 K ........................................... 250
APPENDIX R: Attenuation Function Parameters ...................................................................... 252
APPENDIX S: Tabulated XA Values ......................................................................................... 253
APPENDIX T: Excess Volume Comparison .............................................................................. 258
APPENDIX U: Calculated Thermal Expansivities of Perdeutero Compounds.......................... 259
APPENDIX V: Effect of Method for Determining Isobaric Thermal Expansivity .................... 261
APPENDIX W: Protiated Molar Density Regression Coefficients ............................................ 264
APPENDIX X: Uncertainty Analysis Equations ........................................................................ 265
APPENDIX Y: Uncertainty Analysis Data ................................................................................ 268
vi
CHAPTER 1: Introduction
1.1 Significance and Overarching Objective
Hydrogen bonding is a complicated molecular phenomenon which affects macroscopic
vapor pressure and thermodynamic mixture properties. While not as strong as a covalent bond, the
cumulative molecular attractive effects of hydrogen bonding are significant and result in non-ideal
behavior at the macroscopic level. A unifying thermodynamic theory capable of bridging the
microscopic and macroscopic length scales remains elusive. The challenge of leveraging a
molecular understanding of association for the prediction of bulk properties has created an
intellectual gap between fundamental chemistry and engineering applications.
This failure to reconcile molecular phenomena with macroscopic properties extends
beyond the laboratory, affecting industry as well. Computer simulators used to design and operate
separations processes struggle to model vapor-liquid equilibrium data for associating systems
accurately.1 Furthermore, the thermodynamic models underpinning these simulators suffer from
limited predictive power, due in part to their reliance on fitted parameters which lack a sufficient
connection to the fundamental chemistry occurring in these systems. This limitation is especially
unsettling for industry, considering that these processes are often highly energy and financially
intensive, constituting up to 33% of industrial operating costs.2
Because of this, hydrogen bonding is deserving of study not merely to expand fundamental
understanding at the molecular level, but also with the intent of improving thermodynamic
modeling capabilities. In short, a firm understanding of the molecular level is necessary to make
meaningful and informed predictions of macroscopic properties.
1
1.2 Association: A Molecular Approach
In its most basic form, hydrogen bonding is a short-range and directional interaction that
originates from the covalent bond between a proton and electronegative heteroatom. The
polarization due to the large difference in electronegativity of the atoms results in the hydrogen
atom having a partial positive charge and thereby facilitating interactions with neighboring
molecules with electronegative atoms or unsaturated bonds.3 Hydrogen bonds are characterized by
the angle created by the proton and the two electronegative atoms on either side. Stronger hydrogen
bonds are correlated with increased hydrogen bond linearity.4
Hydrogen bonding can occur within the same molecule (intramolecular) or between
neighboring molecules (intermolecular). When molecules hydrogen bond intermolecularly with
other molecules of the same species, it is referred to as self-association as opposed to cross-
association, which occurs between dissimilar species. Cross-association is also described as
‘solvation’ in the engineering literature, but the term has become less favored recently to avoid
confusion with the other meanings of the term. Compounds can also be classified by their degree
of participation in hydrogen-bonding. Examples are provided in Table 1-1. Hydrogen-bond-donors
contain a proton that is covalently bound to an electronegative atom such as oxygen, whereas
hydrogen-bond-acceptors typically contain electron lone pairs or 𝜋-electrons.
2
Table 1-1: Hydrogen bonding classifications (adapted from Pimentel and McClellan5).
Type Description Examples
molecules with one or more
haloforms, highly halogenated compounds,
I donor groups and no acceptor
acetylenes, protonated amines, heteroaromatics
groups
molecules with one or more
ketones, ethers, esters, olefins, aromatics,
II acceptor groups and no donor
tertiary amines, nitriles, isonitriles
groups
water, alcohols, phenols, inorganic and
molecules with both donor
III carboxylic acids, primary and secondary
groups and acceptor groups
amines
molecules with neither donor saturated hydrocarbons, carbon tetrachloride,
IV
nor acceptor groups carbon disulfide
For self-associating molecules of Type III, such as alcohols, hydrogen bond types can be
classified according to their participation in hydrogen bond networks. Literature has identified the
following hydroxyl types, which are presented in Figure 1:6–8 Alpha hydroxyls (𝛼) correspond to
unassociated monomers. Beta hydroxyl sites (𝛽) participate in hydrogen bonding by accepting a
hydrogen bond through the oxygen lone pairs. Like the isolated alpha hydroxyls, beta sites have
an unassociated O-H bond. Gamma hydroxyls (𝛾) participate only as hydrogen bond donors and
remain unassociated at the oxygen. Delta hydroxyls (𝛿) participate both as hydrogen bond donors
and acceptors. Less common are eta hydroxyls (𝜂) and zeta hydroxyls (𝜁) both of which accept
hydrogen bonds on each of the lone pairs. However, they differ in that eta bonds also donate a
hydrogen bond at the proton.
3
Figure 1-1: Specific hydrogen bonding environments for alcohols and their corresponding
covalent bonds. Alpha hydroxyl sites (α-black) are associated with alcohol monomers. Beta
sites (β-blue) and gamma sites (γ-red) are located on the ends of polymeric chains. Beta
receives a hydrogen bond whereas a gamma O-H donates a hydrogen bond. Delta hydroxyls
(δ-green) receive and donate a hydrogen bond. Eta hydroxyls (η-purple) and zeta hydroxyls (ζ-
orange) each receive two hydrogen bonds; however, eta differs in that it also donates a
hydrogen bond.
Because of their ability to donate and receive hydrogen bonds, self-associating liquids are capable
of forming complex frameworks including linear chains,9 rings,10 coils,11 and supramolecular
structures.12,13 The population of specific oligomers depends on the molecular structure,
temperature and pressure, as well as the concentration.14,15 For alcohols, the most notable structure
is the dimer, which early literature deemed unique from other oligomerizations. This consideration
of uniqueness was rooted in the idea that the dimer was likely a cyclic structure whose polarized
electron pairs enhanced the strength of subsequent additions.16,17 Van Ness et al. proposed that
cyclic oligomerizations beyond that of the dimer were unlikely due to the fast exchange rate of
hydrogen bonds,14 which occur in the order of 10-12 s as reported by pump-probe infrared
spectroscopic measurements.18,19
Debate remains as to the prominence of linear versus cyclic structures.20 Infrared
experiments on ethanol in carbon tetrachloride conducted by Schwager et al. suggested that linear
4
structures predominate.21 More recent spectroscopic work performed on ethanol + cyclohexane
has suggested that cyclic trimers and tetramers dominate at higher alcohol concentrations.22
Viscosity measurements provide a conflicting description and instead suggest that cyclic structures
predominate in alcohols.23 Those findings contradict the results of some molecular dynamics
calculations which support chain polymerization for primary alcohols.24,25 Lack of a consensus
indicates that experiments directed at unraveling alcohol speciation could provide meaningful
clarity.
1.3 Measuring Association: Infrared Spectroscopy
1.3.1 Light and the Infrared Region
Light is an optical phenomenon possessing the characteristics of a wave and a particle, also
known as a photon. We now understand that the visible light we perceive constitutes only a fraction
of the entire electromagnetic spectrum, which spans from ultra-high-frequency gamma rays to
low-frequency radio waves. Light is a powerful tool for the interpretation of molecular behavior.
Of particular interest are light frequencies in the infrared region of the electromagnetic spectrum.
This region is further subdivided into the far-infrared (14.3-50 μm), mid-infrared (2.5-15 μm), and
near-infrared (0.7-2.5 μm). Light absorption from the far-infrared spectrum induces excitation in
the rotational states. In contrast, radiation from the mid and near-infrared region is responsible for
the vibrational excitation of covalently bonded groups of electrons.26 This work will focus our
attention on light from the electromagnetic spectrum's mid-infrared (fundamental infrared) region.
The near infrared region corresponds to overtones of the fundamental vibrations.
1.3.2 Molecular Vibrations and Infrared Absorption
From classical electrodynamics, it is understood that for a molecule to absorb radiation,
two criteria must be satisfied: (1) the frequency of the vibrational oscillation must exactly match
5
that of the incoming radiation, and (2) the vibration must be accompanied by a change in either or
both the magnitude or the direction of the dipole moment.27,28
The first criterion is necessary for a chemical bond to interact with the irradiating light.
Like many molecular phenomena, interatomic vibrations are quantized. Absorption of a photon of
the correct energy excites a vibrational mode in a molecule from the ground state to a higher
vibrational quantum state. Excitation requires a photon of the same frequency as the chemical
bond's vibration which therefore bridges the gap between the vibrational quantum levels matches
exactly according to Bohr’s frequency condition Δ𝐸 = ℎ𝑐𝜈̃. The light that misses this resonant
condition is transmitted unchanged.
The second condition is slightly more nuanced than the first. The vibration of the bond
creates a change in the molecular dipole moment and produces an alternating electric field that
fluctuates at a frequency equal to the vibrational frequency. During vibration, coupling occurs
between changes in the charge distribution of the bond and the oscillation of the incident infrared
electromagnetic field.29 This coupling allows the photons to transfer energy to the vibration
increasing its amplitude. The intensity of that absorbance is directly proportional to the rate of
change of the electrostatic dipole moment with the oscillatory amplitude; in general, the more
significant the gradient in the dipole moment with amplitude, the more intense the absorbance.
This effect is readily observed for the hydrogen bonding of the hydroxyl group which introduces
a noticeable change in the dipole moment.30
1.3.3 The Beer-Lambert-Bouguer Law
A quantitative understanding of the associating species is necessary for a complete
description of hydrogen bonding and careful parameterization of thermodynamic models. To
quantitatively utilize the absorbance spectrum of a chemical compound we need to correlate the
6
concentration of a specific type of chemical bond with its absorbance signal. The Beer-Lambert-
Bouguer law (BLBL) provides such a relationship.
Consider a sample container with parallel faces which contains an absorbing species that
is being irradiated by a monochromatic source. If we neglect reflective losses at the interfacial
surfaces due to differences in the refractive index, it is evident that the intensity of the incident
beam (𝐼0 ) will decrease as it proceeds through the sample. The reduction in the number of photons
passing through a differential slice of the sample per unit time is proportional to the number of
photons available for absorption and the concentration of the absorbing species. The change in
radiant power of monochromatic radiation (𝑑𝑃), which is absorbed at a specific level (𝑃), can be
related to the number of absorbing molecules in a slice of the sample (𝑑𝑎) by a proportionality
factor (𝑘) via Eq. 1-1.
𝑑𝑃
= −𝑘𝑃 Eq. 1-1
𝑑𝑎
Rearrangement and applying the limits of integration produces Eq. 1-2 and Eq. 1-3, respectively.
𝑃 𝑁
𝑑𝑃
∫ = −𝑘 ∫ 𝑑𝑎 Eq. 1-2
𝑃0 𝑃 0
where:
𝑃0 = 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑎𝑛𝑡 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛
𝑁 = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑏𝑠𝑜𝑟𝑏𝑖𝑛𝑔 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑏𝑒𝑎𝑚 𝑝𝑒𝑟 𝑎𝑟𝑒𝑎
𝑃
ln = −𝑘𝑁 Eq. 1-3
𝑃0
Removing the negative sign by inverting the logarithmic term leads to Eq. 1-4.
𝑃0
ln = 𝑘𝐶𝑙 Eq. 1-4
𝑃
7
Converting from natural to common logarithms facilitates quicker quantification of changes in
𝑑𝑚3
orders of magnitude. In this form, 𝑘 is replaced with the molar attenuation coefficient (𝜖, [𝑚𝑜𝑙⋅𝑐𝑚])
𝑃0
and the ratio of log is defined as the absorbance (𝐴, [𝐴. 𝑈. ]) resulting in the BLBL expression
𝑃
(Eq. 1-5).
𝑃0
log = 𝐴 = 𝜖𝐶𝑙 Eq. 1-5
𝑃
The power of this relationship lies in its simplicity and applicability; however, quantification of
the absorbing species concentration at a particular wavenumber relies on knowledge of the sample
path length and the molar attenuation coefficient. While the path length is constant for a given
measurement, there is a variation in the molar attenuation coefficient as a function of wavenumber.
The ability of a particular covalent bond-type to absorb radiation of a certain wavelength
is an intrinsic property and its propensity to vibrational excitation is expressed in the molar
attenuation coefficient. Though peak height was originally used to quantify the molar attenuation
coefficient, more recent work has suggested that peak area is a more accurate means of establishing
the relationship between absorbance and concentration. This is straightforward for isolated
vibrations, but the process becomes more nuanced when vibrational resonances have significant
overlap, as in the case of hydrogen bonding involving hydroxyl groups. This issue will be
discussed in more detail in future chapters. To summarize: Uncertainty in the attenuation
coefficient has hampered quantification efforts directed at alcohol hydrogen bonding and, until
recently, has impeded meaningful progress in the area.
The molar attenuation coefficient is easily determined for materials that do not self-
associate by plotting their absorbance versus concentration at a constant pathlength. The slope of
such a plot is taken as the molar attenuation coefficient for that material. In the case of self-
8
associating materials, such as alcohols, the quantification process is not as straightforward.
Hydrogen-bonding strongly influences the electronic character of covalent bonds, causing large
changes in their transition dipole moments.31 For alcohols, this manifests as broadening and
redshifting of the hydroxyl stretching frequency, accompanied by an increase in absorbance.14,32
The difficulty in describing the wavenumber dependence of the molar attenuation
coefficient have limited accurate quantification of the hydroxyl region. Prior works have attempted
to circumvent this issue by assuming molar attenuation coefficients are constant for all oligomers.
However, quantum calculations and numerous observations suggest that this assumption is likely
incorrect.33 A more fruitful method was described by Asprion et al., where equilibrium constants
were simultaneously determined in conjunction with the molar attenuation coefficients for the
monomer, dimer, and polymer, under the assumption of a 1,2-n association model.34 However, the
work herein shows that the dimer/polymer attribution of attenuation coefficients is flawed and
develops a new method for quantification of hydrogen bonds and the distribution of hydrogen bond
configurations.
1.4 Modeling Association
1.4.1 Activity Coefficients
Phase behavior is governed by the Gibbs energy, 𝐺, which is a combination of enthalpy,
𝐻, and entropy, 𝑆, namely 𝐺 = 𝐻 − 𝑇𝑆. Hydrogen-bonding, much like a chemical reaction,
contributes to both the enthalpy and entropy. The behavior of species in a mixture is determined
by the chemical potential, which is a defined quantity determined by the composition derivative
of the extensive Gibbs energy of a system. Analogous to potential energy for mechanical driving
forces, chemical potential is an indicator of the direction of change that a system will take based
on chemical driving forces. Since chemical potential is a gradient quantity, it is important to
9
understand that the chemical potential exists relative to a standard state, 𝜇𝐴𝑜 . The standard state is
selected by the practitioner, though process design conventions typically use the pure fluid at the
same temperature and pressure as the mixture. Integration of the chemical potential from the
standard state to the mixture state is quantified by the activity of the species in solution, 𝑎𝑖 = 𝑥𝑖 𝛾𝑖 ,
where 𝑥𝑖 is the mole fraction of the component and 𝛾𝑖 is the corresponding activity coefficient.
(𝜇𝑖 − 𝜇𝑖𝑜 )/(𝑅𝑇) = ln 𝑥𝑖 𝛾𝑖 Eq. 1-6
The activity acts as a unitless “correction” measure of the chemical potential relative to the
pure component when the pure standard state is used. For an ideal solution relative to the pure
component standard state the mole fraction is the activity (𝛾𝑖 = 1). The Gibbs energy of mixing is
the Gibbs energy of the mixture relative to the mole fraction weighted Gibbs energies of the
𝑖𝑠
components. For an ideal solution, the Gibbs energy of mixing (Δ𝐺𝑚𝑖𝑥 ) is
𝑖𝑠
Δ𝐺𝑚𝑖𝑥
= ∑ ln 𝑥𝑖 Eq. 1-7
𝑅𝑇
𝑖
The activity coefficient is influenced by phenomena such as the aggregatory behavior in
alcohols. Aggregation reduces the effective number of particles in the solution so the monomer
activity requires a chemical potential correction due to intermolecular interactions and the related
entropy and enthalpy of association.35 Capturing this non-ideal behavior is important for the design
and operation of unit operations such as distillation and extraction.
1.4.2 Modeling Considerations
It is well understood that association causes deviations from ideal behavior, which is
reflected in the vapor-liquid-equilibrium (VLE) data of these systems. To capture these deviations,
Raoult’s Law can be modified to include a fugacity coefficient (𝜑𝑖 ) and an activity coefficient (𝛾𝑖 ).
10
Deviations from ideal gas behavior are described by the fugacity coefficient, whereas an activity
coefficient is included to capture departures from ideal solution behavior.
𝑠𝑎𝑡 𝑠𝑎𝑡
𝑉 𝐿 (𝑃 − 𝑃 𝑠𝑎𝑡 )
𝑦𝑖 𝜑𝑖 𝑃 = 𝑥𝑖 𝛾𝑖 𝜑𝑝𝑢𝑟𝑒 𝑖𝑃 exp ( ) Eq. 1-8
𝑅𝑇
This work models non-idealities with a “gamma-phi” approach, which involves the
modeling of the component activity coefficients. When the standard state is taken as the pure
component at the same temperature and pressure as the mixture, from Eq. 1-9 and Eq. 1-10 it can
be seen that the activity coefficient is related to the excess Gibbs energy per mole of the mixture
𝐸
(𝐺𝑀𝑖𝑥 ), where 𝑛𝑖 and 𝑛𝑇 are the moles of component 𝑖 and the total number of moles,
respectively.36 The excess Gibbs energy is the difference between the Gibbs energy of the mixture
and the Gibbs energy of an ideal solution.
𝐸
𝑛𝑇 𝐺𝑀𝑖𝑥 = 𝑅𝑇 ∑ 𝑛𝑖 ln(𝛾𝑖 ) Eq. 1-9
𝑖
𝐸
𝜕𝑛𝑇 𝐺𝑀𝑖𝑥
𝑅𝑇 ln(𝛾𝑖 ) = ( ) Eq. 1-10
𝜕𝑛𝑖 𝑇,𝑃,𝑛𝑗 (𝑗≠𝑖)
Bala and Lira illustrated that a component’s activity coefficient arises from the contribution
of three separate terms: the combinatorial (𝛾𝑖𝑐𝑜𝑚𝑏 ), the residual (𝛾𝑖𝑟𝑒𝑠𝑖𝑑 ), and association (𝛾𝑖𝑎𝑠𝑠𝑜𝑐 ).37
Logarithms allow the contributions to be summed as seen in Eq. 1-11.
ln(𝛾𝑖 ) = ln(𝛾𝑖𝑐𝑜𝑚𝑏 ) + ln(𝛾𝑖𝑟𝑒𝑠𝑖𝑑 ) + ln(𝛾𝑖𝑎𝑠𝑠𝑜𝑐 ) Eq. 1-11
The combinatorial term represents non-idealities due to entropy, which arise from
molecular differences in size and shape. The residual term arises from energetic non-idealities and
also contains several adjustable parameters.37 Finally, the association term represents non-ideal
11
effects ascribable to intermolecular forces such as hydrogen-bonding and is the focus of this
work.38 Models used to calculate the contribution of the association term to the activity coefficient
fall into three categories: lattice, chemical theory, and perturbation theory. However, for the scope
of this work, henceforth, discussions will be limited to the chemical and perturbation theories.
The chemical theory description of association represents hydrogen bonding with a weak
reversible chemical reaction where chemically distinct aggregates coexist in chemical equilibrium
within an ideally-behaving solution.39 This equilibrium can be described in terms of its equilibrium
constants as well as the temperature and concentration of the system.40 However, for this approach
to be used successfully, the modeler must have advanced knowledge of aggregate stoichiometries,
and the equilibrium constants must be determined from experiments. This limitation was
considered a disadvantage by early work, which claimed that predictions involving chemical
theory were of limited quantitative use.39 Despite the early criticism, the chemical theory was
improved by Campbell et al. who derived expressions for the monomer mole fraction and used it
to describe systems of alcohols and alkanes. By assuming that alcohols aggregate in linear chains,
he was able to develop a closed-form expression for association contribution to the gamma term.41
A statistical mechanics-based description of association is provided by perturbation theory,
which relates specific site-based interactions with bulk fluid behavior.42 One of the most popular
perturbation theories involves the work of Wertheim, who developed a model for association based
on the interaction between acceptor sites (𝐴𝑖 ) and donor sites (𝐷𝑗 ) located on repulsive cores. A
hallmark achievement of Wertheim’s theory was the connection between the excess Helmholtz
energy of association and the concentration of non-bonded sites at equilibrium (𝑋𝐴𝑖 ). From this
relationship, the monomer density could be related to the association strength (Δ 𝐴𝑖 𝐷𝑗 ) between
interacting sites. Despite its rigor, the deterministic nature of Wertheim’s theory does not easily
12
lend itself to experiments.40 A welcomed simplification was developed by Chapman et al., who
reduced Wertheim’s theory to its first-order thermodynamic perturbation (TPT-1),42 which
considers linear chains. Recently, Bala and Lira have demonstrated the equivalence of TPT-1 and
chemical theory for alcohols and provided a simplified form for 𝛾 𝑎𝑠𝑠𝑜𝑐 in terms of the fraction of
non-bonded acceptor sites (𝑋𝐴𝑖 ), monomer density (𝜌0 ) and solution molar density (𝜌).37 Their
derivation assumed no excess volume, a universal packing factor, conventional mixing rules, and
a van der Waals repulsive term for compressibility.
𝑋𝐴 𝜌
ln 𝛾1𝑎𝑠𝑠𝑜𝑐 = 2 ln ( 𝐴,0
) − (1 − 𝑋𝐴,0 ) + ( ) 𝑥1 (1 − 𝑋 𝐴 ) Eq. 1-12
𝑋 𝜌1
𝜌
ln 𝛾2𝑎𝑠𝑠𝑜𝑐 = ( ) 𝑥1 (1 − 𝑋 𝐴 ) Eq. 1-13
𝜌2
1.5 Overview of this work
The introduction has reviewed the concepts of spectroscopy necessary to characterize
concentrations of species and the shifts of vibrational frequencies when hydrogen bonding is
present. Chapter 2 Focuses on the Beer-Lambert-Bouguer law and presents an application that can
be leveraged to readily calculate sample pathlength from spectrophotometer output files.
Quantification requires accurate pathlengths. In Chapter 3, a new methodology is presented to
quantify hydrogen bonding by scaling the raw spectra to obtain integrated peak areas that are
independent of temperature as demonstrated for 1-butanol. Chapter 4 demonstrates measurement
of mid-range IR data for several alcohols and application of spectroscopy for engineering modeling
by determining association strengths from published literature measurements. Due to the
unavailability of the raw spectra, the methods of Chapter 3 cannot be applied, but the spectra are
shown be adequately modeled with two association strengths. A new activity model based on the
13
RTPT method is developed and applied. In Chapter 5, spectra are collected for ethanol and 1-
butanol in cyclohexane. The methods of Chapter 3 are refined to provide more thorough modeling.
The Resummed Thermodynamic Perturbation Theory (RTPT) is shown to represent the
experimental data, not only for the overall association, but also for the distribution of hydrogen
bond types in solution. In Chapter 6, additional spectra are collected spectra are collected extending
the systems to seven primary alcohols, two secondary alcohols and one tertiary alcohol. The
methods of Chapter 3 are further refined, and results are applied for engineering modeling. Chapter
7 demonstrates a relationship between infrared wavenumber and NMR chemical shift for the
hydroxyl group and the mapping of infrared spectra to correlate the NMR spectra. Chapter 8
provides density measurements that were collected in anticipation of NMR spectroscopy that was
not performed within this work due to time limitations. Chapter 9 provides overall conclusions and
offers recommendations for future directions.
14
CHAPTER 2: A MATLAB® Application for Calculation of Cell Pathlength in
Absorption/Transmission Infrared Spectroscopy
2.1 Preface
Fourier transform infrared spectroscopy is an established technique for the qualitative
determination of organic structure. Absorption infrared spectroscopy can be employed
quantitatively using Beer’s law. However, reliable quantitative analysis requires that the
pathlength is known within 1%. Pathlengths based on nominal spacer thickness are not sufficient
for accurate work due to variances in cell assembly and thermal expansion. The “interference
pattern” method provides an accurate determination of cell pathlength but requires plotting of
empty cell spectra and counting of the fringes. This work provides a MATLAB® application with
a graphical user interface implementing an interactive plot where the user selects the region for the
calculation. Then the program automatically computes the cell pathlength providing a rapid
determination of pathlength for practitioners.
2.2 Publisher Permission
Reprinted (Adapted or Reprinted in part) with permission from
Killian Jr., W. G.; Storer, J. A.; Killian Sr., W.; Lira, C. T. A MATLAB Application for Cell
Pathlength in Absorption Transmission Spectroscopy. Spectroscopy 2020, 35 (8), 26–28.
Copyright 2020 MJH Life Sciences
2.3 Introduction
Absorption infrared spectroscopy is an established instrumental method for qualitative
determination of molecular functionality. Common to both academic and industrial laboratory
settings, infrared spectroscopy is primarily used for qualitative identification of functional moieties
via their characteristic resonance frequencies and absorbance intensities. Modern infrared
spectrophotometers utilizing Fourier transforms allow for fast and efficient data collection. This
15
technological advance has enhanced the utility of infrared spectroscopy, making it an expedient
method for qualitative as well as quantitative measurements.
Vibrational excitation occurs when chemical bonds absorb photons whose frequency
matches that of the bond. For this excitation to be infrared active, the magnitude of the dipole
moment vector must change with oscillation. In absorption spectroscopy, the fraction of incident
light absorbed by the sample at a specific wavelength is proportional to the concentration of
absorbing molecules in the beam path. This relationship, which describes the interaction of light
with the chemical medium at a particular frequency, is described by the Beer-Lambert-Bouguer
law (Eq. 2-1)
𝐴𝜈̃,𝑖 = 𝜖𝜈̃,𝑖 𝐶𝑖 𝑙 Eq. 2-1
where 𝜖𝜈̃ is the molar attenuation coefficient (known as the molar extinction coefficient in older
literature), 𝐶𝜈̃ is the concentration of the chromophore and 𝑙 is the pathlength. Provided that the
molar attenuation coefficient and pathlength are known the formula can be rearranged, allowing
for concentration to be calculated from an absorbance peak height or area. Harris 43 provides in-
depth discussion of the interplay of concentration, pathlength, and intensity.
Cell pathlength is typically set using lead or Teflon® spacers sandwiched between two salt
plate windows. According to Meloan,27 quantitative reliability of infrared measurements
necessitates that the pathlength must be known within 1%. Because cells are often disassembled
and reassembled for cleaning and polishing of windows, the pathlength can vary upon reassembly
relative to calibrations and the nominal dimensions spacer dimensions provided by the
manufacturers are not adequate for quantification. Further, cell pathlength may change due to
thermal expansion,44 and in-situ measurements provide the best reliability of pathlength
measurement.28 The pathlength should be calculated during calibration and then the attenuation
16
coefficient is determined based on the known concentration and absorbance measurement
according to Eq. 2-2.
𝑑𝑚3
𝜖[ ] = 𝐴𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 /(𝐶𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 ⋅ 𝑙𝑐𝑎𝑙𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 ) Eq. 2-2
𝑚𝑜𝑙 ⋅ 𝑐𝑚
If the cell is disassembled or reassembled, the pathlength (𝑙𝑠𝑎𝑚𝑝𝑙𝑒 ) should be measured again
before the sample is loaded and then the concentration is determined using Eq. 2-3.
𝑚𝑜𝑙
𝐶𝑠𝑎𝑚𝑝𝑙𝑒 [ ] = 𝐴𝑠𝑎𝑚𝑝𝑙𝑒 /(𝜖𝑙𝑠𝑎𝑚𝑝𝑙𝑒 ) Eq. 2-3
𝑑𝑚3
2.4 Pathlength Measurement Using the Interference Pattern
When the cell is empty, the refractive indices are significantly different for the salt plate
windows and the inert gas (nitrogen, air, etc.) in the sample cell. Thus, a measurable amount of the
irradiating light is internally twice reflected at the gas/salt interfaces within the cell before exiting
on the detector side. Constructive and destructive interference between the irradiating light and
internally reflected light occurs at all wavelengths, resulting in maximum intensities when the
wavelength of light exiting the cell is an integer multiple of twice the cell pathlength. Likewise,
minimums in intensity occur when the wavelengths of exiting light are half integer multiples of
twice the cell pathlength. Wavelengths lying between half and full integer multiples give rise to
intermediate absorbance intensities. The resulting spectrum is comprised of successive oscillations
known as an “interference pattern” which allows for the pathlength to be calculated from the span
of several adjacent minima using Eq. 2-4 where 𝑙 is the cell pathlength in centimeters, 𝑁𝑐 is the
number of complete cycles (fringes), and 𝜈̃𝐻 , 𝜈̃𝐿 are the high and low wavenumbers selected for
the calculation.28
17
1 𝑁𝑐
𝑙 [𝑐𝑚] = ( ) Eq. 2-4
2 𝜈̃𝐻 − 𝜈̃𝐿
2.5 The Pathlength Application
The application can be run on any operating system where MATLAB® is installed. No
additional MATLAB® toolboxes are required. The code is developed for absorption spectroscopy.
The cell should be clean and the windows free of surface imperfections or clouding. Before taking
a scan of the empty cell a background scan should be collected to remove environmental effects.
Next, the empty cell should be scanned, ideally with a dry inert gas in the sample space, and the
instrument output should be saved in a .csv format with the first column containing wavenumbers
and the second column containing absorbance or percent transmittance. The app will disregard any
lines in the file with text.
A screenshot of the application’s graphical user interface (GUI) is shown in Figure 2.1.
The empty cell spectrum is loaded by clicking the Get Data button which opens a standard file
browser allowing the practitioner to select the empty cell .csv file. After the file has loaded, the
file name will be displayed as the title of the plot, e.g. in Figure 2.1, the file is Path.csv. The entire
empty cell spectrum is displayed as a blue line (obscured by the smoothed red line in Figure 2.1),
enabling the user to visually inspect the spectra before selecting the region to be used for the
pathlength calculation. The user can vary the region of the spectrum used for the calculation by
adjusting the maximum and minimum values in the Wavenumber Window box. Data resolution is
calculated and displayed.
A smoothing filter is provided because noise in the spectra can interfere with determination
of the wave minima used for counting cycles. The smoothing filter applies a quadratic spline fitted
to a specified wavenumber span. Setting the smoothing span to the width of the top third of the
18
cycle typically works well. The plot is interactive, thus clicking on the curve displays coordinates,
permitting rapid determination of a reliable span to use for smoothing.
Figure 2-1: Graphical user interface for the calculation of cell pathlength.
Once the desired region has been established, and the span specified, depressing the
Calculate Pathlength button performs the calculation and generates a value for the cell pathlength
in centimeters. The smoothed spectrum is displayed as a red line. The number of fringes used in
the calculation are determined automatically, eliminating tedious counting where user errors may
occur. Circles appearing on the plot of Figure 2-1 denote the fringe minimums detected by the
program and the x’s describe complete cycles (𝑁𝑐 ) relative to the first minimum. The first and last
circle observed on the plot coincide with the wavenumber values used to perform the calculation.
The calculation should be performed two to three times using different wavenumber ranges to
confirm the pathlength. Values should vary by less than 0.2%.28
19
The app is configured so that smoothing can be changed without resetting the selected
wavenumber range. Noise in the spectra can cause incorrect identification of the cycle minima if
the smoothing span is too small. The user can identify this behavior when viewing the circles on
the calculated plot, reset the smoothing filter, and recalculate the pathlength. The complete cycles
within the active plot window are used in the calculation, not the nominal wavenumbers used in
the GUI to select the window.
2.6 Summary and Conclusions
The pathlength application provides a convenient method to accurately and quickly calculate
the pathlength of IR cells. The app is easy to use, and the visualization of the fringe pattern provides
an opportunity for the user to interactively change the selected wavenumber window and
smoothing to use for the pathlength calculation. The app eliminates tedious counting where user
error may occur. The pathlength app is distributed via the MATLAB® Central File Exchange
repository at https://www.mathworks.com/matlabcentral/fileexchange/77353-pathlength and code
is available at https://www.egr.msu.edu/thermoprops/pathlength. Documentation is provided with
the app download for users who may need to edit the data .csv file as well as an example data file
and step-by-step instructions.
20
CHAPTER 3: Quantitative Analysis of Infrared Spectra of Binary Alcohol +
Cyclohexane Solutions with Quantum Chemical Calculations
3.1 Preface
Hydrogen bonding has profound effects on the behavior of molecules. Fourier transform
infrared spectroscopy (FTIR) is commonly used to qualitatively identify hydrogen bonding
moieties present in a chemical sample. However, quantitative analysis of infrared (IR) spectra is
nontrivial for the hydroxyl stretching region where hydrogen bonding is most prominently
expressed in organic alcohols and water. Specifically, the breadth and extreme overlap of the
𝜈̃(𝑂𝐻) stretching bands, and the order of magnitude variability of their IR attenuation coefficients
complicates the analysis. In the present work, sequential molecular dynamics (MD) simulations
and quantum mechanical (QM) calculations are used to develop a function to relate the integrated
IR attenuation coefficient to the vibrational frequencies of hydroxyl bands across the 𝜈̃(𝑂𝐻)
stretching region. This relationship is then used as a guide to develop an attenuation coefficient
scaling function to quantitatively determine concentrations of alcohols in hydrocarbon solution
from experimental IR spectra by integration across the entire hydroxyl frequency range.
The computational work presented in this chapter was the performed in majority by Aseel
Bala and with support from Cesar Plascencia. Experimental measurements and identification of
the functional form of the molar attenuation coefficient were contributed by the author of this
work.
3.2 Publisher Permission
Reprinted (Adapted or Reprinted in part) with permission from
Bala, A. M.; Killian, W. G.; Plascencia, C.; Storer, J. A.; Norfleet, A. T.; Peereboom, L.;
Jackson, J. E.; Lira, C. T. Quantitative Analysis of Infrared Spectra of Binary Alcohol +
21
Cyclohexane Solutions with Quantum Chemical Calculations. J. Phys. Chem. A 2020, 124 (16),
3077–3089. https://doi.org/10.1021/acs.jpca.9b11245.
Copyright 2020 American Chemical Society
3.3 Introduction
Much can be learned from a chemical sample based solely on its interactions with
electromagnetic radiation. In Fourier transform infrared (FTIR) spectroscopy, light from the
infrared region (10-12500 cm-1) is used to excite chemical bond vibrations. Measurements can be
categorized into three main wavenumber ranges: near- (4000-12500 cm-1), mid- (200-4000 cm-1)
and far IR (10-200 cm-1) with most fundamental molecular vibrations occurring in the mid-IR.
Within the harmonic oscillator approximation, energy differences among the vibrations of
different molecules and bonds result from differences in their bond strengths and reduced masses,
leading to characteristic absorptions for specific functional groups. Incident light is absorbed when
the vibrational excitation has an associated transition dipole (net dipole change upon excitation),
and the absorption intensity reflects the magnitude of the transition dipole. Qualitative analysis of
infrared absorption spectra enables structure elucidation of chemical compounds via their
characteristic frequencies and absorbance intensities. However, since the transition dipoles are
strongly modulated by the degree and topology of hydrogen bonding, the attenuation coefficients
(classically known as extinction coefficients) required for accurate quantification of hydroxyl
moieties from their infrared absorptions vary widely.
Quantitative analyses of IR absorption spectra can be used to gain insight into the
concentration of functional groups in a solution. For example, Williams et al.45 explored the
relationship between absolute integrated intensities of the C-H stretching and bending bands of
gas-phase alkanes. Comparing the results from density functional theory (DFT) calculations to
22
experimental IR spectra, these authors found that the numbers of C-H bonds in the molecules
studied were linearly correlated with the integrated intensities of C-H stretching and bending
bands. IR has also found use in chemometrics;46 several researchers have correlated intensities of
various classes of compounds with physical characteristics of the molecules such as numbers of
methylene groups,47 molecular size48 and degree of branching.49
The O-H stretching bands of alcohols and carboxylic acids have been the focus of studies
probing the complex effects of hydrogen bonding.7,8,19,50–55 In alcohols, the O-H sites absorb across
a range of ~3200 cm-1 to 3700 cm-1, roughly displaying two overall absorption regions: a sharp
high frequency band and a broader composite of several overlapping bands at lower frequency.
The formation of hydrogen bonds has long been known to red-shift the O-H stretching frequency
while simultaneously increasing its integrated intensity,56,57 a phenomenon that is easily
understood in terms of differences in the vibrationally modulated dipole oscillations. In studies of
supercritical and liquid methanol, Wu et al.52 found that an isothermal increase in density causes
the integrated O-H peak area to increase and the vibrational band to shift to lower frequencies. As
expected, isobaric heating has the opposite effect. Hydrogen bonding has also been studied in
nozzle sprays,58 but quantitative transference of aggregate distributions to static conditions would
be speculative.
To address the challenges involved in IR analysis of hydroxyl peaks, computational tools
such as molecular dynamics (MD)59 and quantum chemical (QM) simulations45,50,60–65 have been
used to elucidate the effects of hydrogen bonding on IR peak characteristics. MD and QM
calculations offer different balances of computational expense with modeling rigor; the former can
model very large systems at modest computational cost but does not completely capture the
changes in electronic structure induced by hydrogen bonding which are also largely responsible
23
for the dipole variations that define IR absorption intensity. Interaction energies and cluster
distributions are also very sensitive to the potentials chosen; indeed, in a simulation of 10 mol%
CH3OD in CCl4 at 300K, Kwac and Geva66 found that depending on the empirical force field, the
simulated fraction of hydroxyls in monomers (denoted as 𝛼 below) varied widely, from 5.4 to
18.5%, while the fraction of hydroxyls in long chains (denoted as 𝛿 below) varied from 52-75%.
For these reasons, MD is often used in conjunction with QM66–69 rather than as the primary tool
for hydrogen bonding investigation.
A recent development in the computational community is empirical mapping. This
approach, developed by Skinner et al.31,70–73 for water, creates functions or “maps” relating
vibrational frequencies to approximate spectra using MD calculations. In this way, one can obtain
a meaningful fundamental understanding of a system’s IR response without having to use
excessive computational resources. Mesele and Thompson74 extended these techniques to primary
alcohols, developing several “universal” maps that relate the transition frequencies, dipole
derivatives and position matrix elements to the electric field on the atoms.
In this work, we present a combined computational and experimental approach which
leverages the power of simulations to address the challenges of interpreting the infrared spectra of
hydroxyls. In short, we use the qualitative trends produced from large-scale simulations to develop
the shape of an attenuation coefficient function for quantitative liquid phase infrared spectroscopy.
We apply this technique to quantitatively analyze the entire IR hydroxyl band and to calculate the
relative and absolute concentrations of hydroxyl groups in the various contexts (monomers and
oligomers) existing in solution. In the discussions below, references to the hydroxyl vibrational
bands pertain to the vibrations of the covalent O-H bond in a RO-H---OHR hydrogen bond, not
the vibrations of the actual H---O hydrogen bond.56 Also, the term ‘formal concentration’ refers to
24
the concentration of a given compound in solution ignoring speciation into hydrogen bonded (or
other) clusters – i.e. the molecules are counted individually. The term “formal concentration” in
chemistry is synonymous with “apparent concentration” used in chemical engineering.
3.4 Analysis of the O-H Stretching Band
Quantitative interpretation of infrared spectra begins with the Beer-Lambert-Bouguer law,
which relates the observed absorbance of a solute to its concentration in solution according to Eq.
2-1. where 𝐴𝜈̃,𝑖 and 𝐶𝜈̃,𝑖 are the observed absorbance and concentration of each absorbing solute i
respectively. The pathlength, 𝑙, is the thickness of sample that the light passes through and 𝜀𝜈̃,𝑖 is
the molar attenuation coefficient, known in older literature as the absorption or extinction
coefficient, of species i. The attenuation coefficient is a fundamental property of a molecular
transition (e.g. a vibration), relating absorbance of light at a specific frequency to the compound’s
concentration. To simplify the application of IR spectra, 𝜀𝜈̃,𝑖 is usually assumed to be independent
of solute concentration. In common practice with any absorption spectroscopy, solutions of known
concentrations are prepared and analyzed. For a given solute absorbance, the observed peak height
values are then plotted against the experimental solute concentrations of the solutions. The molar
attenuation coefficient is calculated as the slope of this plot, which is ideally linear. The
complication with hydroxylated analytes is that they speciate into hydrogen bonded aggregates
whose absorptions and attenuation coefficients differ greatly from those of the isolated monomers.
3.5 Motivation
While research in this area is extensive, we are unaware of any work that successfully
enables quantification of the entire hydroxyl IR band area for alcohols by relating the integrated
area to the formal alcohol concentration. Quantification of the bond type distribution would
improve understanding of bond cooperativity and effects of temperature on the cluster
25
distributions. These are the insights needed to inform efforts to model phase equilibria.75–77 Indeed,
in the development of engineering models for the association of an alcohol in an inert solvent, the
key quantity that defines the extent of hydrogen bonding is the fraction of hydroxyl protons that
remains nonbonded at equilibrium.
For alcohols, self-association by hydrogen bonding strongly shifts and broadens the
observed IR bands in the -OH region, complicating quantification. Moreover, there is disagreement
in the literature concerning the assignments of this region's vibrational bands to specific structural
features. In early studies,34,51,78 vibrational bands were assigned to hydrogen bonded clusters and
were distinguished based on the size of the cluster. Hall and Wood6 proposed that covalent O-H
bond vibrations should be classified individually according to whether and how they participate in
hydrogen bonding. Their categories, which we adopt here, are as follows: If the O-H moiety is
isolated, neither accepting nor donating a hydrogen atom, it is classified as an α hydroxyl. If it is
accepting one hydrogen atom (i.e. interacting via the lone electron pairs on its oxygen) but not
donating (i.e. the O-H hydrogen has no additional close contacts), it is classified as a 𝛽 hydroxyl.
These and the other four hydroxyl classifications are illustrated in Figure 3-1. Throughout this
work, the hydrogen-bonded molecules are referred to as oligomers, and the oligomers together
with neighboring molecules not involved in the hydrogen bonded aggregate as clusters.
26
Figure 3-1: Classification labels of hydroxyl environments.
The motivation behind the current study is to develop a procedure capable of accurately
determining the fraction of free hydroxyl protons and conduct a thorough quantitative analysis of
the O-H IR bands. To develop a quantitative interpretation of IR spectra, the following procedure
was followed. First, we generated hydrogen-bonding environments using MD simulations of
alcohol + cyclohexane mixtures. Then, probable hydrogen bonds were identified and small clusters
from the MD frames were extracted. Their hydroxyl vibrations were then evaluated for frequency
and intensity using QM. Next, a variety of functional forms for the attenuation coefficient were
proposed to represent the shape of the frequency-dependent absorbance intensity from the QM
calculations. The proposed functional forms were then used to scale datasets of experimental IR
spectra measured over various temperature and alcohol/hydrocarbon composition ranges. To
evaluate the proposed functions for the attenuation coefficient, the experimental hydroxyl regions
of the scaled infrared spectra were integrated, generating parity plots of measured and formal
concentrations. Least squares regression of the parity plot was used to optimize the parameters of
the proposed scaling function. The recommended form of the attenuation coefficient function is
provided below.
27
3.6 Computational Methods
3.6.1 Molecular Dynamics Simulations
Molecular dynamics simulations were carried out using the AMBER 14 package.79 The
AMBER94 force field was implemented with the AM1-BCC charge method with no modifications
to the force field parameters. For each concentration, a cubic box of 1-butanol and cyclohexane
molecules was created using PACKMOL80 using conditions in Table 3-1 and the energy of the
system was minimized within 1500 steps. Next, the box was heated with a 40 ps NVT ensemble,
using a time step of 2 fs, in two stages. The temperature was ramped up from 0 to 283.15 K during
the first 9000 steps, and then equilibrated at that temperature for the 11000 steps. Next, density
was equilibrated for 8 ns using an NPT ensemble at 1 bar with a time step size of 2 fs. The
temperature and pressure were controlled using the Langevin thermostat (with a collision
frequency of 2 fs) and Berendsen barostat respectively; both were implemented with default
parameters. The 2.4 ns production NPT stage was at the same conditions as the equilibration.
During the NVT and NPT simulations, periodic boundary conditions were enforced in the x,y and
z coordinates. The cutoff for non-bonded interactions was set at 8 Å. Beyond 8 Å, the default
AMBER long-range corrections were used; a continuum model was used for Lennard-Jones
interactions and the Particle Mesh Ewald (PME) summation method was used for electrostatic
interactions.
To save computational time, C-H and O-H bond lengths were constrained using the
SHAKE algorithm. This method was repeated for ethanol + cyclohexane at equimolar
compositions using a 10 ns NPT equilibration period and a 3 ns NPT production period. The
simulation details that varied between systems are given in Table 3-1. The purpose of the runs was
to provide hydrogen-bonded oligomers for the quantum calculations, and thus low temperatures
28
were selected for the simulations. As shown in Table 3-1, the equilibration segments were longer
than 8 ns in all cases, giving all the systems ample time to equilibrate.
Table 3-1: MD simulation details of run parameters for 1-butanol + cyclohexane systems
studied in this work.
System Units 1-butanol + cyclohexane ethanol + cyclohexane
Alcohol Mole Fraction --- 0.1016 0.5 0.5
Number of alcohol
--- 26 128 168
molecules
Number of
--- 230 128 168
cyclohexane molecules
Density during
[kg/m3] 785 ± 3 797 ± 3 769 ± 3
production
Temperature [Kelvin] 283.15 283.15 298.15
NPT Equilibration
[ns] 8 8 10
Time
NPT Production Time [ns] 2.4 2.4 3
* Ideal solution densities based on experimental pure component densities are 792, 802, 779
kg/m3, respectively.
The hydrogen bond criteria were defined as an O---O distance < 3.2 Å and an O-H∙∙∙O bond
angle > 130⁰ which is consistent with the definition used by Jeffrey4 to denote strong and
moderately strong hydrogen bonds. Using this definition, hydrogen bonded oligomers of various
sizes were identified, and each hydroxyl was assigned a class based on the labels in Figure 3-1.
The distribution of hydroxyl types, i.e. 𝛽, 𝛾, or 𝛿, for 1-butanol molecules that participate
in trimers during the MD simulation is shown in Figure 2 and serves as a measure of thermal
equilibration. The data for Figure 2 was obtained from 589 frames collected every 40 fs in the 2.4
ps of the production run. Each molecule appeared in a trimer in about 15% of the frames. The
29
distribution was random for the three bond types across all molecules in solution confirming that
bond formation and breaking occurs randomly in the simulation.
Figure 3-2: Bond distribution from an equimolar (i.e. mole fraction 0.5 alcohol) 1-butanol +
cyclohexane MD simulation presented as the number of occurrences collected from trimers
using the production frames. The x-axis represents the unique identifying number of each 1-
butanol molecule. The distribution demonstrates that bond breaking and formation occur at
random during the simulation.
3.6.2 Quantum Mechanics Simulations
A hydrogen-bonded alcohol oligomer and its neighboring molecules, herein referred to as
a cluster, were taken from frames of the MD trajectory. Specifically, all alcohol and cyclohexane
molecules with atoms that fell within 5 Å of the hydrogen-bonded hydroxyl hydrogen atoms in the
oligomer were retained for the QM calculation and served as explicit solvent molecules for the
oligomer. All other molecules beyond this cutoff were excluded. This cutoff effectively ensured
that the sampled hydroxyl environments were representative of the entire frame while minimizing
the computational effort required for quantum calculations. To this end, a constrained geometry
optimization was then performed on the cluster where only the -CH2OH groups in the oligomer
were allowed to relax.
30
Gaussian0981 was used to optimize the -CH2OH groups of each cluster and perform
frequency calculations. The chosen B3LYP82 method combined with the modest 6-31G* basis
set83–85 has been shown60,65 to capture the effects of hydrogen bonding for a reasonable
computational cost. This modest level of theory was selected to survey a large number of samples,
seeking patterns of behavior, rather than absolute quantitative values. Higher level calculations on
a modest subset of the systems studied confirmed the persistence of the behaviors discovered,
verifying that the results were not an artifact due to this relatively low level of theory. The
calculated IR frequencies and attenuation intensities for the O-H stretching vibrations were
screened by first scaling86 the frequencies of each cluster by 0.96 and then collecting all frequency
and intensity pairs with a frequency above 3100 cm-1. The number of collected vibrational modes
was ~24,000 and ~2,600 for the ethanol + cyclohexane and 1-butanol + cyclohexane systems,
respectively.
Initially, the datasets for the two alcohols were examined separately, but the patterns for
these two linear primary alcohols were found to exhibit near-perfect overlap. This behavior is in
accord with the exact overlap of the O-H bands in their gas-phase spectra87,88. The datasets were
therefore combined, plotted as intensity vs. wavenumber, and smoothed by applying a moving
average with a Hann window-type and a window size of 101 cm-1. Various other moving average
window types were considered, but differences were minor.
For purposes of classification, vibrational motions were divided into coupled and
uncoupled categories for analysis based on criteria explained below. Coupling can be substantial
between functional groups that have near-degenerate vibrational frequencies and are close and
suitably oriented. Hydrogen bonding represents the major mechanism of such coupling between
hydroxyls on different alcohols. The classification of IR peaks to structural classes of hydrogen
31
hydroxyls (𝑎, 𝛽, 𝛾, or 𝛿) becomes complicated for strongly coupled O-H vibrations due to
contributions from multiple O-H stretching displacements in a given vibrational mode of a cluster.
Therefore, vibrational modes were categorized as described below, and only assigned to hydrogen
bond O-H classifications for vibrations in which one O-H moiety’s motions were dominant.
Further, only linear clusters were considered for the vibrations analyzed; structures including η
and 𝜁 hydroxyls were neglected because they were found to occur infrequently compared to the
other types of hydroxyls. Table 3-2 shows the average percentage bond distribution among the
vibrations that were structurally assigned. For each bond classification, the value shown was
calculated by averaging the numbers of each O-H bond type over 13,000 frames. While 𝛼, 𝛽, 𝛾,
and 𝛿 hydroxyls occurred in relatively high populations, 𝜂 and 𝜁 constitute less than 1% of the O-
H bonds, on average.
Table 3-2: Average distribution of O-H bond types in % for three systems. Data from 13,000
frames is averaged for each system.
1-Butanol + Cyclohexane Ethanol + Cyclohexane
Hydroxyl Type 𝑥𝐵𝑢𝑂𝐻 = 0.1 𝑥𝐵𝑢𝑂𝐻 = 0.5 𝑥𝐸𝑡𝑂𝐻 = 0.5
𝛼 41.79 20.76 23.29
𝛽 18.54 21.34 21.23
𝛾 19.05 22.59 22.95
𝛿 20.23 34.33 31.18
𝜂 0.27 0.72 0.95
𝜁 0.12 0.27 0.39
32
As further explained below, vibrational localization among the various O-H bond
stretching modes in an oligomer was assessed in terms of the displacement of each O-H hydrogen
atom in a given O-H mode, using the following criteria:
𝑑 2 = Δ𝑥𝐻2 + Δ𝑦𝐻2 + Δ𝑧𝐻2 Eq. 3-1
where Δ𝑥𝐻 , Δ𝑦𝐻 and Δ𝑧𝐻 are the displacements of the hydrogen atom in the x, y, and z directions
respectively as reported in the quantum chemical vibrational analysis. A vibrational mode was
considered to be uncoupled (isolated) if one of the O-H hydrogen atoms had a d2 value at least 0.3
Å2 greater than any other atom in the optimized geometry. If a particular stretching mode fit the
criteria then the active O-H bond was classified (𝑎, 𝛽, 𝑒𝑡𝑐.) according to Wood and Hall’s labeling
scheme. Otherwise, if the motions are more evenly distributed over multiple sites, the hydroxyls
in the cluster remain unclassified and are included in the analysis of coupled vibrations. The 0.3
Å2 criterion used here to differentiate coupled and uncoupled vibrations is arbitrary, but does
provide initial insights into the populations of bond-types that contribute to the different regions
of the O-H stretching band.19,89–91
3.7 Experimental Methods
Anhydrous cyclohexane (lot# SHBJ0085) and 1-butanol (lot# SHBH8917) of 99.5% and
99.8% purity, respectively, were purchased from Sigma-Aldrich. All reagents were dried in a
glovebox under nitrogen atmosphere using 20% w/v Spectrum M194 3- Å 1/16” molecular sieves.
Sieves were prepared by flame heating under vacuum until vessel pressure reached 11.3 Pa gauge,
after which they were allowed to cool and brought to atmospheric pressure with dry house nitrogen.
Sieves were added to all reagents and drying occurred for at least 72 hours before use in all cases
as suggested by Williams and Lawton.92 All glassware was cleaned in acetone and hexane and
oven dried overnight before use. Sample concentrations were prepared volumetrically assuming
33
ideal volumes of mixing. Temperature-dependent volumes were calculated through 𝑉 = ∑𝑖 𝑥𝑖 𝑉𝑖 .
Excess volumes for the mixtures are less than 0.4%93 so the use of ideal mixing volumes is well
within other experimental errors. Liquid molar volume calculations used accepted experimental
values from the NIST ThermoData Engine.94 Experimental liquid density data were regressed with
a polynomial over the experimental temperature range and the polynomial was used when
calculating molar volumes for concentration calculations.
The temperature was monitored during the sample preparation and was taken as the average
reading of two thermometers positioned near the samples. Binary samples consisting of alcohol
and cyclohexane were prepared in 10-mL volumetric flasks of type A precision. Alcohol was
measured using an appropriately sized Hamilton gas-tight syringe. Each concentration was
independently prepared. After sample preparation, the flask was stoppered and inverted twenty
times before a thirty second vortex stir to homogenize. Samples were then transferred to
borosilicate scintillation vials using screw tops with a PTFE liner. Vials were wrapped in Parafilm
before being removed from the glovebox.
Samples were analyzed using a JASCO FT/IR-6600 Spectrometer. The sample
compartment was under a continuous house nitrogen purge. Samples were injected into the cell
from a Luer lock syringe into an airtight valve system consisting of 1/16” O.D. stainless steel
components that was connected to a Specac demountable heatable liquid flow cell (GS20582) with
CaF2 windows and a PTFE spacer (GS20070-0.01MM, GS20070-0.025MM, GS20070-0.1MM,
and GS20070-0.5MM). The cell windows were cleaned with hexane at the end of each day and
the cell apparatus was stored under house nitrogen between uses.
The cell was dried internally before use with low pressure house nitrogen with complete
drying confirmed by FTIR scan. The FTIR sample compartment was purged with house nitrogen
34
at a flowrate of 50 ft3/hr for 30 minutes before the background scan and throughout all
experimental scans. Temperatures were set by a Specac 4000 Series High Stability Temperature
Controller operating a Specac electrical heating jacket (GS20730) with cooling water running
through the jacket at the recommended rate of 0.5 L/min. Observed temperature variations were
less than ±0.1 °C. The sample cell’s path length was measured in centimeters using the fringe
interference method27 depicted in Eq. 2-4.
Ethanol and 1-butanol solutions were scanned at 10 °C increments from 30-60 °C. All
samples reached thermal equilibrium (constant temperature) within six minutes and were allowed
to stabilize for an additional two minutes before scans. The IR method consisted of 128 background
and sample scans at a resolution of 0.5 cm-1. For each composition, the empty chamber background
was collected at ambient temperature and was automatically subtracted from each of the sample
spectra before further processing
3.8 Results and Discussion
3.8.1 Processing and Preliminary Analysis of Experimental IR Spectra
The background-subtracted IR spectra were processed as follows. The hydroxyl band
region was determined to be 3049.9 to 3755.2 cm-1 which is the integration range used in analyses
below. The solvent bands were removed from the spectra by subtracting the concentration-
weighted absorbance of neat cyclohexane at the same temperature and nominal path length as the
sample. The subtraction was tuned by a factor 𝑓 to eliminate residual solvent peaks in the
wavenumber range 1800 – 2500 cm-1 where alcohol did not absorb. The mathematical operation
𝑚𝑜𝑙𝑎𝑟 𝑚𝑜𝑙𝑎𝑟
was a subtraction of 𝑓 ⋅ 𝜌𝑠𝑎𝑚𝑝𝑙𝑒 𝐴𝑐𝑦𝑐𝑙𝑜 (𝜈̃, 𝑇)𝑙𝑐𝑦𝑐𝑙𝑜 /(𝜌𝑐𝑦𝑐𝑙𝑜 𝑙𝑠𝑎𝑚𝑝𝑙𝑒 ) where 𝜌𝑚𝑜𝑙𝑎𝑟 is molar density
and the adjustment was always 0.99 ≤ 𝑓 ≤ 1.01.
35
The processed experimental IR spectra for 1-butanol in cyclohexane in the region of the
hydroxyl stretching band are shown in Figure 3-3 for 𝑥𝐵𝑢𝑂𝐻 = 0.0819. As the temperature
increased, peak one (~3645 cm-1), peak two (~3632 cm-1), and peak three (~3518 cm-1) increased
in absorbance while peak four (~3457 cm-1) and broad peak five (~3341 cm-1) diminished.
Figure 3-3: Spectra for 8.19 mol% n-butanol in cyclohexane after subtraction of the
concentration-weighted contribution of cyclohexane.
While band assignment is a topic of great interest, we defer such discussion to subsequent
publications. In this work, we instead conduct a quantitative analysis of the entire hydroxyl region.
To this end, we begin with an investigation of the physical significance of the absorbance band
area. Figure 3-4 shows the relationship between the total area under the entire O-H absorbance
band and the formal concentration for four solutions at five temperatures calculated as
area/pathlength 𝐴𝐼 /𝑙 = (1/𝑙)∫ 𝐴(𝜈̃)𝑑𝜈̃ where wavenumbers are in cm-1, pathlength is in cm, and
𝐴𝐼 /𝑙 has units of cm-2. We refer to the calculations as ‘unscaled’ because, in later figures, we scale
𝐴𝐼 /𝑙 using an attenuation coefficient function.
36
Figure 3-4: Total absorbance band area of the O-H band for 1-butanol + cyclohexane data as a
function of the formal molarity. Fifty-six spectra were collected at four temperatures (30, 40,
50 and 60 °C). For a given mole fraction of 1-butanol, temperature variations affect the
molarity and distribution of O-H bond moieties, with lower temperatures favoring formation
of dimers and oligomers via hydrogen bonding. Lower temperatures exhibit higher areas at a
given formal concentration.
It is immediately clear from Figure 3-4 that the linearity of the integrated area and
concentration that is traditionally assumed for a given absorbance in the Beer-Lambert-Bouguer
law does not apply for the unscaled overall hydroxyl band. Figure 4 indicates that the molar
attenuation coefficient, ε, must vary strongly at different vibrational frequencies since the
integrated areas for a fixed mole fraction vary with temperature due to the changes in the
distribution of hydrogen bonds. As discussed below, the results of the QM/MM analysis provide
key insights into this variability, resulting in relationships between absorption intensity and
wavenumber.
3.8.2 Results from QM/MM
In this section, the calculated IR characteristics of the alcohol hydroxyl stretch vibrations
are investigated. The simulation results for alcohol hydroxyl stretch vibrations were analyzed in
37
three groupings: (1) a subset of uncoupled vibrations for linear oligomers of the 1-butanol +
cyclohexane system; (2) a subset of uncoupled vibrations for linear oligomers of the ethanol +
cyclohexane system; and (3) all O-H stretching vibrations (coupled and uncoupled and all
structures including cyclics) in both the 1-butanol + cyclohexane and ethanol + cyclohexane
systems. The purpose of the limited study of linear oligomers (monomers, dimers, trimers, and
tetramers) is to explore the relation between hydroxyl classification (𝛼, 𝛽, etc.) and the
frequency/intensity. However, the quantitative attenuation coefficient scaling function is based on
the third grouping which includes hydroxyl stretches for linear, cyclic, and branched oligomers.
Table 3-3: Number of each species and bond analyzed with QM calculations for uncoupled
vibrations of 1-butanol and ethanol molecules. All species larger than a monomer have one 𝛽
and one 𝛾 hydroxyl. Trimers and tetramers also possess one and two 𝛿 hydroxyls respectively.
Cyclic oligomers and those with coupled vibrations are not included.
𝒙𝑩𝒖𝑶𝑯 = 0.1 𝒙𝑩𝒖𝑶𝑯 = 0.5 𝒙𝑬𝒕𝑶𝑯 = 0.5 Total
Monomer 653 366 859 1878
Species
Dimer 153 230 413 796
Trimer 53 52 236 341
Tetramer 46 23 223 292
𝛼 653 366 859 1878
Hydroxyl Type
𝛽 252 305 872 1429
𝛾 252 305 872 1429
𝛿 145 98 682 925
Table 3-3 lists the numbers of each bond and species type analyzed for the uncoupled cases
involving 1-butanol and ethanol (cases 1 and 2). The multiple representations for each hydrogen
bonding type, with a wide range of vibrational frequencies and intensities are averaged in Table
38
3-4. Hydroxyls are included in the table only if all the vibrations of the oligomer hydroxyls are
uncoupled. The reported “intensity” from Gaussian09 is the integrated intensity, 𝐼, in units of
km/mol, which is defined by the equation:
𝐼=∫ 𝜀(𝜈̃)𝑑𝜈̃ Eq. 3-2
𝑏𝑎𝑛𝑑
We use 𝐼 reported by Gaussian09 as a measure of the strength of each hydroxyl’s IR absorbance
(attenuation coefficient), with the understanding that it is proportional to the peak height 𝜀𝑚𝑎𝑥 , and
use only the pattern of the behavior, not the absolute values.
Table 3-4: Average calculated vibrational frequencies and vibrational intensity (I), for
uncoupled hydroxyl stretch vibrations in butanol + cyclohexane.
𝝂̃𝒂𝒗𝒈 (cm-1) 𝑰 (km/mol)
𝒙𝑩𝒖𝑶𝑯 = 𝒙𝑩𝒖𝑶𝑯 = 𝒙𝑬𝒕𝑶𝑯 = 𝒙𝑩𝒖𝑶𝑯 = 𝒙𝑩𝒖𝑶𝑯 = 𝒙𝑬𝒕𝑶𝑯 =
0.1 0.5 0.5 0.1 0.5 0.5
Monomer 𝛼 3602 3587 3589 32 54 40
𝛽 3601 3583 3578 52 72 69
Dimer
𝛾 3479 3471 3468 377 411 395
𝛽 3589 3572 3574 68 96 79
Trimer 𝛾 3418 3414 3423 464 525 497
𝛿 3410 3412 3407 584 537 537
𝛽 3586 3574 3574 68 88 83
Tetramer 𝛾 3436 3426 3418 427 428 425
𝛿 3387 3363 3373 629 676 651
The clusters captured from the MD all have different arrangements, each providing a
different vibrational frequency and absorption intensity. The first observation is that with an
increase in the formal concentration of alcohol in the simulated box, there is a slight red shift in
almost all the vibrational frequencies accompanied by an increase in intensity. In general, a red
shift and increase in intensity is observed as the size of the hydrogen-bonded oligomer increases.
The wavenumber shift is most prominent between dimers and larger oligomers with dimer 𝛽 and
39
𝛾 hydroxyls having vibrational frequencies that are, on average, ~10 and ~50 cm-1 higher than
those of trimers and tetramers respectively. The trend in Table 3-4 shows the intensity increasing
significantly in the order 𝛼, 𝛽, 𝛾, 𝛿.
Figure 3-5 plots the absorption intensity as a function of vibrational frequency for the
combined data points from each of the uncoupled hydroxyl vibrations computed for the two formal
1-butanol concentrations, 𝑥𝐵𝑢𝑂𝐻 = 0.1 and 𝑥𝐵𝑢𝑂𝐻 = 0.5. For improved visual comparison with
experimental data, all QM frequencies that were already scaled were also blue-shifted by 60 cm-1
to center the monomer (-type) vibrational region on the experimental maximum at 3645 cm-1.
When the intensity vs. wavenumber data for the two concentrations was plotted, the plots for the
two concentrations overlapped, indicating that formal concentration has little effect on the 𝜈̃-𝐼
relationship. Therefore, they are plotted together in Figure 3-5 and the different markers denote
the four hydroxyl types studied here: 𝛼 (downward triangles), 𝛽 (upward triangles), 𝛾 (right-facing
triangles), and 𝛿 (left-facing triangles). While the focus of the current discussion concerns 1-
butanol, the same calculations for ethanol overlap as discussed below. Because the calculated
points are dense, the results for each alcohol obscure the results for the other when plotted together.
For clarity, background markers (x’s) are plotted for ethanol in cyclohexane, and later (Figure 3-6)
we show a plot for ethanol with 1-butanol in the background.
40
Figure 3-5: Uncoupled hydroxyl stretching frequencies for two concentrations of 1-butanol +
cyclohexane mixtures (triangles) calculated from QM simulations compared to ethanol
calculations (shown as x). Down, up, right, and left-pointing markers denote 𝛼, 𝛽, 𝛾, and 𝛿
hydroxyl vibrations respectively.
From Figure 3-5, the two contexts in which the O-H hydrogen atom is free, 𝛼 and 𝛽,
overlap completely and are responsible for the sharp higher frequency peak in the O-H stretching
region. Together, these two bond types constitute most of the free hydroxyl protons in solution,
though the current analysis excludes branched oligomers such as those which include 𝜁 hydroxyls.
Significant disagreement exists in the literature concerning the 𝛼-𝛽 overlap in IR spectra. Several
authors have assumed that 𝛽 hydroxyl vibrations do not contribute significantly to the sharp high
frequency peak, instead allocating that band entirely/predominantly to the α vibration.14,95,96 In
these cases, it is assumed that most clusters in solution are cyclic (resulting in few 𝛽 hydroxyls),
that the 𝛽 peak occurs at a different frequency altogether, or that the intensity of the 𝛽 absorbance
is significantly less than that of the 𝛼. However, consistent with more recent work in this area,7,97
at the concentrations examined in this work, the QM calculation averages indicate only a slight
41
red-shift of 𝛽 relative to 𝛼 vibrations, and comparable absorption intensities for both 𝛼 and 𝛽
hydroxyls. Presumably, for these systems, the change in electronic structure when the oxygen atom
accepts another proton only weakly affects the free hydroxyl bond strength. The next observation
is that the uncoupled 𝛾 hydroxyls appear at a lower frequency and, in general, have greater
integrated attenuation coefficients, (approximately 3 to 10 times that of the α and β hydroxyls).
Uncoupled 𝛿 hydroxyls follow the same pattern as the 𝛾 hydroxyls, showing additional red
shifting and integrated attenuation coefficients that are, in general, 3 to 20 times that of the 𝛼 and
𝛽 hydroxyls). The frequency trend is easily explained by recognizing that, as the hydroxyl protons
become more “shared” due to hydrogen bonding, the covalent bond is weakened. As a result, the
potential energy surface describing the hydroxyl stretch becomes broader and shallower, causing
the vibrational frequency of the hydroxyl to red shift. As for the intensity, it is directly proportional
to the square of the transition dipole moment. Thus, the increase in intensity for hydroxyls in
oligomers (β, γ, and δ hydroxyls) versus hydroxyls in monomers (α hydroxyls) arises from
arrangements of the oligomers that increase the transition dipole moment relative to the transition
dipole moment in a monomer. Conversely, the low intensities for the oligomers relative to the
monomer can be rationalized by recognizing that certain arrangements of the oligomer can
decrease the dipole moment relative to the monomer.
The most significant and least obvious finding of this work is that the relationship between
the integrated attenuation coefficient of an O-H bond and its vibrational wavenumber follows a
curve that is independent of the bond category. As well-articulated by an anonymous reviewer:
“The system is dynamic in nature, with a range of hydrogen‐bond distances and angles that are
continually being made and reformed. This leads to a near continuum of net dipoles for each
categorized species. The dimers, trimers, and tetramers, etc., are in a dynamic condition, and the
42
resulting infrared spectrum is a measure of the overall averages.” Previous studies in this area have
recognized patterns in vibrational characteristics. For example, as early as 1956, Huggins and
Pimentel98 noted a red-shift and increase in absorption intensity with increased hydrogen bond
strength. Asprion et al.34 performed curve fitting of alcohol in hydrocarbon mixtures by using
separate attenuation coefficients for monomer, dimer, and polymer, and found an increased red
shift and an increased attenuation coefficient for each. More recently, Mesele and Thompson74
conducted DFT calculations on neat alcohols and showed that the empirical maps that relate
transition frequencies, position matrix elements, dipole derivatives and the electric field are
surprisingly linear. Moreover, these relationships were identical for all four primary alcohols tested
and were predicted to hold for all other alcohols.
Having independently uncovered this pattern, we further explored the vibration/intensity
relation by repeating the described MD + QM procedure for an equimolar mixture of ethanol +
cyclohexane at 298.15 K. As shown in Figure 3-5, the ethanol calculations (shown as x’s) lie
directly under the 1-butanol + cyclohexane calculations. For greater clarity, Figure 3-6 shows the
ethanol + cyclohexane vibrations with butanol + cyclohexane vibrations included as x’s in the
background. The visual similarities between the two mixtures seen in Figure 3-5 and Figure 3-6
give an early indication that the uncovered patterns may be universal for primary alcohols.
Because inclusion of the coupled vibrations is also essential for a complete analysis of the
spectra, the coupled vibrations were also collected from the QM results. Some of the coupled
vibrations have small intensities, reflecting near-negligible transition dipoles. Figure 3-7 combines
Figure 3-5 and Figure 3-6 but also includes the coupled vibrations and the Hann moving average
calculated with a wavenumber window of 101 cm-1. The coupled vibrations strongly broaden the
43
O-H vibrational band at low frequencies, indicating that the attenuation coefficient should become
approximately constant at the low frequency end of the -OH stretching region.
Figure 3-6: Uncoupled hydroxyl stretching for an equimolar ethanol + cyclohexane mixture
(triangles) calculated from QM simulations. Background x’s show the 1-butanol calculations
from Figure 3-5. Down, up, right, and left-pointing markers denote 𝛼, 𝛽, 𝛾, and 𝛿 hydroxyl
vibrations respectively.
44
Figure 3-7: Uncoupled and coupled 1-butanol and ethanol hydroxyl stretching frequencies and
integrated attenuation coefficients calculated from QM simulations (left axis). Uncoupled
bond vibrations of ethanol + cyclohexane and 1-butanol + cyclohexane are shown as x. Circles
are coupled vibrations for both systems. Solid jagged line – Moving weighted average (Hann
window, window size = 101). Solid line (right axis) – integrated attenuation function curve
used to obtain parity plots from experimental data, as explained below.
3.8.3 Scaling and Further Analysis of Experimental IR Spectra
Noting the order of magnitude variations in absorption intensities calculated as a function
of O-H participation in hydrogen bonding, and the resulting nonlinear behavior with concentration
and temperature seen in Figure 3-4, to quantify hydroxyl spectra, a function relating attenuation
coefficient to wavenumber is needed. At first, it would seem that attenuation coefficients for each
category of absorbing species might need to be separately assigned. However, noting the simplicity
of the patterns obtained from quantum simulations a curve that mimics the QM trends was explored
as a scaling function applied to experimental data. The moving average in Figure 3-7 suggests an
attenuation coefficient curve constructed from three line segments smoothed with splines at the
intersections as shown in Figure 3-8. To develop the curve, seven parameters are needed: two for
each line segment and one for the width of the splines. The spline widths are constrained to be the
45
same in order to minimize the number of parameters. Of the seven parameters, six are adjusted to
spectra: 1&2) the intersection of Seg1 and Seg2, (xB, yB); 3) the slope of Seg1, m1; 4&5) the
intersection of Seg2 and Seg3 (xR, yR); and 6) the spline width. The slope of Seg3 is fixed at zero
as suggested by the moving average. It is worth noting that a polynomial function was insufficient
in capturing the curve characteristics well.
Fitting of the above function to experimental data entailed dimensionless scaling of the
integrated attenuation coefficient curve relative to the attenuation coefficient at 3645 cm-1, 𝜓(𝜈̃) =
𝜖(𝜈̃)/𝜖3645. At each frequency, the assumed 𝜓(𝜈̃) was calculated and used to scale the
experimental absorbance by:
𝐴(𝜈̃)
𝐴̃(𝜈̃) = Eq. 3-3
𝜓(𝜈̃)
where 𝐴̃ represents the scaled absorbance at wavenumber 𝜈̃. Following scaling, each spectrum was
integrated to obtain a scaled area 𝐴̃𝑖𝑛𝑡 = ∫ 𝐴̃(𝜈̃)𝑑𝜈̃ . The integrated areas for all scaled spectra
were divided by path length and plotted against alcohol molarity to check agreement with Beer’s
law linearity and to determine k (Eq. 3-4).
𝐴̃𝑖𝑛𝑡 ∫ 𝐴(𝜈̃)/𝜓(𝜈̃)𝑑𝜈̃ 𝜖3645 ∫ (𝐴(𝜈̃)/𝜖(𝜈̃)𝑑𝜈̃
= = = 𝜖3645 ∫ 𝑐(𝜈̃)𝑑𝜈̃ = 𝑘𝑐 Eq. 3-4
𝑙 𝑙 𝑙
In Eq. A-1, 𝜖3645 (APPENDIX A: Detailed Summary of Attenuation Coefficient Function)
has units of length2/mol and k has units of length/mol. Using cm as length units results in 𝜖3645 /𝑘
= 1 cm. A linear parity plot of predicted and experimental concentrations would result when the
numerical values of k and 𝜖3645 match. Iterating, while using cm as path length units and
wavenumber in cm-1, to obtain k with each iteration and using the value to calculate the
dimensionless scaling function 𝜓(𝜈̃) = 𝜖(𝜈̃)/(𝑘(1 cm)) led to convergence where ∫ 𝑐(𝜈̃)𝑑𝜈̃ =
46
𝑐/(1 cm). After convergence, the attenuation coefficient at any wavenumber could be calculated
using 𝜖(𝜈̃) = 𝜖3645 𝜓(𝜈̃). The choice of the reference frequency at 3645 cm-1 for scaling was
arbitrarily selected as it is the location of the maximum of the free hydroxyl peak. Specification of
a different wavenumber would simply scale 𝜓(𝜈̃) relative to that wavenumber. Some key values
for the attenuation curve are 𝜖3645 = 820 dm2/mol, 𝜖𝑅 = 29,600 dm2/mol, the slope of segment
1 is 122 cm·dm2/mol, and the slope of segment 2 is −131 cm·dm2/mol. The complete function is
given with more precision APPENDIX A: Detailed Summary of Attenuation Coefficient Function.
Figure 3-8: Schematic of three-segment attenuation coefficient curve smoothed with splines.
The text discusses the parameters used for the elements and APPENDIX A provides parameter
values.
Figure 3-9 shows plots of the spectra that were scaled by applying the optimized 𝜓(𝜈̃) to
Figure 3-3 via Eq. 3-4. In the resulting scaled spectra, peaks one (~3645 cm-1) and two (~3632 cm-
1
) now dominated, while peaks three (~3518 cm-1) and four (~3457 cm-1) became slightly more
resolved, and peak five (~3341 cm-1) was diminished considerably. All peaks retained the
temperature-dependent behavior observed in the unscaled spectra.
47
Figure 3-9: Scaled absorbance spectra for 1-butanol + cyclohexane at 8.19 mol% 1-butanol.
Spectra from Figure 3-3 are scaled with 𝜓(𝜈̃).
Across these rescaled spectra, Beer’s law is now validated as evidenced by comparing calculated
concentration 𝑐 = (∫ 𝐴̃𝑑𝜈̃)/(𝜖3645 𝑙) with formal concentration for 56 spectra at various
concentrations and four temperatures in Figure 3-10. The coefficient of determination is R2 =
0.9993 for the 56 spectra.
48
Figure 3-10: Scaled absorbance area of the O-H region for 1-butanol + cyclohexane data.
Fifty-six spectra of 1-butanol in cyclohexane were collected at four different temperatures and
at four pathlengths. The dashed line represents parity between experimental formal
concentrations and integrated areas.
The proportionality between the integrated area across the O-H region and the formal
concentration is significantly corrected in the scaled area plot in Figure 3-10 compared to that of
Figure 3-4. To compensate for the effect of temperature on the density, the band areas tabulated in
Table 3-5 include a thermal density correction:
𝐴𝑖𝑛𝑡,𝑟𝑒𝑝𝑜𝑟𝑡𝑒𝑑 𝐴𝑖𝑛𝑡,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝜌𝑟𝑒𝑓
= ( ) Eq. 3-5
𝑙 𝑙 𝜌𝑇
where 𝜌𝑟𝑒𝑓 and 𝜌𝑇 are the mixture densities at a reference temperature (298.15 K) and the
temperature of the experiment, respectively. After thermal correction, the integrated concentration
response of the scaled areas is remarkably uniform regardless of temperature for all the
compositions studied when the coefficient of variance between temperatures is compared as a
percentage in Table 3-5. The coefficient of variance (COV) is calculated using:
49
2
𝑠𝑡𝑑𝑑𝑒𝑣 ∑(𝐴𝑟𝑒𝑎 − 𝑀𝑒𝑎𝑛 𝐴𝑟𝑒𝑎)
𝐶𝑂𝑉 = = (√ ) /(𝑀𝑒𝑎𝑛 𝐴𝑟𝑒𝑎) Eq. 3-6
𝑀𝑒𝑎𝑛 𝐴𝑟𝑒𝑎 𝑠−1
where s is the number of temperature data points at each mole fraction.
Table 3-5: Examples of integrated areas and percent coefficient of variation (COV) for the
O-H absorbance and scaled absorbance.
Nominal Thermally Corrected
Mole fraction Cell
T [°C] Unscaled Scaled
x (1-butanol) Pathlength
[cm] Area / l [cm-2] COV Area / l [cm-2] COV
30 45100 1558
40 42030 1549
0.2005 0.002 11% 1.1%
50 38790 1537
60 35180 1520
30 16700 648.0
40 15110 656.2
0.0819 0.010 16% 1.1%
50 13300 660.3
60 11410 664.4
30 2609 170.9
40 1919 172.6
0.0218 0.050 45% 0.6%
50 1331 172.8
60 864.9 173.3
30 798.0 91.20
40 502.5 91.92
0.0112 0.100 55% 1.1%
50 324.5 91.91
60 222.8 89.81
Figure 3-10 and Table 3-5 demonstrate that, with the scaled spectra, the Beer-Lambert-
Bouguer law can be applied successfully to the entire hydroxyl absorption region permitting
50
quantification of the hydroxyl band to determine the alcohol concentration in solution. Stated
another way, the various free or H-bonded hydroxyl sites in a scaled spectrum are respectively
interpreted with the same weighting, just as NMR spectra show absorbances with areas directly
proportional to the amounts of the respective sites. To our knowledge, no other work has performed
a comparable quantitative analysis of the O-H region due to its known complexity. Currently, work
is under way to explore the analysis of experimental spectra from binary mixtures of ethanol in
cyclohexane using the same method. Preliminary data indicate that the functional form of the molar
attenuation coefficient observed for 1-butanol may be applied. The work presented here provides
a powerful tool to map IR band absorptions to concentration in solution, effectively placing all O-
H sites, regardless of their context, on a quantitatively equal footing.
3.9 Summary and Conclusions
In this work, the hydroxyl vibrational band was analyzed quantitatively for alcohol +
hydrocarbon mixtures using insights from QM simulations. First, MD simulations were carried
out to generate sample environments around hydroxyl groups. Then, selected clusters were further
analyzed with QM calculations to gain an understanding of their vibrational spectroscopic
characteristics. Evaluation of isolated vibrations on solvated linear oligomers provided insight on
the relations between bonding and vibrational frequency/intensity, but coupled vibrations were
included when developing the attenuation coefficient curve. Inspired by trends seen in the plots of
QM calculated absorption intensities vs. wavenumbers, an attenuation coefficient scaling function
was devised to normalize experimental spectra across the whole O-H region. This attenuation
coefficient curve was calculated using three line segments smoothed with splines at the
intersections. When experimental spectra for butanol-cyclohexane samples at various
concentrations and temperatures were scaled using this effective attenuation coefficient function,
51
the integrated areas of the whole O-H stretching region were found to quantitatively reflect the
known alcohol concentrations. Quantitation across the notoriously variable hydrogen bonding
region of the infrared spectrum, historically analyzed in qualitative terms or only in local segments,
represents a new and powerful tool. Ongoing work will explore extensions of this tool to additional
functional groups and mixtures, and their use to gain insight into physical properties and phase
behavior.
3.10 Acknowledgements
This work was supported by the National Science Foundation under Grant No. 1603705
and USDA National Institute of Food and Agriculture, Hatch/Multi-State project MICL04192.
This work was supported in part by Michigan State University through computational resources
provided by the Institute for Cyber-Enabled Research. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the author(s) and do not necessarily reflect
the views of the funding agencies. Helpful discussions with Chun-Min Chang of the MSU Institute
for Cyber-Enabled Research are gratefully acknowledged.
52
CHAPTER 4:Parameterization of a RTPT Association Activity Coefficient
Model using Spectroscopic Data
4.1 Preface
Self-associating species exhibit highly nonideal vapor-liquid phase behavior in many
mixtures, which is challenging to model. The most successful models, such as the statistical
associating fluid theory (SAFT) equations, are based on Wertheim’s first thermodynamic
perturbation theory (TPT-1). However, despite its success, the traditional TPT-1 implementations
of SAFT lack the requisite flexibility to account for the cooperative effects of hydrogen bonding
observed in alcohol + hydrocarbon systems. The resummed thermodynamic perturbation theory
(RTPT) provides an improved representation of these systems by considering hydrogen bond
cooperativity. While more robust than its TPT-1 predecessor, RTPT has only been validated
against computer simulations. In this work, we develop a form of the RTPT association
contribution to the activity coefficient and fit RTPT parameters to experimental spectroscopic data.
The agreement between the RTPT model and experimental infrared spectroscopy data is striking
despite requiring only one additional parameter compared to TPT-1. Calculated enthalpies indicate
the occurrence of positive hydrogen bond cooperativity in all the systems examined. Association
constants fitted to spectroscopic data are used to calculate activity coefficient contributions using
a Flory combinatorial term. For completeness, regressions of vapor-liquid equilibrium data are
included and utilize a non-random two-liquid (NRTL) model residual contribution.
4.2 Publisher Permission
Reprinted (Adapted or Reprinted in part) with permission from
Killian, W. G.; Bala, A. M.; Lira, C. T. Parameterization of a RTPT Association Activity
Coefficient Model Using Spectroscopic Data. Fluid Phase Equilib. 2022, No. 554, 113299.
https://doi.org/10.1016/j.fluid.2021.113299.
53
Copyright 2021 Elsevier
4.3 Introduction
Hydrogen bonding significantly influences the physical behavior of solutions. When
comparing associating fluids with those of similar molecular weight and atomic constitution that
do not associate, associating species have higher boiling points and enthalpies of vaporization, and
exhibit strongly nonideal behavior in mixtures, which is challenging to model. Since the early
1990s, association modeling using Wertheim's statistical mechanics99–102 has been integrated into
several equations of state (EOS), such as cubic plus association (CPA), as well as the statistical
associating fluid theory (SAFT) family of equations.42,103,104
The concept of hydrogen bond cooperativity has been proposed by a number of
spectroscopic studies when data are available over a range of 0 to 20 mol %.7,105–107 Similarly,
quantum mechanical calculations have shown that formation of the first hydrogen bond exhibits a
smaller magnitude association constant than subsequent hydrogen bonds (positive cooperativity)
.61,108–111 Marshall and Chapman recently developed the two-constant Wertheim approach for
cooperative bonding known as resummed thermodynamic perturbation theory (RTPT).112 The
study yielded an equation for the Helmholtz free energy and the model was compared to Monte
Carlo simulations but has not yet been evaluated with experimental data. Experimental methods
such as nuclear magnetic resonance (NMR) or Fourier Transform infrared (FTIR) spectroscopy
are commonly used to measure the extent of hydrogen bonding.105,111,113,77,78,34,114,53,115 Several
investigators have concluded that a more complete description of the spectroscopic data for
mixtures which contain alcohols requires two association constants7,34,77,78,116 due to cooperative
hydrogen bonding. The two-constant models also provide an improved representation of phase
equilibria and multicomponent heat of mixing.117–120
54
This work extends the RTPT model of Marshall and Chapman112 to an activity coefficient
model expression and to demonstrate the strength of the resulting model by comparing to infrared
(IR) spectroscopic data for binary alcohol + hydrocarbon systems. To this end, we fit the
association constants to IR data from literature, predict the association contribution to the activity
coefficients, and demonstrate fitting of the residual contribution to phase equilibria data. This
manuscript is organized as follows: First, a brief background is provided on the use of spectroscopy
for the determination of association constants. Then, the TPT-1 and RTPT methods are
summarized, and an overview of the methods used for solving the RTPT material balances is given.
We then derive the RTPT contribution of association to activity coefficients and outline the
procedures used for regressing infrared spectroscopy data and for modeling VLE data. Finally, we
discuss the results of the regressions and apply the fitted RTPT model to phase equilibria data as
proposed by Bala et al.121
4.4 Background
4.4.1 Hydrogen Bonding and IR Spectroscopy
Infrared spectroscopy provides quantification of hydrogen bonding by relating integrated
peak areas linearly to concentration through the Beer-Lambert law using an integrated molar
attenuation coefficient122. The fundamental stretching frequency of an alcohol’s hydroxyl group,
𝜈̃(𝑂𝐻), appears in the infrared region 3000-3800 cm-1 in the form of three peaks, a sharp higher
frequency peak and two broad lower frequency peaks. The absorbances are sensitive to local
molecular environments making IR an ideal probe for the measurement of hydrogen bonding. The
position of a particular alcohol molecule within the larger oligomer, as categorized in Figure 4-1,
largely determines the frequency at which that residue’s hydroxyl group will absorb.122 Alpha (𝛼)
hydroxyls are synonymous with unassociated monomer alcohol molecules. Located at one end of
55
linear aggregates are beta (𝛽) hydroxyls which also possess a non-bonded hydroxyl proton. As
recently as the early 2000s, confusion existed about whether the sharp peak that appears near 3650
cm-1 included a contribution from the unbound hydroxyl group (𝛽) at the end of linear aggregates.
von Solms et al.115 compared monomer fractions calculated using spectroscopic data from several
publications where the 3650 cm-1 peak was interpreted as monomer. Current interpretations
recognize that IR vibrational bands are more appropriately attributed to bonds7,53,89,123 as first
proposed by Hall and Wood,6 rather than to entire hydrogen bonded species. Furthermore, it is
clear today that 𝛼 and 𝛽 hydroxyls absorb in the same region with similar absorption intensity.122
At the opposite end of hydrogen-bonded chains are gamma (𝛾) hydroxyls which have a
bound proton and a non-hydrogen-bonded oxygen. Located between the 𝛽 and 𝛾 hydroxyls are the
delta-type (𝛿) hydroxyls which participate in two interactions; one hydrogen bond is donated at
the hydroxyl proton, and one hydrogen bond is accepted at the oxygen. The participation of 𝛿 in
two hydrogen bonds significantly weakens the parent covalent bond causing it to appear at lower
wavenumbers (red-shifted) relative to the other bond types with a significantly larger attenuation
coefficient.
Figure 4-1: Nomenclature for various bonding motifs found in alcohol solutions.
Older literature often interpreted red-shifted absorbances that appeared in response to
increased hydrogen bonding in terms of associated species95,115 (i.e., dimer, polymer) rather than
the bond types depicted in Figure 4-1. One such example from the work of Asprion et al.34,124 is
56
shown in Figure 4-2 and features a species-based evaluation. The following examples illustrate the
distinction between a species-based interpretation and one established using bond types. A linear
dimer contains one 𝛾 and one 𝛽 hydroxyl, and both contribute to the absorbance spectrum
producing two peaks instead of a single dimer peak. Similarly, a linear oligomer comprised of 𝑛
alcohol residues has one 𝛾, one 𝛽, and (𝑛 − 2) 𝛿 hydroxyls, and will contribute to peaks in each
of the three vibrational regions, indicating that chains contribute to the region labeled as monomer
in Figure 4-2. Finally, a cyclic aggregate without branching would contain only 𝛿 hydroxyls. The
availability of integrated peak areas from the work of Asprion et al., permits the measurements to
be reinterpreted in this work.
Figure 4-2: Curve fitting example from Asprion et al.,34 which assigns certain regions of the
infrared spectrum to aggregates of a specified size. In the figure, the absorbance (A´) is divided
by cell pathlength (d) and a tilde is used to denote the mole fraction of alcohol. The monomer
(Mo), dimer (Di), and polymer (Po) designations are not used in this work. Reprinted from34
with permission from Elsevier.
4.4.2 Wertheim’s Thermodynamic Perturbation Theory
Wertheim developed a novel approach to association which relied on a material balance
between acceptor and donor sites on molecules. The formation of a hydrogen bond, therefore,
57
consumes one acceptor site and one donor site. The simplest application of Wertheim's theory is
TPT-1, where bonding is determined by independent probabilities. The potential for generalized
application of Wertheim’s method was recognized by Chapman et al.,125 leading to the SAFT EOS.
The SAFT model uses a spherical reference fluid, modeling species as irreversibly bonded chains
and modeling reversible association between species with an association constant. The resulting
statistical associating fluid theory (SAFT) equation of state was extended by Gross and Sadowski
who replaced SAFT’s spherical molecule reference fluid with that of a hard-chain reference fluid
for the dispersion contribution resulting in the perturbed-chain statistical associating fluid theory
(PC-SAFT).103,126 The association term remains relative to a hard-sphere reference fluid. Three
component-specific parameters are used in PC-SAFT to relate to the reference fluid: 1) a segment
number (𝑚); 2) a segment diameter (𝜎); and 3) a segment energy (𝜖𝑖 /𝑘). Each association pair
𝐴𝐷
requires two association parameters: 1) an effective bonding volume (𝜅𝑖𝑗 ); and 2) a bonding
association energy (𝜖𝑖𝑗𝐴𝐷 /𝑘). Parameters for pure components are typically obtained by regression
of pure-component saturated vapor pressure and saturated-liquid density data.
The use of bonding sites permits different bonding schemes. Typically alcohols are
modeled using the 2B bonding scheme, which allocates two association sites to the host molecule;
one acceptor site on the oxygen and one donor site on the hydroxyl proton.126 The 2B scheme
restricts bonding to linear chains. In the typical TPT-1 implementation, a single association
constant, Δ𝑖𝑗 , is assumed for all oligomerizations with the form:
3 𝐴𝐷
Δ𝑖𝑗 = 𝑁𝐴 𝑑𝑖𝑗 𝑔𝑖𝑗 (𝑑)𝜅𝑖𝑗 (exp (𝜖𝑖𝑗𝐴𝐷 /(𝑘𝑇)) − 1) ) Eq. 4-1
where Δ𝑖𝑗 in this manuscript is in cm3/mol and 𝑑𝑖𝑗 is the temperature-dependent segment pair
diameter in Angstroms and Avogadro’s number, 𝑁𝐴, is included to as a unit conversion factor. The
58
hard sphere radial distribution function at contact, 𝑔𝑖𝑗 (𝑑) , is given by eq A7 of Gross and
Sadowski103, converted here to use molar density.
2
1 𝑑𝑖 𝑑𝑗 3𝜁2 𝑑𝑖 𝑑𝑗 2𝜁22
𝑔𝑖𝑗 (𝑑) = +( ) +( ) Eq. 4-2
(1 − 𝜁3 ) 𝑑𝑖 + 𝑑𝑗 (1 − 𝜁3 )2 𝑑𝑖 + 𝑑𝑗 (1 − 𝜁3 )3
𝜁𝑙 = (𝜋/6)𝑁𝐴 𝜌 ∑𝑖 𝑥𝑖 𝑚𝑖 𝑑𝑖𝑙 ; 𝑙 ∈ {2,3} Eq. 4-3
𝜖𝑖
𝑑𝑖 = 𝜎𝑖 [1 − 0.12 exp (−3 ( ))] Eq. 4-4
𝑘𝑇
4.4.3 The Resummed Thermodynamic Perturbation Theory
Sear and Jackson introduced a method for cooperative bonding deriving equations from
statistical mechanics.127 More recently, Marshall and Chapman provided a more rigorous
derivation and demonstrated that the primary difference between RTPT framework and the final
equation of Sear and Jackson’s work is the temperature dependence given by the Mayer term
instead of a Boltzmann term.112 To our knowledge, neither approach has been compared with
spectroscopic data.
In the RTPT approach, one association constant, Δ2 , represents the bonding of an acceptor
and donor site to form a dimer species. A second association constant, Δ𝑁, represents chain
formation of n-mers (𝑛 ≥ 3) and uses the same value for trimers and longer chains. Both
𝐴𝐷
association constants are given by the form in Eq. 4-1 but have distinct values for 𝜅𝑖𝑗 and 𝜖𝑖𝑗𝐴𝐷 /𝑘.
The expression for Helmholtz energy provided by Marshall and Chapman112 is
𝐴𝑎𝑠𝑠𝑜𝑐 𝜌 𝜌𝑜 𝐴
(𝑥1 𝜌𝑋 𝐴 )2 𝛥𝑐 (𝑜)
= 𝑥1 𝜌 𝑙𝑛 − 2𝑥1 𝜌𝑋 + + 𝑥1 𝜌 − Eq. 4-5
𝑅𝑇 𝑥1 𝜌 𝜌𝑜 𝑁𝐴 𝑉
59
As explained in APPENDIX D: Conversion of Extensive Helmholtz Energy to Molar, the
notation in this work uses molar Helmholtz energy, and molar densities. The quantity 𝜌1 = 𝑥1 𝜌 is
the apparent molar density of alcohol in the mixture. Here, 𝜌1 𝑋 𝐴 = 𝑥1 𝜌𝑋 𝐴 is the sum of the molar
densities of bonding species with unbonded acceptor site 𝐴, which is the apparent molar density
multiplied by the fraction of acceptor sites that are unbound, 𝑋 𝐴 . The density of unbound donor
sites is 𝑥1 𝜌𝑋 𝐷 . The density of alcohol monomer is 𝜌𝑜 where both the acceptor and donor sites are
unbound. By material balance, in a binary solution of alcohol + hydrocarbon, 𝑋 𝐴 = 𝑋 𝐷 . Because
the unbound donors and acceptors are equal in a binary mixture of alcohol + hydrocarbon, the
acceptor site balances are written herein even though most discussion will refer to free hydrogens
which are hydrogen bond donors.
The notation of Marshall and Chapman112 is transformed in this work to accommodate
empirical fitting of bonding volumes as part of the association constant, and thus we define the
(1) (2)
dimer association constant Δ2 ≡ 𝑁𝐴 𝑓𝐴𝐵 Δ, and the n-mer association constant Δ𝑁 ≡ 𝑁𝐴 𝑓𝐴𝐵 Δ,
where the right-hand side is Marshall and Chapman’s notation and the left-hand side is used here.
Adapting notation to this work, the last term of Eq. 4-5 is
Δ𝑐 (𝑜) (𝑥1 𝜌𝑋 𝐴 )2 Δ2
= Eq. 4-6
𝑁𝐴 𝑉 1 + (Δ2 − Δ𝑁 )𝜌𝑜
Note that Δ𝑐 (𝑜) is single term representing a perturbation and not an association constant.
The bonding equilibrium balance equations are obtained by the derivatives of Eq. 4-5 with respect
to 𝑥1 𝜌𝑋 𝐴 and 𝜌𝑜 as provided by Marshall and Chapman112 as eqs (17) and (18) in their work.
Adapting the notation to this work, these become
𝑥1 𝜌𝑋 𝐴 𝑥1 𝜌𝑋 𝐴 Δ2
−1= Eq. 4-7
𝜌𝑜 1 + (Δ2 − Δ𝑁 )𝜌𝑜
60
2
𝑥1 𝜌 𝑥1 𝜌𝑋 𝐴 (Δ2 − Δ𝑁 )(𝑥1 𝜌𝑋 𝐴 )2 Δ2
=( ) − Eq. 4-8
𝜌𝑜 𝜌𝑜 (1 + (Δ2 − Δ𝑁 )𝜌𝑜 )2
(see APPENDIX E: Key Material Balance Equationsfor the derivation manipulating Eq. 4-7 and
Eq. 4-8). Eq. 4-7 can be rearranged to identify the contributions of 𝛼 and 𝛽 hydroxyls to the free
site density.
𝑥1 𝜌𝑋 𝐴 𝛥2 𝜌𝑜
𝑥1 𝜌𝑋 𝐴 = 𝜌𝑜 + = 𝜌𝛼 + 𝜌𝛽 ;
1 + (Δ2 − Δ𝑁 )𝜌𝑜
Eq. 4-9
𝑥1 𝜌𝑋 𝐴 𝛥2 𝜌𝑜
𝜌𝛽 =
1 + (Δ2 − Δ𝑁 )𝜌𝑜
4.5 Methods and Derivations
4.5.1 Comparison of RTPT with Chemical Theory
Until the late 1980s, hydrogen bonding was described exclusively using chemical theory
and it continues to be used today.61,41,128,129 The RTPT model possesses striking similarities to the
2-constant Kretschmer-Wiebe model118,128, which is not apparent by the equations provided here.
A comparison is provided in the APPENDIX C: Relation of RTPT to Kretschmer-Wiebe. There
are four distinct differences between the equilibrium constants in the Wertheim method of
Chapman et al. compared to traditional chemical theory. First, the association constants relate the
bonding energy to the equilibrium constant via the Mayer function instead of the Boltzmann
function. Secondly, the Wertheim equilibrium constant exhibits a composition dependence
represented by the radial distribution function. As shown in APPENDIX C: Relation of RTPT to
Kretschmer-Wiebe, at a fixed temperature, if the composition dependence of the radial distribution
function is omitted and a Boltzmann term is used in place of the Mayer term, the association
constant reduces to an equivalent chemical theory concentration-based equilibrium constant, 𝐾𝑐 .
The third distinction from chemical theory is most important, being that the approach represents
61
association as a site interaction rather than a chemical reaction between species. The approach uses
probabilities of bonding and eliminates the need to write out all the chemical reactions between
species. Finally, the fourth distinction is that the Wertheim framework can be readily extended to
higher order perturbations.
4.5.2 Solution Procedure for XA
Marshall and Chapman solved a cubic equation for monomer density 𝜌𝑜 and then 𝑋 𝐴 . Here
we present an iterative method that simplifies the calculation by implementing a quadratic equation
instead. The monomer density is related to the free site fraction via the following quadratic
equation derived in the APPENDIX E: Key Material Balance Equations.
2𝑥1 𝜌𝑋 𝐴
𝜌0 = Eq. 4-10
1 + 𝑥1 𝜌Δ𝑁 𝑋𝐴 + √(1 + 𝑥1 𝜌Δ𝑁 𝑋𝐴 )2 + 4(Δ2 − Δ𝑁 )𝑥1 𝜌𝑋𝐴
Note that fraction of monomer 𝜌0 /(𝑥1 𝜌) remains finite even at infinite dilution of component (1)
where the limiting value is unity. The fraction of sites free is shown in the APPENDIX E: Key
Material Balance Equations to be
1
𝑋𝐴 = Eq. 4-11
1 + 𝑥1 𝜌Δ2 𝑋𝐴 /𝑠 2
𝑠 = 1 + (Δ2 − Δ𝑁 )𝜌0 Eq. 4-12
TPT-1 can be obtained from RTPT by setting Δ2 = Δ𝑁 , resulting in (𝑠 = 1) in Eq. 4-11 and Eq.
4-10 becomes 𝜌0 = 𝑥1 𝜌(𝑋 𝐴 )2 after inserting Eq. 4-11.
Excess volumes for liquid systems are very small and typically ignored in many common
models such as Flory’s equation and Scatchard-Hildebrand. Therefore, in this work, the ideal
solution density was assumed and the partial molar volume at all compositions is equivalent to the
62
pure component molar volume. Pure component densities were regressed for experimental
literature data and mixture densities and volumes were calculated using
1 𝑥1 𝑥2
= + Eq. 4-13
𝜌 𝜌𝑝𝑢𝑟𝑒,1 𝜌𝑝𝑢𝑟𝑒,2
1
𝑉𝑖 = 𝑉𝑝𝑢𝑟𝑒 𝑖 = Eq. 4-14
𝜌𝑝𝑢𝑟𝑒,𝑖
The procedure to solve iteratively for 𝑋 𝐴 uses several steps at a given apparent compositon
𝑥1 . First, the pure component liquid densities are calculated at the experimental temperature via a
polynomial fitted to experimental pure component densities. The mixture density is evaluated in
Eq. 4-13 and the association constants are evaluated via Eq. 4-1, Eq. 4-2, Eq. 4-3, and Eq. 4-4.
To initialize iterative soluion to Eq. 4-10, Eq. 4-11, and Eq. 4-12, the free site fraction 𝑋 𝐴
is assumed and the monomer density is calculated by Eq. 4-10. The free site fractions are refined
in Eq. 4-11 and Eq. 4-12 using the assumed value of 𝑋 𝐴 . A successive substitution reinserts 𝑋 𝐴
into Eq. 4-10, Eq. 4-11, and Eq. 4-12 until the set of values converges. The values of 𝜌𝑜 and 𝑋 𝐴
can then be used in other calculations as shown in later sections.
4.5.3 VLE Methods and Derivation of the RTPT Activity Coefficient Model
Vapor-liquid equilbria (VLE) can be characterized using the gamma-phi approach where
the fugacity coefficient (𝜑̂𝑘 ) and activity coefficient (𝛾𝑘 ) describe the non-ideality of the vapor
and liquid phases, respectively,
𝑦𝑘 𝜑̂𝑘 𝑃 = 𝑥𝑘 𝛾𝑘 𝑃𝑘𝑠𝑎𝑡 Eq. 4-15
Traditionally, the activity coefficient combines the association contribution with a
combinatorial and residual term, as seen in Eq. 4-16. The combinatorial term is entropically based
63
and accounts for differences in the size and shape of molecules, while the residual expression
captures the energetics of molecular interactions and other remaining effects.
ln 𝛾𝑘 = ln 𝛾𝑘𝑐𝑜𝑚𝑏 + ln 𝛾𝑘𝑟𝑒𝑠 + ln 𝛾𝑘𝑎𝑠𝑠𝑜𝑐 Eq. 4-16
Here, we use the Flory term for the combinatorial contribution
𝑉𝑘 𝑉𝑘
ln 𝛾𝑘𝑐𝑜𝑚𝑏 = ln + (1 − ) Eq. 4-17
𝑉 𝑉
and the NRTL is used for the residual term. For a binary mixture, this is given by:
2
𝜏𝑘𝑗 𝐺𝑘𝑗 𝐺𝑗𝑘
ln 𝛾𝑘𝑟𝑒𝑠 = 𝑥𝑗2 [ 2 + 𝜏𝑗𝑘 (𝑥 + 𝑥 𝐺 ) ] ; 𝑘 ≠ 𝑗
(𝑥𝑘 𝐺𝑘𝑗 + 𝑥𝑗 ) 𝑘 𝑗 𝑗𝑘 Eq. 4-18
𝐺𝑙𝑚 = exp(−𝛼𝑙𝑚 𝜏𝑙𝑚 ) ; 𝜏𝑙𝑚 = 𝑎𝑙𝑚 + 𝑏𝑙𝑚 /𝑇
where the empirical parameters 𝑎𝑙𝑚 , 𝑏𝑙𝑚 , 𝛼𝑙𝑚 = 𝛼𝑚𝑙 are adjusted to experimental data.
Recognizing that vapor phase deviations from an ideal gas are typically smaller than a
couple of percent for the conditions used herein, we represent the fugacity coefficients using the
Hayden-O’Connell equation of state.
The association contribution, ln 𝛾𝑘𝑎𝑠𝑠𝑜𝑐 , is the key term in Eq. 4-16 and the focus of this
work. As shown by Bala et al.37 when the species standard states are at the same T an P as the
mixture, liquid-phase activity coefficients are given by
1 𝜕𝐴𝑎𝑠𝑠𝑜𝑐 𝐴𝑎𝑠𝑠𝑜𝑐
ln 𝛾𝑘𝑎𝑠𝑠𝑜𝑐 = ( ) | − |
𝑅𝑇 𝜕𝑛𝑘 𝑇,𝑃,{𝑛 𝑅𝑇 𝑝𝑢𝑟𝑒 𝑘
𝑗≠𝑘 }
assoc
𝑚𝑖𝑥 Eq. 4-19
𝑃 𝜕𝑉 𝑃assoc
+ ( ) − 𝑉
𝑅𝑇 𝜕𝑛𝑘 𝑇,𝑃,𝑛 𝑅𝑇 𝑘
{𝑗≠𝑘}
1 𝜕𝐴𝑎𝑠𝑠𝑜𝑐 𝑉𝑘
( ) = ln 𝜑̂𝑘𝑎𝑠𝑠𝑜𝑐 − 𝑍 𝑎𝑠𝑠𝑜𝑐 Eq. 4-20
𝑅𝑇 𝜕𝑛𝑘 𝑇,𝑃,{𝑛 𝑉
𝑗≠𝑘 }
64
The activity coefficients are obtained by differentiation of Eq. 4-5 as summarized in APPENDIX
G: Activity Coefficients. The derivations in the original paper are unnecessarily constrained to
zero excess volume as explained by Bala, et al.130 and are accordingly revised here. The formulas
for a binary mixture, where component (1) is the alcohol and component (2) is the hydrocarbon,
are:
𝜌𝑜 𝜌𝑝𝑢𝑟𝑒 1
ln 𝛾1𝑎𝑠𝑠𝑜𝑐 = ln ( ) − (1 − 𝑋𝑝𝑢𝑟𝑒𝐴
1)
𝑥1 𝜌 𝜌𝑜,𝑝𝑢𝑟𝑒 1
𝑉1 𝜕 ln 𝑔11
+ 𝑥1 (1 − 𝑋 𝐴 ) ( (1 + ( ) )
𝑉 𝜕 ln 𝜌 𝑇,{𝑛 }
𝑖 Eq. 4-21
𝜕 ln 𝑔11
−𝜌( ) )
𝜕𝜌1 𝑇,𝑉,𝑛
2
𝑉2 𝜕 ln 𝑔11 𝜕 ln 𝑔11
ln 𝛾2𝑎𝑠𝑠𝑜𝑐 = 𝑥1 (1 − 𝑋 𝐴 ) ( (1 + ( ) )−𝜌( ) ) Eq. 4-22
𝑉 𝜕 ln 𝜌 𝑇,{𝑛 } 𝜕𝜌2 𝑇,𝑉,𝑛
𝑖 1
where 𝜌𝑖 represents molar density and the derivative terms are given by
𝜕 ln 𝑔11 1 𝜁3 𝑑1 3𝜁2 6𝜁2 𝜁3
( ) = ( 2
+ ( )( 2
+ )
𝜕 ln 𝜌 𝑇,{𝑛 𝑔11 (1 − 𝜁3 ) 2 (1 − 𝜁3 ) (1 − 𝜁3 )3
𝑖≠𝑘 }
Eq. 4-23
𝑑1 2 4𝜁22 6𝜁22 𝜁3
+( ) ( + ))
2 (1 − 𝜁3 )3 (1 − 𝜁3 )4
𝜕 ln 𝑔11
𝜌( )
𝜕𝜌𝑘 𝑇,𝑉,{𝑛
𝑖≠𝑘 }
1 (𝑛𝜁3,𝑛𝑘 ) 𝑑1 3(𝑛𝜁2,𝑛𝑘 ) 6𝜁2 (𝑛𝜁3,𝑛𝑘 )
= ( 2
+ ( )( + ) Eq. 4-24
𝑔11 (1 − 𝜁3 ) 2 (1 − 𝜁3 )2 (1 − 𝜁3 )3
𝑑1 2 4𝜁22 (𝑛𝜁2,𝑛𝑘 ) 6𝜁22 (𝑛𝜁3,𝑛𝑘 )
+( ) ( + ))
2 (1 − 𝜁3 )3 (1 − 𝜁3 )4
where
65
𝜕𝜁
(𝑛𝜁𝑙,𝑛𝑘 ) ≡ 𝑛 (𝜕𝑛𝑙 ) = (𝜋/6)𝑁𝐴 𝜌𝑚𝑘 𝑑𝑘𝑙 ; 𝑙 ∈ {2,3} Eq. 4-25
𝑘 𝑇,𝑉,{𝑛𝑖≠𝑘 }
To generate activity coefficients, the molar volumes, densities, and 𝑋 𝐴 are first calculated
at the mixture concentration and for the pure alcohol using the method of Section 4.5.2 above.
Then the radial distribution function derivatives and activity coefficients are calculated using Eq.
4-21 to Eq. 4-25.
For the binary case presented here, the form of the activity coefficient expressions can be
related to those resulting from TPT-1 presented by Bala and Lira37 if the general forms of the
monomer and radial distribution function derivative are used. The differences of the activity
coefficient expressions between RTPT and TPT-1 are in the relation between 𝜌𝑜 and 𝑋 𝐴 , rather
than the way that these quantities appear in the activity coefficient relations. For TPT-1, since 𝜌𝑜 =
𝑥1 𝜌(𝑋 𝐴 )2, the leading logarithm term appears as
2
𝜌𝑜 𝜌𝑝𝑢𝑟𝑒 1 𝑋𝐴 𝐴
ln ( ) = ln ( 𝐴 ) = 2 ln(𝑋 𝐴 /𝑋𝑝𝑢𝑟𝑒 1) Eq. 4-26
𝑥1 𝜌 𝜌𝑜,𝑝𝑢𝑟𝑒 1 𝑋𝑝𝑢𝑟𝑒 1
where it is written in the previous work as a sum over the acceptor and donor instead using the
factor of two on the acceptor expression.
The contribution of association to the excess Helmholtz energy is derived in APPENDIX
F: Excess Helmholtz Energy and is
𝑎𝑠𝑠𝑜𝑐
𝐴𝐸 𝜌𝑜 𝜌𝑝𝑢𝑟𝑒 1 𝐴 𝐴
( ) = 𝑥1 ln ( ) + 𝑥1 (𝑋𝑝𝑢𝑟𝑒 1−𝑋 ) Eq. 4-27
𝑅𝑇 𝜌1 𝜌𝑜,𝑝𝑢𝑟𝑒 1
66
Eq. 4-27 can be converted to the form typically cited for TPT-1 by inserting the TPT-1 monomer
density 𝜌𝑜 = 𝑥1 𝜌(𝑋 𝐴 )2. For practitioners, the infinite dilution values are important. Using Eq.
4-21 to Eq. 4-25, it can be shown that the activity coefficients at infinite dilution are given by
𝜌𝑝𝑢𝑟𝑒 1
ln(𝛾1𝑎𝑠𝑠𝑜𝑐 )∞ = ln ( ) − (1 − 𝑋𝑝𝑢𝑟𝑒𝐴
1)
𝜌𝑜,𝑝𝑢𝑟𝑒 1 Eq. 4-28
𝑉2 𝜕 ln 𝑔11
ln(𝛾2𝑎𝑠𝑠𝑜𝑐 )∞ = (1 − 𝑋𝑝𝑢𝑟𝑒
𝐴
1) ( (1 + ( ) | )
𝑉1 𝜕 ln 𝜌 𝑇,{𝑛 }
𝑖 𝑥2 =0
Eq. 4-29
𝜕 ln 𝑔11
− 𝜌1 ( ) | )
𝜕𝜌2 𝑇,𝑉,𝑛
1 𝑥2 =0
Note that ln(𝛾1𝑎𝑠𝑠𝑜𝑐 )∞ is independent of solvent and that ln(𝛾2𝑎𝑠𝑠𝑜𝑐 )∞ is related to the volume ratio
of the components. For a strongly associating component (1) in an inert solvent (2)
𝜌𝑝𝑢𝑟𝑒 1
ln(𝛾1𝑎𝑠𝑠𝑜𝑐 )∞ ~ ln ( ) Eq. 4-30
𝜌𝑜,𝑝𝑢𝑟𝑒 1
ln(𝛾2𝑎𝑠𝑠𝑜𝑐 )∞ ~ 𝑉2 /𝑉1 Eq. 4-31
When component (1) associates strongly, 𝜌𝑜,𝑝𝑢𝑟𝑒 1 is a small number and thus 𝛾1𝑎𝑠𝑠𝑜𝑐
𝐴 𝑎𝑠𝑠𝑜𝑐
becomes infinite when 𝑋𝑝𝑢𝑟𝑒 1 goes to zero but 𝛾2 has a limiting value. The radial distribution
function derivatives do not cancel in ln(𝛾2𝑎𝑠𝑠𝑜𝑐 )∞ unless the pure species parameters are extremely
similar, but the magnitude of the difference in terms is typically a minor contribution.
4.5.4 Parameterization of RTPT using Spectroscopic Data
To obtain association parameter values for the RTPT activity coefficient, we use
spectroscopic data for alcohol + alkane systems presented by Asprion et al.34,124 However, we
reinterpret the integrated monomer areas (see Figure 4-2) as the sum of 𝛼 and 𝛽 hydroxyls.
67
Fitting the spectroscopic data using TPT-1 requires the determination of three parameters:
the integrated Beer-Lambert molar attenuation coefficient (𝜀𝐵𝐿 ) and two parameters used to
𝐴𝐷
calculate Δ𝑖𝑗 : the effective association site volume (𝜅𝑖𝑗 ) and the association energy (𝜖𝑖𝑗𝐴𝐷 ). As
discussed earlier, a more complete description is provided by RTPT, which uses two association
constants of different values, Δ2 and Δ𝑁 , to account for hydrogen bond cooperativity. Our
implementation of RTPT requires only one additional parameter compared to TPT-1 for a total of
four parameters: the integrated molar attenuation coefficient, an association energy for the
formation of the dimer (𝜖2𝐴𝐷 ), an association energy for oligomers comprised of three or more
alcohol residues (𝜖𝑁𝐴𝐷 ), and an effective association site volume (𝜅 𝐴𝐷 ) which is the same for the
dimer and n-mer. In principle, 𝜅 𝐴𝐷 could differ for the dimer and n-mer, but we achieved
satisfactory fits using a common value and thus cannot justify an additional parameter. For the
pure component parameters used in Eq. 4-1, we selected the PC-SAFT values from the work of
Gross and Sadowski126 which are included in Table 4-1.
Table 4-1: PC-SAFT molecular parameters used in this work.
𝝐𝒊
Component 𝒎𝒊 𝝈𝒊 [Å] [𝑲]
𝒌
methanol 1.52550 3.23 188.90
ethanol 2.38270 3.1771 198.24
1-propanol 2.99970 3.2522 233.40
2-propanol 3.09290 3.2085 208.42
1-butanol 2.75150 3.6139 259.59
1-pentanol 3.62600 3.4508 247.28
1-hexanol 3.51460 3.6735 262.32
phenol 3.09089 3.4438 315.03
n-hexane 3.05760 3.7983 236.77
cyclohexane 2.53030 3.8499 278.11
68
To establish a relationship between absorbance and concentration, the Beer-Lambert law
in Eq. 4-32 is used.
𝐴𝐼 = 𝜀𝐵𝐿 𝑙𝐶 = 𝜀𝐵𝐿 𝑙𝑥1 𝜌𝑋 𝐴 Eq. 4-32
In this equation, 𝐴𝐼 is the integrated absorbance, 𝜀𝐵𝐿 is the integrated molar attenuation
coefficient, 𝑙 is the optical cell pathlength (instead of the variable 𝑑 used by Asprion), and 𝐶 is the
analyte concentration. Asprion’s tabulated AI/l values for ‘monomer’ were reinterpreted as
unbound hydrogens (𝛼 + 𝛽). The integrated molar attenuation coefficient was obtained by
rearranging the Beer-Lambert expression as seen in Eq. 4-33. In this form the concentration, or
equivalently, the molar density of unbonded hydroxyl protons, was expressed as 𝑥1 𝜌𝑋 𝐴 as
described earlier.
(𝐴𝐼 /𝑙)
𝜀𝐵𝐿 = Eq. 4-33
𝑥1 𝜌𝑋𝐴
The regression procedure for the association parameters was carried out as follows. First,
at each experimental composition, a value of 𝐴𝐼 /𝑙 was obtained from Asprion’s data. Then, using
guessed parameter values for the association constants, a corresponding value for the free site
density, 𝑥1 𝜌𝑋 𝐴 , was calculated using methods of Section 4.5.2. Next, we calculated the value of
𝜀𝐵𝐿 . In other studies, 𝜀𝐵𝐿 is sometimes determined from dilute measurements where association is
insignificant34, but that approach requires arbitrary selection of the dilute range. Instead, we used
a two-stage regression. During stage-1, each isotherm (𝑇𝑗 ) was regressed individually by adjusting
parameters to minimize the objective function in Eq. 4-34, which fits a value of 𝜀𝐵𝐿,𝑗 and one or
two Δ values for TPT-1 or RTPT, respectively.
𝑑𝑎𝑡𝑎 2
(𝐴𝐼 /𝑙)
𝑜𝑏𝑗 = ∑ (( ) − 𝜀𝐵𝐿,𝑗 ) Eq. 4-34
𝑥1 𝜌𝑋𝐴 𝑖
𝑖=1
69
Here, 𝜀𝐵𝐿,𝑗 is the mean value of the integrated molar attenuation coefficient across the
composition range for data at a single temperature. Once this stage is complete, this value is
averaged again for the three temperatures studied to obtain the parameter, 𝜀𝐵𝐿,𝑠1 . During stage-2,
the mean value of the stage-1 integrated molar attenuation coefficient (𝜀𝐵𝐿,𝑠1 ) was fixed for each
solute-solvent pair and replaced 𝜀𝐵𝐿,𝑗 in Eq. 4-34 while optimizing the 𝜅 𝐴𝐷 , 𝜖2𝐴𝐷 and 𝜖𝑁𝐴𝐷 for all
temperatures. Details of the regressions are provided by flowsheets in the APPENDIX H:
Regression Flow Diagram.
Regression of parameters for the Mayer form of the association constant presents
challenges like the well-studied Boltzmann/Arrhenius form. The pre-exponential and exponential
parameters in Eq. 4-1 are strongly interacting, and the implicit use of infinite temperature as a
reference temperature is problematic for regression. The proper way to obtain parameters and
confidence intervals is through the use of a reference temperature near the experimental
temperatures.131,132 Without the use of a reference temperature, the asymptotic confidence interval
for 𝜅 𝐴𝐷 spanned zero which is unphysical. Instead, a modified form of the association constants,
given in Eq. 4-35 and Eq. 4-36, was used which included a temporary parameter (𝑝) and a
temporary reference temperature (𝑇𝑟 ) which was set to 313.05 K.
𝜖𝑁𝐴𝐷 𝜖2𝐴𝐷
Δ2 = 𝑁𝐴 𝑑3 𝑔11 (𝑑)𝑝 exp ( ) (exp ( )−1) Eq. 4-35
𝑘𝑇𝑟 𝑘𝑇
3
𝜖𝑁𝐴𝐷 𝜖𝑁𝐴𝐷
Δ𝑁 = NA 𝑑 𝑔11 (𝑑)𝑝 exp ( ) (exp ( )−1) Eq. 4-36
𝑘𝑇𝑟 𝑘𝑇
Values for 𝑝, 𝜖2𝐴𝐷 , 𝜖𝑁𝐴𝐷 were obtained by regression after which we calculated 𝜅 𝐴𝐷 =
𝑝 exp(𝜖𝑁𝐴𝐷 /(𝑘𝑇𝑟 )). Because the 𝜅 𝐴𝐷 and Δ values are dependent on temporary parameters, the
95% confidence interval for these quantities were determined by the bootstrap (replacement by
70
resampling) method133, which involved calculating and sorting the values from 1000 bootstrap
fitting trials and selecting the 25th and 975th values as the lower and upper bounds of the confidence
interval, respectively. Due to the interacting parameters and nonlinearity, the optimum parameters
are not centered in the confidence intervals.
4.6 Results and Discussion
The primary goal of this work is to provide the derivation and implementation of an
association term that is consistent with the framework of RTPT as introduced by Marshall and
Chapman. In this section, we begin by demonstrating that TPT-1 is incapable of representing the
infrared data. The publication of this work134 included incorrect parameter values due to a coding
error that was discovered after publishing the work. The figures and tables in this chapter are
revised from the publication to provide results from the correct parameter values. The plots are
𝐴𝐷 𝐴𝐷
indistinguishable, and the values of 𝜖(2 𝑜𝑟 𝑁) are nearly the same but the 𝜅 values are noticeably
different.
4.6.1 Comparison of TPT-1 with RTPT
To compare the TPT-1 and RTPT models, a stage-1 regression was performed, and a
representative result featuring ethanol + cyclohexane is displayed in Figure 4-3. The inclusion of
one additional association energy term in RTPT for the formation of n-mers provides excellent
representation of the absorption while TPT-1 lacks the flexibility to capture the curvature of the
experimental data for ethanol above 2 mol%. Similar trends were observed in the other primary
alcohols as well as 2-propanol and phenol. Furthermore, the integrated molar attenuation
coefficients obtained from the RTPT stage-1 regression shows a smaller coefficient of variation
(3%) compared to TPT-1 (≥ 18%), which additionally exhibits an unexpectedly strong temperature
dependence (Figure 4-4). As part of a final effort to fit the data using TPT-1, we performed a stage-
71
2 regression of TPT-1 using the average 𝜀𝐵𝐿,𝑠1 obtained from the stage-1 regression of RTPT. This
also resulted in an unsatisfactory fit as seen in Figure 4-5. Therefore, all subsequent regressions
were performed with RTPT.
Figure 4-3: Stage-1 regression using RTPT and TPT-1 for ethanol + cyclohexane binary
system. RTPT fits the data well (–), while TPT-1 (– –) is unable to replicate the curvature of
the experimental data.
Figure 4-4: Average integrated molar attenuation coefficients,εBL,all j, which resulted from the
stage-1 regression of ethanol + cyclohexane data using TPT-1 and RTPT.
72
Figure 4-5: Stage-2 regression of TPT-1 (–) using stage-1 RTPT integrated molar attenuation
coefficient for ethanol + cyclohexane system demonstrating inability of TPT-1 to fit the data as
explained in the text.
Stage-2 regression was performed on each solute-solvent pair as seen in Figure 4-6 to Figure 4-10,
where Figure 4-10 presents an example of a satisfactory fit with RTPT for 1-hexanol +
cyclohexane. A complete list of parameters obtained from the regressions as well as their 95%
confidence intervals are presented in Table 4-2. The regressed values are coupled to the pure
component parameters summarized in Table 4-1, and should not be directly transferred to other
pure component parameters.
73
Figure 4-6: RTPT fit compared to spectroscopic data with parameters regressed using the two-
stage method for the systems of ethanol + n-hexane (left) and ethanol + cyclohexane (right) at
three temperatures.
Figure 4-7: RTPT fit compared to spectroscopic data with parameters regressed using the two-
stage method for the systems of 1-propanol + n-hexane (left) and 2-propanol + n-hexane
(right) at three temperatures.
74
Figure 4-8: RTPT fit compared to spectroscopic data with parameters regressed using the two-
stage method for the systems of 1-butanol + n-hexane (left) and 1-hexanol + cyclohexane
(right) at three temperatures.
Figure 4-9:RTPT fit compared to spectroscopic data with parameters regressed using the two-
stage method for the systems of 1-pentanol + n-hexane (left) and phenol + n-hexane (right) at
three temperatures.
75
Figure 4-10:RTPT fit compared to spectroscopic data with parameters regressed using the two-
stage method for the systems of 1-hexanol + n-hexane (left) and 1-hexanol + cyclohexane
(right) at three temperatures.
76
Table 4-2: Regressed association parameters using RTPT and the two-stage regression method.
𝜿𝑨𝑫 , [cm3/mol] 𝝐𝑨𝑫
𝟐 /𝒌 , [K] 𝝐𝑨𝑫
𝑵 /𝒌 , [K]
Solute Solvent Value 95% C.I. Value 95% C.I. Value 95% C.I.
n-hexane 6.413E-02 2.586E-02 - 1.745E-01 1568 1241 - 1834 2435 2141 - 2697
methanol
cyclohexane 8.264E-01 1.383E-01 - 1.632E+00 659.7 349.3 - 1221 1693 1500 - 2226
n-hexane 1.606E-01 6.234E-02 - 4.710E-01 1355 1041 - 1647 2220 1907 - 2502
ethanol
cyclohexane 3.153E-03 8.299E-04 - 9.722E-03 2320 2021 - 2659 3341 3000 - 3739
1-propanol n-hexane 1.049E-02 4.238E-03 - 2.306E-02 2117 1851 - 2412 2930 2689 - 3202
2-propanol n-hexane 1.388E-02 6.356E-03 - 3.260E-02 2111 1860 - 2317 2849 2590 - 3078
n-hexane 4.087E-02 2.041E-02 - 7.603E-02 1528 1334 - 1701 2425 2237 - 2634
1-butanol
cyclohexane 1.884E-02 6.401E-03 - 3.932E-02 1801 1554 -2149 2630 2408 - 2938
1-pentanol n-hexane 8.840E-03 3.393E-03 - 2.490E-02 2244 1947 - 2517 2984 2679 - 3271
n-hexane 1.996E-02 1.227-02 - 3.190E-02 1817 1683 - 1966 2659 2520 - 2804
1-hexanol
cyclohexane 1.262E-02 8.164E-03 - 2.021E-02 1903 1770 - 2044 2741 2603 - 2876
phenol n-hexane 6.549E-02 3.534E-02 – 1.605E-01 1668 1390 - 1865 2289 1612 - 2369
77
Values presented in Table 4-2 for the association parameters and those presented in a later section
(Table 4-3) are slightly different than published values due to an error in calculating the
preexponential term. This error has been addressed and all values presented in this document have
been corrected.
4.6.2 Delta Comparison with Solvent Trends
To better understand the influence of the inert solvent on association, RTPT Δ-values were
calculated for each of the binary pairs at 298.15 K at an alcohol mole fraction of one. A plot of Δ2
and Δ𝑁 versus carbon chain length is provided in Eq. 4-11 and Eq. 4-12 for the first six primary
alcohols. The results of the error analysis are included to convey the bootstrap 95% confidence
intervals. We encountered some challenges with regressing the methanol data, which is reflected
in the deviation of the parameter values and confidence intervals from trends observed in the other
systems. A more detailed discussion of these challenges is provided later. The Δ values associated
with the formation of the n-mer were larger in magnitude than the corresponding dimer value in
all cases. This is consistent with the understanding of positive cooperability which postulates that,
once the first hydrogen bond has formed, subsequent hydrogen bonds become more energetically
favorable, hence Δ𝑁 ≫ Δ2 .110,111,119
The solvent appears to have a minimal effect on Δ2 as the data from both solvents is
clustered around a mean value near 1024 cm3/mol. Except for 1-pentanol the value of Δ2 appears
relatively constant and virtually independent of solvent. The Δ𝑁 values displayed a less consistent
trend than those obtained for Δ2 . Values obtained in n-hexane are generally larger than those
obtained in cyclohexane. Average values are 15710 cm3/mol and 16630 cm3/mol for solvents n-
hexane and cyclohexane, respectively. Δ𝑁 increases by about 20% between 1-propanol and 1-
78
hexanol when n-hexane is the solvent. Based on these results, solvent interactions appear to have
a greater influence on the formation of oligomers larger than dimers.
Figure 4-11: Δ2 versus carbon chain length Figure 4-12: Δ𝑁 versus carbon chain length
with error bars for primary alcohols methanol with error bars for primary alcohols methanol
through 1-hexanol in cyclohexane and n- through 1-hexanol in cyclohexane and n-
hexane at 298.15 K and pure alcohol hexane at 298.15 K and pure alcohol
concentration. concentration.
The methanol + alkane systems show an unusual trend in regressed parameters compared
to larger alcohols: the 𝜅 𝐴𝐷 value is relatively large and the 𝜖2𝐴𝐷𝑜𝑟 𝑁 values are relatively small.
Figure 4-13 shows a decrease in the free hydrogen 𝐴𝐼 /𝑙 values between 8 and 14 mol%; this trend
was observed for both methanol systems. Similar behavior was observed for both ethanol systems
and 1-butanol + n-hexane at high concentrations, though the decrease in the latter was minimal.
As the concentration of hydroxyls is increased, the 𝜈̃(𝑂𝐻) region becomes congested with
overlapping absorbances from several types of hydrogen bonds thereby making definitive
assignment difficult. This issue is not uncommon and has been addressed by Meijer135, Barlow et
al.,136 and Wandschneider et al.61 who note that in these situations the problem is under-constrained
and comparable fits can be obtained from several different interpretations depending on the initial
79
guess. We attribute the decrease in the 𝐴𝐼 /𝑙 values to the absorbance peak fitting procedure when
subtracting the overlap of the 𝛾 (designated by Asprion as dimer) and (𝛿) (designated by Asprion
as polymer) with the free hydrogen (𝛼 + 𝛽) peak resulting in an underestimation of the free
hydrogen population. This is a possible explanation for the challenges encountered in fitting the
methanol systems and the unusual values of regressed 𝜅 𝐴𝐷 and 𝜖2𝐴𝐷𝑜𝑟 𝑁 compared to other systems.
These observations highlight an opportunity to revisit the partitioning of the hydroxyl region in
future spectral measurements using the scaling methods of Bala et al.122
Figure 4-13: RTPT fit compared to spectroscopic data with parameters regressed using the two-
stage method for the systems of methanol + n-hexane (left) and methanol + cyclohexane (right)
at three temperatures. Poor quality data for is observed at high alcohol concentrations for both
solvents, resulting in inferior model parameters.
4.6.3 Enthalpy of Positive Cooperativity
To better understand the thermodynamics of association, the enthalpies of association for
the dimer (Δ𝐻2 ) and n-mer (Δ𝐻𝑁 ) were calculated using Δ2 𝑜𝑟 𝑁 values obtained from stage-2 of
the RTPT regression and the van’t Hoff relation.
80
𝑑 ln Δ2 𝑜𝑟 𝑁
Δ𝐻2 𝑜𝑟 𝑁 = 𝑅 Eq. 4-37
1
𝑑 (𝑇 )
To provide a basis for comparison, all enthalpies were calculated using Δ2 𝑜𝑟 𝑁 values at 298.15 K
and pure alcohol composition. Table 4-3 summarizes the results and shows that the dimer
association enthalpy is about 70-75% of the n-mer value. The average enthalpy for the formation
of a dimer across all systems studied was -16.6 kJ/mol. This value is 6.9 kJ/mol smaller in
magnitude than the average n-mer enthalpy of -23.6 kJ/mol. The calculated enthalpies do not show
a clear trend with carbon number.
Comparison with literature must be made against chemical theory approaches that use two
concentration-based association constants (𝐾𝐶 ) since no other experimental RTPT fits exist. To
our knowledge, the only work meeting this criteria for primary alcohols is Kretchmer and Wiebe120
who modeled alcohol vapors using dimers and tetramers. They reported values of -16.7 kJ/mol for
Δ𝐻2 and values of -92.4, -84.1, and -94.6 kJ/mol for cyclic hydrogen-bonded tetramers of
methanol, ethanol, and 2-propanol, respectively. Subtracting their dimer value from a tetramer and
dividing the remainder by the three remaining bonds results in average n-mer enthalpies, Δ𝐻𝑁 , of
(-92.4-(-16.7))/3 = -25.3, -22.5, and -26.0 kJ/mol per hydrogen bond, respectively. These values
are consistent with the average dimer value of -16.6 kJ/mol and n-mer value of -23.6 kJ/mol per
hydrogen bond presented in this work. These findings further highlight the differences between
the hydrogen bonds of dimers and those formed in higher-order oligomers.
81
Table 4-3: Enthalpies of association based on extrapolation to
purity.
298.15 K, 𝒙𝒂𝒍𝒄𝒐𝒉𝒐𝒍 = 𝟏
−𝚫𝑯𝟐 −𝚫𝑯𝑵 −𝚫𝑯𝟐
Solute Solvent
−𝚫𝑯𝑵
[kJ/mol] [kJ/mol]
n-hexane 15.37 22.52 0.683
methanol
cyclohexane 8.427 16.39 0.514
n-hexane 13.38 20.47 0.654
ethanol
cyclohexane 21.29 29.77 0.715
1-propanol n-hexane 19.58 26.32 0.744
2-propanol n-hexane 19.60 25.73 0.762
n-hexane 14.72 22.09 0.666
1-butanol
cyclohexane 16.94 23.80 0.712
1-pentanol n-hexane 20.48 26.62 0.769
n-hexane 16.90 23.88 0.708
1-hexanol
cyclohexane 17.62 24.56 0.717
phenol n-hexane 15.40 20.52 0.750
4.6.4 Activity Coefficient Contributions
After fitting association parameters to the spectroscopic data, the association contribution to the
activity coefficients can be calculated using Eq. 4-21 and Eq. 4-22. This contribution can then be
combined with the residual and combinatorial contributions using Eq. 4-16 to complete the activity
coefficient calculation. For the mixtures studied here, the activity coefficient is dominated by the
82
association contribution and the combinatorial contribution is minor. To illustrate the importance
of the association effects, we omit the residual term entirely for the mixtures shown in this section
and predict the activity coefficient using the association parameters shown in Table 4-2 for 𝛾 𝑎𝑠𝑠𝑜𝑐
and Flory’s model for 𝛾 𝑐𝑜𝑚𝑏 .
Figure 4-14 shows an excellent prediction of the activity coefficients for the system 1-
hexanol + cyclohexane using only the association and combinatorial contributions. For this system,
the combinatorial contribution is so small that it is not discernable when plotted relative to the
association contribution, thus only the sum of the contributions is plotted. The prediction of the
activity coefficient from only the association contribution is striking; no empirical adjustment of a
residual contribution is included. Because the activity coefficient represents the behavior of the
mixture relative to the pure species, the infinite dilution activity coefficient for alcohol is
dependent on the extent of hydrogen bonding in the pure alcohol (Eq. 4-28). The predicted
behavior of the activity coefficients for the associating component are of particular interest in
Figure 4-14. The 1-hexanol infinite dilution activity coefficient is approached with a slope that
decreases in magnitude as infinite dilution is approached as illustrated by the inset plot. Clearly,
the infinite dilution value is not approached asymptotically from below but rather from the side,
and the slope change occurs below about 2 mol% alcohol (inset of Figure 4-14).
83
Figure 4-14: Activity coefficient data at 70 °C for the system of 1-hexanol + cyclohexane
overlaid with the predicted activity coefficients resulting from the sum of the RTPT association
term and the Flory combinatorial. Experimental data is from Svoboda et al.137
The change of slope is due to the smaller association constant of the dimer compared to the n-mer
and cannot be modeled with a TPT-1 single association constant which approaches infinite dilution
with a large magnitude slope.
The association and combinatorial contributions are displayed alongside experimental data
in Figure 4-15 for the methanol + cyclohexane system. For this system, the combinatorial
contribution is discernable on the plot, but still minor. While the representation of the experimental
data is satisfactory for most of the mole fractions, there is a noticeable overprediction of the infinite
dilution value for methanol. We believe that this behavior is attributable to the challenges
discussed previously regarding the spectroscopic data for this system. The decrease in tabulated
absorbance above 8 mol% reported for this system effectively depresses the calculated values of
𝑋 𝐴 as mole fraction increases, leading to an overrepresentation of the degree of hydrogen bonding
when extrapolated to pure alcohol. This leads to an overprediction of the activity coefficient at
infinite dilution. Similar behavior occurs in the ethanol systems and thus uncertainties for these
84
systems may be larger than the reported confidence intervals. Because the concentration of
hydroxyls is more significant for shorter alcohols, hydrogen bonding is more prominent compared
to longer alcohols at the same alcohol mole fraction, and interference from the 𝛾 and 𝛿 hydroxyls
in the absorbance spectra appears at lower mole fractions for a given temperature in these systems,
complicating accurate determination of monomer.
Figure 4-15: Contributions to the overall activity coefficient (–) for methanol (blue) +
cyclohexane (black) system at 55 °C compared to the data of Morachevskii.138 The solid line is
the sum of the combinatorial term (– - –) and predicted association term (– –). Infinite dilution
values were calculated by linearly regressed data provided by Lazzaroni et al.139 and denoted
with red symbols.
4.6.5 Extension to Vapor Liquid Equilibrium
For chemical process design, accurate representation of phase equilibria data is critical.
The association contribution of Eq. 4-16 is dominant but some adjustment is necessary for precise
modeling. The residual term is small but integral to precise process engineering modeling. Asprion
et al.117 selected the UNIQUAC method for accurate fitting. In this work, we demonstrate that the
combination of RTPT with the NRTL residual model results in precise fitting of experimental data
over industrially relevant temperature ranges. Figure 12 demonstrates fitting for the system 1-
85
hexanol + cyclohexane, and Figure 4-17 demonstrates fitting for 1-butanol + cyclohexane. The
vapor phase virial coefficient Hayden-O’Connell parameter, 𝜂, used for regression is 2.2 for 1-
butanol and 1-hexanol. The non-randomness parameter, 𝛼𝑖𝑗 , was set to 0.3 for both systems. The
residual adjustment is minor in both cases and the fits over the temperature ranges are excellent
using only a small temperature-independent NRTL contribution, representing a correction to the
combinatorial contribution at infinite temperature. The temperature dependence of the VLE data
is captured entirely by the association contribution predicted from the spectroscopic fits, and no
temperature-dependent residual contribution is needed.
Figure 4-16: Results of Aspen regression using association parameters obtained from
experimental infrared data of Asprion for 1-hexanol + cyclohexane. Experimental data are
from Svoboda et al.137
86
Figure 4-17: Regression results for 1-butanol + cyclohexane system using the association
parameters obtained from the data of Asprion. Experimental data are from Smirnova and
Kurtunina.140
4.7 Summary and Conclusions
This work derives a form of the association contribution to the activity coefficient that is
consistent with the resummed thermodynamic perturbation of Wertheim’s theory developed by
Marshall and Chapman. RTPT accounts for positive cooperativity in hydrogen bonding and
expresses association effects using two constants, one for the formation of a dimer and another for
all subsequent bonds formed. Further, we reinterpret Asprion’s spectroscopic measurements in
terms of hydrogen bonding motifs and evaluate the RTPT and TPT-1 thermodynamic models.
RTPT provides a better description of the data when compared to TPT-1 using only one additional
fitted parameter. The regression resulted in the prediction of enthalpy which averages -16.6 kJ/mol
for dimer and subsequent hydrogen bonds have an average per bond enthalpy of -23.6 kJ/mol.
Both enthalpies are consistent with values of Kretschmer and Weibe. The association constants for
the dimer are found to be approximately 1024 cm3/mol in both n-hexane and cyclohexane at 298.15
K. The average association constants for n-mer are 15710 cm3/mol and 16630 cm3/mol for solvents
87
n-hexane and cyclohexane, respectively. The Δ𝑁 exhibits a 20% increase between 1-propanol and
1-hexanol when n-hexane is the solvent but are independent of carbon number in cyclohexane. A
form for the association activity coefficient was derived in a manner consistent with RTPT.
Furthermore, the parameters obtained from the regression were used to satisfactorily predict
activity coefficients and phase behavior for selected binary alcohol + hydrocarbon systems.
Examination of the experimental spectroscopic data for three systems exhibit a decrease in
absorbance at concentrations above 8 mol%, suggesting that uncertainty beyond the statistical
results is probable, and improved methods of curve fitting are required for future measurements.
4.8 Acknowledgements
This material is based upon work supported by the National Science Foundation under
Grant No. 1603705 USDA National Institute of Food and Agriculture, Hatch/Multi-State project
MICL04192. Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views of the funding agencies.
We thank Norbert Asprion for sharing of the tabulated monomer data from his thesis and the
reviewers for helpful comments.
88
CHAPTER 5:Infrared Quantification of Ethanol and 1-Butanol Hydrogen
Bonded Hydroxyl Distributions in Cyclohexane
5.1 Preface
Quantifying the mid-range infrared hydroxyl stretch absorbance region has traditionally
been a challenge due to the wavenumber dependence of the attenuation coefficient. Interpretation
often assigns a single attenuation coefficient to each type of hydrogen-bonded aggregate. This
work leverages a recently developed technique of scaling hydroxyl stretching absorbances in the
mid-infrared region with a continuous attenuation coefficient function that produces integrated
areas, which directly correlate to hydroxyl concentrations. After scaling, the hydroxyl absorbance
is fitted with five curves, of which four are dominant. These four curves represent unique hydroxyl
configurations and translate to specific aggregate structures. The technique is applied to ethanol
and 1-butanol. The resulting population distributions of hydrogen-bonded hydroxyl configurations
are compared with the resummed thermodynamic perturbation theory (RTPT) model for linear
chains as a function of concentration and temperature. The model is demonstrated to capture the
critical features of the distributions.
5.2 Publisher Permission
Reprinted (Adapted or Reprinted in part) with permission from
Killian, W. G.; Bala, A. M.; Norfleet, A. T.; Peereboom, L.; Jackson, J. E.; Lira, C. T. Infrared
Quantification of Ethanol and 1-Butanol Hydrogen Bonded Hydroxyl Distributions in
Cyclohexane. Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 2023,
285, 121837. https://doi.org/10.1016/j.saa.2022.121837.
Copyright 2022 Elsevier
89
5.3 Introduction
Given its broad importance in the chemical and biological sciences, hydrogen bonding has
been the subject of many notable works over the century since its recognition.4,5,142 Yet, despite
great strides in mapping its molecular behavior, a complete and predictive understanding of the
bulk effects of hydrogen bonding remains elusive. Quantification and characterization of the
specific hydrogen-bonded aggregates are needed to test theoretical knowledge and practical
applications such as modeling the phase equilibria of liquid mixtures. For associating compounds
such as alcohols, effective representation of phase equilibria necessitates accounting for hydrogen
bonding, typically in the form of association parameters. These model parameters depend strongly
on the chemical and physical interactions of the molecules, motivating quantitative spectroscopic
analysis to probe them.115
For thermodynamic modeling of an associating substance, a fundamental property is the
extent of its association at equilibrium. For alcohols, this can be expressed in the fraction of
hydroxyls that remain unbonded (i.e., not involved in hydrogen bonding).115 Different techniques,
including nuclear magnetic resonance (NMR), near-infrared spectroscopy (NIR), and mid-infrared
spectroscopy (MIR), have been used to assess this ‘monomer fraction’. The application of NMR
is limited for quantification due to its long measurement timescale, which collapses all hydroxyl
resonances to a single chemical shift value. Thus, essential details are unavailable via NMR as the
chemical shifts of individual hydrogen-bonded clusters (dimers and oligomers) are unknown. As
for NIR spectroscopy, while the measurement timescale is appropriate the spectra show heavily
overlapped vibrational overtone bands, which are difficult to analyze.143
Mid-infrared spectroscopy (henceforth, IR) remains one of the most popular spectroscopic
methods for studying the self-association of alcohols14,34,54,105,106,123,144–147 since it provides a high
90
level of detail in the 𝜈̃(𝑂𝐻) stretch region. However, in an IR spectrum of a sample containing
alcohol molecules in monomeric and aggregated arrangements, correctly assigning the constituent
absorbances among the fundamental 𝜈̃(𝑂𝐻) bands has proved challenging. More problematic is
that quantitative analysis of all resonances in the overall hydroxyl region (3000-3750 cm-1) has
been hampered by an inability to determine the wavelength dependence of the molar attenuation
coefficient. The hydroxyl region is also strongly temperature-dependent, diminishing confidence
in the peak integration necessary to quantify unbonded hydroxyls. To address this issue, Bala et
al.122 introduced a wavelength-dependent molar attenuation coefficient function with a form based
on trends computed via quantum chemical simulations. The function was then optimized and
refined using variable concentration and temperature experimental data to provide a temperature-
independent integrated hydroxyl absorbance. The present work aims to apply curve-fitting to the
absorbances in the scaled spectra to obtain populations of specific hydroxyl types from the
alcohol’s 𝜈̃(𝑂𝐻) absorbance. Further, we compare the experimental populations with predictions
from Marshall and Chapman’s resummed thermodynamic perturbation theory (RTPT),112,134
which captures essential features of the observed hydroxyl distributions.
5.4 Background
There are many infrared-active vibrational modes present in alcohols; however, the most
useful mode for the study of alcohol self-association is the hydroxyl stretching vibration, ν̃(OH),
which appears in the 3000 to 3750 cm-1 region of the IR spectrum. The effect on this covalent bond
upon forming a hydrogen bond is readily observed. Specifically, participation in a hydrogen bond
redistributes the electron density within the O-H bond, thereby lowering and broadening its
absorption frequency and increasing its infrared attenuation constant. Even small changes in
alcohol concentration or measurement temperature result in significant variations among the free
91
and hydrogen-bonded hydroxyl populations in the landscape of the 𝜈̃(𝑂𝐻) region. This sensitivity
to association makes it an ideal reporter for quantifying the monomer fraction. IR enables the
various hydrogen bonded O-H sites to be categorized and enumerated according to their positions
within a particular oligomer.7,53,89,123 This approach was first proposed by Hall and Wood6 and has
led to the definition of the distinct hydrogen-bonded hydroxyl configurations shown in Figure 5-1:
alpha (α), beta (β), gamma (𝛾), and delta (𝛿).106 Lastly, eta (𝜂), and zeta (𝜁) hydroxyls are also
present in alcohol-containing solutions and have been identified in molecular dynamics
simulations.148
Figure 5-1: Hydroxyl types observed in liquid alcohols. The number of hydrogen bonds
received by the oxygen (Oi) and donated by the proton (Hj) for each hydroxyl type is denoted
in superscript.
When alcohol is dilute (less than approximately 0.2 mol%) in a non-hydrogen-bond-
forming (inert) solvent such as cyclohexane, the alcohol spectra exhibit a sharp singlet near 3640
cm-1 with a low wavenumber shoulder. As the concentration of alcohol is increased, dimers begin
to form, producing 𝛽 and 𝛾 hydroxyls. The 𝛽 hydroxyls (O1H0 as shown in Figure 5-1, where the
superscripts indicate the number of participations in hydrogen bonds) are located at the terminus
of a hydrogen-bonded chain and receive a hydrogen bond at the oxygen while the hydroxyl proton
remains free. A 𝛾 hydroxyl is located on the chain end opposite of 𝛽 and consists of a non-
92
hydrogen-bonded oxygen attached to a hydrogen-bonded proton (O0H1). Molecular dynamics and
quantum mechanics simulations106,122 of alcohol clusters indicate that the frequency of 𝛼 and 𝛽
hydroxyls appear in the same wavenumber region of the infrared spectrum and contribute to the
sharper peak at ~3640 cm-1 while 𝛾 hydroxyls are visible near 3500 cm-1.
As the concentration of alcohol is increased further, the 𝛾 absorbance increases, and a broad
peak emerges in the absorbance spectra centered near 3325 cm-1 as the population of chains
increases. The 𝛿 hydroxyls, which appear in the center of chains, contribute to this broad
absorbance; their resonance is the most red-shifted relative to the non-bonded 𝛼 sites. In linear
oligomers, 𝛿 hydroxyls are hydrogen-bonded at the oxygen and the hydrogen (O1H1), an
arrangement that significantly polarizes the electron density within the parent O-H bond. In
addition, 𝛿 hydroxyls feature a broad Gaussian-like149 distribution of frequencies since each
covalent bond is weakened to a different extent depending on their position or geometric
orientation within the “floppy” hydrogen-bonded chain.
Collectively, the 𝜈̃(𝑂𝐻) region is sensitive to subtle changes in hydrogen bonding induced
by altering the alcohol concentration or sample temperature. Temperature is a measure of the
average kinetic energy of the molecules. Thus, an increase in temperature indicates that more
alcohol molecules are able to escape the hydrogen bonding potential. This manifests as a decrease
in 𝛿 hydroxyls coupled with a concurrent increase in the ‘free peak’ produced by 𝛼 and 𝛽
hydroxyls.
Positively cooperative hydrogen bonding occurs due to the polarization of the O-H bond,
allowing the hydroxyl proton to become a more favorable hydrogen bond donor. Partial charges
within the dimer are stabilized by adding a monomer to form a trimer. This significant difference
in the bonding interaction of the dimer relative to other oligomerizations has been noted by other
93
researchers7,61,105–107 and warrants special consideration through the use of two association
parameters; one describing the effects of dimer formation and a second to other capture the
formation of larger oligomers.
5.5 Methods
5.5.1 Infrared Measurements
Anhydrous ethanol, 1-butanol, and cyclohexane were obtained from Sigma-Aldrich at
purities of 99.5%, 99.8%, and 99.5%, respectively. Ethanol and 1-butanol were stored over
activated 3 Å molecular sieves for at least 72 h before use, and no further purification was
performed. Solutions of alcohol + cyclohexane were prepared volumetrically using type-A
glassware and Hamilton gas-tight syringes in a glovebox under a nitrogen atmosphere.
Experimental concentrations were calculated by assuming ideal solution behavior, which neglects
the excess volume of mixing. Pure component molar densities for mixing solutions were calculated
using a second-order polynomial fitted to pure component data from the NIST ThermoData
Engine.150 The coefficient of determination was higher than 0.99 for all components.
Infrared spectra were collected in absorbance mode using a Jasco FT/IR-6600
spectrophotometer equipped with a Ge/KBr beam splitter and DLaTGS detector. Before the
background spectrum was collected, the sample compartment was purged with nitrogen for 30
min. The nitrogen purge was maintained for all subsequent measurements. All measured spectra,
including the background, were collected using 128 scans at 2 cm-1 resolution using a Specac
demountable liquid flow cell (model GS20582) equipped with CaF2 windows. Cell temperature
was controlled with a Specac electrically heated jacket and model 4000 high stability temperature
controller in 10 °C increments from 30 to 60 °C with an uncertainty of +/- 0.05 °C. Cold tap water
was supplied to the heating jacket at the manufacturer's recommended flow rate of 500 mL/min.
94
The sample was introduced to the flow cell using an external valve system that connected the
sample syringe with a Luer-lock fitting, preventing the introduction of environmental moisture.
Teflon spacers were selected to produce absorbances of ≤ 1 A.U. in the region 3050 cm-1
to 3800 cm-1. The pathlength must be known within 1% to relate the absorbance reliably to
concentration.27 As such, the manufacturer-provided nominal spacer thickness dimensions were of
inadequate precision. Instead, pathlengths were calculated for each sample using a MATLAB®
application151 based on the fringe interference spectrum of the empty cell taken over a 300 cm -1
interval.
5.5.2 MD/QM Simulations and Scaling Methods
To guide the interpretation and curve fitting of the spectra, we selected peak locations to
represent the key hydroxyl types found in molecular dynamics simulations. In our previous
work,121 clusters were extracted from molecular dynamics simulations and analyzed using
quantum mechanical simulations at the B3LYP level of theory with the 6-31G* basis set to
interrogate the infrared vibrations associated with the O-H bond. Other details regarding the
molecular dynamics and quantum mechanics simulations are provided by Bala et al.121,122
After removing solvent absorbance, spectra are often fitted with curves for the different
hydroxyl configurations where each curve or wavenumber range is assigned a single attenuation
coefficient.34,106,149 A major development from our previous work is the Beer’s law scaling of the
mid-range IR absorption spectra to obtain temperature-independent integrated hydroxyl
absorbances for ethanol and 1-butanol at concentrations of up to 20 mol% alcohol in
cyclohexane.122 The scaling applies a function that changes with wavenumber, whose shape was
developed from trends in quantum chemical calculations and refined empirically using multiple
concentration and temperature spectra. This scaling compensates for the significant increase in
95
attenuation coefficient displayed by the lower frequency hydrogen-bonded species. The scaled
spectra are analyzed via curve-fitting, and their integrated absorptions are compared quantitatively.
5.5.3 Spectra Preprocessing and Regression of Molar Attenuation Coefficient
Previously, we reported on spectra for 1-butanol up to 20 mol% in cyclohexane;122 this
work includes two additional concentrations: 25 mol% and 30 mol%. Ethanol solutions up to 30
mol% in cyclohexane were prepared, but evaporative losses prevented measurements at 60 °C at
concentrations above 10 mol% ethanol. Because concentration changes due to thermal expansion
of solutions, we use mole fraction to quantify composition in reported results. The collected spectra
were subjected to a series of preprocessing steps before the molar attenuation coefficient was
regressed. Baseline correction and the subtraction of absorption signals attributable to the
cyclohexane in the region of interest – ν̃(3050 cm-1 to 3800 cm-1) – were performed using methods
detailed elsewhere.152 Additional spectra adjustments, such as smoothing, were not performed at
any point. The overall hydroxyl region was scaled using methods presented by Bala et al.122 and
summarized in Section 5.5.2. Parity plots comparing measured and correlated concentrations for
both alcohols and the scaling constants are available in APPENDIX K: Parity Plots.
5.5.4 Curve-Fitting
The absorbance profile of liquid phase vibrations combines characteristics of a gas-phase
Lorentzian shape and the Gaussian character exhibited by solids.135 To account for these features,
a product pseudo-Voigt shape was chosen for three reasons: the model has a physical basis135; it
contains fewer parameters than a sum-Voigt profile, and it is commonly applied to spectra obtained
from the liquid phase.153 The form of Eq. 5-1 was adapted from Kruger et al.154 and requires four
fitted parameters: peak height (𝐻), peak center (𝜈̃0 ), Gaussian width (𝑊𝐺 ), and Lorentzian width
(𝑊𝐿 ).
96
𝐻
𝐴= ⋅ 𝑒𝑥𝑝 (−𝑊𝐺2 (𝜈̃𝑖 − 𝜈̃𝑜 )2 ) Eq. 5-1
1+ 𝑊𝐿2 (𝜈̃𝑖 − 𝜈̃0 )2
Within a single temperature, the scaled spectra were fitted with six product pseudo-Voigt
curves: five for the 𝜈̃(𝑂𝐻) region hydroxyls (𝛼, 𝛽, 𝛾, 𝛿, and unassigned) and one for the 𝜈̃(𝐶𝐻)
region. The curve-fitting process began with the most dilute alcohol samples and proceeded to
higher alcohol concentrations. Optimized parameters from each regression were used as the initial
values for the next highest concentration of alcohol.
To isolate the hydroxyl absorption, the pseudo-Voigt curve associated with the alcohol’s
alkyl group 𝜈̃(𝐶𝐻) was subtracted from the spectrum. This band was centered in the region of
~2950 cm-1 and extended into the low-frequency region of the broad 𝜈̃(𝑂𝐻). It became visible
only after subtracting the absorbance of the solvent.
Curve areas were restricted to positive values and optimized simultaneously by a nonlinear
least-squares method. Integration of the curve fit areas was achieved using a trapezoidal Riemann
sum of the fitted curve with 2 cm-1 intervals over the range of 3050 cm-1 to 3800 cm-1.
5.5.5 Modeling Hydrogen Bonding with RTPT
Conflicting theories exist in the literature regarding the relative prominence of chains and
rings in alcohols.155 Here, we consider a model for cooperative bonding limited to linear chains.
Recently, the Wertheim resummed thermodynamic perturbation theory (RTPT) developed by
Marshall and Chapman112 has been shown to agree with infrared spectroscopic measurements from
Asprion.34,134 RTPT considers cooperative bonding, distinguishing between association strengths
for the first hydrogen bond between two monomers to form a dimer (Δ2 ) and association strengths
for subsequent hydrogen bonds in an oligomer (Δ𝑁 ). Association strengths (Δ𝑛 ) are similar to
concentration-based equilibrium constants but include composition dependence. The molar or
97
number density of chains of length i in solution is related to the molar or number densities of the
constituent species,
𝜌𝑖 = Δ𝑛 𝜌𝑖−1 𝜌𝛼 Eq. 5-2
which represents the addition of monomer, α, to a chain of length 𝑖 − 1, and Δ𝑛 depends on whether
a dimer, Δ𝑛 = Δ2 or an n-mer, Δ𝑛 = Δ𝑁 , is formed. The sum of molar densities for 𝛼 hydroxyls
and 𝛽 hydroxyls divided by the molar density of alcohol in the solution produces the fraction of
non-bonded hydroxyl protons, 𝑋 𝐷 = 𝑁𝛼+𝛽 /𝑁𝑎 where 𝑁𝑎 represents the number of alcohol
molecules. The RTPT model uses a balance on proton acceptor and proton donor sites to calculate
the densities of hydroxyl configuration. The association model used here assumes a 2B bonding
scheme156 for the hydroxyl, which translates to one proton acceptor site (A) on the oxygen and one
proton donor site (D) attributed to the hydroxyl proton. When alcohol is the only associating
species, for the RTPT model, the density of acceptor sites equals the density of donor sites 𝜌𝐷 =
𝜌 𝐴 and the fraction of nonbonded donor sites is equivalent to the fraction of non-bonded sites
𝑋 𝐷 = 𝑋 𝐴 . The total density of donor sites is related to the fraction of non-bonded donor sites via
𝜌𝐷 = 𝑥𝑎 𝜌𝑋 𝐷 where 𝑥𝑎 is the mole fraction of alcohol, and 𝜌 is the molar density of the mixture.
Wertheim statistical mechanics models hydrogen bonding association strengths using adjustable
parameters for the bonding volume (𝜅 𝐴𝐷 ), which represents the volume of a square well site on a
molecule, and the square-well depth, 𝜖 𝐴𝐷 /𝑘. In this work we regress a single value for the bonding
volume for dimers and oligomers, and we regress two bonding energies, 𝜖2𝐴𝐷 /𝑘 or 𝜖𝑁𝐴𝐷 /𝑘, in the
Mayer f-function according to
3
𝜖2𝐴𝐷𝑜𝑟 𝑁
Δ2 𝑜𝑟 𝑁 = 𝑁𝐴 𝑑𝑎𝑎 𝑔𝑎𝑎 (𝑑)𝜅 𝐴𝐷 (exp ( ) − 1) Eq. 5-3
𝑘𝑇
98
where the subscripts 𝑎𝑎 denote the bonding between two alcohol molecules. The association
strength is often expressed in molecular units, but here we use molar units (volume/mole). To
convert molecular parameters into molar units, the expression includes Avogadro’s constant (𝑁𝐴 ).
The compositional dependence of Δ2 𝑜𝑟 𝑁 is ascribed to the variation in the hard-sphere radial
distribution function at contact (𝑔aa (𝑑)), which depends on the solution packing fraction (𝜁𝑙 ) and
the temperature-variant segment diameter of the associating fluid (𝑑𝑎𝑎 ). The effect of temperature
on the segment diameter is calculated via Eq. 5-6 using a segment diameter (𝜎𝑖 ) and the depth of
the pair potential (𝜖𝑖 /𝑘) for each species present in the solution. The pure component parameters
used for this regression are from PC-SAFT as fitted to pure component data by Gross and
Sadowski126 and are provided in Table 5-1.
2
1 𝑑𝑖 𝑑𝑗 3𝜁2 𝑑𝑖 𝑑𝑗 2𝜁22
𝑔𝑖𝑗 (𝑑) = +( ) +( ) Eq. 5-4
(1 − 𝜁3 ) 𝑑𝑖 + 𝑑𝑗 (1 − 𝜁3 )2 𝑑𝑖 + 𝑑𝑗 (1 − 𝜁3 )3
𝜁𝑙 = (𝜋/6)𝑁𝐴 𝜌 ∑𝑖 𝑥𝑖 𝑚𝑖 𝑑𝑖𝑙 ; 𝑙 ∈ {2,3} Eq. 5-5
𝜖𝑖
𝑑𝑖 = 𝜎𝑖 [1 − 0.12 exp (−3 ( ))] Eq. 5-6
𝑘𝑇
Killian et al. demonstrated that Δ2 and Δ𝑁 are related to the fraction of non-bonded hydroxyl
protons (𝑋 𝐴 ) through the monomer density (𝜌𝛼 ) according to Eq. 5-7.134
2𝑥𝑎 𝜌𝑋 𝐴
𝜌𝛼 = Eq. 5-7
1 + 𝑥𝑎 𝜌Δ𝑁 𝑋𝐴 + √(1 + 𝑥𝑎 𝜌Δ𝑁 𝑋𝐴 )2 + 4(Δ2 − Δ𝑁 )𝑥𝑎 𝜌𝑋𝐴
The fraction of free hydroxyls is simultaneously related to the monomer density using Eq. 5-8
and Eq. 5-9.
99
1
𝑋𝐴 = Eq. 5-8
1 + 𝑥𝑎 𝜌Δ2 𝑋𝐴 /𝑠 2
𝑠 = 1 + (Δ2 − Δ𝑁 )𝜌𝛼 Eq. 5-9
Table 5-1: PC-SAFT pure component parameters.
𝝐𝒊
Component 𝒎𝒊 𝝈𝒊 , [Å] , [𝑲]
𝒌
Ethanol 2.3827 3.1771 198.24
1-Butanol 2.7515 3.6139 259.59
Cyclohexane 2.5303 3.8499 278.11
5.6 Results and Discussion
5.6.1 Frequency Distributions of Hydrogen Bonds from QM Calculations
The MD and QM simulations for ethanol + cyclohexane (50 mol% ethanol) and 1-butanol
+ cyclohexane (10 & 50 mol% 1-butanol) mixtures provide the distribution of vibrational
frequencies for 𝛼, 𝛽, 𝛾, and 𝛿 hydroxyls shown in Figure 5-2. Findings from the two binary systems
are plotted together because separate plots showed no discernable differences between them. The
vibrations of each hydroxyl type were binned in 5 cm-1 wide bins and normalized and the resulting
lines were smoothed with a 7-bin moving average and normalized to display the relative
populations for each hydroxyl type as a function of wavenumber. These processing steps were
carried out to refine the shape of the distributions and display relative information about them more
clearly. The peak heights should not be used to compare hydroxyl types because the sample size
is insufficient to reflect the quantitative distribution at the simulated concentrations. However, the
peak shapes provide some qualitative information and a relative description of the vibrational
frequencies of different hydroxyl types. In agreement with other studies,7,52,59,60,157 Figure 5-2
demonstrates that 𝛼 and 𝛽 hydroxyls appear at the same frequency and show significant overlap,
100
confirming that the 𝛽-hydroxyl contribution to the sharp high-frequency O-H peak should not be
ignored. These calculations provide a supporting basis for assigning peak areas to specific
hydroxyl configurations.
Figure 5-2: Smoothed normalized distributions of hydroxyl types from QM simulations of 1-
butanol + cyclohexane, and ethanol + cyclohexane mixtures. Vibrational frequencies of
hydroxyl types based on sample size (𝑛) for 𝛼 (n=1878), 𝛽 (n=1430), 𝛾 (n=1430), and 𝛿
(n=927) hydroxyls are binned in 5 wavenumber bins and smoothed with a 7-bin moving
average.
5.6.2 Assigning Curves to Specific Hydroxyl Configurations
The assignment of the curve-fit areas to specific hydroxyl types was inspired by the
distributions observed in the MD/QM analysis, as shown in Figure 5-3. In keeping with the
recommendations of Meier135, a minimum number of curves were used to partition the 𝜈̃(𝑂𝐻)
region. As a result of the scaling process, the integrated 𝜈̃(𝑂𝐻) curves are equal to the
concentration of hydroxyl groups with units of molarity.122
101
Figure 5-3: Raw infrared spectrum of 20 mol% 1-butanol in cyclohexane at 30 °C (left).
Beer’s law scaled and curve-fit infrared absorbance spectrum for 20 mol% 1-butanol in
cyclohexane at 30 °C (right). Scaling reduces the contribution of the delta hydroxyls around
3350 cm-1 relative to the non-hydrogen-bonded-hydroxyl-proton absorbance near 3640 cm-1.
The scaled peak areas are directly proportional to concentration.
Since the resonances of 𝛼 and 𝛽 hydroxyls are similar,122 their individual contributions to
the peak area cannot be readily discerned from the absorbance spectra. In addition, absorbances
from the stable anti and gauche rotamers of ethanol158 and 1-butanol159 also appear in this region
(~3645 cm-1). The barrier to rotation is low for the C-O axis and is easily surmounted at ambient
temperature,160 further complicating definitive α/β band assignment.149 In the following sections,
we will refer to the concentration of 𝛼 and 𝛽 hydroxyls collectively as the sum of non-hydrogen-
bonded hydroxyl protons.105,106 The 𝛾 and 𝛿 hydroxyls appear at lower wavenumbers relative to
the non-hydrogen-bonded hydroxyl protons. Additionally, a minor curve was required to fill out
the spectrum; since we could not attribute it with confidence, it is listed as unassigned. The area
of the unassigned curve is small compared to others – typically less than 1% of the total area for
both alcohols - and was therefore omitted when allocating integrated areas to specific hydroxyl
configurations. Additionally, occurrences of 𝜂 and 𝜁 in our molecular simulations of alcohol +
102
cyclohexane mixtures were less than 1%, even for mixtures of up to 50 mol% alcohol.121 Therefore
the fitting of additional peaks to the spectra was not justify. The influence of the alcohol 𝜈̃(𝐶𝐻)
can be observed on the right, centered near 2950 cm-1 (Figure 5-3).
In addition to curve areas, we also collected metrics on the curves. The variation in peak
center with respect to concentration and temperature for ethanol and 1-butanol in cyclohexane are
shown in Figure 5-4. The peak centers of 𝛼 and 𝛽 vary insignificantly with concentration and
temperature, which is to be expected since the absorbance band is narrow. However, there is a shift
in the peak centers for the 𝛾 and 𝛿 hydroxyls in both alcohol systems. The shift is most pronounced
below 10 mol% for the γ hydroxyls and below 5 mol% for the δ hydroxyls and the unassigned
hydroxyls for both alcohols. In both instances, the peak centers shift to lower wavenumbers with
increasing concentration. Similar behavior was reported by Asprion et al.34 at low alcohol
concentrations. The peak centers also monotonically shifted to lower wavenumbers as the
temperature decreased within a particular mole fraction though this shift is less pronounced.
Figure 5-4: Location of peak center for ethanol (left) and 1-butanol (right) in cyclohexane as a
function of mole fraction and experimental temperature. The groupings are (left to right) (𝛼 +
103
𝛽) = black/blue, (𝛾) = orange, (unassigned) = cyan, and (𝛿) = green at temperatures 30 °C
(◊), 40 °C (○), 50 °C (∆), and 60 °C (□).
5.6.3 Hydroxyl Populations – Temperature and Concentration Effects
It is well understood that temperature profoundly affects the extent of hydrogen bonding.143
Higher temperatures represent an increase in kinetic energy of the molecules in the system, which
decreases their participation in hydrogen bonding. The effect of temperature and alcohol
concentration on the hydrogen bond populations is evident in Figure 5-5. For a fixed mole fraction,
temperature increases induced a reduction in δ hydroxyl concentration for both alcohols. The
population change was quantifiable since the total integrated hydroxyl area was independent of
temperature.122 This reduction in 𝛿 hydroxyls was offset by an increase in the concentration of
non-hydrogen-bonded hydroxyls (𝛼 + 𝛽) and 𝛾 hydroxyls. For example, the dissociation of one
of the hydrogen bonds in a trimer produces a new 𝛼 hydroxyl at the expense of a 𝛿 hydroxyl.
Dissociation of a 𝛿 hydroxyl in the middle of a chain would produce a 𝛽 hydroxyl and a 𝛾 hydroxyl.
104
Figure 5-5: Hydroxyl populations for ethanol (top row) and 1-butanol (bottom row) at 30 °C
(lefthand column) and 50 °C (righthand column). The provided dashed lines connect the data
and aid the observation of trends.
Like temperature, variation in alcohol concentration resulted in significant changes to the
hydroxyl populations. Below 1 mol% alcohol, the contribution of the 𝛿 hydroxyls was modest.
Yet, as the solution became more concentrated in alcohol, the population of 𝛿 hydroxyls became
105
significant, increasing with alcohol concentration and becoming the most prominent type in the
solution above approximately 3-5 mol% alcohol at 30 °C and above approximately 7-8 mol%
alcohol at 60 °C.
5.6.4 Regression and Statistics
The association energies and bonding volume for Eq. 5-3 are regressed using the 𝑋 𝐴 data
simultaneously at all temperatures. The regression minimized the objective function
2
𝑋𝑖𝐴 (𝑚𝑜𝑑𝑒𝑙)−𝑋𝑖𝐴 (𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙)
∑𝑑𝑎𝑡𝑢𝑚 𝑖 [ ] . Since the majority of the uncertainty in the peak area
𝑋𝑖𝐴 (𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙)
occurs at low concentrations of alcohol where complete subtraction of the solvent is most difficult,
this type of objective function places more emphasis on the higher alcohol concentrations where
the overall magnitude of 𝑋 𝐴 was smaller.
Since the pre-exponential and exponential terms of Eq. 5-3 are strongly coupled, an
intermediate variable 𝑝 was regressed using Eq. 5-10 and Eq. 5-11 which are modified from Eq.
5-3 to include a reference temperature (𝑇𝑟𝑒𝑓 ).131 A value 𝑇𝑟𝑒𝑓 = 298.15 K was selected as it falls
near the experimental range yet is not an explicitly measured temperature.134 Following regression,
𝜅 𝐴𝐷 values were obtained via Eq. 5-12.
3
𝜖𝑁𝐴𝐷 𝜖2𝐴𝐷
Δ2 = 𝑁𝐴 𝑑𝑎𝑎 𝑔𝑎𝑎 (𝑑) (𝑝 ⋅ 𝑒𝑥𝑝 ( )) (𝑒𝑥𝑝 ( ) − 1) Eq. 5-10
𝑘𝑇𝑟𝑒𝑓 𝑘𝑇
3
𝜖𝑁𝐴𝐷 𝜖𝑁𝐴𝐷
Δ𝑁 = 𝑁𝐴 𝑑𝑎𝑎 𝑔𝑎𝑎 (𝑑) (𝑝 ⋅ 𝑒𝑥𝑝 ( )) (𝑒𝑥𝑝 ( ) − 1) Eq. 5-11
𝑘𝑇𝑟𝑒𝑓 𝑘𝑇
𝜖𝑁𝐴𝐷
𝜅 𝐴𝐷 = 𝑝 ⋅ exp ( ) Eq. 5-12
𝑘 𝑇𝑟𝑒𝑓
106
To provide the 95% confidence interval for the parameters, bootstrapping (replacement by
resampling) was employed.133 The regression was performed 1000 times using randomly sampled
experimental data. Once sorted, the 25th and 975th values were selected as the lower and upper
limits of the confidence interval, respectively. A summary of fitted parameters, along with the
respective 95% confidence interval, is given in Table 5-2. A similar analysis using the Δ2 and Δ𝑁
values is provided in Table 5-3.
Table 5-2: RTPT association constant regression values.
𝐴𝐷 𝐴𝐷
𝜖𝑑𝑖𝑚𝑒𝑟 (95% 𝜖𝑛𝑚𝑒𝑟 (95%
System 𝜅 𝐴𝐷 (95% C.I.) , [𝐾] , [𝐾]
𝑘 C.I.), [K] 𝑘 C.I.), [K]
5.347E-03 to 1877-
Ethanol 1.12E-02 2115 2847 2634-3076
2.180E-02 2355
8.586E-04 to 2545-
1-Butanol 1.49E-03 2715 3330 3174-3499
2.443E-03 2884
Results of the regression can be seen in Figure 5-6. The fraction of non-hydrogen-bonded
hydroxyl protons increases with temperature for both alcohols, and lower alcohol concentrations
correlate with higher values of 𝑋 𝐴 . These trends make physical sense as 𝛽 hydroxyls comprise a
more significant percentage of smaller chains, and alcohol molecules increasingly prefer to remain
unassociated in situations where they are highly diluted and at higher temperatures. Interestingly,
the approach to 𝑋 𝐴 = 1 from higher to lower alcohol mole fractions is not asymptotic, and the data
moves from an upward concavity to a downward concavity near 1 mol% as seen in the bottom row
plots of Figure 5-6 and also observed by Asprion et al.34
𝐴 𝐴
Figure 5-7 provides a depiction of the residual values calculated as 𝑋𝑚𝑜𝑑𝑒𝑙 − 𝑋𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 .
The largest deviation values occur in the dilute range at alcohol concentrations less than ~4 mol
%. While the magnitudes of the residuals are most significant in this concentration range, when
107
considered on a percentage basis, they constitute less than 4%. The scatter in the dilute region
occurs because overall areas are small, and the resulting areas are sensitive to baseline adjustments
and the removal of cyclohexane signals from the hydroxyl stretching absorbance. However, as
seen in the lower row plots of Figure 5-6, this region exhibits a noticeable change in slope as the
alcohol becomes more dilute and 𝑋 𝐴 approaches one.
Figure 5-6: (Top) Fraction of non-hydrogen-bonded hydroxyl protons from curve-fitting the
infrared spectra overlayed by the XA values obtained by regression of the resummed
thermodynamic perturbation theory (RTPT) constants to all four experimental temperatures for
108
ethanol (left) and 1-butanol (right). (Bottom) Enlargement of top plots to show behavior in the
dilute region for ethanol (left) and 1-butanol (right).
𝑋 𝐴 change rapidly, with the largest overprediction occurring at the highest temperatures. In the
interval of 10 mol% to 30 mol%, the model tended to underpredict 𝑋 𝐴 for both alcohols by 0.01-
0.02, and the most considerable deviations occur at the higher temperatures of 50°C and 60°C.
Figure 5-7: Residual values from regression calculated using (XA(model) – XA(experimental)).
109
Figure 5-8: Superimposed XA data for ethanol and 1-butanol in conjunction with the RTPT
model.
When the experimental values of 𝑋 𝐴 are superimposed in Figure 5-8, it is difficult to
discern the subtle differences between ethanol and 1-butanol, particularly when the concentration
of alcohol is less than 10 mol%. Janeček and Paricaud,148 using molecular simulation, observed
similarities in 𝑋 𝐴 vs. 𝑥𝑎 for a series of alcohols, consistent with Figure 5-8, suggesting that 𝜌Δ
values may be the same for all alcohols. While a model assuming the same 𝜌Δ values may be a
valid first approximation, the confidence intervals for the fitted energy parameters do not overlap,
so we present an analysis using the individual fits.
5.6.5 Comparison with Previous Work
Since RTPT has only recently been applied to experimental data, comparisons with other
works over sufficient concentration and temperature ranges are limited and typically involve
chemical theory.7,54,75,78 Recently, we demonstrated134 the application of RTPT to the data of
Asprion et al.34 and Table 4-2 contains those parameters. The association constants which
correspond with those parameters at 298.15 K are summarized in Table 5-3 in the rows labelled
‘Asprion’ for comparison with this work. Fits to Asprion’s data provides dimer constants that are
110
smaller compared to the current work and n-mer constants that are larger. The publication for this
chapter134 presents the refitted values listed in for ethanol and 1-butanol. The published manuscript
for Chapter 4 listed incorrect parameter values as explained and corrected in Chapter 4.
Table 5-3: RTPT association constants for pure alcohol. The rows labeled ‘Asprion’ are fits to
the data of Asprion et al.34 as explained in the text.
(95%
3 (95% C.I.),
Δ2 , [𝑐𝑚 C.I.), Δ𝑁 , [𝑐𝑚3
Alcohol Temperature [𝑐𝑚3 / ref.
/𝑚𝑜𝑙] [𝑐𝑚3 / /𝑚𝑜𝑙]
𝑚𝑜𝑙]
𝑚𝑜𝑙]
646.89 – 8330.1 – This
772.02 8629.8
904.54 8956.8 Work
30 °C
257.52 – 11789 -
429.17 12454 Asprion
744.25 13131
Ethanol
335.81 – 3229.0 – This
379.24 3413.6
433.37 3559.8 Work
60 °C
116.29 – 3695.0 –
198.41 4253.5 Asprion
349.73 4910.2
1218.1 – 10115 - This
1361.6 10362
1519.9 10610 Work
30 °C
674.25 – 12434 –
841.21 13007 Asprion
1015.4 13465
1-Butanol
505.32 – 3418.0 – This
561.35 3558.9
619.93 3678.9 Work
60 °C
369.30 – 4856.8 –
454.13 5497.5 Asprion
529.08 5982.4
RTPT predictions of 𝛼 + 𝛽 hydroxyl concentrations obtained from the data of Asprion are
plotted in conjunction with experimental data and modeling results of this work in Figure 5-9.
Parameters obtained from fitting the data of Asprion et al.34 result in lower 𝛼 + 𝛽 concentrations
at all temperatures and the difference between the two sets becomes most noticeable above 2 mol%
alcohol.
111
Figure 5-9: Experimental concentration of 𝛼 + 𝛽 hydroxyls overlayed with predictions of the
RTPT model using association parameters from this work (- -) and modified parameters from
the reinterpreted work of Asprion et al.134 (⋅⋅) for ethanol (left) and 1-butanol (right) at
temperatures 60°C (green), 50°C (red), 40°C (blue), and 30°C (black).
While the parameters obtained from the reinterpretation of Asprion’s data are similar in
magnitude to those obtained in this work it is important to discuss some differences in approach
between our current method and our reinterpretation of Asprion’s work, where we utilized a table
of integrated areas - obtained from the assignment of five pseudo-Voigt curves - as the basis for
our analysis. When curve-fitting is undertaken using unscaled spectra (Figure 5-3 left), modeling
the lower wavenumber hydroxyl absorbances requires tall, broad curves, which in turn produce
significant overlap with the 𝛼 + 𝛽 region and may contribute to a lower free-end contribution.
Examining the integrated free-end absorbance areas after curve fitting, some of the data of Asprion
et al.34 and Reilly et al.53 show a plateau or maxima in the free-end values, which occurs at high
alcohol concentrations, low temperatures, or with short alcohols. By scaling before curve fitting
in this work, smaller peaks are needed to fit absorbance in the 𝛾 and 𝛿 region, resulting in less
overlap of peak tails with the free-end region. Even where plateaus or maxima are not observed in
Asprion’s skimmed data, we believe that the differences in data processing explain the larger
112
values of 𝑋 𝐴 obtained in this work at high concentrations. We do not observe maxima in our values
of 𝑋 𝐴 using the scaling method applied above.
Since the pseudo-Voigt profile is a symmetric function, an additional complication is
introduced, which impacts the relationship between integrated area and concentration. Recently
we observed via quantum mechanics calculations and experimental data regression that the molar
attenuation coefficient increases continuously for the fundamental hydroxyl region as a function
of decreasing wavenumber. Therefore, curve-fitting infrared data prior to scaling with Beer’s law
would require using asymmetric peaks to account for the continuous wavenumber dependence of
the molar attenuation coefficient. In principle, a linear function could be multiplied by the pseudo-
Voigt function.
5.6.6 Experimental and Modeled Hydroxyl Populations
If hydroxyls are assumed to aggregate in linear chains, the monomer density (𝜌𝛼 ) can be
used in conjunction with Δ2 and Δ𝑁 to calculate the populations of each unique hydroxyl type. Eq.
16 of the supplemental material of Killian et al.134 is
Δ2 𝜌𝛼2
𝑥𝑎 𝜌𝑋 𝐴 = 𝜌𝛼 + Eq. 5-13
1 − Δ𝑁 𝜌𝛼
Recognizing that the density of non-hydrogen-bonded hydroxyl protons results from either 𝛼 or 𝛽
hydroxyls, the second term on the right represents the 𝛽 hydroxyls. Because every linear chain
with a 𝛽-hydroxyl on one end must have a 𝛾-hydroxyl on the opposing end, then
Δ2 𝜌𝛼2
𝜌𝛾 = 𝜌𝛽 = Eq. 5-14
1 − Δ𝑁 𝜌𝛼
The overall balance is given in APPENDIX E: Key Material Balance Equations.
113
2Δ2 𝜌𝛼2 Δ2 Δ𝑁 𝜌𝛼3
𝑥𝑎 𝜌 = 𝜌𝛼 + + Eq. 5-15
1 − Δ𝑁 𝜌𝛼 (1 − Δ𝑁 𝜌𝛼 )2
Recognizing that the second term on the right is the sum (𝜌𝛽 + 𝜌𝛾 ), then, the final term is 𝜌𝛿
where
Δ2 Δ𝑁 𝜌𝛼3
𝜌𝛿 = Eq. 5-16
(1 − Δ𝑁 𝜌𝛼 )2
Using the fit of non-hydrogen-bonded 𝛼 + 𝛽 hydroxyls represented by 𝑋 𝐴 in Figure 5-6, the
parameters provide a model of the distribution of the hydroxyl type populations. The
concentrations from Figure 5-5 are replotted in Figure 5-10 as fractions of alcohol appearing in a
particular hydroxyl type, along with the RTPT model predictions. The fraction of alcohol in
monomer 𝛼 hydroxyls decreases with increasing alcohol concentration. For ethanol and 1-butanol,
the experimental fraction of 𝛾 hydroxyls increases up to 5 mol% alcohol and then remains
relatively constant. The fraction of 𝛿 hydroxyls increases most rapidly at alcohol concentrations
less than 10 mol% for both alcohols.
114
Figure 5-10: Fraction of alcohol molecules involved in specific types of hydrogen-bonding
overlayed with the resummed thermodynamic perturbation theory (RTPT) predictions for
ethanol (top row) and 1-butanol (bottom row) at 30 °C (lefthand column) and 50 °C (righthand
column).
The RTPT predictions tend to underestimate the fraction of 𝛾 hydroxyls and overestimate the
fraction of 𝛿 hydroxyls at all temperatures, as seen in Figure 5-10. At 50°C, the underprediction
of the 𝛾 hydroxyls is more apparent, as is the overprediction of the 𝛿 hydroxyls. The presence of
cyclic species would shift the distribution from 𝛾 to 𝛿 hydroxyls that are already over-predicted.
A possible explanation is branching due to 𝜁 and 𝜂 hydroxyls. Because the oxygen atom of these
115
configurations accepts two protons, the number of 𝛾 hydroxyls increases at each branch point
relative to a linear chain with one 𝛾 hydroxyl. Since 𝛿 hydroxyls are the most prevalent
configuration, then the formation of an 𝜂 or 𝜁 hydroxyl would likely come at the expense of a 𝛿.
Paolantoni et al.123 report evidence of branching in pure 1-octanol using IR, and Janeček and
Paricaud148 report branching in molecular simulations. However, the ability of the RTPT linear
chain model to capture the key features is striking.
5.6.7 Enthalpies
There are several considerations when determining the strength of a hydrogen bond,
namely the donor and acceptor strength and its immediate environment. Since the polarity of the
solvent plays a significant role in the strength of a hydrogen bond, enthalpies obtained in one
medium often do not translate to another diluent. An indirect comparison can be made using the
fitted association constants. Enthalpies of association for the dimer (−Δ𝐻2 ) and the n-mer (−Δ𝐻𝑁 )
can be calculated using their respective association constants and the van’t Hoff relation as
expressed in Eq. 5-17. The values are provided in Table 5-4 and compared to values obtained by
refitting data of Asprion et al.34
𝑑 𝑙𝑛 Δ2 𝑜𝑟 𝑁
Δ𝐻2 𝑜𝑟 𝑁 = 𝑅
1 Eq. 5-17
𝑑( )
𝑇[𝐾]
For ethanol, enthalpy values obtained from this work were ~7% and ~14% different for the dimer
and n-mer, respectively, when compared to our fit of the same system from Asprion et al.134 For
1-butanol, the enthalpies differed by ~37% and ~23% for the dimer and n-mer, respectively.
Calculations performed by Wandschneider et al.61 using B3LYP demonstrated that cooperativity
accounted for roughly 7 kJ/mol, which is similar to the difference between ΔH2 and ΔH𝑁 seen in
the experimental values.
116
Table 5-4: Hydrogen-bonding enthalpies for ethanol and 1-butanol in cyclohexane compared to
values from fitting data of Asprion et al.34
𝑥alcohol = 1 and 25 °C
kJ 𝑘𝐽 −ΔH2
System −ΔH2 , [ ] −ΔH𝑁 , [ ] Source
mol 𝑚𝑜𝑙 −ΔH𝑁
Ethanol + 19.6 25.7 0.76 This work
Cyclohexane 21.3 29.8 0.72 fit Asprion
1-Butanol + 24.5 29.6 0.83 This work
Cyclohexane 16.9 23.8 0.71 fit Asprion
While the enthalpies derived from RTPT appear reasonable, we sought an alternative
means of determining the enthalpy to better evaluate our results. Hare and Sorensen161 detailed a
method of calculating the hydrogen bond dispersion energy from Raman intensities. More
recently, Paolantoni et al.123 adapted this approach to infrared and leveraged it to calculate the
enthalpy associated with a hydroxyl transitioning from an 𝛼/𝛽 to other bonding configurations (𝛾
or 𝛿). This method relies on the assumption that the absorbance peak heights in the hydroxyl region
follow a Boltzmann distribution. Calculation of Δ𝐻2 𝑜𝑟 𝑁 from the slope of the integrated
experimental hydroxyl populations were performed via Eq. 5-18. This approach is independent of
the thermodynamic model.
The method was applied to each mole fraction and are plotted in Figure 5-11. We are less
confident in the enthalpy values below 5 mol% alcohol since neither the 𝛾 or 𝛿 populations are
significant in this range, and we have therefore omitted them from the plot. For Δ𝐻2 the enthalpies
for both systems generally increase with increasing alcohol concentration and are mostly clustered
between 3-10 kJ/mol, which is near the 7 kJ/mol suggested by Paolantoni et al.123 for 1-octanol.
The increase in enthalpy with respect to increasing alcohol concentration is less pronounced for
Δ𝐻𝑁 and both alcohols appear to reach similar values near 31 kJ/mol which is roughly 13%
different than the value obtained by Paolantoni et al.123 for 1-octanol (27.2 kJ/mol).
117
𝐴𝑖𝑛𝑡 (𝛼 + 𝛽) 𝐴𝑖𝑛𝑡 (𝛼 + 𝛽)
𝑑 ln ( ) 𝑑 ln ( )
𝐴𝑖𝑛𝑡 (𝛾) 𝐴𝑖𝑛𝑡 (𝛿)
Δ𝐻2 = 𝑅 ; Δ𝐻𝑁 = 𝑅 Eq. 5-18
1 1
𝑑( ) 𝑑( )
𝑇[𝐾] 𝑇[𝐾]
Figure 5-11: Calculated ∆H2 and ∆HN derived from the relationship of Paolantoni et al. using
integrated peak areas.
5.6.8 Average Oligomer Chain Length
A valuable attribute of RTPT is that the average oligomer size (𝑛ave ) can be calculated
using Eq. 5-19, provided the assumption of linear association.156 This calculation produces the
tandem plots seen in Figure 5-12. The average chain length increases at lower temperatures and
higher alcohol concentrations for ethanol and 1-butanol. The increase in oligomer size with
increasing concentration is most apparent at 30 °C. At 40 °C to 60 °C, the change in oligomer
length occurs most rapidly at alcohol concentrations less than 10 mol%. A global maximum in the
average chain length was not predicted from the model for either alcohol for any of the
compositions studied here.
118
𝑛ave = 1/𝑋 𝐴 Eq. 5-19
Since the 𝑋 𝐴 values for both alcohols are similar, and we plotted them together in Figure
5-8; it is not surprising that the calculated average chain lengths are also very similar. This
similarity occurs despite the apparent differences in the regressed thermodynamic parameters as
well as the Δ2 and Δ𝑁 values. The small discrepancy in chain length between ethanol and 1-butanol
never exceeds one in the concentration ranges examined.
Figure 5-12: Variation in the average oligomer size (𝑛𝑎𝑣𝑒 ) as a function of alcohol
composition and temperature for ethanol and 1-butanol superimposed with the prediction from
the RTPT model over the entire experimental concentration range (left) and with emphasis on
the dilute region (right). Experimental values are used, and the average aggregate size is
greater at higher alcohol concentrations and lower temperatures.
5.7 Summary and Conclusions
Infrared spectra of the hydroxyl stretching region for ethanol + cyclohexane and 1-butanol
+ cyclohexane were collected at 30°C, 40°C, 50°C, and 60°C. The spectra were scaled so that
the integrated peak area was proportional to concentration. The scaled spectra were curve-fit to
partition the 𝜈̃(𝑂𝐻) region into the four principal hydroxyl configurations (𝛼, 𝛽, 𝛾, and 𝛿), but we
were unable to justify additional curves for 𝜂 and 𝜁 hydroxyls. Since it is difficult to conclusively
119
separate 𝛼 and 𝛽 contributions due to rotational isomerism about the C-O-H bond, they were
combined and regressed collectively as the fraction of non-hydrogen-bonded hydroxyl protons
(𝑋 𝐴 ) to obtain association parameters for the resummed thermodynamic perturbation theory
(RTPT), which assumes linear oligomerizations without branching. 𝑋 𝐴 was similar between the
two alcohols at concentrations less than 10 mol % alcohol and the values for both alcohols were
sigmoidal in shape when plotted as a function of mole fraction. We compared RTPT model
predictions with the curve-fit experimental hydroxyl populations and observed good agreement,
which is consistent with the dominance of linear oligomers in the concentration region examined.
The current work provides higher values of 𝑋 𝐴 above 2 mol% when compared to refit data of
Asprion et al.,34 which utilized the 𝛼 + 𝛽 interpretation of the high frequency peak.
Hydrogen bond cooperativity was observed for alcohols ethanol and 1-butanol in
cyclohexane, and that is reflected in the enthalpy values, which are lower for the dimer than the n-
mer in both cases. Calculated enthalpies are comparable to other works investigating the
phenomenon of positive cooperativity in self-associating alcohols and fall within the commonly
accepted range of hydrogen bonding enthalpies (17 to 34 kJ/mol). The regressed association
parameters facilitate calculation of the distribution of aggregate lengths present in the binary
system using 𝑋 𝐴 . The average chain length was calculated for both alcohols and was found to be
within one unit for all investigated concentrations.
5.8 Acknowledgments
This material is based upon work supported by the National Science Foundation under
Grant No. 1603705, USDA National Institute of Food and Agriculture, and Hatch/Multi-State
projects MICL02585 and MICL04192. Any opinions, findings, and conclusions or
120
recommendations expressed in this material are those of the author(s) and do not necessarily reflect
the views of the funding agencies.
121
CHAPTER 6:Modeling Phase Equilibria Using Infrared Spectroscopy
6.1 Introduction
The ability to correlate the molecular properties of materials to their phase equilibria in
mixtures remains an ongoing modeling challenge for industry.1 Interest in climate responsible
feedstocks has increased the need to model the polar biorenewable compounds. The
thermodynamic behavior of these highly polar compounds is governed by strong intermolecular
interactions such as hydrogen bonding or electrostatic interactions involving permanent or fixed
dipoles which contribute to the highly non-ideal behavior of these materials. To correctly describe
or predict the complex phase behavior of these systems, the attractive contribution of association
to the system non-ideality must be explicitly represented. This is the approach taken by the most
modern equations of state (EOS) such as Statistical Associating Fluid Theory (SAFT), Elliot,
Suresh, and Donohue (ESD) and Cubic Plus Association (CPA). However, parameterization of the
association term remains challenging.
Spectroscopic techniques such as nuclear magnetic resonance (NMR) spectroscopy77,155,162
and Fourier transform infrared spectroscopy34,61,114,117,149,163 (FTIR) have emerged as the two most
popular means of probing intermolecular interactions. However, spectroscopic interpretation is not
straightforward. Numerous investigators have interpreted the NMR chemical shift of the hydroxyl
resonance as an indicator of the extent of hydrogen bonding. However, attributing the observed
chemical shift into its constituent contributions requires that the chemical shift of all hydroxyl
protons be known for each type of hydrogen bond. Unfortunately, this is practically impossible
because the only bonding configuration for which the shift can be measured directly is the free
alcohol at infinite dilution. Often, shifts of the bonded species are adjustable parameters fitted
along with an assumed association scheme.
122
Similar controversy has surrounded infrared measurements. To quantify the concentrations
of specific oligomers the Beer-Lambert law must be invoked which necessitates knowledge of the
molar attenuation coefficient for a particular bond configuration and a specific frequency. In the
past this was achieved in a fashion analogous to the NMR measurements where an association
model was assumed, and the equilibrium constants and molar attenuation coefficients were
regressed along to the spectra using molar attenuation coefficients as adjustable parameters34 or a
scaling was implemented of the temperature dependence of the integrated areas on either side of a
isosbestic point.114 Recently the work of Bala et al.122 applied trends implied by quantum
mechanics calculations to develop a functional form of the molar attenuation coefficient with a
wavelength dependency that decouples the association model from the fitting of attenuation
constants.
More recently, this work was extended by Killian et al.164 to ethanol and 1-butanol in
cyclohexane and the populations of specific hydrogen-bond configurations were determined from
the scaled infrared absorbance of the hydroxyl region. These populations enabled the
parameterization of association constants for the resummed thermodynamic perturbation theory
(RTPT) of Marshall and Chapman112 by fitting the fraction of non-bonded hydroxyl protons (𝑋 𝐴 ).
This marked the first time that 𝑋 𝐴 from spectroscopic data had been used to parameterize RTPT
and demonstrated that two association constants were required to correctly describe hydrogen
bonding cooperativity. In this work we build upon our previous contributions and present results
for 10 alcohols in cyclohexane.
The fitted RTPT model is used to calculate activity coefficients and model phase equilibria
as demonstrated by Killian et. al.134 Insight from infinite dilution activity coefficients guides the
123
parameter selection, and the resulting model improves the representation of phase equilibria and
the excess enthalpy.
6.2 Background
6.2.1 Thermodynamic Perturbation Theory of Wertheim
For purposes of engineering thermodynamic modeling, simplified, robust approaches
capable of representing the complex hydrogen bonding molecular interactions are essential.
Among the most popular approaches are rooted in the work of Wertheim which describes
association as an interaction between acceptor and donor sites assigned to the molecules. The
interaction between acceptor and donor sites is governed by an association strength (Δ𝑖𝑗 ). The
association strength differs from an equilibrium constant because it can include a composition
dependence due to changes in packing fraction. Several forms exist for Δ𝑖𝑗 with the most common
being those of PC-SAFT (Eq. 6-1) and ESD (Eq. 6-2) which differ in the pure component values
need to parameterize the radial distribution function (𝑔𝑖𝑗 ).
𝑐𝑚3 𝐴𝐷
𝜖𝑖𝑗𝐴𝐷
Δ𝑖𝑗 [ ] = 𝑁𝐴 𝑑(𝜎)3 𝑔𝑖𝑗 𝜅𝑖𝑗 (𝑒𝑥𝑝 ( ) − 1) Eq. 6-1
𝑚𝑜𝑙 𝑘𝑇
𝑐𝑚3 𝐴𝐷
𝜖𝑖𝑗𝐴𝐷
Δ𝑖𝑗 [ ] = 𝑔𝑖𝑗 𝜅𝑖𝑗 (𝑒𝑥𝑝 ( ) − 1) Eq. 6-2
𝑚𝑜𝑙 𝑘𝑇
Both models have similar forms for the association strength which requires two association
𝐴𝐷
parameters are required per Δ𝑖𝑗 : an effective bonding volume (𝜅𝑖𝑗 ) and a bonding association
𝐴𝐷
𝜖𝑖𝑗
energy ( ). In addition, the PC-SAFT and ESD models differ in the molecular size parameters
𝑘
and their implementation for the 𝑔𝑖𝑗 . While the full implementation of RTPT regardless of model
124
affords the flexibility of using up to four parameters, we previously demonstrated164 that that
bonding volumes can be set equal (𝜅2𝐴𝐷 = 𝜅𝑁𝐴𝐷 ) while adjusting the energy parameters
independently.
6.3 Methods and Modeling
6.3.1 Sample Preparation and Measurement
Materials were purchased from Sigma-Aldrich at the purity listed in Table 6-1. Reagents
were used without purification and were stored over activated 3 Å molecular sieves in a glovebox
for at least two weeks prior to use.
Table 6-1: Chemical source and purity.
Name Item Purity Anhydrous Supplier
methanol 322415-100ML 99.8% yes Sigma-Aldrich
ethanol 459836-100ML 99.5% yes Sigma-Aldrich
1-propanol 279544-100ML 99.7% yes Sigma-Aldrich
1-butanol 281549-100ML 99.8 % yes Sigma-Aldrich
1-pentanol 138975-100ML 99.1% no Sigma-Aldrich
1-hexanol 471402-100ML 99.1% yes Sigma-Aldrich
2-propanol 278475-100ML 99.5 % yes Sigma-Aldrich
2-butanol 294810-100ML 99.5% yes Sigma-Aldrich
i-butanol 294829-100ML 99.5% yes Sigma-Aldrich
t-butanol 471712-100ML 99.5% yes Sigma-Aldrich
cyclohexane 227048-100ML 99.5% yes Sigma-Aldrich
125
Solutions were prepared in a glove box under a nitrogen atmosphere to preclude the
introduction of moisture. Because excess molar volumes in these systems are typically less than
0.5% (see APPENDIX T: Excess Volume Comparison), samples were prepared assuming ideal
solution behavior and the partial molar volume of each component was assumed to be equal to the
pure component molar volume at all concentrations (Eq. 6-3). Pure component molar volume was
obtained via regression of a polynomial to experimental data from the NIST database150 and
1
𝑉𝑖 = 𝑉𝑝𝑢𝑟𝑒 𝑖 = Eq. 6-3
𝜌𝑝𝑢𝑟𝑒 𝑖
1 𝑥1 𝑥2
= + Eq. 6-4
𝜌 𝜌𝑝𝑢𝑟𝑒 𝑐1 𝜌𝑝𝑢𝑟𝑒 𝑐2
mixture densities were obtained via Eq. 6-4. In these equations, 𝜌 is molar density. Measurements
were performed at 30 °C, 40 °C, 50 °C, and 60 °C for all systems except for methanol +
cyclohexane which was measured at 20 °C, 30 °C, 40 °C, and 50 °C.
6.3.2 Beer-Lambert-Bouguer Law Scaling and Quantification of XA
The methods of scaling developed by Bala et al.122 was used, except that a refined fitting
method was implemented. In the original work, two regressions were nested: an outer non-linear
loop regressed the scaling parameters except for the absorption coefficient at 3645 cm-1; a nested
linear regression determined the absorption coefficient at 3645 cm-1. In the refined method, and
single non-linear regression is performed using the objective function is
𝐴𝑠𝑐𝑎𝑙𝑒𝑑,𝑖 2
𝑂𝐵𝐽 = ∑ (𝑘𝑚𝑒𝑎𝑛 − ) Eq. 6-5
𝐶𝑖
𝑎𝑙𝑙 𝐶𝑖 𝑎𝑡
𝑎𝑙𝑙 𝑇
126
𝐴𝑠𝑐𝑎𝑙𝑒𝑑,𝑖
where 𝑘𝑚𝑒𝑎𝑛 = (∑𝑖 ) /𝑛𝑖 . 𝐴𝑠𝑐𝑎𝑙𝑒𝑑.𝑖 is the alcohol stretch integrated area after scaling, 𝐶𝑖 is
𝐶𝑖
the alcohol concentration, the index 𝑖 runs over all concentrations and temperatures, and 𝑛𝑖 is the
𝑓
number of measurements. During regression, the value of 𝜖𝐵 of Figure 3-8 was constrained to a
value of 1 while 𝜈̃𝐵 , 𝜈̃𝑅 , 𝜖𝑅 , the spline width and the slope of segment 1 were varied. After
𝑓
regression, the value is assigned, 𝜖𝐵 = 𝑘𝑚𝑒𝑎𝑛 . Parameter values for the attenuation functions are
summarized in APPENDIX R: Attenuation Function Parameters.
Figure 6-1: Functional form of the integrated molar attenuation coefficient optimized for each
alcohol (left) required to obtain agreement between integrated hydroxyl absorbance area and
alcohol concentration (right).
The curve-fitting methodology of Killian et al.164 was implemented to obtain experimental 𝑋 𝐴
values representing the fraction of nonhydrogen bonded protons for ten alcohol + cyclohexane
binary and the values of 𝑋 𝐴 are summarized in APPENDIX S: Tabulated XA Values.
6.3.3 Association Parameter Regression using the Fraction of non-Bonded Hydroxyl Protons
The advantage of the RTPT association model compared to the TPT-1 model is the
description of cooperative self-association. The RTPT approach implements a different enthalpy
of formation of the dimer compared to other oligomers. This difference results in two unique
127
association strengths; one for the dimer and one for all successive oligomerizations. Recently we
demonstrated134 how the fraction of non-bonded hydroxyls (𝑋 𝐴 ) can be related to the dimer (Δ2 )
and n-mer (Δ𝑁 ) association constants through the molar density of monomer (𝜌0 ) as seen in Eq.
6-6 and Eq. 6-7. The equations are simultaneously solved iteratively using the adjustable
parameters 𝜖𝑖𝑗𝐴𝐷 and 𝜅𝑖𝑗
𝐴𝐷
to optimize the association strengths to fit the experimental values of 𝑋 𝐴
versus composition data at all experimental temperatures. Regression used previously published
methods.164
2𝑥1 𝜌𝑋 𝐴
𝜌0 = Eq. 6-6
1 + 𝑥1 𝜌Δ2 𝑋𝐴 + √(1 + 𝑥1 𝜌Δ𝑁 𝑋𝐴 )2 + 4(Δ2 − Δ𝑁 )𝑥1 𝜌𝑋𝐴
𝐴
(𝜌0 (Δ2 − Δ𝑁 ) + 1)2
𝑋 = 2 2 Eq. 6-7
𝜌0 (Δ2 − Δ2𝑁 ) + Δ2 (−2𝜌02 Δ𝑁 + 2𝜌0 + 𝜌𝑥1 𝑋𝐴 ) − 2𝜌0 Δ𝑁 + 1
6.3.4 Modeling of Phase Equilibria
The relationship between a liquid and its vapor can be represented using the gamma-phi
corrections to Raoult’s law (Eq. 6-8) where non idealities of the liquid and vapor phases are
described using an activity coefficient (𝛾𝑖 ) and fugacity coefficient (𝜙𝑖 ), respectively.
𝑦𝑖 𝜙𝑖 𝑃 = 𝑥𝑖 𝛾𝑖 𝑃𝑖∗ Eq. 6-8
Commonly, nonidealities represented by the activity coefficient are attributed to three
contributions: combinatorial, residual, and association. The association term encompasses the non-
idealities introduced by molecular aggregation. The combinatorial term is used to describe the
entropic effects resulting from mixing components of differing molecular size and shape. Finally,
the residual term considers molecular energetics as well as compensating for any remaining non-
idealities which are otherwise not accurately represented. The parameters in for the residual term
are regressed to experimental phase equilibria data and excess enthalpy data.
128
6.3.5 Modeling the Association Contribution
In this work we utilize the RTPT form of the association contribution.134 For the self-
associating component, the association contribution of the activity coefficient (𝛾1𝑎𝑠𝑠𝑜𝑐 ) takes the
form of Eq. 6-9. For the non-associating component (𝛾2𝑎𝑠𝑠𝑜𝑐 ) the equation takes the form of Eq.
6-10.
𝜌𝑜 𝑉1 𝜌𝑜,𝑝𝑢𝑟𝑒 1
ln 𝛾1𝑎𝑠𝑠𝑜𝑐 = ln ( ) + 𝑥1 (1 − 𝑋 𝐴 ) ( ) − ln ( 𝐴
) − (1 − 𝑋𝑝𝑢𝑟𝑒 1) Eq. 6-9
𝑥1 𝜌 𝑉 𝜌𝑝𝑢𝑟𝑒 1
𝑉2
ln 𝛾2𝑎𝑠𝑠𝑜𝑐 = 𝑥1 (1 − 𝑋 𝐴 ) ( ) Eq. 6-10
𝑉
The infinite dilution forms of these expressions are shown in Eq. 6-11 and Eq. 6-12 for the
associating and non-associating component, respectively.
𝜌𝑝𝑢𝑟𝑒 1
ln(𝛾1𝑎𝑠𝑠𝑜𝑐 )∞ = ln ( 𝐴
) − (1 − 𝑋𝑝𝑢𝑟𝑒 1) Eq. 6-11
𝜌𝑜,𝑝𝑢𝑟𝑒 1
𝑉2
ln(𝛾2𝑎𝑠𝑠𝑜𝑐 )∞ = (1 − 𝑋𝑝𝑢𝑟𝑒
𝐴
1) ( ) Eq. 6-12
𝑉1
6.3.6 Experimental Limiting Activity Coefficients
The infinite dilution activity coefficient provides significant insight about bounds for
deviations from ideality for the activity coefficient (partial molar excess Gibbs energy) and the
partial molar excess enthalpy. The infinite dilution activity coefficient is closely related to the
solvation Gibbs energy and provides an experimental measurement of that property.165 Equally
important to capturing the correct limiting value of the activity coefficient is replicating the
temperature dependence via the partial molar enthalpies. Consideration of these bounds is required
to keep model results in a range of experimental behavior.
129
We were interested in the trends exhibited by the limiting activity coefficients for each
alcohol + cyclohexane pair. Selected data from the collection of Lazzaroni et al.139 are plotted
below in Figure 6-2 along with the corresponding 95% confidence interval. Plots for cyclohexane
and 2-propanol are in APPENDIX N: Limiting Activity Coefficient Regressions. Previously we
reported that the association contribution is dominant especially at low concentrations of alcohol.
Conceptually, this dominance occurs because the infinite dilution activity coefficient quantifies
the solvation in the mixed state relative to the pure standard state. Because alcohol molecules are
significantly hydrogen bonded in the pure liquid, and nonbonded at infinite dilution, the large
difference in 𝑋 𝐴 creates a large association contribution to the infinite dilution activity coefficient.
130
Figure 6-2: Linear regression of the natural logarithm of the limiting activity coefficient data
of Lazzaroni et al.139 with respect to inverse temperature for methanol (upper-left), ethanol
(upper-right), 1-butanol (middle-left), 1-pentanol (middle-right), 1-hexanol (lower-left), and 2-
propanol (lower-right) in cyclohexane.
131
Eq. 6-13 provides the relation between the infinite dilution activity coefficient and the infinite
𝐸,∞
dilution partial molar excess enthalpies (𝐻𝑖 ). Values were calculated according using the slope
of the line of best fit (𝑏𝑟𝑒𝑔 ) through the a plot of ln 𝛾 ∞ versus 1/𝑇. For primary alcohols with
𝐸,∞
available infinite dilution activity coefficients in cyclohexane, calculated values of (𝐻𝑖 ) are
displayed in Figure 6-3. The experimental limiting partial molar excess enthalpy values for the
alcohols have a mean value of approximately 20 kJ/mol.
𝐸,∞ 𝜕(ln 𝛾𝑖∞ )
𝐻𝑖 = 𝑅( ) = 𝑅𝑏𝑟𝑒𝑔 Eq. 6-13
𝜕(1/𝑇) 𝑃,{𝑛
𝑗 ≠𝑖}
Figure 6-3: Limiting partial molar enthalpies for primary and secondary alcohols in
cyclohexane and cyclohexane in primary and secondary alcohols at 318.15 K. Values were
calculated according to Eq. 6-13.
6.3.7 Modeling the Combinatorial and Residual Contributions
The combinatorial contribution was represented with modified Flory’s (Eq. 6-14) and the
binary form of NRTL was utilized for the residual term (Eq. 6-13). For Eq. 6-16, the empirical
132
parameters 𝑎𝑖𝑗 , 𝑏𝑖𝑗 are adjusted to the experimental P-xy, T-xy, and 𝐻 𝐸 data. The alpha values
𝛼𝑖𝑗 = 𝛼𝑗𝑖 were set to 0.3 for both components except for the methanol + n-hexane system where
𝛼𝑖𝑗 = 𝛼𝑗𝑖 = 0.25.
2 2
𝜌𝑠𝑜𝑙𝑛 3 𝜌𝑠𝑜𝑙𝑛 3
ln 𝛾𝑖𝑐𝑜𝑚𝑏 = ln [( ) ] +1 −( ) Eq. 6-14
𝜌𝑖 𝜌𝑖
2
𝜏𝑖𝑗 𝐺𝑖𝑗 𝐺𝑗𝑖
ln 𝛾𝑖𝑟𝑒𝑠𝑖𝑑 = 𝑥𝑗2 [ 2 + 𝜏𝑗𝑖 (𝑥 + 𝑥 𝐺 ) ] ; 𝑖 ≠ 𝑗 Eq. 6-15
(𝑥𝑖 𝐺𝑖𝑗 + 𝑥𝑗 ) 𝑖 𝑗 𝑗𝑖
𝐺𝑖𝑗 = exp(−𝛼𝑖𝑗 𝜏𝑖𝑗 ) ; 𝜏𝑖𝑗 = 𝑎𝑖𝑗 + 𝑏𝑖𝑗 /𝑇 Eq. 6-16
6.4 Results and Discussion
6.4.1 Regression of XA
Regression of the association parameters also requires a selection of the method to
represent 𝑔𝑖𝑗 . Three models were considered: PC-SAFT, ESD, and 𝑔𝑖𝑗 = constant. For the PC-
SAFT method, various parameters have been used in literature for segment reference diameter, 𝜎,
dispersion energy, 𝜖, and segment number 𝑚. Parameters of Gross and Sadowski, Tamouza et al.
and NguyenHuynh and coworkers are shown in Figure 6-4126,166–168 The values for 𝑚 and 𝜎 are
quite different, but the molecular size 𝑚𝑑 3 calculated at 298.15 K (Figure 6-4 lower left) are very
similar, a trend noted by Bala et al.130 The values of Gross and Sadowski were used for methanol.
Two values are presented for ethanol. The values shown by the red squares with white x are the
recommended values, but these do not fit a systematic trend and therefore the group contribution
method of NguyenHuynh et al. was applied. Note that both sets of ethanol parameters result in a
very similar molecular volume. The value has a very minor effect on the segment diameter, 𝑑, and
molecular size. PC-SAFT parameters are summarized in
133
along with the selected ESD parameters.169 The 𝑋 𝐴 used in the regression are provided in
APPENDIX F: Tabulated 𝑋 𝐴 values.
Figure 6-4: Segment numbers, segment diameters, dispersion energies and molecular size from
NguyenHuynh, Gross and Sadowski, and Tamouza et al. as described in the text.126,166–168
134
Table 6-2: PC-SAFT and ESD parameters used for the radial distribution function at contact.
Parameter 𝒎 𝝈 (Å𝟑 ) 𝝐/𝒌 [K] b (cm3/mol)
Methanol 1.5255 3.230 188.9 19.962
Ethanol 1.7464 3.567 280.03 24.743
1-Propanol 2.1326 3.6605 275.288 30.005
1-Butanol 2.5188 3.7162 272.391 35.762
1-Pentanol 2.905 3.7533 270.4 40.732
1-Hexanol 3.2912 3.7798 268.921 46.45
Cyclohexane 2.5303 3.8499 278.11 35.030
The infinite dilution values for the activity coefficients are closely related with the value
of 𝑋 𝐴 in pure alcohol as shown by Eq. 6-11. The 𝑋 𝐴 data were regressed with both PC-SAFT and
ESD models using experimental pure component volumes and the ideal volume of mixing. The
resulting infinite dilution activity coefficient values obtained for each component are plotted in
Figure 6-5. Both models give alcohol infinite dilution activity coefficients for the association
contribution that are larger than the experimental infinite dilution value for 1-propanol through 1-
hexanol. The ESD model gives values too high for all the alcohols. Considering only the infinite
dilution association contribution for cyclohexane, the PC-SAFT model is acceptable. Also plotted
in the figure are the values resulting from 𝑔𝑖𝑗 = 1 for the radial distribution at contact,
implemented by using Eq. 6-2 an approach which has been used in the GCA-EOS170, and by Hao
and Chen.171 As noted by Bala and Lira,37 this approximation becomes mathematically equivalent
to traditional chemical theory when combined with the original Flory equation for the
combinatorial contribution.
Because the RTPT model is sensitive to the choice of the radial distribution function at
contact, we applied 𝑔𝑖𝑗 = 1 resulting in in the values given by the black circles of Figure 6-5. This
approach maximizes the transferability of RTPT since a numerically equivalent association
135
contribution can be obtained for any model which adopts the fitted parameters when the density
and temperature are identical.
Figure 6-5 Comparison on experimental infinite dilution activity coefficients with values
obtained by fitting the PC-SAFT and ESD radial distribution at contact using experimental
pure component densities.
Regressing Eq. 6-2 using 𝑔𝑖𝑗 = 1, the statistically optimal parameter for regression of the
XA data fails to accurately capture the limiting value of the excess enthalpy for the alcohol. As seen
in Figure 6-6 (bottom) for the 1-butanol + cyclohexane system, the slope of the modeled values is
steeper than the corresponding experimental values. Therefore, the residual term would need to be
negative to counter this behavior. This was observed for all the binary systems and the partial
molar excess enthalpy for each component is plotted in Figure 6-7 and are ~5-7 kJ/mol larger for
each alcohol than those calculated from the infinite dilution activity coefficients (Figure 6-3).
Therefore, using the statistically optimal values would require an exothermic residual contribution
to counteract the endothermic overprediction of the excess enthalpy by the association term.
Commonly, the residual contribution to the excess enthalpy is expected to be endothermic for a
136
dispersion interaction of dissimilar molecules. The values for cyclohexane were slightly smaller
than the experimental values by approximately 1 kJ/mol.
Figure 6-6: Regression of XA with unconstrained parameter values for 1-butanol in
cyclohexane (top). Experimental172 and modeled molar enthalpy ascribed to the association
contribution which results from unconstrained parameters for 1-butanol + cyclohexane system
at 318.15 K (lower).
137
Figure 6-7: Limiting partial molar enthalpies for alcohol + cyclohexane systems at 318.15 K.
Values were calculated using the statistically optimal parameter set for the regression of XA.
To meet the need for representing the experimental excess enthalpy, the n-mer value was
constrained to 𝜖𝑁𝐴𝐷 /𝑘 = 2500 K, resulting in a compromise of the quality of the 𝑋 𝐴 (Figure 6-8).
As a result, the residual contribution was positive across the entire composition range. The reason
for the need to constrain the enthalpy is not currently known.
During regression, unconstrained regression of a separate 𝜅2𝐴𝐷 was considered. However,
a strong correlation was noted upon the corresponding value of 𝜖2𝐴𝐷 /𝑘. For the resulting fits, we
anecdotally noted that the value of Δ2 was similar from the various fits. Thus, in keeping with our
previous work the 𝜅 𝐴𝐷 terms were set equal to one-another (𝜅2𝐴𝐷 = 𝜅𝑁𝐴𝐷 ).134 Thus, even with the
second association strength in the model, only two association parameter values are fitted, like
TPT-1. Confidence intervals were determined using the bootstrap methods with 1000 trials and a
reference temperature of 298.15 K as explained in the previous work.164
138
𝐴𝐷
𝜖𝑁
Figure 6-8: Comparison of unconstrained (--) versus = 2500 𝐾 constrained (-) regression
𝑘
of XA data for 1-hexanol + cyclohexane.
6.4.2 Association Parameters
Association parameters obtained from the constrained 𝑋 𝐴 regression are presented in Table
6-3 and the corresponding Δ2 and Δ𝑁 are plotted in Figure 6-10 at 318.15 K. The plots include the
95% confidence limits from the bootstrap method. The bonding volume and bonding energy are
strongly correlated and for determination of uncertainty of association strength, the lower/upper
limit of each parameter should not be used together to determine the lower/upper limit of the
association energy. Instead, the pairs of optimized parameters from each bootstrap fitting trial were
used together to calculate association strengths at 318.15 K for each trial, which were subsequently
sorted and then the 95% confidence limits were determined from the 25th and 975th values.
139
Table 6-3: Association parameters and 95% confidence interval obtained from constrained
optimization.
𝒄𝒎𝟑 𝝐𝑨𝑫 𝝐𝑨𝑫
Alcohol 𝜿𝑨𝑫 , [ ] 95% CI 𝟐
, [𝑲] 95% CI 𝑵
, [𝑲]
𝒎𝒐𝒍 𝒌 𝒌
methanol 2.5327 2.3794 – 2.6688 1677.2 1529.0-1814.3 2500
ethanol 1.8410 1.8011 -1.8762 1676.2 1598.1 – 1761.8 2500
1-propanol 2.0550 1.9863 – 2.1184 1699.6 1623.3 – 1768.5 2500
1-butanol 2.0527 1.9950 – 2.1044 1767.5 1721.0 – 1812.8 2500
i-butanol 1.9592 1.8655 – 2.0301 1817.1 1718.1 – 1890.2 2500
1-pentanol 2.2745 2.1802 – 2.3599 1434.7 1311.4 – 1551.5 2500
1-hexanol 2.2657 2.2029 – 2.3241 1759.6 1674.2 – 1831.4 2500
2-propanol 2.2756 2.1693 – 2.3505 1614.0 1528.0 – 1690.0 2500
2-butanol 1.6536 1.6019 – 1.7097 1723.9 1634.2 – 1801.5 2500
t-butanol 1.3283 1.2876 – 1.3612 1948.1 1874.3 – 2017.4 2500
140
Figure 6-9: Association parameter comparison with 95% confidence interval for 𝜅 𝐴𝐷 (left) and
𝜖2 𝐴𝐷
(right).
𝑘
Figure 6-10: Delta comparison with 95% confidence interval for the dimer (left) and the n-mer
(right) at 318.15 K.
6.4.3 Mapping RTPT onto TPT-1
Mapping of the fitted RTPT association contribution to TPT-1 is of interest for two reasons:
1) the mapping provides an engineering approximation to mimic the association behavior of the
141
fit to the spectroscopic data because most engineering association models are coded using TPT-1
and this mapping provides transfer of the spectroscopically determined association contribution
for use in existing code; and 2) the method permits demonstration of the benefits of using RTPT
𝑎𝑠𝑠𝑜𝑐,∞
by comparing similar magnitude contributions when the ln 𝛾𝑎𝑙𝑐𝑜ℎ𝑜𝑙 values from the two methods
are nearly the same.
Using RTPT parameters obtained from regression of the experimental infrared data, 𝛾 ∞
values were generated between 290 – 420 K for both the alcohol and non-hydrogen-bonding
solvent cyclohexane. Then, TPT-1 𝜅 𝐴𝐷 and 𝜖 𝐴𝐷 /𝑘 were regressed simultaneously for both
components to obtain values that minimized the error between TPT-1 predicted and RTPT
generated limiting activity coefficients according to Eq. 6-17.
𝑂𝐵𝐽 = ∑ ∑ (ln 𝛾𝑖∞,𝑎𝑠𝑠𝑜𝑐,𝑅𝑇𝑃𝑇 − ln 𝛾𝑖∞,𝑎𝑠𝑠𝑜𝑐,𝑇𝑃𝑇1 )𝑗 Eq. 6-17
𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑗
A graphical example of this is provided in Figure 6-11 for ethanol + cyclohexane. Note that the
alcohol infinite dilution activity coefficient is fitted better than the cyclohexane activity coefficient
because the objective function favors fitting of the larger activity coefficient. Similar plots are
available in APPENDIX O: Mapping RTPT onto TPT-1 for the other binary systems. TPT-1
parameters obtained from this mapping process are available in Table 6-4. The association energy
for TPT-1 is larger than either RTPT association energy.
142
Figure 6-11: Mapping the RTPT predicted limiting activity coefficients to the two-parameter
form of TPT-1 for ethanol and cyclohexane.
Table 6-4: TPT-1 association parameters obtained by mapping from RTPT limiting activity
coefficients.
𝜿𝑨𝑫 , 𝝐𝑨𝑫
Compound , [K]
[cm3/mol] 𝒌
methanol 0.064448 3633.50
ethanol 0.92537 2670.94
1-propanol 0.91623 2704.14
1-butanol 0.84923 2723.85
1-pentanol 0.67127 2816.11
1-hexanol 0.80767 2763.15
i-butanol 0.83051 2716.96
2-propanol 0.94781 2724.52
2-butanol 0.59068 2762.06
t-butanol 0.54158 2725.75
143
Figure 6-12: Comparison of XA regression for TPT-1 (--) and RTPT (-) for 1-hexanol +
cyclohexane system.
A comparison of TPT-1 and RTPT for the 1-hexanol + cyclohexane system is shown in
Figure 6-12. While TPT-1 may match the RTPT limiting activity coefficients to an acceptable
degree, it is unable to capture the curvature of the experimental 𝑋 𝐴 data in the dilute or
concentrated alcohol region, and calculated values of 𝑋 𝐴 do not match the values from
spectroscopy. This mismatch may affect mixture calculations with components with other
association sites where the bond populations are more important for the statistics of cross
associations.
For comparisons of the models with phase equilibria and excess enthalpy, the TPT-1
association parameters were used to repeat regressions that had been conducted for RTPT. The
same data and weights were used and thus the comparisons reflect the abilities of the models fitted
to the same data using the same 𝑔𝑖𝑗 = 1.
6.4.4 Regressed Phase Behavior and Excess Enthalpy
The following values of the Hayden-O’Connell parameters were used for the vapor phase
nonidealities are summarized in Table 6-5: Summary of Hayden-O’Connell parameters used for
144
phase equilibria regression. In some cases, parameters were estimated based on trends in existing
values.
Table 6-5: Summary of Hayden-O’Connell parameters used for phase equilibria regression.
Alcohol 𝜼 Alcohol 𝜼
Methanol 1.63 1-Hexanol 2.2
Ethanol 1.4 2-Propanol 1.32
1-Propanol 1.4 2-Butanol 1.75
1-Butanol 2.2 Isobutanol 1.9
1-Pentanol 2.2 t-Butanol 1.0
Selected results demonstrating the ability of RTPT and comparing to TPT-1 are available
in Figure 6-13 to Figure 6-16 and the remaining systems are available in this APPENDIX P: Phase
Equilibria and Excess Enthalpy.
One improvement of RTPT relative to TPT-1 is that the bubble line slope is larger in
magnitude over a wider composition range as seen in the 1-propanol, t-butanol, and methanol
systems, and more closely matches the experimental data. This behavior occurs because the
association contribution to the activity coefficient is relatively flat as infinite dilution of the alcohol
is approached as shown in Figure 6-17 (bottom left). This extends the composition range of the
large value of the activity coefficient.
A second advantage of RTPT becomes apparent when considering the excess enthalpy.
The overall shape and temperature dependence of the excess enthalpy is improved when RTPT is
used. An inadequate temperature dependence from TPT-1 is evident even though the association
energy is larger than either association energy of RTPT.
Of particular interest is the behavior of the limiting values of the excess enthalpy and
activity coefficients which is demonstrated in Figure 6-17. Experimental excess enthalpy data
145
exhibits a pronounced rollover below 1 mol% alcohol for 1-butanol. This subtlety is captured by
RTPT. The RTPT model also reproduces the crossover of the values from the two temperatures at
low mole fractions.
Figure 6-13: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)173 and
𝐻 𝐸 (right)174 for 1-propanol in cyclohexane.
For the methanol + cyclohexane system in Figure 6-16, the TPT-1 model has been
previously noted to have difficulty fitting the liquid-liquid upper critical solution temperature and
the excess enthalpy simultaneously.130 Similar challenges are encountered with RTPT, and for this
work we favored the fit of excess enthalpy for both models. The improvement in representation of
excess enthalpy is significant for RTPT.
146
Figure 6-14: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)140,175 and
𝐻 𝐸 (right)172 for 1-butanol in cyclohexane.
Figure 6-15: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)176,177 and
𝐻 𝐸 (right)178 for t-butanol in cyclohexane.
147
Figure 6-16: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)179,180 and
𝐻 𝐸 (right)181,182 for methanol in cyclohexane.
148
Figure 6-17: Limiting excess enthalpy data172 for 1-butanol + cyclohexane system at 298.14
and 318.14 K overlayed with the RTPT regression results (top). Activity coefficient data172 for
1-butanol + cyclohexane system at 318.14 K overlayed with the RTPT regression results in the
dilute region (bottom left) and across the composition range with combinatorial and residual
contributions (bottom right).
149
Table 6-6: NRTL residual contribution parameters obtained via regression.
Component i Component j 𝒂𝒊𝒋 𝒃𝒊𝒋 𝒂𝒋𝒊 𝒃𝒋𝒊 𝜶𝒊𝒋 = 𝜶𝒋𝒊
Methanol Cyclohexane -2.7704 1182.16 0.91635 -444.28 0.3
Methanol n-Hexane -0.57056 657.24 -0.78994 -95.841 0.25
Ethanol Cyclohexane 0.76289 251.46 -0.87517 5.3007 0.3
1-Propanol Cyclohexane 2.7896 -563.00 -2.1271 486.91 0.3
1-Butanol Cyclohexane 1.8510 -383.49 -1.6993 431.18 0.3
1-Pentanol Cyclohexane 3.2920 -615.50 -2.3097 470.99 0.3
1-Hexanol Cyclohexane 3.1474 -520.49 -2.1549 402.07 0.3
i-Butanol Cyclohexane 2.9990 -378.95 -1.9335 280.44 0.3
2-Propanol Cyclohexane 3.5046 -658.62 -2.6118 553.51 0.3
2-Butanol Cyclohexane 3.4572 -514.31 -2.4332 423.13 0.3
t-Butanol Cyclohexane 0.23868 473.92 -0.83368 -63.126 0.3
6.4.5 Analysis of Ternary System
Extension of binary interactions to multicomponent mixtures is important for industrial
modeling. The ternary system of methanol + n-hexane + cyclohexane was predicted using
parameters obtained from binary regressions of methanol + cyclohexane and methanol + n-hexane.
Since methanol + n-hexane was not measuring using infrared the association parameters from
methanol + cyclohexane were used. Individually, these systems are challenging to represent given
the liquid-liquid phase split which occurs in both binary systems and the sensitivity of the liquid-
liquid region to temperature. Furthermore, infrared measurements were performed only to ~10
mol% methanol since higher concentrations would have resulted in a liquid-liquid phase split. As
seen in Figure 6-16 RTPT can accurately represent the LLE and the temperature dependence of
the ternary binodal if the temperatures are not within 4 K of the upper critical solution temperatures
150
of the constituent binaries. The slope of the tie lines of the ternary is well matched by RTPT (Figure
6-18).
Figure 6-18: RTPT representation of ternary LLE in methanol + n-hexane + cyclohexane
system at 293.15 K (left) and 303.15 K (right).183
6.4.6 The Shape of the Contributions to the Excess Enthalpy
Analogous to the activity coefficient, the 𝐻 𝐸 can be subdivided into association,
combinatorial, and residual contributions as presented in Figure 6-19. The remaining systems are
available in APPENDIX P: Phase Equilibria and Excess Enthalpy. The total excess enthalpy shows
a slight asymmetry where the apex is shifted to lower alcohol mole fractions. In all cases examined
the association contribution is strongly skewed to the lefthand side of the diagram contributing
significantly at dilute concentrations of alcohol and seems to be responsible for the overall
asymmetry. The magnitude of the combinatorial contribution is insignificant. However, another
interesting observation considers the shape of the residual contribution which is symmetric for
most of the systems.
151
Figure 6-19: Contributions to 𝐻 𝐸 for i-butanol (upper left), 1-pentanol (upper right), 2-
propanol (lower left), and 1-hexanol (lower right) in cyclohexane at 318.15 K.
6.4.7 Analysis of Residual Contribution
Taken individual the 𝑎𝑖𝑗 /𝑎𝑗𝑖 , 𝑏𝑖𝑗 /𝑏𝑗𝑖 , and 𝛼𝑖𝑗 /𝛼𝑗𝑖 parameters, it is difficult to discern
behavior trends. Using the rightmost expression in Eq. 6-16 we see that 𝜏𝑖𝑗 and 𝜏𝑗𝑖 can be calculated
from the 𝑎 and 𝑏 values. This can in turn be used to evaluate the contribution of the NRTL residual
to 𝛾𝑖∞ ; calculated via Eq. 6-18 and Eq. 6-19. Generally, the residual contribution to the limiting
152
activity coefficient decreased for both components as the size of the alcohol carbon chain increased
though the values for the cyclohexane are more scattered.
ln 𝛾1𝑟𝑒𝑠𝑖𝑑,∞ = 𝜏𝑖𝑗 ⋅ exp(−αij 𝜏𝑖𝑗 ) + 𝜏𝑗𝑖 Eq. 6-18
ln 𝛾2𝑟𝑒𝑠𝑖𝑑,∞ = 𝜏𝑗𝑖 ⋅ exp(−αji 𝜏𝑗𝑖 ) + 𝜏𝑖𝑗 Eq. 6-19
Figure 6-20: Contribution of the NRTL residual term to the limiting activity coefficient for
alcohols in cyclohexane (solid) and cyclohexane in alcohols (open) calculated at 318.15 K.
153
Figure 6-21: Ratio of Δ𝑁 /Δ2 values for primary (square), branched primary (diamond)
secondary (circle), and tertiary (triangle) alcohols in cyclohexane at 318.15 K.
Δ
The ratio of the association constants ( Δ𝑁 ) at 318.15 K is shown in Figure 6-21. The n-mer
2
association constant is ~10-15 times that of the dimer for primary alcohols except for 1-pentanol.
The value of the ratio varies with temperature. Both secondary alcohols are not easily differentiated
from their primary analogs. The deviation of the 1-pentanol ratio from the other systems is apparent
and warrants further investigation.
6.5 Summary and Conclusions
This chapter confirms and extends conclusions of previous work where we demonstrated
that two association parameters are needed to model spectroscopic data and the association term
is the dominant contribution to solution nonideality.
The statistically optimum parameters result in infinite dilution partial molar enthalpies that
are 5-7 kJ/mol too large. Thus, the value ϵAD 𝑁 /𝑘 = 2500 𝐾 was used and two association
parameters (𝜅 𝐴𝐷 and 𝜖2𝐴𝐷 /𝑘) were adjusted.
154
While the values of the association parameters are unique for each alcohol, the values of Δ2
and Δ𝑁 are similar for primary alcohols. Future work may show that a reasonable approximation
is to use a universal value for all shorter chain primary alcohols. The association strengths for 2-
propanol were like those for primary alcohols, while the association strengths for 2-butanol were
smaller.
The ratio of association constant of the n-mer was found to be roughly ten to twelve times
greater than the dimer for primary alcohols at 318.15 K suggesting that formation of the dimer is
the most difficult step thermodynamically. To obtain agreement with experimental excess
enthalpies we utilized literature measurements of the infinite dilution activity coefficients to
provide a realistic boundary on the parameter optimization. This constraint improved the
representation of the excess enthalpy while still providing an acceptable regression of the infrared
data and partial molar excess enthalpy and allowed for a positive NRTL residual contribution.
Despite the asymmetry of the excess enthalpy, it was found that the residual contribution was
symmetric for many of the systems.
The shape of RTPT activity coefficients and partial molar enthalpies has a slope that
decreases in magnitude as infinite dilution of the alcohol is approached. The shape improves the
fitting of VLE and excess enthalpy.
As an engineering approximation, TPT-1 can be fitted to infinite dilution activity
coefficients predicted by RTPT. The resulting TPT-1 does not match the fraction of non-bonded
hydroxyls obtained from the infrared measurements, and thus should not be regarded as
representing the bonding distributions in solution. The temperature dependence of RTPT is
superior to TPT-1. RTPT can replicate the limiting behavior of the excess enthalpy and the activity
155
coefficients. However, TPT-1 provides a first approximation with an engineering application
where the level of detail provided by RTPT is not required.
Because of the use of 𝑔𝑖𝑗 = 1 for both RTPT and the mapped fitting of TPT-1, the resulting
association model is independent of other pure component parameters and should be transferable
to any engineering model. Only two association parameters were fitted for both models.
156
CHAPTER 7:Relation of Hydroxyl NMR Chemical Shift to Infrared
Vibrational Frequency
7.1 Introduction
Spectroscopy is an important experimental method for probing hydrogen bonding
interactions. Both mid-range infrared spectroscopy (IR) and nuclear magnetic resonance (NMR)
spectroscopy have been used to investigate hydrogen bonding. However, each technique is
accompanied by unique strengths and limitations. Recently, we have been refining spectroscopic
techniques for studying alcohols in hydrocarbons for improvement of engineering models121,122,164
and this work focuses on two alcohols systems. Understanding of alcohol mixtures provides a base
for furthering understanding in more complex systems.196
Quantifying hydrogen bonding for small alcohols using IR at concentrations above 30
mol% is challenging since the absorbance of the hydroxyl group exceeds one absorbance unit.
NMR is more accommodating for high alcohol concentrations and with a coaxial sample tube can
be used to measure pure alcohol. NMR’s ability to measure more concentrated samples is offset
by the level of detail which the measurement can provide. For alcohols dissolved in a nonpolar
solvent the lifetime of a hydrogen bond is much shorter than the measurement timescale of NMR,
thus the chemical shift of the hydroxyl proton is an average of a many shielding environments
present in the sample. The diversity of environments experienced by hydroxyl protons in the
solution collapses to a single chemical shift. Mid-range infrared measurements occur on a
timescale sufficient to capture different environments, and the resulting spectra can be used to
calculate the populations of hydroxyl types.164
While the two methods rely on fundamentally different phenomena (magnetic moment
relaxation for NMR and transition dipole moment for IR), properly interpreting the measurements
of a single sample should lead the researcher to the same quantification of bonding. Quantum
157
calculations provide both a predicted chemical shift and vibrational stretch for hydroxyl bonds in
selected environments. The integrated area in the scaled IR vibrational spectra is temperature-
dependent, and one method of characterizing modulations in hydrogen bonding configurations
uses the correlation of integrated areas on either side of a naturally occurring isosbestic point using
difference spectra relative to a reference temperature.106,123 Recently, we have demonstrated that
the IR spectra can be scaled to provide a temperature-independent integrated area under the
hydroxyl region.122 This work demonstrates that the scaled and integrated absorbance signal of an
alcohol’s hydroxyl group can be mapped using quantum calculations as a guide to correlate the
NMR shift at any concentration and temperature.
7.2 Background
Hydroxyl configurations in hydrogen bonds are characterized by Greek letters.164 An
unbonded hydroxyl is denoted as 𝛼. A hydroxyl accepting a proton on the oxygen with a free
proton and labeled 𝛽. A hydroxyl donating a proton with an unbonded oxygen is denoted 𝛾. A
hydroxyl accepting a single proton and donating a proton is 𝛿. Gutowsky and Saike197 proposed
that the observed chemical shift of a hydroxyl proton relative to tetramethylsilane (TMS) can be
related to the fraction of the free and complexed species in the system. For hydrogen bonding
𝑁𝑀𝑅 𝑁𝑀𝑅
alcohols in an inert solvent this relationship takes the form of Eq. 7-177 where 𝛿𝑜𝑏𝑠 , 𝛿𝑓𝑒𝑛𝑑𝑠 ,
𝑁𝑀𝑅 𝑁𝑀𝑅
𝛿𝑐𝑦𝑐𝑙𝑖𝑐 , and 𝛿𝑐ℎ𝑎𝑖𝑛 are the observed chemical shift, the chemical shift of the free-hydroxyls (𝛼/𝛽),
the chemical shift of all members of a cyclic species, and the chemical shift of the non-free-end
molecules in a linear chain, respectively. In the notation of Gutowski and Saike used in this
equation, the 𝛿𝑖𝑁𝑀𝑅 𝑠 represent the NMR shifts, not the hydroxyl labeling. The concentration of
monomers (𝛼 hydroxyls), cyclic aggregates of size 𝑖 and linear clusters of length 𝑖, and the total
concentration of alcohol are denoted as 𝐶𝑀 , 𝐶𝑖𝑐 , 𝐶𝑖𝑙 , and 𝐶𝑎𝑙𝑐 , respectively.
158
∞
𝐶𝑀 𝑁𝑀𝑅 𝑛𝐶𝑖𝑐 𝑁𝑀𝑅 𝐶𝑖𝑙 𝑁𝑀𝑅
𝐶𝑖𝑙
𝑁𝑀𝑅
𝛿𝑜𝑏𝑠 = 𝛿 + ∑ [( )𝛿 +( )𝛿 + ((𝑖 − 1) ) 𝛿 𝑁𝑀𝑅 ]
𝐶𝑎𝑙𝑐 𝑓𝑒𝑛𝑑𝑠 𝐶𝑎𝑙𝑐 𝑐𝑦𝑐𝑙𝑖𝑐 𝐶𝑎𝑙𝑐 𝑓𝑒𝑛𝑑𝑠 𝐶𝑎𝑙𝑐 𝑐ℎ𝑎𝑖𝑛 Eq. 7-1
𝑖=2
One important assumption is that the chemical shifts are fixed constants for each hydroxyl
configuration. A second assumption is that there is no change in volume upon formation of a
hydrogen bond. Recently, we have developed a new method to quantify bond configurations using
mid-range IR spectroscopy which provides an opportunity to evaluate the use of constants for each
hydroxyl configuration.164 Previous chapters indicate that hydrogen bonded hydroxyls present a
range of wavenumbers in the infrared spectrum that can be attributed to a near-continuum of
species which reflect overlapping frequency distributions of the bond configurations. This
behavior is supported by the quantum chemical calculations of Bala121 for ethanol + cyclohexane
and 1-butanol + cyclohexane and provides the inspiration for the relationship proposed here. A
scaling of the IR spectra provides temperature-independent integrated areas of the hydroxyl stretch
region, permitting quantification of the total hydroxyl content in solution,122 and after curve-fitting,
the distribution of configurations for primary alcohols is modeled satisfactorily by the assumption
of linear chains and two association strengths.164 This work provides a method to map from the
broad hydroxyl distribution at a given concentration to a single value for the NMR shift.
7.3 Methods
7.3.1 Mapping Overview
Bala performed quantum calculations for hydroxyl configurations that were sampled from
classical molecular dynamics simulations.121 The IR stretching frequency and NMR shift were
calculated using the B3LYP level of theory and 6-31G* basis set. Though the above is a simple
approximation, higher levels of theory exhibit the same trends.122 Configurations were calculated
159
and plotted in Figure 7-1. Also shown in the figure are the bond configurations identified for each
hydroxyl.
Figure 7-1: Quantum calculation results from Bala121 demonstrating the relationship between
the hydroxyl proton chemical shift and the hydroxyl infrared absorbance frequency for 1-
butanol + cyclohexane and ethanol + cyclohexane. The second-degree polynomial function of
best fit relating these two quantities is represented by the dashed line which serves as the
initial value for the optimization.
The mapping process overview is illustrated in Figure 7-2. Scaling has been performed as
described previously,164 and it should be noticed that the ordinate is subsequently normalized using
division with the sample concentration. Because the normalized area under the scaled spectra is
independent of temperature, the fractional area for a subset of wavenumbers is related to the
fraction of hydroxyls in that wavenumber range. IR spectra in the lower figure can be mapped to
an NMR shift using Figure 7-1. Selecting a finite wavenumber range provides an integrated area
that can be mapped by using the average wavenumber for the finite wavenumber range. By
summing the contributions weighted by the fractional area, a single value is obtained for the NMR
shift of the solution.
160
Figure 7-2: Schematic providing a visual description of the method being proposed herein.
Fractional areas produced by integration of the scaled infrared spectra and multiplied by the
corresponding mapping function value and summed for all wavenumber intervals to produce a
single chemical shift value.
7.3.2 Mapping Implementation
High quality NMR data are obtained from Bala121 and Karachewski et. al77. Because the
NMR data are not available at the same temperatures and concentrations as the infrared
measurements, modified Akima interpolation was used on the experimental NMR data to replicate
the mole fractions and temperatures used in our infrared experiments.
161
Since the QM calculations require scaling, we optimized the mapping relationship using
the polynomial coefficients as the initial guess. The mapping equation was allowed to shift in the
y-direction by values optimized according to the objective function calculated as 𝑜𝑏𝑗 =
2
∑𝑛𝑖((𝛿𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑,𝑖 − 𝛿𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,𝑖 )/𝛿𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,𝑖 ) . Values for the mapping function are 𝛿 𝑁𝑀𝑅 =
2
−2.511𝐸 − 05𝜈̃𝑎𝑣𝑒 + 0.1581𝜈̃ − 243.10 + 𝑜𝑓𝑓𝑠𝑒𝑡.
7.4 Results
IR data are available up to approximately 30 mol% alcohol to keep the absorbance below
two, and thus the mapping is limited to this composition range. For the 1-butanol + cyclohexane
system the mapping function was optimized for each temperature (Figure 7-3 top) using the y-
direction shift required to achieve the best fit (Figure 7-3 bottom left). The NMR shift changes by
about 4 ppm over the concentration range while the optimized offset increases by only 0.12 ppm
with increasing temperature over the 30 °C range. For the 2-propanol + cyclohexane system a
single offset was regressed to all temperatures simultaneously (Figure 7-4 Top).
7.5 Discussion
It is impressive that such a simple relation between IR and NMR can produce such results.
In the course of this work, we also considered a linear relation for the mapping function, but the
polynomial provided better mapping. The mapping function derived based on 1-butanol and
ethanol was applied to NMR data for both 1-butanol and 2-propanol. We note that the polynomial
function has a maximum near 3100 cm-1 but is essentially flat in this region. Improvements may
be possible via a broader range of species for the mapping function and particularly more
calculations for larger oligomers which exhibit vibrations at the lowest wavenumbers. Study of a
broader range of alcohols is also suggested. While the shape of the mapping function may be
improved, we hypothesize that refinements will have similar shape to the one presented. The
162
mapping process requires that the alcohol hydroxyl region in the infrared spectra is scaled prior to
mapping. The success of the mapping is further evidence of the validity of the scaling method.
Another observation from Figure 7-1 is that the 𝛾 bonds on one end of a chain should have
a different average NMR shift value compared to 𝛿 bonds. Much of the distribution of 𝛾 bonds is
obscured in Figure 7-1 and the original work should be consulted for more detail.122 Both the 𝛽
and 𝛼 hydroxyls should be considered as unbonded. When curve fitting the scaled IR spectra up
to about 30 mol% the peak centers are nearly constant,164 suggesting that an average shift value
may be possible to propose for each alcohol configuration when fitting NMR data rather than using
adjustable constants for hydroxyl configurations. By reducing the adjustable constants, it may be
possible to increase confidence in the association constants used to fit the NMR shifts. More work
is required to evaluate the mapping function to evaluate whether it is appropriate for a wider class
of alcohols and cyclic structures. The MD simulations of primary alcohols show minimal cyclic
structures.122,148
163
Figure 7-3: Interpolated experimental NMR data for 1-butanol in cyclohexane (o) overlayed
with the regressed values of the optimized mapping function for each temperature (--) (top).
Vertical shift of mapping function that was required to produce an optimized result as a function
of temperature (bottom left). Absolute error between calculated and experimental values (bottom
right).
164
Figure 7-4: Interpolated experimental NMR data for 2-propanol in cyclohexane (o) overlayed
with the values from the regressed mapping function (--) (top). Absolute error between
calculated and experimental values (bottom).
7.6 Conclusions
This work provided a mapping relationship between the chemical shift of an alcohol
hydroxyl proton and its wavenumber absorbance using quantum calculations to guide
development. This mapping function was optimized using experimental infrared and NMR data
for the systems 1-butanol + cyclohexane and 2-propanol + cyclohexane.
165
CHAPTER 8: Densities of Selected Deuterated Solvents
8.1 Preface
Deuterated organic solvents are used frequently in modern nuclear magnetic resonance
(NMR) experiments; however, the densities of these liquids are seldom available for conditions
outside 298.15 K. To address this shortcoming, we collected density data for twelve common NMR
solvents, including dichloromethane-d2, toluene-d8, pyridine-d5, ethanol-d6, tetrahydrofuran-d8,
dimethyl sulfoxide-d6, benzene-d6, acetone-d6, methanol-d4, cyclohexane-d12, acetonitrile-d3, and
chloroform-d. Temperature-dependent liquid density values for dichloromethane-d2, toluene-d8,
pyridine-d5, dimethyl sulfoxide-d6, ethanol-d6, and tetrahydrofuran-d8 were reported for the first
time. Our measurements provide an expanded temperature range for benzene-d6, chloroform-d,
methanol-d5, cyclohexane-d12, acetonitrile-d3, and acetone-d6, for which some literature values
were available. The coefficient of isobaric thermal expansivity and molar volume of each
component was calculated, and the latter quantity was compared with its protiated form.
8.2 Publisher Permission
Reprinted (Adapted or Reprinted in part) with permission from
Densities of Selected Deuterated Solvents
William G. Killian, Andrew T. Norfleet, and Carl T. Lira
Journal of Chemical & Engineering Data 2022 67 (4), 893-901
DOI: 10.1021/acs.jced.1c00990.
Copyright 2022 American Chemical Society
8.3 Introduction
Nuclear magnetic resonance (NMR) spectroscopy is a versatile analysis technique for the
chemical, biological, and engineering sciences. Modern NMR spectrometers utilize increasingly
166
powerful magnets to probe the intricacies of molecular structure and behavior. Deuterated solvents
are typically used to solubilize analytes since 2H locks the spectrometer and produces a signal that
is unobtrusive to that of a 1H nucleus. When leveraging NMR to probe intermolecular interactions
or evaluate molecular diffusion,198 knowledge of the relationship between temperature and density
for the analyte and the solvent is necessary. Unfortunately, deuterated solvent density information
is limited near ambient temperatures or unavailable, even for the most frequently encountered
perdeutero solvents. In this work, we report density for twelve deuterated NMR solvents, including
dichloromethane-d2, toluene-d8, pyridine-d5, ethanol-d6, tetrahydrofuran-d8, and dimethyl
sulfoxide-d6, benzene-d6, acetone-d6, methanol-d4, cyclohexane-d12, acetonitrile-d3, and
chloroform-d at atmospheric pressure and temperatures between 273.15 K and 368.15 K.
Temperature-dependent liquid densities and isobaric thermal expansivities are reported for the first
time over this temperature interval for dichloromethane-d2, toluene-d8, pyridine-d5, dimethyl
sulfoxide-d6, ethanol-d6, and tetrahydrofuran-d8. Our measurements provide an expanded
temperature range for benzene-d6,199–203 chloroform-d,203–206 methanol-d5,207,208 cyclohexane-
d12,209,210 acetonitrile-d3,211,212 and acetone-d6.213
8.4 Experimental Methods
8.4.1 Components
The solvent suppliers, purity, and deuteration percentages are listed in Table 8-1. Materials
were received in sealed glass ampoules and were used without further purification, except for
chloroform-d, which was stored over activated 3 Å molecular sieves for two weeks prior to
measurement. The molecular sieves were activated by heating in a kiln for five days at 975 K
before cooling to room temperature under a vacuum. Molecular masses were obtained for each
deuterated compound from the NIST Chemistry WebBook,150 except for tetrahydrofuran-d8, which
167
was calculated from its protiated form using Eq. 8-1 where the molecular mass of the hydrogen-
containing analog (𝑀𝐻 ) was added to the product of the number of deuterium (𝑁𝐷 ) and the
difference in molar mass between hydrogen and deuterium (ΔDH ) taken as 1.006277 g/mol.
𝑀𝐷 = 𝑀𝐻 + 𝑁𝐷 (Δ𝐷𝐻 ) Eq. 8-1
Table 8-1: Compound specifications.
% atom % Water,
Compound Manufacturer % Purity, (Method) M/[g/mol]
D (Method)
acetone-d6 Isotec 99.96% ---- ---- 64.1161
acetonitrile-d3 Aldrich 99.5% ---- ---- 44.0704
Cambridge
benzene-d6 Isotope 99.95%* 100%, (GC/MSb)* 0.0059%, (KFc)* 84.1488
Laboratories
chloroform-d Sigma-Aldrich 99.82%* 100%, (GCa)* 0.0018%, (KFc)* 120.3840
a c
cyclohexane-d12 Sigma-Aldrich 99.6%* 99%, (GC )* 0.00%, (KF )* 96.2334
Cambridge
dichloromethane-d2 Isotope 99.965%* 100%, (GC/MSb)* 0.0042%, (KFc)* 86.945
Laboratories
dimethyl sulfoxide-d6 Aldrich 99.93%* 99.96%, (GCa)* 0.0058%, (KFc)* 84.170
a c
ethanol-d6 Sigma-Aldrich 99.7%* 99%, (GC )* 0.0304%, (KF )* 52.1054
methanol-d4 Sigma-Aldrich 99.981%* 99.9%, (GCa)* 0.0014%, (KFc)* 36.0665
pyridine-d5 Aldrich 99% ---- ---- 84.1307
c
tetrahydrofuran-d8 Sigma-Aldrich 99.82%* ---- 0.01%, (KF )* 80.155
a c
toluene-d8 Sigma-Aldrich 99.80%* 100%, (GC )* 0.0022%, (KF )* 100.1877
a
Gas chromatography
b
Gas chromatography-mass spectrometry
c
Karl Fischer titration
* information obtained from the manufacturer's certificate of analysis
8.4.2 Calibration and Measurements
Measurements were performed using an Anton-Parr DMA45 vibrating tube densimeter,
calibrated in five-degree increments from 278.15 K to 368.15 K using ultrapure Milli-Q® water
and Drierite®-desiccated air at atmospheric pressure. Accepted densities for water were obtained
from the NIST Chemistry WebBook, SRD 69.150 Air density was calculated using equation Eq.
8-2 at atmospheric pressure for each experimental temperature expressed in units of Kelvin. Eq.
8-2 was obtained from the densimeter instrument documentation. The atmospheric pressure (𝑃)
168
was monitored throughout the calibration with a NIST-Traceable® barostat, which had an
expanded measurement uncertainty with 0.95 level of confidence of U(P) = 0.62 hPa. Constants
A and B were computed for each temperature using Eq. 8-3 and Eq. 8-4, respectively, where 𝜏 is
the experimental period of oscillation.
A 12 L Thermo Fisher Scientific A25 bath served as the reservoir for a 50% (v/v) mixture
of water and ethylene glycol, which flowed to the densimeter at a rate of 20 L/min via a Thermo
Fischer Scientific AC200 immersion circulator. Heat loss between the reservoir and the densimeter
was minimized by encapsulating the lines with 1-inch-thick pipe insulation, which provided an
approximate R-Value of 6.1. The bath temperature sensor was calibrated using a NIST-traceable
calibrated Hart Scientific 1510 platinum resistance probe and 1529-R display. The circulating fluid
temperature was measured at the entrance to the densimeter, and it differed by less than 0.05 K
from the bath setpoint temperature. The measurements are reported at the bath temperature with a
standard uncertainty of u(T) = 0.05 K. Analyte density was calculated via Eq. 8-5 using the
measured period of oscillation, the calibration constants 𝐴 and 𝐵, and the bath setpoint
temperature.
g 0.0012930 𝑃[MPa]
𝜌𝑎𝑖𝑟 [ 3
]= ( ) Eq. 8-2
cm 1 + 0.00367 ∗ (𝑇[Kelvin] − 273.15) 0.101325
sec2 cm3 𝜏2 − 𝜏2 Eq. 8-3
𝐴[ ] = 𝜌𝑤𝑎𝑡𝑒𝑟 − 𝜌𝑎𝑖𝑟
g 𝑤𝑎𝑡𝑒𝑟 𝑎𝑖𝑟
𝐵[sec 2 ] = τ2air − (𝐴 ⋅ 𝜌𝑎𝑖𝑟 ) Eq. 8-4
2
g 𝜏𝑠𝑎𝑚𝑝𝑙𝑒 −𝐵 Eq. 8-5
𝜌𝑠𝑎𝑚𝑝𝑙𝑒 [ 3 ] =
cm 𝐴
169
Analytes were transferred from their primary container to a stoppered Schlenk flask in a
glovebox under a nitrogen atmosphere to minimize moisture contamination. The freeze-pump-
thaw method was used to dislodge dissolved gasses from the sample. Once the sample was frozen
using liquid nitrogen, a vacuum was applied until the vessel’s pressure was less than 3.33 Pa. This
process was repeated three times, after which the contents of the flask were brought to atmospheric
pressure with dry nitrogen.
Measurements were systematically performed in five-degree increments, which spanned
the analyte’s melting point or 273.15 K to its boiling point or 368.15 K at atmospheric pressure.
Following degassing, 0.7 cm3 of the sample was transferred by pipette into the densimeter’s inlet.
After the circulator reached the desired temperature, the sample was allowed to thermally
equilibrate for five minutes before the period of oscillation was recorded. Values were calculated
as the arithmetic mean of twenty periods. After a set of measurements for a particular compound
were completed, the sample chamber was cleaned with ethanol and dried with pressurized air
produced by the densimeter’s built-in pump. Upon completion of the measurements, the density
of Milli-Q® water was remeasured at each temperature. The check resulted in identical density
values as the calibration, demonstrating reproducibility.
8.4.3 Regression of Protiated Solvent Data for Comparison
Specific density data were collected from NIST Chemistry WebBook150 for the protiated
form of each analyte. Accepted values were regressed over the experimental temperature range
using a second-order polynomial. The regression coefficients were then used with the experimental
temperatures to calculate the specific density for the protiated components to compare with our
measured values. Coefficients for these expressions can be found in the APPENDIX W: Protiated
Molar Density Regression Coefficients.
170
8.5 Results
Densities (𝜌/(g/cm3 )) for all measured perdeutero compounds are available in Table 8-2
and Table 8-3 along with their respective combined expanded uncertainty 𝑈(𝜌). To estimate the
impact of purity on the measured density values a contaminant with a molecular mass 12 g/mol
less than the deuterated compound was assumed in all cases. For compounds where the purity was
not listed a conservative estimate of 99% was assumed which led to a larger 𝑈(𝜌). Sample purity
was found to be the most significant contribution to the uncertainty of the density values. To gauge
the effect of percent deuteration on the measured density values, the ratio of molecular weight of
the completely protiated species to the molecular weight of the completely deuterated analog was
used. Detailed calculations of 𝑈(𝜌) are provided in APPENDIX X: Uncertainty Analysis
Equations and APPENDIX Y: Uncertainty Analysis Data.
171
Table 8-2: Specific densities of perdeutero compounds at 1.013 MPaa (Part I).
cyclohexane-
toluene-d8 benzene-d6 tetrahydrofuran-d8 pydridine-d5 chloroform-d
T/[K] d12
ρ/[g/cm3] ρ/[g/cm3] ρ/[g/cm3] ρ/[g/cm3] ρ/[g/cm3] ρ/[g/cm3]
278.15 0.9602 1.0088 1.0635 1.5311
283.15 0.9032 0.9551 0.9609 1.0029 1.0581 1.5216
288.15 0.8978 0.9500 0.9551 0.9968 1.0528 1.5122
293.15 0.8924 0.9450 0.9493 0.9908 1.0475 1.5027
298.15 0.8870 0.9400 0.9436 0.9847 1.0421 1.4932
303.15 0.8815 0.9348 0.9379 0.9785 1.0368 1.4838
308.15 0.8760 0.9296 0.9319 0.9721 1.0313 1.4737
313.15 0.8705 0.9246 0.9262 0.9660 1.0261 1.4640
318.15 0.8650 0.9194 0.9204 0.9597 1.0207 1.4544
323.15 0.8593 0.9143 0.9145 0.9533 1.0153 1.4444
328.15 0.8536 0.9089 0.9085 0.9469 1.0098 1.4344
333.15 0.8479 0.9037 0.9026 0.9405 1.0043
338.15 0.8421 0.8984 0.8966 0.9988
343.15 0.8363 0.8932 0.8906 0.9933
348.15 0.8879 0.8846 0.9878
353.15 0.8824 0.9822
358.15 0.8770 0.9766
363.15 0.8716 0.9712
368.15 0.9655
𝑈(𝜌) 0.0022 0.0003 0.0003 0.0029 0.0031 0.0004
a
Temperature standard uncertainty is u(T) = 0.05 K. A coverage factor of k = 1.96 was used to estimate the
combined expanded uncertainties for each component density 𝑈(𝜌).
172
Table 8-3: Specific densities of perdeutero compounds at 1.013 MPaa (Part II).
dichloromethane- acetonitrile- dimethyl
acetone-d6 ethanol-d5 methanol-d4
T/[K] d2 d3 sulfoxide-d6
ρ/[g/cm3] ρ/[g/cm3] ρ/[g/cm3] ρ/[g/cm3] ρ/[g/cm3] ρ/[g/cm3]
278.15 0.9034 1.3868 0.9100 0.8569 0.9088
283.15 0.8972 1.3775 0.9052 0.8512 0.9035
288.15 0.8909 1.3683 0.9003 0.8454 0.8982
293.15 0.8845 1.3590 0.8955 0.8396 0.8928
298.15 0.8782 1.3495 0.8906 0.8338 0.8875 1.1837
303.15 0.8717 1.3401 0.8857 0.8280 0.8821 1.1783
308.15 0.8653 1.3304 0.8806 0.8221 0.8767 1.1727
313.15 0.8588 1.3208 0.8757 0.8163 0.8714 1.1674
318.15 0.8522 0.8707 0.8104 0.8659 1.1620
323.15 0.8456 0.8656 0.8044 0.8604 1.1565
328.15 0.8388 0.8603 0.7983 0.8547 1.1510
333.15 0.8550 0.7922 0.8491 1.1455
338.15 0.8497 0.7861 0.8433 1.1400
343.15 0.8442 1.1345
348.15 0.8387 1.1291
353.15 0.8330 1.1235
358.15 1.1181
363.15 1.1127
368.15 1.1073
𝑈(𝜌) 0.0032 0.0004 0.0040 0.0044 0.0006 0.0003
a
Temperature standard uncertainty is u(T) = 0.05 K. A coverage factor of k = 1.96 was used to estimate the
combined expanded uncertainties for each component density 𝑈(𝜌).
173
8.6 Discussion
8.6.1 Density Comparison with Literature
A second-order polynomial was used to correlate the temperature dependency of the
measured specific densities using Eq. 8-6, where the temperature is expressed in units of Kelvin.
The regression coefficients are provided in Table 8-4.
g
𝜌[ ] = 𝐶1 𝑇 2 + 𝐶2 𝑇 + 𝐶3 Eq. 8-6
cm3
For the six solvents with temperature-dependent literature data, a graphical comparison of
literature values with those from this work are presented in Figure 8-1, Figure 8-2, and Figure 8-3.
Values for acetonitrile-d3 were obtained from figure 4 of Sassi et al.211 using WebPlotDigitizer214
open source software. For benzene-d6 this work agrees with Dixon and Schiessler199 and Dymond
et al.200 within 0.03% and within 0.3% of other data. For chloroform-d, values from this work are
approximately 0.1% to 0.16% higher than literature. For cyclohexane-d12 this work agrees within
0.05% with Matsuo and van Hook,209 and is up to 0.4% higher than other literature values.
Methanol-d4 densities are withing 0.02% of Bender and van Hook,207 and are up to 0.7% higher
than Kudryavtsev, et al.208 For acetone-d6 our density values are approximately 1.2% higher than
those reported by Szydlowski et al.,213 for unknown reasons. Acetonitrile-d3 densities from this
work vary systematically and are as much as 0.35% lower to 0.05% higher than values presented
by Sassi et al.211. In summary, the current data agree within 0.4% with at least one other researcher
for five of the six systems where literature data are available.
174
Table 8-4: Specific density regression coefficients.
C1 [g/cm3K2] C2 [g/cm3K]
C3 [g/cm3] R2
/10-6 /10-3
ethanol-d6 -1.1833 -0.27586 1.0781 0.99997
chloroform-d -1.2505 -1.1744 1.9545 0.99999
pyridine-d5 -0.36677 -0.85098 1.3285 0.99999
toluene-d8 -0.50221 -0.71875 1.1989 0.99999
dichloromethane-d2 -1.2128 -1.1696 1.8060 0.99998
benzene-d6 -0.53085 -0.83683 1.2403 0.99999
cyclohexane-d12 -0.84209 -0.58643 1.1367 0.99999
acetone-d6 -1.0443 -0.65683 1.1669 0.99999
methanol-d4 -0.73126 -0.63752 1.1426 0.99998
acetonitrile-d3 -0.73245 -0.72757 1.1159 0.99999
dimethyl sulfoxide-d6 -0.0025562 -1.0919 1.5095 0.99999
tetrahydrofuran-d8 -0.90612 -0.68908 1.2706 0.99999
*underbar indicates the last significant digit
Figure 8-1: Comparison of experimental data with literature values from this work (t.w.)
denoted by (■) and literature for benzene-d6 (left) 199–203 and chloroform-d (right).203–206 The
percent deuteration is reported in parenthesis unless it was not listed (NL) by the author(s).
175
Figure 8-2: Comparison of experimental data with literature values from this work (t.w.) denoted
by (■) and literature for acetone-d6 (left) 213 and methanol-d5 (right).207,208
Figure 8-3: Comparison of experimental data with literature values from this work (t.w.)
denoted by (■) and literature for cyclohexane-d12 (left)199,210 and acetonitrile-d3(right).211 The
percent deuteration is reported in parenthesis unless it was not listed (NL) by the author(s).
8.6.2 Prediction of Density for Perdeutero Compounds and Molar Volume Comparison
As early as 1936, McClean and Adams203 discussed estimating a deuterated liquid's density
using the density of its protiated analog. Based on the assumption that the molar volume of
176
hydrogen and deuterium forms were equal, they derived the form of Eq. 8-7 where (𝜌𝐷,𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 )
is the estimated specific density of the deuterated compound, (𝜌𝐻 ) is the specific density of the
protiated compound, (𝑁𝐷 ) is the number of deuterium atoms per molecule, (𝑀𝑤𝐻 ) is the molecular
mass of the hydrogen analog, and (Δ𝐷𝐻 ) is the difference in atomic mass between hydrogen and
deuterium, taken as 1.006277g/mol. We have subsequently modified the original expression to
account for the fraction of hydrogen atoms in the sample that have been replaced by deuterium
(𝜉).
The modified McClean/Adams expression was used to predict specific densities at each
experimental temperature using the 𝜉-values listed in the second column of Table 8-5. The mean
percentage error (MPE) was calculated for each component using Eq. 8-8 where (𝑛𝑇 ) is total
number of measurements performed across the temperature range. The results are provided in the
third column of Table 8-5.
𝑔
𝜌𝐷,𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 [ ] = 𝜌𝐻 [1 + (𝜉𝑁𝐷 𝛥𝐷𝐻 )/𝑀𝑤𝐻 ] Eq. 8-7
𝑐𝑚3
𝑛𝑇
100% |𝜌𝐷,𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,𝑖 − 𝜌𝐷,𝑐𝑎𝑙𝑐𝑢𝑎𝑙𝑡𝑒𝑑,𝑖 |
𝑀𝑃𝐸 = ( )∑ Eq. 8-8
𝑛𝑇 𝜌𝐷,𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,𝑖
𝑖=1
Interestingly, the modified McClean/Adams relation provides an acceptable estimation of
specific density for many of the components. MPE values are less than 0.45% for all compounds
except acetone-d6 which exhibited a MPE of ~1.3%. Generally, compounds that contained more
deuterium atoms were associated with a larger MPE. The difference in mass between hydrogen
and deuterium is responsible for most of the increase in specific density from protiated compounds
to their deuterated isotopologues.
177
Table 8-5: Error analysis of the modified McClean/Adams formula.
Fraction of
Mean Percentage
Compound Deuterated Atoms
Error
(𝝃)
acetone-d6 0.9996 1.339%
acetonitrile-d3 0.995 0.0389%
benzene-d6 0.9996 0.257%
chloroform-d 0.9982 0.131%
cyclohexane-d12 0.996 0.258%
dichloromethane-d2 0.9996 0.161%
dimethyl sulfoxide-d6 0.9993 0.259%
ethanol-d6 0.997 0.278%
methanol-d4 0.99981 0.207%
pyridine-d5 0.99 0.231%
tetrahydrofuran-d8 0.9982 0.403%
toluene-d8 0.9980 0.258%
The absence of complete agreement between measurement and prediction suggests that we
reexamine the volumetric effects of deuterium substitution (molar volume isotope effect), which
were neglected in the modified McClean/Adams relationship. These effects are best illustrated by
comparing the molar volumes of a deuterated species (𝑉𝑀,𝐷 ) with its protiated analog (𝑉𝑀,𝐻 ) as
seen in Figure 8-4 to Figure 8-9. The subplots illustrate the percent difference in molar volume
between protiated and deuterated which was calculated according to Eq. 8-9 using the average
molar volume as the basis for the calculation.
2 ∗ |𝑉𝑀,𝐷 − 𝑉𝑀.𝐻 |
% Difference = 100% ( ) Eq. 8-9
𝑉𝑀,𝐷 + 𝑉𝑀.𝐻
The molar volume of all measured components increases with rising temperature, a pattern
consistent with the behavior of protiated liquids. Molar volumes became more similar at higher
178
temperatures for benzene/benzene-d6, cyclohexane/cyclohexane-d12,
tetrahydrofuran/tetrahydrofuran-d8, toluene/toluene-d8, pyridine/pyridine-d5, methanol/methanol-
d4, ethanol/ethanol-d6, and dimethyl sulfoxide/dimethyl sulfoxide-d6. Acetone/acetone-d6 and
chloroform/chloroform-d exhibited the opposite trend as a greater discrepancy in molar volumes
was observed at higher temperatures. A consistent trend was not observed for
acetonitrile/acetonitrile-d3 and dichloromethane/dichloromethane-d2. Generally, differences
between protiated and deuterated molar volumes were less than 0.45% across the temperature
region, except for acetone/acetone-d6, in which the protiated form differed from the deuterated
form by as much as 1.38%.
Deuterium substitution decreases the zero-point vibrational energy of the C-D bond relative
to a C-H bond. This decrease in energy produces a C-D bond which is roughly 0.005 Å shorter
than a comparable C-H bond.215–217 Incidentally, intermolecular forces such as hydrogen bonding
are strengthened, leading to decreased intermolecular distances.218 Collectively, these effects
produce a smaller molar volume for the deuterated material in the materials studied here, except
for acetonitrile-d3, whose slight percent difference does not provide a definitive conclusion.
179
Figure 8-4: Comparison of protiated and deuterated molar volumes for acetone/acetone-d6 (left)
and acetonitrile/acetonitrile-d3 (right).
Figure 8-5: Comparison of protiated and deuterated molar volumes for benzene/benzene-d6 (left)
and chloroform/chloroform-d (right).
180
Figure 8-6: Comparison of protiated and deuterated molar volumes for
cyclohexane/cyclohexane-d12 (left) and dichloromethane/dichloromethane-d2 (right).
Figure 8-7: Comparison of protiated and deuterated molar volumes for dimethyl
sulfoxide/dimethyl sulfoxide-d6 (left) and ethanol/ethanol-d6 (right).
181
Figure 8-8: Comparison of protiated and deuterated molar volumes for methanol/methanol-d4
(left) and pyridine/pyridine-d5 (right).
Figure 8-9: Comparison of protiated and deuterated molar volumes for
tetrahydrofuran/tetrahydrofuran-d8 (left) and toluene/toluene-d8 (right).
182
8.6.3 Isobaric Thermal Expansivity
Isobaric thermal expansivity (𝛼𝑃 ) describes the volumetric response of a substance to
changes in temperature at constant pressure. Isobaric thermal expansivity can be measured
1 𝜕𝜌
𝛼𝑃 = − ( ) Eq. 8-10
𝜌 𝜕𝑇 𝑃
experimentally219,220; however, it is more common to calculate it indirectly using specific density
data according to Eq. 8-10; yet, obtaining meaningful values from the indirect method necessitates
a particular approach. Historically 𝛼𝑃 has been calculated using the derivative of an imposed fit of
𝜌 (𝑇) data (i.e. a polynomial), yet recent work has shown that this approach produces 𝛼𝑃 -values
which exhibit an imprecise temperature dependence.221 Rather, the partial differential should be
evaluated using the central difference method according to Eq. 8-11, where the change in the
experimental density is assessed numerically over a temperature interval (Δ𝑇) of 5 K.221–223 At the
low and high-temperature 𝜌 (𝑇) endpoints, 𝛼𝑃 was calculated using the forward (Eq. 8-12) or
backward (Eq. 8-13) single-sided difference method, respectively. Experimental values of density
were used when applying these formulas.
1 𝜌(𝑇 + Δ𝑇) − 𝜌(𝑇 − Δ𝑇)
𝛼𝑃 [K −1 ] = − ( ) Eq. 8-11
𝜌(𝑇) 2Δ𝑇
1 𝜌(𝑇 + Δ𝑇) − 𝜌(𝑇)
𝛼𝑃 [K −1 ] = − ( ) Eq. 8-12
𝜌(𝑇) Δ𝑇
1 𝜌(𝑇) − 𝜌(𝑇 − Δ𝑇)
𝛼𝑃 [K −1 ] = − ( ) Eq. 8-13
𝜌(𝑇) Δ𝑇
Calculated values of the coefficient of isobaric thermal expansivity from the experimental
density data are depicted in Figure 8-10. The 𝛼𝑃 increases with temperature and was regressed
183
using Eq. 8-14. Polynomial coefficients are presented in Table 8-6 and values of the coefficient of
isobaric thermal expansivity from the resulting polynomial are plotted in as hashed lines in Figure
8-10. This chapter’s appendix demonstrates that values calculated from analytically differentiating
the density polynomial are imprecise near the ends of the experimental temperature range.
𝛼𝑃 [K −1 ] = 𝐶4 𝑇 2 + 𝐶5 𝑇 + 𝐶6 Eq. 8-14
Figure 8-10: Temperature dependency of the coefficient of isobaric thermal expansion (left)
dichloromethane-d2 (square), chloroform-d (diamond), cyclohexane-d12 (pentagon), benzene-
d6 (circle), toluene-d8 (triangle), pyridine-d5 (inverted triangle), (right) acetone-d6 (square),
acetonitrile-d3 (diamond), dimethyl sulfoxide-d6 (pentagon), methanol-d4 (circle), ethanol-d6
(triangle), and tetrahydrofuran-d8 (inverted triangle).
184
Table 8-6: Isobaric thermal expansivity regression coefficients.
C4 [K-3] C5 [K-2] C6 [K-1]
R2
/10-8 /10-5 /10-3
ethanol-d6 3.7793 -1.9694 3.6127 0.992
chloroform-d 1.5413 -0.60449 1.7173 0.978
pyridine-d5 1.0340 -0.48487 1.5554 0.980
toluene-d8 0.83694 -0.30557 1.2526 0.976
dichloromethane-d2 -0.27635 0.51448 0.11464 0.950
benzene-d6 1.6940 -0.80352 2.1149 0.979
cyclohexane-d12 2.7276 -1.3494 2.8262 0.995
acetone-d6 3.6277 -1.7398 3.4161 0.997
methanol-d4 4.1043 -2.2088 4.1431 0.992
acetonitrile-d3 1.8732 -0.77178 2.0310 0.991
dimethyl sulfoxide-d6 -0.90891 0.68402 -0.31254 0.820
tetrahydrofuran-d8 0.90010 -0.20088 1.0469 0.988
*underbar indicates the last significant digit
8.7 Summary and Conclusions
Density values were measured for twelve deuterated solvents commonly encountered in
NMR spectroscopy. Temperature-dependent specific densities of dichloromethane-d2, toluene-d8,
pyridine-d5, ethanol-d6, tetrahydrofuran-d8, and dimethyl sulfoxide-d6 were reported for the first
time at near ambient temperatures. Comparisons were made for benzene-d6, acetone-d6, methanol-
d4, cyclohexane-d12, acetonitrile-d3, and chloroform-d, for which there were some literature values
available. Across the temperature range used in this study, all compounds exhibited decreasing
specific density in response to temperature increases. The coefficient of isobaric thermal
expansivity was calculated and found to increase with temperature for all components.
8.8 Acknowledgements
This material is based upon work supported by the National Science Foundation under
Grant No. 1603705 USDA National Institute of Food and Agriculture, Hatch/Multi-State project
185
MICL04192. Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views of the funding agencies.
186
CHAPTER 9:Conclusions and Future Directions
9.1 Conclusions
This work has resulted in several significant findings.
1. A functional form of the mid-range infrared attenuation was developed for scaling spectra
that resulted in temperature-independent integrated areas for the hydroxyl stretch. The
attenuation function differs in magnitude by approximately a factor of 30 for strongly
bonded hydroxyls relative to free hydroxyls. This is the first method proposed for
conversion of the hydroxyl stretch region to obtain a temperature-independent integrated
area.
2. The scaling of the IR spectra does not require an assumed association model for
quantification of hydroxyl region.
3. The scaled IR spectra can be mapped to NMR spectra using a mapping function based on
quantum calculations, as further evidence of the validity of the scaling method.
4. All alcohols in this study exhibited similar attenuation functions.
5. Curve fitting of the hydroxyl region can be achieved by use of four primary peaks and one
minor peak for hydroxyls, and one peak to subtract the -CH overlap. Two of the hydroxyl
peaks represent the 𝛼 and 𝛽 hydroxyls and the other peaks represent the 𝛾 and 𝛿 hydroxyls.
6. The scaled spectra permit quantification of the fraction of non-hydrogen bonded
hydroxyls, 𝑋 𝐴 by the fraction of the peak areas for the 𝛼 and 𝛽 hydroxyls relative to the
total hydroxyl area. This analysis is independent of any association model for the first time.
7. The values of 𝑋 𝐴 vs. alcohol mole fraction are very similar for the homologous series of
alcohols from ethanol to hexanol, inclusive up to 30 mol% alcohol.
187
8. Behavior of 𝑋 𝐴 for secondary alcohols compared to primary alcohol was inconclusive,
with 2-propanol showing behavior like the primary alcohols, but 2-butanol showing less
bonding.
9. Two association strengths are necessary for fitting of the infrared spectra.
10. Hydrogen bond type distributions from curve fitting the scaled spectra using four principal
hydroxyl peaks are well-represented by the of the RTPT model over the temperature range
of 30-60 °C.
11. An activity coefficient model was developed based on the RTPT model.
12. The fits of the spectra using the RTPT model were implemented as the association
contribution in the NTRL-PA methodology, resulting in improved representation
compared to TPT-1 for the dilute alcohol region and improved temperature dependence of
the enthalpy of mixing. By constraining the bonding energy of the n-mer to 𝜖𝑁𝐴𝐷 /𝑘 =2500
K, the same number of adjustable parameters are used compared to TPT-1 with improved
results.
13. Mapping of RTPT to TPT-1 was demonstrated to permit approximate modeling of alcohols
using existing phase equilibria code which is primarily coded using TPT-1.
14. Molar densities of deuterated solvents are within about one percent of the density of the
nondeuterated form with the only exception being acetone-d6. Quantification will be useful
for future variable temperature NMR studies.
9.2 Future work
There are numerous avenues of research that would be extensions of the work presented in this
dissertation, and they are briefly summarized in the points below with some of the accompanying
challenges.
188
1. The approaches presented in this work could be applied to additional alcohol systems. The
number of secondary and tertiary alcohol systems in this study was small, and primary
alcohols investigated did not exceed six carbons. Longer alcohols would have a lower
hydroxyl concentration and with an increase in carbon chain length of the alcohol the
competition between associative and dispersive interactions would become more
significant. Like the systems examined in this work a similar investigation could be
undertaken for diol compounds. Other non-hydrogen bond forming solvent could also be
explored.
2. An extension of the infrared scaling methods would involve primary, secondary, and
tertiary amines. Like alcohols, amines self-aggregate and differences in infrared
absorbance due to these aggregates has been well documented in literature occurring in the
neighborhood of 3300 cm-1. NMR data is present for amine systems in cyclohexane for
comparison.224
3. The current work utilizes 2B (one acceptor site and one donor site) description of the
hydroxyl which ignores the influence of rings and branching and assumes equality between
the number of acceptor and donor sites. It is not unreasonable to assume that the smaller
chain alcohols such as methanol and possibly ethanol would likely see an improvement by
using a hydroxyl model which assigns two acceptor sites to the oxygen and one donor site
to the proton (3B model). More fundamentally, the free electron pairs on the oxygen are
not held in energetically identical orbitals. However, this extension would require
derivation of cooperative bonding in a more general way, considering that our current
RTPT model assumes that the number of acceptor and donor sites are equal. This line of
reasoning could be extended to water which has additional complications compared to
189
methanol. Not only is it known to form up to four hydrogen bonds (4C model – 2 acceptor
sites and 2 donor sites) but it is difficult to dissolve in a non-hydrogen bond forming solvent
for purposes of infrared measurements. This reduces solvent candidates to polar aprotic
solvents such as acetonitrile which can cross associate with water.
4. The generalization of RTPT to multicomponent system would be powerful. Another
important direction involves cross association such as between a ketone and alcohol. The
carbonyl absorbance is well resolved in the infrared region. Incorporation of a ketone
would be an accessible addition since it only serves as an acceptor and could likely be
represented with a single association site. A similar argument could be made for a nitrile
5. The relationship between the hydroxyl stretching frequency and the NMR chemical shift
of the hydroxyl proton would benefit from additional NMR data. Preliminary findings
suggest that the form of the relationship is valid and does not vary significantly between
primary and secondary alcohols. Ideally, the NMR measurements should be performed at
temperatures and compositions that are near those measured in the infrared so that minimal
interpolation is needed.
6. From an engineering modeling perspective, there would be value in exploring the
feasibility of one set of association parameters for all the primary alcohols from methanol
to 1-hexane.
190
BIBLIOGRAPHY
(1) Lira, C. T.; Elliott, J. R.; Gupta, S.; Chapman, W. G. Wertheim’s Association Theory for
Phase Equilibrium Modeling in Chemical Engineering Practice. Ind. Eng. Chem. Res. in
press. https://doi.org/10.1021/acs.icer.2c02058.
(2) Kontogeorgis, G. M.; Folas, G. K. Thermodynamic Models for Industrial Applications:
From Classical and Advanced Mixing Rules to Association Theories; Wiley: Hoboken, N.J.,
2010.
(3) Yoshida, Z.; Ishibe, N.; Ozoe, H. Intermolecular Hydrogen Bond Involving a .Pi. Base as
the Proton Acceptor. X. Hydrogen Bonding of Phenol with Acetylenes and Allenes. J. Am.
Chem. Soc. 1972, 94 (14), 4948–4952. https://doi.org/10.1021/ja00769a026.
(4) Jeffrey, G. A. An Introduction to Hydrogen Bonding; Oxford University Press, 1997.
(5) Pimentel, G. C.; McClellan, A. L. The Hydrogen Bond, 2nd Edition.; A Series of Books in
Chemistry; W.H. Freeman and Company: San Francisco, 1960.
(6) Hall, A.; Wood, J. L. Origin of the Structure of the OH Stretching Band of Some Phenols in
Solution. Spectrochim Acta A Mol Biomol Spectrosc 1967, 23 (10), 2657–2667.
https://doi.org/10.1016/0584-8539(67)80157-0.
(7) Choperena, A.; Painter, P. An Infrared Spectroscopic Study of Hydrogen Bonding in Ethyl
Phenol: A Model System for Polymer Phenolics. Vib. Spectrosc. 2009, 51 (1), 110–118.
https://doi.org/10.1016/j.vibspec.2008.11.008.
(8) Ohta, K.; Tominaga, K. Vibrational Population Relaxation of Hydrogen-Bonded Phenol
Complexes in Solution: Investigation by Ultrafast Infrared Pump–Probe Spectroscopy.
Chemical Physics 2007, 341 (1), 310–319.
https://doi.org/10.1016/j.chemphys.2007.07.025.
(9) Singh, L. P.; Richert, R. Watching Hydrogen-Bonded Structures in an Alcohol Convert from
Rings to Chains. Phys. Rev. Lett. 2012, 109 (16), 167802.
https://doi.org/10.1103/PhysRevLett.109.167802.
(10) Zimmermann, D.; Häber, T.; Schaal, H.; Suhm, M. A. Hydrogen Bonded Rings, Chains and
Lassos: The Case of t-Butyl Alcohol Clusters. Molecular Physics 2001, 99 (5), 413–425.
https://doi.org/10.1080/00268970010017009.
(11) Yagai, S.; Kubota, S.; Saito, H.; Unoike, K.; Karatsu, T.; Kitamura, A.; Ajayaghosh, A.;
Kanesato, M.; Kikkawa, Y. Reversible Transformation between Rings and Coils in a
Dynamic Hydrogen-Bonded Self-Assembly. J. Am. Chem. Soc. 2009, 131 (15), 5408–5410.
https://doi.org/10.1021/ja9005892.
(12) Račkauskaitė, D.; Bergquist, K.-E.; Shi, Q.; Sundin, A.; Butkus, E.; Wärnmark, K.; Orentas,
E. A Remarkably Complex Supramolecular Hydrogen-Bonded Decameric Capsule Formed
191
from an Enantiopure C2-Symmetric Monomer by Solvent-Responsive Aggregation. J. Am.
Chem. Soc. 2015, 137 (33), 10536–10546. https://doi.org/10.1021/jacs.5b03160.
(13) Hecksher, T.; Jakobsen, B. Communication: Supramolecular Structures in Monohydroxy
Alcohols: Insights from Shear-Mechanical Studies of a Systematic Series of Octanol
Structural Isomers. J. Chem. Phys. 2014, 141 (10), 101104.
https://doi.org/10.1063/1.4895095.
(14) Van Ness, H. C.; Van Winkle, J.; Richtol, H. H.; Hollinger, H. B. Infrared Spectra and the
Thermodynamics of Alcohol-Hydrocarbon Systems. J. Phys. Chem. 1967, 71 (5), 1483–
1494. https://doi.org/10.1021/j100864a046.
(15) Kirsch, J. L.; Coffin, D. R. Infrared and Nuclear Magnetic Resonance Studies of Hydrogen
Bonding in Aliphatic Alcohol Systems. J. Phys. Chem. 1976, 80 (22), 2448–2451.
https://doi.org/10.1021/j100563a003.
(16) Bellamy, L. J.; Pace, R. J. Hydrogen Bonding by Alcohols and Phenols—I. The Nature of
the Hydrogen Bond in Alcohol Dimers and Polymers. Spectrochimica Acta 1966, 22 (3),
525–533. https://doi.org/10.1016/0371-1951(66)80083-8.
(17) Cohen, A. D.; Reid, C. Hydrogen Bonding by NMR. The Journal of Chemical Physics 1956,
25 (4), 790–791. https://doi.org/10.1063/1.1743072.
(18) Woutersen, S.; Emmerichs, U.; Bakker, H. J. A Femtosecond Midinfrared Pump–Probe
Study of Hydrogen-Bonding in Ethanol. The Journal of Chemical Physics 1997, 107 (5),
1483–1490. https://doi.org/10.1063/1.474501.
(19) Mazur, K.; Bonn, M.; Hunger, J. Hydrogen Bond Dynamics in Primary Alcohols: A
Femtosecond Infrared Study. J. Phys. Chem. B 2015, 119 (4), 1558–1566.
https://doi.org/10.1021/jp509816q.
(20) Ludwig, R. The Structure of Liquid Methanol. ChemPhysChem 2005, 6 (7), 1369–1375.
https://doi.org/10.1002/cphc.200400663.
(21) Schwager, F.; Marand, E.; Davis, R. M. Determination of Self-Association Equilibrium
Constants of Ethanol by FTIR Spectroscopy. J. Phys. Chem. 1996, 100 (50), 19268–19272.
https://doi.org/10.1021/jp9613448.
(22) Balanay, M. P.; Kim, D. H.; Fan, H. Revisiting the Formation of Cyclic Clusters in Liquid
Ethanol. J. Chem. Phys. 2016, 144 (15), 154302. https://doi.org/10.1063/1.4945809.
(23) Thomas, L. H. 371. Viscosity and Molecular Association. Part V. The Association Model,
and Hydrogen-Bond Enthalpies. Journal of the Chemical Society (Resumed) 1963, 1995–
2002.
(24) Lehtola, J.; Hakala, M.; Hämäläinen, K. Structure of Liquid Linear Alcohols. J. Phys. Chem.
B 2010, 114 (19), 6426–6436. https://doi.org/10.1021/jp909894y.
192
(25) Jindal, A.; Vasudevan, S. Geometry of OH⋯O Interactions in the Liquid State of Linear
Alcohols from Ab Initio Molecular Dynamics Simulations. Phys. Chem. Chem. Phys. 2020,
22 (12), 6690–6697. https://doi.org/10.1039/D0CP00435A.
(26) Ewing, G. W. Instrumental Methods of Chemical Analysis; McGraw-Hill: New York, 1969.
(27) Meloan, C. Elementary Infrared Spectroscopy; Macmillan Pub. Co.: Collier Macmillan:
New York, NY, 1963.
(28) Potts, W. J. Chemical Infrared Spectroscopy: Volume 1, Techniques; John Wley & Sons,
1963.
(29) Silverstein, R. M.; Webster, F. X.; Kiemle, D. J.; Bryce, D. L. Spectrometric Identification
of Organic Compounds, Eighth edition.; Wiley: Hoboken, NJ, 2015.
(30) Vinogradov, S. N.; Linnell, R. H. Hydrogen Bonding; Van Nostrand Reinhold: New York,
1971.
(31) Schmidt, J. R.; Corcelli, S. A.; Skinner, J. L. Pronounced Non-Condon Effects in the
Ultrafast Infrared Spectroscopy of Water. The Journal of Chemical Physics 2005, 123 (4),
044513. https://doi.org/10.1063/1.1961472.
(32) Coggeshall, N. D. Electrostatic Interaction in Hydrogen Bonding. The Journal of Chemical
Physics 1950, 18 (7), 978–983. https://doi.org/10.1063/1.1747822.
(33) Tolbin, A. Yu.; Pushkarev, V. E.; Tomilova, L. G. A Mathematical Analysis of Deviations
from Linearity of Beer’s Law. Chemical Physics Letters 2018, 706, 520–524.
https://doi.org/10.1016/j.cplett.2018.06.056.
(34) Asprion, N.; Hasse, H.; Maurer, G. FT-IR Spectroscopic Investigations of Hydrogen
Bonding in Alcohol-Hydrocarbon Solutions. Fluid Phase Equilib. 2001, 186 (1–2), 1–25.
https://doi.org/10.1016/s0378-3812(01)00363-6.
(35) Anslyn, E. V.; Dougherty, D. A. Modern Physical Organic Chemistry; University Science:
Sausalito, CA, 2006.
(36) Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New
Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE
Journal 1975, 21 (1), 116–128. https://doi.org/10.1002/aic.690210115.
(37) Bala, A. M.; Lira, C. T. Relation of Wertheim Association Constants to Concentration-
Based Equilibrium Constants for Mixtures with Chain-Forming Components. Fluid Phase
Equilib. 2016, 430, 47–56. https://doi.org/10.1016/j.fluid.2016.09.015.
(38) Elliott, J. R.; Lira, C. T. Introductory Chemical Engineering Thermodynamics, Second
edition.; Prentice-Hall international series in the physical and chemical engineering
sciences; Prentice Hall: Upper Saddle River, NJ, 2012.
193
(39) Prausnitz, J. M.; Lichtenthaler, R. N.; Azevedo, E. G. de. Molecular Thermodynamics of
Fluid-Phase Equilibria; Pearson Education, 1998.
(40) Economou, I. G.; Donohue, M. D. Chemical, Quasi-Chemical and Perturbation Theories for
Associating Fluids. AIChE J. 1991, 37 (12), 1875–1894.
https://doi.org/10.1002/aic.690371212.
(41) Campbell, S. W. Chemical Theory for Mixtures Containing Any Number of Alcohols. Fluid
Phase Equilib. 1994, 102 (1), 61–84. https://doi.org/10.1016/0378-3812(94)87091-8.
(42) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State
for Associating Liquids. Ind Eng Chem Res 1990, 29 (8), 1709–1721.
https://doi.org/10.1021/ie00104a021.
(43) Harris, D. C. Quantitative Chemical Analysis, 7th edition.; W. H. Freeman: New York,
2006.
(44) Amunson, K. E.; Anderson, B. A.; Kubelka, J. Temperature Effects on the Optical Path
Length of Infrared Liquid Transmission Cells. Appl Spectrosc 2011, 65 (11), 1307–1313.
https://doi.org/10.1366/11-06405.
(45) Williams, S. D.; Johnson, T. J.; Sharpe, S. W.; Yavelak, V.; Oates, R. P.; Brauer, C. S.
Quantitative Vapor-Phase IR Intensities and DFT Computations to Predict Absolute IR
Spectra Based on Molecular Structure: I. Alkanes. Journal of Quantitative Spectroscopy
and Radiative Transfer 2013, 129, 298–307. https://doi.org/10.1016/j.jqsrt.2013.07.005.
(46) Kumar, N.; Bansal, A.; Sarma, G. S.; Rawal, R. K. Chemometrics Tools Used in Analytical
Chemistry: An Overview. Talanta 2014, 123, 186–199.
https://doi.org/10.1016/j.talanta.2014.02.003.
(47) Jones, R. N. The Intensities of the Infra-Red Absorption Bands of n-Paraffin Hydrocarbons.
Spectrochimica Acta 1957, 9 (3), 235–251. https://doi.org/10.1016/0371-1951(57)80137-4.
(48) Wexler, A. S. Infrared Determination of Structural Units in Organic Compounds by
Integrated Intensity Measurements: Alkanes, Alkenes and Monosubstituted Alkyl
Benzenes. Spectrochimica Acta 1965, 21 (10), 1725–1742. https://doi.org/10.1016/0371-
1951(65)80085-6.
(49) Řeřicha, R.; Jarolímek, P.; Horák, M. Determination of the Degree of Branching in Alkanes
by Infrared Spectroscopy. II. Variation of the Intensity of the Antisymmetrical Stretching
Vibration Bands of C-H Bonds in CH3 and CH2 Groups. Collect. Czech. Chem. Commun.,
CCCC 1967, 32 (5), 1903–1912. https://doi.org/10.1135/cccc19671903.
(50) Murdoch, K. M.; Ferris, T. D.; Wright, J. C.; Farrar, T. C. Infrared Spectroscopy of Ethanol
Clusters in Ethanol–Hexane Binary Solutions. The Journal of Chemical Physics 2002, 116
(13), 5717–5724. https://doi.org/10.1063/1.1458931.
194
(51) Tucker, E. E.; Becker, E. D. Alcohol Association Studies. II. Vapor Pressure, 220-MHz
Proton Magnetic Resonance, and Infrared Investigations of Tert-Butyl Alcohol Association
in Hexadecane. J. Phys. Chem. 1973, 77 (14), 1783–1795.
https://doi.org/10.1021/j100633a012.
(52) Wu, X.; Chen, Y.; Yamaguchi, T. Hydrogen Bonding in Methanol Studied by Infrared
Spectroscopy. Journal of Molecular Spectroscopy 2007, 246 (2), 187–191.
https://doi.org/10.1016/j.jms.2007.09.012.
(53) Reilly, J. T.; Thomas, A.; Gibson, A. R.; Luebehusen, C. Y.; Donohue, M. D. Analysis of
the Self-Association of Aliphatic Alcohols Using Fourier Transform Infrared (FT-IR)
Spectroscopy. Ind. Eng. Chem. Res. 2013, 52 (40), 14456–14462.
https://doi.org/10.1021/ie302174r.
(54) Coggeshall, N. D.; Saier, E. L. Infrared Absorption Study of Hydrogen Bonding Equilibria.
J. Am. Chem. Soc. 1951, 73 (11), 5414–5418. https://doi.org/10.1021/ja01155a118.
(55) Choperena, A.; Painter, P. Hydrogen Bonding in Polymers: Effect of Temperature on the
OH Stretching Bands of Poly(Vinylphenol). Macromolecules 2009, 42 (16), 6159–6165.
https://doi.org/10.1021/ma900928z.
(56) Sheppard, N. Infrared Spectroscopy and Hydrogen Bonding - Band-Widths and Frequency
Shifts. In Hydrogen Bonding: Papers Presented at the Symposium on Hydrogen Bonding
Held at Ljubljana, 29 July–3 August 1957; Hadži, D., Ed.; Pergamon, 1959; pp 85–105.
https://doi.org/10.1016/B978-0-08-009140-2.50013-0.
(57) Liddel, U.; Becker, E. D. Infra-Red Spectroscopic Studies of Hydrogen Bonding in
Methanol, Ethanol, and t-Butanol. Spectrochimica Acta 1957, 10 (1), 70–84.
https://doi.org/10.1016/0371-1951(57)80165-9.
(58) Laksmono, H.; Tanimura, S.; Allen, H. C.; Wilemski, G.; Zahniser, M. S.; Shorter, J. H.;
Nelson, D. D.; McManus, J. B.; Wyslouzil, B. E. Monomer, Clusters, Liquid: An Integrated
Spectroscopic Study of Methanol Condensation. Phys. Chem. Chem. Phys. 2011, 13 (13),
5855. https://doi.org/10.1039/c0cp02485f.
(59) Andanson, J.-M.; Soetens, J.-C.; Tassaing, T.; Besnard, M. Hydrogen Bonding in
Supercritical Tert-Butanol Assessed by Vibrational Spectroscopies and Molecular-
Dynamics Simulations. The Journal of chemical physics 2005, 122 (17), 174512.
https://doi.org/10.1063/1.1886730.
(60) Doroshenko, I.; Pogorelov, V.; Sablinskas, V.; Balevicius, V. Matrix-Isolation Study of
Cluster Formation in Methanol: O–H Stretching Region. Journal of Molecular Liquids
2010, 157 (2–3), 142–145. https://doi.org/10.1016/j.molliq.2010.09.003.
(61) Wandschneider, D.; Michalik, M.; Heintz, A. Spectroscopic and Thermodynamic Studies
of Liquid N-Butanol+n-Hexane and +cyclohexane Mixtures Based on Quantum Mechanical
Ab Initio Calculations of n-Butanol Clusters. J. Mol. Liq. 2006, 125 (1), 2–13.
https://doi.org/10.1016/j.molliq.2005.11.011.
195
(62) Hansen, P. E.; Spanget-Larsen, J. On Prediction of OH Stretching Frequencies in
Intramolecularly Hydrogen Bonded Systems. Journal of Molecular Structure 2012, 1018,
8–13. https://doi.org/10.1016/j.molstruc.2012.01.011.
(63) Koné, M.; Illien, B.; Laurence, C.; Graton, J. Can Quantum-Mechanical Calculations Yield
Reasonable Estimates of Hydrogen-Bonding Acceptor Strength? The Case of Hydrogen-
Bonded Complexes of Methanol. The Journal of Physical Chemistry A 2011, 115 (47),
13975–13985. https://doi.org/10.1021/jp209200w.
(64) Ohno, K.; Shimoaka, T.; Akai, N.; Katsumoto, Y. Relationship between the Broad OH
Stretching Band of Methanol and Hydrogen-Bonding Patterns in the Liquid Phase. J. Phys.
Chem. A 2008, 112 (32), 7342–7348. https://doi.org/10.1021/jp800995m.
(65) Spanget-Larsen, J.; Hansen, B. K. V.; Hansen, P. E. OH Stretching Frequencies in Systems
with Intramolecular Hydrogen Bonds: Harmonic and Anharmonic Analyses. Chemical
Physics 2011, 389 (1–3), 107–115. https://doi.org/10.1016/j.chemphys.2011.09.011.
(66) Kwac, K.; Geva, E. A Mixed Quantum-Classical Molecular Dynamics Study of the
Hydroxyl Stretch in Methanol/Carbon Tetrachloride Mixtures: Equilibrium Hydrogen-Bond
Structure and Dynamics at the Ground State and the Infrared Absorption Spectrum. J. Phys.
Chem. B 2011, 115 (29), 9184–9194. https://doi.org/10.1021/jp204245z.
(67) Staib, A.; Borgis, D. A Quantum Multi-Mode Molecular Dynamics Approach to the
Vibrational Spectroscopy of Solvated Hydrogen-Bonded Complexes. Chemical Physics
Letters 1997, 271 (4), 232–240. https://doi.org/10.1016/S0009-2614(97)00470-3.
(68) Ghosh, M. K.; Lee, J.; Choi, C. H.; Cho, M. Direct Simulations of Anharmonic Infrared
Spectra Using Quantum Mechanical/Effective Fragment Potential Molecular Dynamics
(QM/EFP-MD): Methanol in Water. J. Phys. Chem. A 2012, 116 (36), 8965–8971.
https://doi.org/10.1021/jp306807v.
(69) Wang, J.; Boyd, R. J.; Laaksonen, A. A Hybrid Quantum Mechanical Force Field Molecular
Dynamics Simulation of Liquid Methanol: Vibrational Frequency Shifts as a Probe of the
Quantum Mechanical/Molecular Mechanical Coupling. The Journal of Chemical Physics
1996, 104 (18), 7261–7269. https://doi.org/10.1063/1.471439.
(70) Corcelli, S. A.; Skinner, J. L. Infrared and Raman Line Shapes of Dilute HOD in Liquid
H2O and D2O from 10 to 90 °C. J. Phys. Chem. A 2005, 109 (28), 6154–6165.
https://doi.org/10.1021/jp0506540.
(71) Corcelli, S. A.; Lawrence, C. P.; Skinner, J. L. Combined Electronic Structure/Molecular
Dynamics Approach for Ultrafast Infrared Spectroscopy of Dilute HOD in Liquid H2O and
D2O. The Journal of Chemical Physics 2004, 120 (17), 8107–8117.
https://doi.org/10.1063/1.1683072.
(72) Auer, B.; Kumar, R.; Schmidt, J. R.; Skinner, J. L. Hydrogen Bonding and Raman, IR, and
2D-IR Spectroscopy of Dilute HOD in Liquid D₂O. Proceedings of the National Academy
of Sciences of the United States of America 2007, 104 (36), 14215–14220.
196
(73) Gruenbaum, S. M.; Tainter, C. J.; Shi, L.; Ni, Y.; Skinner, J. L. Robustness of Frequency,
Transition Dipole, and Coupling Maps for Water Vibrational Spectroscopy. J. Chem.
Theory Comput. 2013, 9 (7), 3109–3117. https://doi.org/10.1021/ct400292q.
(74) Mesele, O. O.; Thompson, W. H. A “Universal” Spectroscopic Map for the OH Stretching
Mode in Alcohols. J. Phys. Chem. A 2017, 121 (31), 5823–5833.
https://doi.org/10.1021/acs.jpca.7b05836.
(75) Kretschmer, C. B.; Wiebe, R. Thermodynamics of Alcohol‐Hydrocarbon Mixtures. J.
Chem. Phys. 1954, 22 (10), 1697–1701. https://doi.org/10.1063/1.1739878.
(76) Deng, D.; Li, H.; Yao, J.; Han, S. Simple Local Composition Model for 1H NMR Chemical
Shift of Mixtures. Chemical Physics Letters 2003, 376 (1), 125–129.
https://doi.org/10.1016/S0009-2614(03)00996-5.
(77) Karachewski, A. M.; McNiel, M. M.; Eckert, C. A. A Study of Hydrogen Bonding in
Alcohol Solutions Using NMR Spectroscopy. Ind. Eng. Chem. Res. 1989, 28 (3), 315–324.
https://doi.org/10.1021/ie00087a010.
(78) Whetsel, K. B.; Lady, J. H. Self-Association of Phenol in Nonpolar Solvents. In
Spectrometry of Fuels; Friedel, R. A., Ed.; Plenum Press: New York, 1970; pp 259–279.
(79) Case, D. A.; Babin, V.; Berryman, J. T.; Betz, R. M.; Cai, Q.; Cerutti, D. S.; Cheatham III,
T. E.; Darden, T. A.; Duke, R. E.; Gohlke, H.; Goetz, A. W.; Gusarov, S.; Homeyer, N.;
Janowski, P.; Kaus, J.; Kolossváry, I.; Kovalenko, A.; Lee, T. S.; LeGrand, S.; Luchko, T.;
Luo, R.; Madej, B.; Merz, K. M.; Paesani, F.; Roe, D. R.; Roitberg, A.; Sagui, C.; Salomon-
Ferrer, R.; Seabra, G.; Simmerling, C. L.; Smith, W.; Swails, J.; Walker, R. C.; Wang, J.;
Wolf, R. M.; Wu, X.; Kollman, P. A. AMBER 14, 2014.
(80) Martínez, L.; Andrade, R.; Birgin, E. G.; Martínez, J. M. PACKMOL: A Package for
Building Initial Configurations for Molecular Dynamics Simulations. Journal of
Computational Chemistry 30 (13), 2157–2164. https://doi.org/10.1002/jcc.21224.
(81) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J.
R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; Li, X.; Caricato, M.;
Marenich, A.; Bloino, J.; Janesko, B. G.; Gomperts, R.; Mennucci, B.; Hratchian, H. P.;
Ortiz, J. V.; Izmaylov, A. F.; Sonnenberg, J. L.; Williams-Young, D.; Ding, F.; Lipparini,
F.; Egidi, F.; Goings, J.; Peng, B.; Henderson, T.; Ranasinghe, D.; Zakrzewski, V. G.; Gao,
J.; Rega, N.; Zheng, G.; Liang, W.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa,
J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Throssell, K.;
Montgomery Jr., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.;
Kudin, K. N.; Staroverov, V. N.; Keith, T.; Kobayashi, R.; Normand, J.; Raghavachari, K.;
Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Millam, J. M.; Klene, M.;
Adamo, C.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.;
Foresman, J. B.; Fox, D. J. Gaussian 09, 2013.
197
(82) Becke, A. D. Density-Functional Thermochemistry. V. Systematic Optimization of
Exchange-Correlation Functionals. The Journal of Chemical Physics 1997, 107 (20), 8554–
8560. https://doi.org/10.1063/1.475007.
(83) Ditchfield, R.; Hehre, W. J.; Pople, J. A. Self‐Consistent Molecular‐Orbital Methods. IX.
An Extended Gaussian‐Type Basis for Molecular‐Orbital Studies of Organic Molecules.
The Journal of Chemical Physics 1971, 54 (2), 724–728. https://doi.org/10.1063/1.1674902.
(84) Hehre, W. J.; Ditchfield, R.; Pople, J. A. Self—Consistent Molecular Orbital Methods. XII.
Further Extensions of Gaussian—Type Basis Sets for Use in Molecular Orbital Studies of
Organic Molecules. The Journal of Chemical Physics 1972, 56 (5), 2257–2261.
https://doi.org/10.1063/1.1677527.
(85) Hariharan, P. C.; Pople, J. A. The Influence of Polarization Functions on Molecular Orbital
Hydrogenation Energies. Theoretica Chimica Acta 1973, 28, 213–222.
https://doi.org/10.1007/BF00533485.
(86) NIST Computational Chemistry Comparison and Benchmark Database; NIST Standard
Reference Database Number 101; Release 19, April 2018, Editor: Russell D. Johnson III;
Http://Cccbdb.Nist.Gov/; DOI:10.18434/T47C7Z.
(87) 1-Butanol (NIST Chemistry WebBook, SRD 69). NIST Chemistry WebBook, SRD 69.
https://webbook.nist.gov/cgi/cbook.cgi?ID=71-36-3&Type=IR-SPEC&Index=QUANT-
IR,1 (accessed 2020-02-12).
(88) Ethanol (NIST Chemistry WebBook, SRD 69). NIST Chemistry WebBook, SRD 69.
https://webbook.nist.gov/cgi/cbook.cgi?ID=C64175&Type=IR-SPEC&Index=29#Refs
(accessed 2020-02-12).
(89) Shinokita, K.; Cunha, A. V.; Jansen, T. L. C.; Pshenichnikov, M. S. Hydrogen Bond
Dynamics in Bulk Alcohols. J. Chem. Phys. 2015, 142 (21), 212450.
https://doi.org/10.1063/1.4921574.
(90) Zheng, R.; Sun, Y.; Shi, Q. Theoretical Study of the Infrared and Raman Line Shapes of
Liquid Methanol. Phys. Chem. Chem. Phys. 2011, 13 (6), 2027–2035.
https://doi.org/10.1039/C0CP01145B.
(91) Torii, H. Time-Domain Calculations of the Infrared and Polarized Raman Spectra of
Tetraalanine in Aqueous Solution. J. Phys. Chem. B 2007, 111 (19), 5434–5444.
https://doi.org/10.1021/jp070301w.
(92) Williams, D. B. G.; Lawton, M. Drying of Organic Solvents: Quantitative Evaluation of the
Efficiency of Several Desiccants. J. Org. Chem. 2010, 75 (24), 8351–8354.
https://doi.org/10.1021/jo101589h.
(93) González, B.; Calvar, N.; Domínguez, Á.; Tojo, J. Dynamic Viscosities of Binary Mixtures
of Cycloalkanes with Primary Alcohols at T=(293.15, 298.15, and 303.15)K: New
198
UNIFAC-VISCO Interaction Parameters. The Journal of Chemical Thermodynamics 2007,
39 (2), 322–334. https://doi.org/10.1016/j.jct.2006.06.008.
(94) NIST Standard Reference Database 103a, NIST ThermoData Engine, Pure Components,
Https://Www.Nist.Gov/Mml/Acmd/Trc/Thermodata-Engine/Srd-Nist-Tde-103a.
https://www.nist.gov/mml/acmd/trc/thermodata-engine/srd-nist-tde-103a.
(95) Fletcher, A. N.; Heller, C. A. Self-Association of Alcohols in Nonpolar Solvents. J. Phys.
Chem. 1967, 71 (12), 3742–3756. https://doi.org/10.1021/j100871a005.
(96) Shinomiya, K.; Shinomiya, T. An Equilibrium Model of the Self-Association of 1- and 3-
Pentanols in Heptane. The Chemical Society of Japan 1990, 63, 1093–1097.
(97) Kontogeorgis, G. M.; Tsivintzelis, I.; von Solms, N.; Grenner, A.; Bøgh, D.; Frost, M.;
Knage-Rasmussen, A.; Economou, I. G. Use of Monomer Fraction Data in the
Parametrization of Association Theories. Fluid Phase Equilibria 2010, 296 (2), 219–229.
https://doi.org/10.1016/j.fluid.2010.05.028.
(98) Huggins, C. M.; Pimentel, G. C. Systematics of the Infrared Spectral Properties of Hydrogen
Bonding Systems: Frequency Shift, Half Width and Intensity. J. Phys. Chem. 1956, 60 (12),
1615–1619. https://doi.org/10.1021/j150546a004.
(99) Wertheim, M. S. Fluids with Highly Directional Attractive Forces.1. Statistical
Thermodynamics. Journal of Statistical Physics 1984, 35 (1–2), 19–34.
https://doi.org/10.1007/bf01017362.
(100) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 2. Thermodynamic
Perturbation-Theory and Integral-Equations. Journal of Statistical Physics 1984, 35 (1–2),
35–47. https://doi.org/10.1007/bf01017363.
(101) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 4. Equilibrium
Polymerization. Journal of Statistical Physics 1986, 42 (3–4), 477–492.
https://doi.org/10.1007/bf01127722.
(102) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 3. Multiple Attraction
Sites. Journal of Statistical Physics 1986, 42 (3–4), 459–476.
https://doi.org/10.1007/bf01127721.
(103) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a
Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40 (4), 1244–1260.
https://doi.org/10.1021/ie0003887.
(104) Gross, J.; Sadowski, G. Modeling Polymer Systems Using the Perturbed-Chain Statistical
Associating Fluid Theory Equation of State. Ind. Eng. Chem. Res. 2002, 41 (5), 1084–1093.
https://doi.org/10.1021/ie010449g.
(105) Gupta, R. B.; Brinkley, R. L. Hydrogen-Bond Cooperativity in 1-Alkanol + n-Alkane Binary
Mixtures. AIChE Journal 1998, 44 (1), 207–213. https://doi.org/10.1002/aic.690440122.
199
(106) Palombo, F.; Sassi, P.; Paolantoni, M.; Morresi, A.; Cataliotti, R. S. Comparison of
Hydrogen Bonding in 1-Octanol and 2-Octanol as Probed by Spectroscopic Techniques. J.
Phys. Chem. B 2006, 110 (36), 18017–18025. https://doi.org/10.1021/jp062614h.
(107) Maes, G.; Smets, J. Hydrogen Bond Cooperativity: A Quantitative Study Using Matrix-
Isolation FT-IR Spectroscopy. J. Phys. Chem. 1993, 97 (9), 1818–1825.
https://doi.org/10.1021/j100111a017.
(108) Umer, M.; Leonhard, K. Ab Initio Calculations of Thermochemical Properties of Methanol
Clusters. J. Phys. Chem. A 2013, 117 (7), 1569–1582. https://doi.org/10.1021/jp308908j.
(109) Umer, M.; Albers, K.; Sadowski, G.; Leonhard, K. PC-SAFT Parameters from Ab Initio
Calculations. Fluid Phase Equilib. 2014, 362, 41–50.
https://doi.org/10.1016/j.fluid.2013.08.037.
(110) Kar, T.; Scheiner, S. Comparison of Cooperativity in CH···O and OH···O Hydrogen Bonds.
J. Phys. Chem. A 2004, 108 (42), 9161–9168. https://doi.org/10.1021/jp048546l.
(111) Dominelli-Whiteley, N.; Brown, J. J.; Muchowska, K. B.; Mati, I. K.; Adam, C.; Hubbard,
T. A.; Elmi, A.; Brown, A. J.; Bell, I. A. W.; Cockroft, S. L. Strong Short-Range
Cooperativity in Hydrogen-Bond Chains. Angew. Chem. Int. Ed. 2017, 56 (26), 7658–7662.
https://doi.org/10.1002/anie.201703757.
(112) Marshall, B. D.; Chapman, W. G. Resummed Thermodynamic Perturbation Theory for
Bond Cooperativity in Associating Fluids. J. Chem. Phys. 2013, 139 (21), 214106.
https://doi.org/10.1063/1.4834637.
(113) Brinkley, R. L.; Gupta, R. B. Hydrogen Bonding with Aromatic Rings. Aiche Journal 2001,
47 (4), 948–953. https://doi.org/10.1002/aic.690470417.
(114) Palombo, F.; Paolantoni, M.; Sassi, P.; Morresi, A.; Cataliotti, R. S. Spectroscopic Studies
of the “Free” OH Stretching Bands in Liquid Alcohols. Journal of Molecular Liquids 2006,
125 (2–3), 139–146.
(115) von Solms, N.; Michelsen, M. L.; Passos, C. P.; Derawi, S. O.; Kontogeorgis, G. M.
Investigating Models for Associating Fluids Using Spectroscopy. Ind. Eng. Chem. Res.
2006, 45 (15), 5368–5374. https://doi.org/10.1021/ie051341u.
(116) Haskell, R. W.; Hollinger, H. B.; Van Ness, H. C. Chemical Model as Applied to Associated
Liquid Solutions. Ethanol-Heptane System. J. Phys. Chem. 1968, 72 (13), 4534–4543.
https://doi.org/10.1021/j100859a028.
(117) Asprion, N.; Hasse, H.; Maurer, G. Application of IR-Spectroscopy in Thermodynamic
Investigations of Associating Solutions. Fluid Phase Equilib. 2003, 205 (2), 195–214.
https://doi.org/10.1016/S0378-3812(02)00181-4.
200
(118) Nagata, I.; Ogasawara, Y. Prediction of Ternary Excess Enthalpies from Binary Data.
Thermochimica Acta 1982, 52 (1–3), 155–168. https://doi.org/10.1016/0040-
6031(82)85193-9.
(119) Hofman, T. Thermodynamics of Association of Pure Alcohols. Fluid Phase Equilibria
1990, 55 (1–2), 39–57. https://doi.org/10.1016/0378-3812(90)85003-S.
(120) Kretschmer, C. B.; Wiebe, R. Pressure-Volume-Temperature Relationships of Alcohol
Vapors 2. J. Am. Chem. Soc. 1954, 76 (9), 2579–2583. https://doi.org/10.1021/ja01638a082.
(121) Bala Ahmed, A. M. Fundamental Studies and Engineering Modeling of Hydrogen Bonding.
Ph.D. Thesis, Michigan State University, East Lansing, MI, 2018.
(122) Bala, A. M.; Killian, W. G.; Plascencia, C.; Storer, J. A.; Norfleet, A. T.; Peereboom, L.;
Jackson, J. E.; Lira, C. T. Quantitative Analysis of Infrared Spectra of Binary Alcohol +
Cyclohexane Solutions with Quantum Chemical Calculations. J. Phys. Chem. A 2020, 124
(16), 3077–3089. https://doi.org/10.1021/acs.jpca.9b11245.
(123) Paolantoni, M.; Sassi, P.; Morresi, A.; Cataliotti, R. S. Infrared Study of 1-Octanol Liquid
Structure. Chem. Phys. 2005, 310 (1–3), 169–178.
https://doi.org/10.1016/j.chemphys.2004.10.027.
(124) Asprion, N. Anwendung Der Spektroskopie in Thermodynamischen Untersuchungen
Assoziierender Lösungen, Dissertation, C,. Ph.D. Thesis, 1996.
(125) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT - Equation-of-State
Solution Model for Associating Fluids. Fluid Phase Equilibria 1989, 52, 31–38.
https://doi.org/10.1016/0378-3812(89)80308-5.
(126) Gross, J.; Sadowski, G. Application of the Perturbed-Chain SAFT Equation of State to
Associating Systems. Ind. Eng. Chem. Res. 2002, 41 (22), 5510–5515.
https://doi.org/10.1021/ie010954d.
(127) Sear, R. P.; Jackson, G. Thermodynamic Perturbation Theory for Association with Bond
Cooperativity. J. Chem. Phys. 1996, 105 (3), 1113–1120. https://doi.org/10.1063/1.471955.
(128) Acree, W. Thermodynamic Properties of Nonelectrolyte Solutions; Academic Press Inc.:
Orlando, 1984.
(129) Apelblat, A. Erratum to “The Concept of Associated Solutions in Historical Development.
Part 1. The 1884-1984 Period.” J. Mol. Liq. 2007, 130 (1–3), 133–162.
https://doi.org/10.1016/j.molliq.2006.09.001.
(130) Bala, A. M.; Liu, R.; Peereboom, L.; Lira, C. T. Applications of an Association Activity
Coefficient Model, NRTL-PA, to Alcohol-Containing Mixtures. Ind. Eng. Chem. Res. in
press. https://doi.org/10.1021/acs.icer.2c01415.
201
(131) Himmelblau, D. M. Process Analysis by Statistical Methods; John Wiley & Sons, Inc: New
York, NY, 1970.
(132) Dolan, K. D.; Yang, L.; Trampel, C. P. Nonlinear Regression Technique to Estimate Kinetic
Parameters and Confidence Intervals in Unsteady-State Conduction-Heated Foods. Journal
of Food Engineering 2007, 80 (2), 581–593.
https://doi.org/10.1016/j.jfoodeng.2006.06.023.
(133) Mishra, D. K.; Dolan, K. D.; Yang, L. Bootstrap Confidence Intervals for the Kinetic
Parameters of Degradation of Anthocyanins in Grape Pomace: Bootstrap Confidence
Intervals for Kinetic Parameter Estimation. J. Food Process Eng 2011, 34 (4), 1220–1233.
https://doi.org/10.1111/j.1745-4530.2009.00425.x.
(134) Killian, W. G.; Bala, A. M.; Lira, C. T. Parameterization of a RTPT Association Activity
Coefficient Model Using Spectroscopic Data. Fluid Phase Equilib. 2022, No. 554, 113299.
https://doi.org/10.1016/j.fluid.2021.113299.
(135) Meier, R. J. On Art and Science in Curve-Fitting Vibrational Spectra. Vib Spectrosc 2005,
39 (2), 266–269. https://doi.org/10.1016/j.vibspec.2005.03.003.
(136) Barlow, S. J.; Bondarenko, G. V.; Gorbaty, Y. E.; Yamaguchi, T.; Poliakoff, M. An IR
Study of Hydrogen Bonding in Liquid and Supercritical Alcohols. J. Phys. Chem. A 2002,
106 (43), 10452–10460. https://doi.org/10.1021/jp0135095.
(137) Svoboda, V.; Holub, R.; Pick, J. Liquid-Vapour Equilibrium. XLVII. The System
Cyclohexane-1-Hexanol. Collect. Czech. Chem. Commun. 1971, 36 (6), 2331–2338.
https://doi.org/10.1135/cccc19712331.
(138) Morachevsky, A. G. Investigation of Solution-Vapor Equilibrium in the Benzene-
Cyclohexane-Methylalcohol System. Vestn. Leningr. Univ., 12, Ser. Fiz. Khim. 1957, 1,
118–126.
(139) Lazzaroni, M. J.; Bush, D.; Eckert, C. A.; Frank, T. C.; Gupta, S.; Olson, J. D. Revision of
MOSCED Parameters and Extension to Solid Solubility Calculations. Industrial &
Engineering Chemistry Research 2005, 44 (11), 4075–4083.
https://doi.org/10.1021/ie049122g.
(140) Smirnova, N. A.; Kurtynina, L. M. Thermodynamic Functions of Mixing for a Number of
Binary Alcohol-Hydrocarbon Solutions. Zh. Fiz. Khim. 1969, 43, 1883.
(141) Marshall, B. D.; Haghmoradi, A.; Chapman, W. G. Resummed Thermodynamic
Perturbation Theory for Bond Cooperativity in Associating Fluids with Small Bond Angles:
Effects of Steric Hindrance and Ring Formation. The Journal of Chemical Physics 2014,
140 (16), 164101. https://doi.org/10.1063/1.4871307.
(142) Pauling, L. The Nature of the Chemical Bond and the Structure of Molecules and Crystals:
An Introduction to Modern Structural Chemistry, 3. ed., 17. print.; Cornell Univ. Press:
Ithaca, NY, 2010.
202
(143) Czarnecki, M. A. Effect of Temperature and Concentration on Self-Association of Octan-
1-Ol Studied by Two-Dimensional Fourier Transform Near-Infrared Correlation
Spectroscopy. J. Phys. Chem. A 2000, 104 (27), 6356–6361.
https://doi.org/10.1021/jp000407q.
(144) Asprion, N.; Hasse, H.; Maurer, G. Thermodynamic and IR Spectroscopic Studies of
Solutions with Simultaneous Association and Solvation. Fluid Phase Equilibria 2003, 208
(1–2), 23–51. https://doi.org/10.1016/S0378-3812(02)00317-5.
(145) Becker, E. D. Infrared Studies of Hydrogen Bonding in Alcohol-Base Systems. Spectrochim
Acta 1961, 17 (4), 436–447. https://doi.org/10.1016/0371-1951(61)80095-7.
(146) Frohlich, H. Using Infrared Spectroscopy Measurements To Study Intermolecular
Hydrogen Bonding: Calculating the Degree of Association, Equilibrium Constant, and Bond
Energy for Hydrogen Bonding in Benzyl Alcohol and Phenol. J. Chem. Educ. 1993, 70 (1),
A3. https://doi.org/10.1021/ed070pA3.
(147) Iwahashi, M.; Hayashi, Y.; Hachiya, N.; Matsuzawa, H.; Kobayashi, H. Self-Association of
Octan-1-Ol in the Pure Liquid State and in Decane Solutions as Observed by Viscosity, Self-
Diffusion, Nuclear Magnetic Resonance and near-Infrared Spectroscopy Measurements.
Faraday Trans. 1993, 89 (4), 707. https://doi.org/10.1039/ft9938900707.
(148) Janeček, J.; Paricaud, P. Size Distribution of Associated Clusters in Liquid Alcohols:
Interpretation of Simulation Results in the Frame of SAFT Approach. J. Chem. Phys. 2013,
139 (17), 174502. https://doi.org/10.1063/1.4827107.
(149) Sassi, P.; Palombo, F.; Cataliotti, R. S.; Paolantoni, M.; Morresi, A. Distributions of H-
Bonding Aggregates in Tert-Butyl Alcohol: The Pure Liquid and Its Alkane Mixtures. J.
Phys. Chem. A 2007, 111 (27), 6020–6027. https://doi.org/10.1021/jp071609q.
(150) Linstrom, P. NIST Chemistry WebBook, NIST Standard Reference Database 69, 1997.
https://doi.org/10.18434/T4D303.
(151) Killian Jr., W. G.; Storer, J. A.; Killian Sr., W.; Lira, C. T. A MATLAB Application for
Cell Pathlength in Absorption Transmission Spectroscopy. Spectroscopy 2020, 35 (8), 26–
28.
(152) Peter R. Griffiths; James A. de Haseth. Fourier Transform Infrared Spectrometry, 2nd ed.;
Chemical Analysis: A Series of Monographs on Analytical Chemistry and It’s Applications;
John Wiley & Sons, Inc, 2007.
(153) Bradley, M. Curve Fitting in Raman and IR Spectroscopy. Application Note, 50733;
Thermo Fischer, Madison, WI, USA, 2007.
(154) Kruger, F. J.; Schwarz, C. E.; du Preez, L. J.; Burger, A. J. Monomer Fraction Data of Dilute
Alcohol/Acetone Systems Measured with Transmission Fourier Transform Infrared
Spectroscopy. Fluid Phase Equilib. 2015, 400, 87–94.
https://doi.org/10.1016/j.fluid.2015.05.010.
203
(155) Karachewski, A. M.; Howell, W. J.; Eckert, C. A. Development of the AVEC Model for
Associating Mixtures Using NMR Spectroscopy. AIChE J. 1991, 37 (1), 65–73.
https://doi.org/10.1002/aic.690370106.
(156) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating
Molecules. Ind. Eng. Chem. Res. 1990, 29 (11), 2284–2294.
https://doi.org/10.1021/ie00107a014.
(157) Vaskivskyi, Ye.; Doroshenko, I.; Chernolevska, Ye.; Pogorelov, V.; Pitsevich, G.
Spectroscopic Studies of Clusterization of Methanol Molecules Isolated in a Nitrogen
Matrix. Low Temperature Physics 2017, 43 (12), 1415–1419.
https://doi.org/10.1063/1.5012794.
(158) Bakke, J. M.; Bjerkeseth, L. H. Rotational Isomerism around the C–O Bond in Saturated
Acyclic Alcohols—Assignment of the Experimental IR Hydroxyl Stretch Bands Using
Semiempirical MO and Ab Initio Calculations. J. Mol. Struct. 1997, 407 (1), 27–38.
https://doi.org/10.1016/S0022-2860(96)09726-8.
(159) Czarnecki, M. A.; Wojtków, D.; Haufa, K. Rotational Isomerism of Butanols: Infrared, near-
Infrared and DFT Study. Chem. Phys. Lett. 2006, 431 (4), 294–299.
https://doi.org/10.1016/j.cplett.2006.09.096.
(160) van der Maas, J. H.; Lutz, E. T. G. Structural Information from OH Stretching Frequencies
Monohydric Saturated Alcohols. Spectrochim Acta Part A: Molecular Spectroscopy 1974,
30 (10), 2005–2019. https://doi.org/10.1016/0584-8539(74)80047-4.
(161) Hare, D. E.; Sorensen, C. M. Raman Spectroscopic Study of Dilute HOD in Liquid H2O in
the Temperature Range − 31.5 to 160 °C. J. Chem. Phys. 1990, 93 (10), 6954–6961.
https://doi.org/10.1063/1.459472.
(162) Eckert, C. A.; McNiel, M. M.; Scott, B. A.; Halas, L. A. NMR Measurements of Chemical
Theory Equilibrium Constants for Hydrogen-Bonded Solutions. AIChE J. 1986, 32 (5),
820–828. https://doi.org/10.1002/aic.690320512.
(163) Kwaśniewicz, M.; Czarnecki, M. A. The Effect of Chain Length on Mid-Infrared and Near-
Infrared Spectra of Aliphatic 1-Alcohols. Appl Spectrosc 2018, 72 (2), 288–296.
https://doi.org/10.1177/0003702817732253.
(164) Killian, W. G.; Bala, A. M.; Norfleet, A. T.; Peereboom, L.; Jackson, J. E.; Lira, C. T.
Infrared Quantification of Ethanol and 1-Butanol Hydrogen Bond Distributions in
Cyclohexane. Spectrochim Acta A Mol Biomol Spectrosc In press.
https://doi.org/10.1016/j.saa.2022.121837.
(165) Roese, S. N.; Heintz, J. D.; Uzat, C. B.; Schmidt, A. J.; Margulis, G. V.; Sabatino, S. J.;
Paluch, A. S. Assessment of the SM12, SM8, and SMD Solvation Models for Predicting
Limiting Activity Coefficients at 298.15 K. Processes 2020, 8 (5), 623.
https://doi.org/10.3390/pr8050623.
204
(166) Tamouza, S.; Passarello, J.-P.; Tobaly, P.; de Hemptinne, J.-C. Application to Binary
Mixtures of a Group Contribution SAFT EOS (GC-SAFT). Fluid Phase Equilibria 2005,
228–229, 409–419. https://doi.org/10.1016/j.fluid.2004.10.003.
(167) NguyenHuynh, D.; Mai, C. T. Q. Application of the Modified Group Contribution PC-SAFT
to Carboxylic Acids and Their Mixtures. Ind. Eng. Chem. Res. 2019, 58 (20), 8923–8934.
https://doi.org/10.1021/acs.iecr.9b02052.
(168) NguyenHuynh, D. Correlation and Prediction of Liquid–Liquid Equilibria for
Alcohol/Hydrocarbon Mixtures Using PC-SAFT Equation of State at High Pressure up to
150 MPa. Fluid Phase Equilibria 2016, 425, 206–214.
https://doi.org/10.1016/j.fluid.2016.06.002.
(169) Elliott, J. R.; Diky, V.; Knotts, T. A.; Wilding, W. Vincent. The Properties of Gases and
Liquids, 6th ed.; McGraw-Hill Education, 2022.
(170) Damaceno, D. S.; Perederic, O. A.; Ceriani, R.; Kontogeorgis, G. M.; Gani, R. Improvement
of Predictive Tools for Vapor-Liquid Equilibrium Based on Group Contribution Methods
Applied to Lipid Technology. Fluid Phase Equilibria 2017.
https://doi.org/10.1016/j.fluid.2017.12.009.
(171) Hao, Y.; Chen, C. Nonrandom Two‐liquid Activity Coefficient Model with Association
Theory. AIChE J 2021, 67 (1), e17061. https://doi.org/10.1002/aic.17061.
(172) French, H. T. Thermodynamic Functions for the Systems 1-Butanol, 2-Butanol, Andt-
Butanol + Cyclohexane. J Solution Chem 1983, 12 (12), 869–887.
https://doi.org/10.1007/BF00643927.
(173) Kurihara, K.; Uchiyama, M.; Kojima, K. Isothermal Vapor−Liquid Equilibria for Benzene
+ Cyclohexane + 1-Propanol and for Three Constituent Binary Systems. J. Chem. Eng. Data
1997, 42 (1), 149–154. https://doi.org/10.1021/je9602475.
(174) Löwen, B.; Schulz, S. Excess Molar Enthalpies of Cyclohexane + N-Alcohols at 283.15,
298.15, 323.15, 343.15 and 363.15 K and at a Pressure of 0.4 MPa. Thermochimica Acta
1995, 265, 63–71. https://doi.org/10.1016/0040-6031(94)02510-U.
(175) Voňka, P.; Svoboda, V.; Strubl, K.; Holub, R. Liquid-Vapour Equilibrium. XLVI. System
Cyclohexane-1-Butanol at 50 and 70 °C. Collect. Czech. Chem. Commun. 1971, 36 (1), 18–
34. https://doi.org/10.1135/cccc19710018.
(176) Triday, J. Q.; Veas, C. Vapor-Liquid Equilibria for the System Cyclohexane-Tert-Butyl
Alcohol. J. Chem. Eng. Data 1985, 30 (2), 171–173. https://doi.org/10.1021/je00040a011.
(177) Buchowski, H.; Bartel, L. Vapor Pressure and Excess Gibbs Function If Carbinols-
Cyclohexane Mixtures. Polish Journal of Chemistry 1978, 52 (2417).
(178) Veselý, F.; Dohnal, V.; Valentová, M.; Pick, J. Concentration and Temperature Dependence
of Heats of Mixing of 1-Butanol, 2-Butanol, and 2-Methyl-2-Propanol with Cyclohexane.
205
Collect. Czech. Chem. Commun. 1983, 48 (12), 3482–3494.
https://doi.org/10.1135/cccc19833482.
(179) Marinichev, A. N.; Susarev, M. P. Liquid-Vapor Equilibrium in the System Methanol–
Cyclohexane at 35, 45, and 55 C and 760 Mm Hg. Zh. Prikl. Khim.(Leningrad) 1965, 38,
1619–1621.
(180) Katayama, H. Liquid–Liquid Equilibria of Two Ternary Systems: Methanol–Cyclohexane
Including 1,3-Dioxolane or 1,4-Dioxane in the Range of 277.79–308.64 K. Fluid Phase
Equilib. 1999, 164 (1), 83–95. https://doi.org/10.1016/S0378-3812(99)00241-1.
(181) Dai, M.; Chao, J.-P. Heats of Mixing of the Partially Miscible Liquid System Cyclohexane
+ Methanol. Fluid Phase Equilib. 1985, 23 (2), 315–319. https://doi.org/10.1016/0378-
3812(85)90014-7.
(182) Gmehling, J.; Krentscher, B. ELDATA: The International Electronic Journal of Physico-
Chemical Data 1995, 1, 181–190.
(183) Góral, M.; Oracz, P.; Warycha, S. Vapour–Liquid Equilibria: XIII. The Ternary System
Cyclohexane–Methanol–Hexane at 303.15 K. Fluid Phase Equilibria 1998, 152 (1), 109–
120. https://doi.org/10.1016/S0378-3812(98)00383-5.
(184) Nagai, Y.; Isii, N. Studies on Volatility of Fuels Containing Ethyl Alcohol. Proceedings of
the Imperial Academy 1935, 11 (1), 23–25. https://doi.org/10.2183/pjab1912.11.23.
(185) Stokes, R. H.; Adamson, M. Thermodynamic Functions for the System Ethanol +
Cyclohexane from Vapour Pressures and Enthalpies of Mixing. J. Chem. Soc., Faraday
Trans. 1 1977, 73 (0), 1232–1238. https://doi.org/10.1039/F19777301232.
(186) Moreau, A.; Martín, M. C.; Chamorro, C. R.; Segovia, J. J. Thermodynamic
Characterization of Second Generation Biofuels: Vapour–Liquid Equilibria and Excess
Enthalpies of the Binary Mixtures 1-Pentanol and Cyclohexane or Toluene. Fluid Phase
Equilibria 2012, 317, 127–131. https://doi.org/10.1016/j.fluid.2012.01.007.
(187) Veselý, F.; Dohnal, V.; Brich, Z. Excess Enthalpies of 1-Hexanol or 1-Octanol +
Cyclohexane. International Data Service Selected Data on Mixtures Ser. A 1987, No. 15,
179–182.
(188) Nagata, I.; Ohta, T.; Uchiyama, Y. Excess Gibbs Free Energies for Binary Systems.
Isopropyl Alcohol with Benzene, Cyclohexane, Methylcyclohexane. J. Chem. Eng. Data
1973, 18 (1), 54–59. https://doi.org/10.1021/je60056a011.
(189) Storonkin, A. V.; Morachevskii, A. G. Liquid-Vapor Equilibrium in the
Benzene+cyclohexane +isopropyl Alcohol System. Zhurnal Fizicheskoi Khimii 1956, 30
(6), 1297–1307.
206
(190) Haase, R.; Tillmann, W. Mixing Properties of the Liquid System 2-Propanol + Cyclohexane.
Zeitschrift für Physikalische Chemie 1995, 189 (1), 81–88.
https://doi.org/10.1524/zpch.1995.189.Part_1.081.
(191) Veselý, F.; Uchytil, P.; Zábranský, M.; Pick, J. Heats of Mixing of Cyclohexane with 1-
Propanol and 2-Propanol. Collect. Czech. Chem. Commun. 1979, 44 (10), 2869–2881.
https://doi.org/10.1135/cccc19792869.
(192) Gierycz, P.; Kosowski, A.; Swietlik, R. Vapor−Liquid Equilibria in Binary Systems Formed
by Cyclohexane with Alcohols. J. Chem. Eng. Data 2009, 54 (11), 2996–3001.
https://doi.org/10.1021/je900050z.
(193) Nataraj, V.; Rao, M. R. Isobaric Vapor-Liquid Equilibrium Data for the Binary Systems
Benzene-Cyclohexane, Benzene-Isobutanol, and Cyclohexane-Isobutanol. Indian J.
Technol. 1967, 5, 212–215.
(194) Vittal Prasad, T. E.; Reddi Prasad Naidu, B.; Madhukiran, D.; Prasad, D. H. L. Boiling
Temperature Measurements on the Binary Mixtures of Cyclohexane with Some Alcohols
and Chlorohydrocarbons. J. Chem. Eng. Data 2001, 46 (2), 414–416.
https://doi.org/10.1021/je000139p.
(195) Belousov, V. P.; Kurtynina, L. M.; Kozulyaev, A. A. VII. Mischungswärme Des
Cyclohexans Mit Propanol, n-Butanol, Isobutanol Und Dekanol. Fiz. Khim. 1970, 163–166.
(196) Böhmer, R.; Gainaru, C.; Richert, R. Structure and Dynamics of Monohydroxy Alcohols—
Milestones towards Their Microscopic Understanding, 100 Years after Debye. Physics
Reports 2014, 545 (4), 125–195. https://doi.org/10.1016/j.physrep.2014.07.005.
(197) Gutowsky, H. S.; Saika, A. Dissociation, Chemical Exchange, and the Proton Magnetic
Resonance in Some Aqueous Electrolytes. The Journal of Chemical Physics 1953, 21 (10),
1688–1694. https://doi.org/10.1063/1.1698644.
(198) Li, D.; Kagan, G.; Hopson, R.; Williard, P. G. Formula Weight Prediction by Internal
Reference Diffusion-Ordered NMR Spectroscopy (DOSY). J. Am. Chem. Soc. 2009, 131
(15), 5627–5634. https://doi.org/10.1021/ja810154u.
(199) Dixon, J. A.; Schiessler, R. W. Synthesis and Properties of Deuterocarbons. Benzene-D6
and Cyclohexane-D12. J. Am. Chem. Soc. 1954, 76 (8), 2197–2199.
(200) Dymond, J. H.; Glen, N.; Robertson, J.; Isdale, J. D. (P, ϱ, T) for {(1−x)C6H6 + XC6D6}
and {(1−x)C6H6 + XC6F6} in the Range 298 to 373 K and 0.1 to 400 MPa. J. Chem.
Thermodyn. 1982, 14 (12), 1149–1158. https://doi.org/10.1016/0021-9614(82)90038-6.
(201) Ingold, C. K.; Raisin, C. G.; Wilson, C. L. 212. Structure of Benzene. Part II. Direct
Introduction of Deuterium into Benzene and the Physical Properties of
Hexadeuterobenzene. J. Chem. Soc. 1936, 915–925.
207
(202) Klit, A.; Langseth, A. Über die darstellung von deuteriobenzol. Z. Phys. Chem. (Leipzig)
1936, 176A (1), 65–80.
(203) McClean, A.; Adams, R. Succinic-α-D2, α’-D2 Acid and Its Derivatives. J. Am. Chem. Soc.
1936, 58 (5), 804–810.
(204) Handa, Y. P. Molar Excess Volumes of Acetonitrile + Chloroform and of Acetonitrile +
Deuterochloroform. J. Chem. Thermodyn. 1977, 9 (2), 117–120.
https://doi.org/10.1016/0021-9614(77)90076-3.
(205) Rabinovich, I. B.; Lobashov, A. A.; Kucheryavyi, V. I. A Negative Isotope Effect in the
Viscosity of Deutero Compounds. Russ. J. Phys. Chem. 1960, 34 (10), 1046–1047.
(206) Breuer, F. W. Chloroform-d (Deuteriochloroform)1. J. Am. Chem. Soc. 1935, 57 (11), 2236–
2237. https://doi.org/10.1021/ja01314a058.
(207) Bender, T. M.; Alexander van Hook, W. Molar Volumes of CH3OH, CH3OD, and CD3OD,
and Apparent or Excess Molar Volumes for (Methanol-OH + Water), (Methanol-OD +
Deuterium Oxide), and (Methanol-OH + Methanol-OD), at 288.15, 298.15, and 313.15 K.
J. Chem. Thermodyn. 1988, 20 (9), 1109–1114. https://doi.org/10.1016/0021-
9614(88)90118-8.
(208) Kudryavtsev, S. G.; Strakhov, A. N.; Ershova, O. V. Volume Properties of Deuterated
Water-Methanol Systems at 278-318 K. Russ. J. Phys. Chem. 1986, 60 (9), 1319–1321.
(209) Matsuo, S.; Van Hook, W. A. Isothermal Compressibility of Benzene, Deuteriobenzene
(C6D6), Cyclohexane, and Deuteriocyclohexane (c-C6D12), and Their Mixtures from 0.1
to 35 Mpa at 288, 298, and 313 K. J. Phys. Chem. 1984, 88 (5), 1032–1040.
https://doi.org/10.1021/j150649a037.
(210) Nikolaev, P. N.; Rabinovich, I. B.; Gal’perin, V. A.; Tsvetkov, V. G. Isotopic Effect in the
Heat Capacity and Compressibility of Perdeuterocyclohexane. Zh. Fiz Khim 1966, 40 (5),
586–590.
(211) Sassi, P.; Morresi, A.; Paliani, G.; Cataliotti, R. S. Differences in the Dynamic Properties of
Liquid CH3CN and CD3CN above 40 °C Revealed by Rayleigh–Brillouin Scattering
Spectroscopy. J. Raman Spectrosc. 1999, 30 (7), 501–506.
https://doi.org/10.1002/(SICI)1097-4555(199907)30:7<501::AID-JRS402>3.0.CO;2-2.
(212) Schroeder, J.; Schiemann, V. H.; Sharko, P. T.; Jonas, J. Raman Study of Vibrational
Dephasing in Liquid CH3CN and CD3CN. J. Chem. Phys. 1977, 66 (7), 3215–3226.
https://doi.org/10.1063/1.434296.
(213) Szydlowski, J.; Gomes de Azevedo, R.; Rebelo, L. P. N.; Esperança, J. M. S. S.; Guedes,
H. J. R. Deuterium Isotope Differences in 2-Propanone, (CH3)2CO/(CD3)2CO: A High-
Pressure Sound-Speed, Density, and Heat Capacities Study. J. Chem. Thermodyn. 2005, 37
(7), 671–683. https://doi.org/10.1016/j.jct.2004.11.001.
208
(214) Rohatgi, A. WebPlotDigitizer, 2021. https://automeris.io/WebPlotDigitizer.
(215) Bartell, L. S.; Roskos, R. R. Isotope Effects on Molar Volume and Surface Tension: Simple
Theoretical Model and Experimental Data for Hydrocarbons. J. Chem. Phys. 1966, 44 (2),
457–463. https://doi.org/10.1063/1.1726709.
(216) Bates, F. S.; Keith, H. D.; McWhan, D. B. Isotope Effect on the Melting Temperature of
Nonpolar Polymers. Macromolecules 1987, 20 (12), 3065–3070.
https://doi.org/10.1021/ma00178a021.
(217) Van Hook, W. A. Isotope Effects in Condensed Phases, the Benzene Example. Influence of
Anharmonicity; Harmonic and Anharmonic Potential Surfaces and Their Isotope
Independence. Molar Volume Effects in Isotopic Benzenes. J. Chem. Phys. 1985, 83 (8),
4107–4117. https://doi.org/10.1063/1.449842.
(218) Scheiner, S.; Čuma, M. Relative Stability of Hydrogen and Deuterium Bonds. J. Am. Chem.
Soc. 1996, 118 (6), 1511–1521. https://doi.org/10.1021/ja9530376.
(219) Pruzan, P.; Minassian, L. T.; Figuiere, P.; Szwarc, H. High Pressure Calorimetry as Applied
to Piezothermal Analysis. Rev. Sci. Instrum. 1976, 47 (1), 66–71.
https://doi.org/10.1063/1.1134493.
(220) Randzio, S. L.; Grolier, J. E.; Quint, J. R. An Isothermal Scanning Calorimeter Controlled
by Linear Pressure Variations from 0.1 to 400 MPa. Calibration and Comparison with the
Piezothermal Technique. Rev. Sci. Instrum. 1994, 65 (4), 960–965.
https://doi.org/10.1063/1.1144926.
(221) Cerdeiriña, C. A.; Tovar, C. A.; González-Salgado, D.; Carballo, E.; Romaní, L. Isobaric
Thermal Expansivity and Thermophysical Characterization of Liquids and Liquid Mixtures.
Phys. Chem. Chem. Phys. 2001, 3 (23), 5230–5236. https://doi.org/10.1039/B104891K.
(222) Zúñiga-Moreno, A.; Galicia-Luna, L. A. Densities, Isothermal Compressibilities, and
Isobaric Thermal Expansivities of Hexan-2-Ol, Octan-1-Ol, and Decan-1-Ol from (313 to
363) K and Pressures up to 22 MPa. J. Chem. Eng. Data 2007, 52 (5), 1773–1783.
https://doi.org/10.1021/je700145e.
(223) Daridon, J.-L.; Nasri, D.; Bazile, J.-P. Computation of Isobaric Thermal Expansivity from
Liquid Density Measurements. Application to Toluene. J. Chem. Eng. Data 2021, 66 (10),
3961–3976. https://doi.org/10.1021/acs.jced.1c00634.
(224) Murphy, R. A.; Davis, J. C. Proton Magnetic Resonance Study of Hydrogen Bonding in
Aliphatic Secondary Amines. J. Phys. Chem. 1968, 72 (9), 3111–3116.
https://doi.org/10.1021/j100855a006.
209
APPENDIX A: Detailed Summary of Attenuation Coefficient Function
Refer to Figure 3-8 of the chapter for a schematic indicating location of the variables. The
integrated attenuation coefficient function is represented by three lines with the intersections
smoothed with cubic splines. The fitted/constrained parameters are indicated below using
superscript 𝔣. The fitted function is expressed in terms of lines and splines for convenience.
Wavenumbers are in cm-1 and the attenuation coefficient is in dm2/mol. Equations for the
attenuation coefficient curve and coefficients are:
Line Segment One
𝜖 = 𝑚1𝔣 𝜈̃ + 𝑏1 Eq. A-1
Line Segment Two
𝜖 = 𝑚2 𝜈̃ + 𝑏2 Eq. A-2
𝜖𝑅𝔣 − 𝜖𝐵𝔣
𝑚2 = Eq. A-3
𝜈̃𝑅𝔣 − 𝜈̃𝐵𝔣
Line Segment Three (𝑚3𝔣 = 0)
𝜖 = 𝜖𝑅𝔣 Eq. A-4
𝔣 𝔣
Splines are determined by (𝜈̃1, 𝜈̃2 ) = 𝜈̃𝐵 ± Δ𝔣 and (𝜈̃3, 𝜈̃4 ) = 𝜈̃𝑅 ± Δ𝔣 , and the corresponding y value
determined from the appropriate line.
Spline One
𝑐1 = 𝑚2 ∗ (𝜈̃1 − 𝜈̃2 ) − (𝜖1 − 𝜖2 ) Eq. A-5
210
𝑑1 = −𝑚1 ∗ (𝜈̃1 − 𝜈̃2 ) + (𝜖1 − 𝜖2 ) Eq. A-6
𝜈̃ − 𝜈̃2
𝑡1 (𝜈̃) = Eq. A-7
𝜈̃1 − 𝜈̃2
𝜖 = (1 − 𝑡1 (𝜈̃))𝜖2 + 𝑡1 (𝜈̃)𝜖1 + 𝑡1 (𝜈̃)(1 − 𝑡1 (𝜈̃)) ((1 − 𝑡1 (𝜈̃))𝑐1 + 𝑡1 (𝜈̃)𝑑1 ) Eq. A-8
Spline Two
𝑐2 = −(𝜖3 − 𝜖𝑅𝔣 ) Eq. A-9
𝑑2 = −𝑚2 ∗ (𝜈̃3 − 𝜈̃4 ) + (𝜖3 − 𝜖𝑅𝔣 ) Eq. A-10
𝜈̃ − 𝜈̃4
𝑡2 (𝜈̃) = Eq. A-11
𝜈̃3 − 𝜈̃4
𝜖 = (1 − 𝑡2 (𝜈̃))𝜖𝑅𝔣 + 𝑡2 (𝜈̃)𝜖3
Eq. A-12
+ 𝑡2 (𝜈̃)(1 − 𝑡2 (𝜈̃)) ((1 − 𝑡2 (𝜈̃))𝑐2 + 𝑡2 (𝜈̃)𝑑2 )
Parameters are provided with up to eight significant digits because calculations of the attenuation
coefficient curve involve taking the differences of large numbers. Parameter values are:
𝑚1𝔣 = 121.51223 cm·dm2/mol 𝑏1 = -443631.92 dm2/mol
𝑚2 = -130.83618 cm·dm2/mol 𝑏2 = 477672.97 dm2/mol
𝑚3𝔣 = 0 cm·dm2/mol 𝑏3 = 𝜈̃𝑅𝔣 = 29639.512 dm2/mol
𝜈̃𝑅𝔣 = 3424.3849 cm-1 𝜖𝑅𝔣 = 29639.512 dm2/mol
𝜈̃𝐵𝔣 = 3650.9240 cm-1 𝜖𝐵𝔣 = 5.140849E-3 dm2/mol
211
𝜖3645 = 819.56086 dm2/mol Δ𝔣 = 8.3506 cm-1
212
APPENDIX B: Processing of Spectra
Infrared spectra were collected using the following method. (1) The infrared spectrometer
was switched on and allowed to warm up for thirty minutes and house nitrogen was fed to the
sample chamber at 50 ft3/hr. The house nitrogen purge was continued at this rate for all
experiments. (2) A background was taken consisting of 128 scans at 0.5 cm-1 resolution. This
spectrum was automatically removed from all subsequent measurements. (3) The temperature-
controlled cell was removed from the glovebox and was positioned within the sample chamber.
(4) Cooling water was supplied from the tap to the cell at a flow rate of 0.5 L/min. Sample supply
lines from the cell were connected to a Luer lock three-way valve system operated outside the
sample chamber. The cell temperature probe and thermal regulation cables were attached to the
cell and the temperature controller which was located outside the sample chamber. The sample
chamber lid was closed. (5) The cell was heated to 100 °C and maintained for five minutes before
being cooled to 30 °C. House nitrogen was supplied to the cell throughout this process to purge
trace moisture from the system. (6) The absorbance of empty cell was measured at 30 °C with 128
scans at 0.5 cm-1 resolution under a constant nitrogen purge. (7) Sample was removed from the
glove box and transferred to a 5 mL glass syringe with a Luer lock fitting. The syringe was
connected to the valve system and the three-way valve was adjusted so that the nitrogen purge was
stopped, and sample could be injected. Sample was injected until it was observed in the outlet
collection container. The outlet value remained open to maintain atmospheric pressure within the
cell. (8) The sample thermally equilibrated for ten minutes before its absorbance was collected
using 128 scans at 0.5 cm-1 resolution. This was done at each of the four experimental
temperatures. (9) When multiple samples were evaluated within the same day, the sample was
forced from the cell using nitrogen and steps (5)-(8) were repeated. At the end of each day the cell
213
was disassembled, and the windows were rinsed with hexane. The windows and cell body were
stored under nitrogen between uses.
Three observations were made regarding the absorbance spectra: (OB1) Solvent
subtraction based on calculated concentration was insufficient for removing the solvent
contribution to the solution spectrum; (OB2) It was observed that cyclohexane absorbance varied
with lot number; (OB3) Despite the careful sample and cell handling, trace contamination caused
small sharp peaks between 3600 cm-1 and 3940 cm-1 which were time and temperature invariant
on a given day. OB1 was addressed in the manuscript. OB2 was addressed by ensuring that the
solvent spectra used for subtraction was consistent with the lot number used for constructing the
solution.
The origin of OB3 was not identified but could be due to impurities in the house nitrogen
because the contaminant peaks were sharp as expected for a gas phase impurity. OB3 was present
spectra with small pathlengths and was remedied by quantifying the effect of the contamination in
the following way. The empty cell spectrum was fitted with a decaying sine wave in the region of
the contamination. Subtracting the empty cell spectrum from the sine wave fit, provided a ‘noise’
spectrum, which was weighted to subtract from the sample spectrum. This method successfully
removed the effects of the contamination from the sample spectra.
214
APPENDIX C: Relation of RTPT to Kretschmer-Wiebe
The equations for 2-K Kretchmer-Wiebe are presented by Nagata and Ogasawara118 and
Acree128 without derivation. The model is derived by Bala121 and called the 2-K CLAM model in
the thesis as an extension of the Kretschmer-Wiebe which Bala calls the 1-K CLAM model. We
provide an illustration of the similarities of 2-K Kretchmer-Wiebe model with Marshall and
Chapman112 resummed perturbation theory RTPT. To illustrate similarities with RTPT, we convert
the notation of Bala and Marshall/Chapman to use the notation of this manuscript.
Recognize the notational differences which are presented in Table C-1:
Table C-1: Notational differences between this work and that of other authors.
This work Marshall112 Bala121
Alcohol density 𝑥𝑎𝑙𝑐 𝜌 𝜌 𝐶𝑎𝑙𝑐
Alcohol Monomer 𝜌𝑜 𝜌𝑜 𝐶𝑀
concentration
Dimer Association Δ2 (1) 𝐾𝐶,2
𝑓𝐴𝐵 Δ
constant
N-mer Association Δ𝑁 (2) 𝐾𝐶,𝑁𝑚𝑒𝑟
𝑓𝐴𝐵 Δ
constant
To derive the 2-K Kretchmer-Wiebe model using the chemical theory approach, we start with an
apparent mole balance written in terms of concentration by dividing both sides by total volume, 𝑉,
𝑛𝑎𝑙𝑐 𝑛𝑎𝑙𝑐,𝑜 + 2𝑛𝑎𝑙𝑐,2 + 3𝑛𝑎𝑙𝑐,3 + ⋯ Eq. C-1
=
𝑉 𝑉
∞
𝑥𝑎𝑙𝑐 𝜌 = 𝜌𝑜 + 2𝜌𝑎𝑙𝑐,2 + 3𝜌𝑎𝑙𝑐,3 + ⋯ = 𝜌𝑜 + ∑ 𝑖𝜌𝑎𝑙𝑐,𝑖 Eq. C-2
𝑖=2
215
To simplify notation, monomers and dimers are designated with o and D subscripts respectively,
and all subsequent oligomers are referred to by the number of alcohol molecules they contain.
Defining two equilibrium constants, one for the formation of dimers and another for all subsequent
oligomers (note that by assuming volume does not change with association, the molar volume for
an oligomer is a multiple of the monomer molar volume, 𝑉𝑎𝑙𝑐,𝑁 = 𝑁𝑉𝑎𝑙𝑐,𝑜 ):
𝜌𝑎𝑙𝑐,2 Φ𝑎𝑙𝑐,2 /2𝑉𝑎𝑙𝑐,𝑜 1 Φ𝑎𝑙𝑐,2
Δ2 = 2
= 2 = 2 𝑉𝑎𝑙𝑐,𝑜 = 𝐾𝐶,2_(𝐾𝑊) 𝑉𝑎𝑙𝑐,𝑜 Eq. C-3
𝜌𝑜 (Φ𝑎𝑙𝑐,𝑜 /𝑉𝑎𝑙𝑐,𝑜 ) 2 Φ𝑎𝑙𝑐,𝑜
𝜌𝑎𝑙𝑐,𝑁 (𝑁 − 1) Φ𝑎𝑙𝑐,𝑁
Δ𝑁 = = 𝑉 = 𝐾𝐶,𝑁_(𝐾𝑊) 𝑉𝑎𝑙𝑐,𝑜 for 𝑁 > 2 Eq. C-4
𝜌𝑎𝑙𝑐,(𝑁−1) 𝜌𝑜 𝑁 Φ𝑎𝑙𝑐,𝑁−1 Φ𝑎𝑙𝑐,𝑜 𝑎𝑙𝑐,𝑜
where the volume fraction notation provides mapping to the Kretschmer-Wiebe equilibria and
𝐾𝐶,2_(𝐾𝑊) and 𝐾𝐶,𝑁_(𝐾𝑊) are the Kretschmer-Wiebe association constants. 𝑉𝑎𝑙𝑐,𝑜 is the same as the
pure alcohol molar volume. Excess volume is assumed to be negligible. For the last equality of
Eq. C-3 and Eq. C-4, the composition dependence of the Wertheim Δ and the minor differences
between the Mayer and Arrhenius temperature dependence must be disregarded. Combining Eq.
C-3 and Eq. C-4 and rearranging yields
𝜌𝑎𝑙𝑐,𝑁 = Δ2 Δ𝑁−2
𝑁 𝜌𝑜
𝑁
for 𝑁 > 1 Eq. C-5
Eq. C-5 can be substituted into the mole balance in Eq. C-2, and rearrangement of Eq. C-6 results
in equation Eq. C-7.
∞
𝜌𝑎𝑙𝑐 = 𝜌𝑜 + ∑ 𝑁Δ2 Δ𝑁−2 𝑁 2
𝑁 𝜌𝑜 = 𝜌𝑜 + 2Δ2 𝜌𝑜 + 3Δ2 Δ𝑁 𝜌𝑜 + ⋯
3
Eq. C-6
𝑁=2
Δ2
𝑥𝑎𝑙𝑐 𝜌 = 𝜌𝑜 (1 + 2Δ2 𝜌𝑜 + {3(Δ𝑁 𝜌𝑜 )2 + 4(Δ𝑁 𝜌𝑜 )3 + ⋯ }) Eq. C-7
Δ𝑁
216
Recognizing an opportunity to use a converging series for the inner braces, Eq. C-7 can be written
as the closed form of the complete series with subtraction of the first two terms that do not appear
in the braces.
Δ2 1
𝑥𝑎𝑙𝑐 𝜌 = 𝜌𝑜 (1 + 2Δ2 𝜌𝑜 + ( − 1 − 2Δ𝑁 𝜌𝑜 )) Eq. C-8
Δ𝑁 (1 − Δ𝑁 𝜌𝑜 )2
Δ2 1
𝑥𝑎𝑙𝑐 𝜌 = 𝜌𝑜 (1 + ( − 1)) Eq. C-9
Δ𝑁 (1 − Δ𝑁 𝜌𝑜 )2
Bala shows that the equation written in terms of 𝐾𝐶,2 = 𝐾𝐶,2_(𝐾𝑊) 𝑉𝑎𝑙𝑐,𝑜 and 𝐾𝐶,𝑁 =
𝐾𝐶,𝑁_(𝐾𝑊) 𝑉𝑎𝑙𝑐,𝑜 , writing in terms of volume fractions, 𝑥𝑎𝑙𝑐 𝜌𝑉𝑎𝑙𝑐,𝑜 = Φ𝑎𝑙𝑐 and 𝜌𝑜 𝑉𝑎𝑙𝑐,𝑜 = Φ𝑜
𝐾𝐶,2 1
Φ𝑎𝑙𝑐 = Φ𝑜 (1 + ( 2 − 1)) Eq. C-10
𝐾𝐶,𝑁 (1 − 𝐾
𝐶,𝑁 Φo /𝑉𝑎𝑙𝑐,𝑜 )
Bala shows this can be solved for any apparent concentration 𝑥𝑎𝑙𝑐 𝜌 as a cubic equation in monomer
volume fraction, Φ𝑜 .
We now show that Eq. C-9 is the same as Marshall and Chapman eq (20) respecting the
approximations set forth. Converting notation, Chapman eq. (20) is
𝑥𝑎𝑙𝑐 𝜌 2𝜌𝑜 Δ2 (1 − 𝜌𝑜 Δ𝑁 ) 𝜌𝑜2 Δ2 Δ𝑁 Eq. C-11
= 1+ +
𝜌𝑜 (1 − 𝜌𝑜 Δ𝑁 )2 (1 − 𝜌𝑜 Δ𝑁 )2
𝑥𝑎𝑙𝑐 𝜌 2𝜌𝑜 Δ2 2𝜌𝑜2 Δ2 Δ𝑁 𝜌𝑜2 Δ2 Δ𝑁 Eq. C-12
=1+ − +
𝜌𝑜 (1 − 𝜌𝑜 Δ𝑁 )2 (1 − 𝜌𝑜 Δ𝑁 )2 (1 − 𝜌𝑜 Δ𝑁 )2
𝑥𝑎𝑙𝑐 𝜌 2𝜌𝑜 Δ2 𝜌𝑜2 Δ2 Δ𝑁 Eq. C-13
=1+ −
𝜌𝑜 (1 − 𝜌𝑜 Δ𝑁 )2 (1 − 𝜌𝑜 Δ𝑁 )2
Adding and subtracting 1/Δ𝑁 results in equation Eq. C-14.
217
𝑥𝑎𝑙𝑐 𝜌 2𝜌𝑜 − 𝜌𝑜2 Δ𝑁 1 1
= 1 + Δ2 ( + − ) Eq. C-14
𝜌𝑜 (1 − 𝜌𝑜 Δ𝑁 )2 ΔN ΔN
Creating common denominators for the first to terms in parenthesis
𝑥𝑎𝑙𝑐 𝜌 Δ𝑁 (2𝜌𝑜 − 𝜌𝑜2 Δ𝑁 ) 1 − 2𝜌𝑜 Δ𝑁 + 𝜌𝑜2 Δ2𝑁 1 Eq. C-15
= 1 + Δ2 ( + − )
𝜌𝑜 Δ𝑁 (1 − 𝜌𝑜 Δ𝑁 )2 Δ𝑁 (1 − 𝜌𝑜 Δ𝑁 )2 ΔN
𝑥𝑎𝑙𝑐 𝜌 1 1 Eq. C-16
= 1 + Δ2 ( − )
𝜌𝑜 ΔN (1 − 𝜌𝑜 Δ𝑁 )2 ΔN
𝑥𝑎𝑙𝑐 𝜌 Δ2 1 Eq. C-17
=1+ ( − 1)
𝜌𝑜 ΔN (1 − 𝜌𝑜 Δ𝑁 )2
This equation matches Eq. C-9. The RTPT model and the 2-K Kretschmer-Wiebe model are
distinctly different in the way that they are derived and the details of the association constants, but
the similarities are striking. The RTPT model has a small composition dependence of the
association constant and the Mayer term for temperature-dependence instead of the Arrhenius term
typically used for the chemical theory approach. Certainly, the Wertheim framework has capability
for higher order perturbations, but the forms are very similar at this level of Wertheim theory
except for the details of the association constants.
218
APPENDIX D: Conversion of Extensive Helmholtz Energy to Molar
The notation of Wertheim99–102 uses extensive variables and most publications that begin
from the Helmholtz energy continue the use of extensive variables and number density. However,
in engineering, particularly with equations of state, the use of intensive properties is predominant.
In this section we provide the conversion of the extensive Helmholtz energy to use molar
properties. We demonstrate the conversion for the resummed thermodynamic perturbation theory
(RTPT) derived by Marshall and Chapman112,141 as an extension of Wertheim’s original TPT-1
model. The current conversion is applied to a binary system in which component (1) has one h-
bond acceptor and one h-bond donor, referred to as the 2B scheme, and component (2) has no sites.
To distinguish between number and molar properties, specifically density, we use a tilde
accent to denote the former and no ornaments to denote the latter. Density in this work always
represents an intensive particle density or molar density, not the reciprocal of extensive volume.
Therefore, 𝜌̃ = 𝑁/𝑉 and 𝜌 have units of particles/length3 and mol/length3 respectively and 𝜌̃ =
𝑁𝐴 𝜌. The molar density of associating component (1) is given by 𝜌1 = 𝑥1 𝜌 whether mixed or pure,
whereas 𝑉1represents only the pure molar volume of component (1). Finally, the density of the
monomer species of the associating species (1) is denoted by 𝜌0 . An underbar denotes extensive
quantities, such as the extensive volume 𝑉, except in the case of number of molecules, 𝑁, and
number of moles, 𝑛. All other symbols used in this work are consistent with those of Marshall and
Chapman.112,141
For the alcohol component (1) with associating sites 𝐴 and 𝐵 in a nonbonding solvent, the
Helmholtz energy in extensive units is written
219
𝐴𝑎𝑠𝑠𝑜𝑐 𝐴𝑎𝑠𝑠𝑜𝑐 𝜌̃ 𝐴𝑎𝑠𝑠𝑜𝑐 𝜌
= =
𝑉𝑘𝐵 𝑇 𝑁𝑘𝐵 𝑇 𝑘𝐵 𝑇
Eq. D-1
𝜌̃𝑜 𝜎̃Γ−𝐴 𝜎̃Γ−𝐵 Δ𝑐 (𝑜)
= 𝜌̃1 ln − 𝜎̃Γ−𝐴 − 𝜎̃Γ−𝐵 + + 𝜌̃1 −
𝜌̃1 𝜌̃𝑜 𝑉
which has units of (length)−3 . Here, 𝜎Γ−𝑖 is the sum of the densities of bonding species with
unbonded site (𝑖). The term Δ𝑐 (𝑜) is the associative contribution to the fundamental graph sum.
Dividing by 𝑁𝐴 converts to molar densities,
𝐴𝑎𝑠𝑠𝑜𝑐 𝜌 𝜌𝑜 𝜎Γ−𝐴 𝜎Γ−𝐵 Δ𝑐 (𝑜)
= 𝑥1 𝜌 ln − 𝜎Γ−𝐴 − 𝜎Γ−𝐵 + + 𝜌1 − Eq. D-2
𝑅𝑇 𝜌1 𝜌𝑜 𝑁𝐴 𝑉
which has units of molar density.
For the 2B case here where only component (1) associates, the sites are typically designated
as an acceptor site (A) and donor site (D). 𝑋 𝐴 and 𝑋 𝐷 are the fraction of the respective sites that
are unbonded, which are equal in the case of one associating species in an inert solvent. The density
of unbonded sites is 𝜎Γ−𝐴 = 𝜎Γ−𝐵 = 𝑥1 𝜌𝑋 𝐴 = 𝑥1 𝜌𝑋 𝐷 so we can simplify
𝐴𝑎𝑠𝑠𝑜𝑐 𝜌 𝜌𝑜 (𝜎Γ−𝐴 )2 Δ𝑐 (𝑜)
= 𝜌1 ln − 2𝜎Γ−𝐴 + + 𝑥1 𝜌 − Eq. D-3
𝑅𝑇 𝜌1 𝜌𝑜 𝑁𝐴 𝑉
𝐴𝑎𝑠𝑠𝑜𝑐 𝜌 𝜌𝑜 (𝑥1 𝜌𝑋 𝐴 )2 𝛥𝑐 (𝑜)
= 𝑥1 𝜌 𝑙𝑛 − 2𝑥1 𝜌𝑋 𝐴 + + 𝑥1 𝜌 − Eq. D-4
𝑅𝑇 𝑥1 𝜌 𝜌𝑜 𝑁𝐴 𝑉
The notation of Marshall and Chapman112 is transformed in this work to accommodate
empirical fitting of bonding volumes as part of the association constant and thus we define the
(1) (2)
dimer association constant Δ2 ≡ 𝑁𝐴 𝑓𝐴𝐵 Δ, and the n-mer association constant Δ𝑁 ≡ 𝑁𝐴 𝑓𝐴𝐵 Δ,
where the right-hand side is Marshall and Chapman notation and the left-hand side is used here.
This work implements the PC-SAFT form of the association constant,
220
𝐴𝐷
Δ(2 or 𝑁) = 𝑑 3 𝑔(𝑑)𝜅(2 or 𝑁) (exp (𝜖(2 or 𝑁) /(𝑘𝑇)) − 1) Eq. D-5
where 𝑑 is the alcohol temperature-dependent segment diameter and the radial distribution
𝐴𝐷
function 𝑔(𝑑) depends on d, composition, and density. The variables 𝜅(2 or 𝑁) and 𝜖(2 or 𝑁) are
adjusted to experimental data. Converting the notation of Marshall and Chapman, the last term of
Eq. D-3 produces
Δ𝑐 (𝑜) (𝜎Γ−𝐴 )2 Δ2 (𝑥1 𝜌𝑋 𝐴 )2 Δ2
= = Eq. D-6
𝑁𝐴 𝑉 1 + (Δ2 − Δ𝑁 )𝜌𝑜 1 + (Δ2 − Δ𝑁 )𝜌𝑜
where Δ2 and Δ𝑁 have units of length3/mol. Also, from Eq. D-4
𝐴𝑎𝑠𝑠𝑜𝑐 𝜌𝑜 2𝑥1 𝜌𝑋 𝐴 (𝑥1 𝜌𝑋 𝐴 )2 Δ𝑐 (𝑜)
= 𝑥1 ln − + + 𝑥1 − Eq. D-7
𝑅𝑇 𝑥1 𝜌 𝜌 𝜌𝑜 𝜌 𝑁𝐴 𝑉𝜌
The extensive number density derivative of Eq. D-1 is common in Wertheim statistics
publications, which is equivalent to the molar density derivative of Eq. D-2. Note that
differentiation of Eq. D-2 with a molar density is equivalent to multiplying Eq. D-2 by 𝑁𝐴 to obtain
equation (1) and then differentiating with 𝑑𝜌̃ = 𝑁𝐴 𝑑𝜌. Consideration of Eq. D-3 together with Eq.
D-6 as a 𝑓(𝑇, 𝜌1 , 𝜌𝑜 , 𝜎𝐴 ) provides a powerful way to extract the equilibrium relations by using the
lumped variables. Gibbs energy is obtained from the density derivative of Eq. 4-36
1 𝜕(𝐴𝑎𝑠𝑠𝑜𝑐 𝜌) 𝜌 𝜕𝐴𝑎𝑠𝑠𝑜𝑐 𝐴𝑎𝑠𝑠𝑜𝑐 𝐴𝑎𝑠𝑠𝑜𝑐 + (𝑃𝑉)𝑎𝑠𝑠𝑜𝑐
( ) = ( ) + =
𝑅𝑇 𝜕𝜌 𝑅𝑇 𝜕𝜌 𝑇
𝑅𝑇 𝑅𝑇
𝑇
𝑎𝑠𝑠𝑜𝑐
Eq. D-8
𝐺
=
𝑅𝑇
The association contribution to the chemical potential of component (𝑘) is obtained with the
derivative
221
𝜕(𝑛𝐴) 𝑉 𝜕(𝑛𝐴) 𝜕(𝐴𝜌)
𝜇𝑘𝑎𝑠𝑠𝑜𝑐 = ( ) = ( ) =( ) Eq. D-9
𝜕𝑛𝑘 𝑇,𝑉,{𝑛 𝑉 𝜕𝑛𝑘 𝑇,𝑉,{𝑛 𝜕𝜌𝑘 𝑇,𝑉,{𝑛
𝑗≠𝑘 } 𝑗≠𝑘 } 𝑗≠𝑘 }
By the expansion rule,
𝜕(𝐴𝑎𝑠𝑠𝑜𝑐 𝜌)
𝜇1𝑎𝑠𝑠𝑜𝑐 = ( )
𝜕𝜌1 𝑇,𝑉,{𝑛𝑗≠1 }
𝜕(𝐴𝑎𝑠𝑠𝑜𝑐 𝜌) 𝜕(𝐴𝑎𝑠𝑠𝑜𝑐 𝜌) 𝜕𝜌𝑜
=( ) +( ) ( )
𝜕𝜌1 𝑇,𝑉,{𝑛
𝜕𝜌𝑜 𝜕𝜌1 𝑇,𝑉 Eq. D-10
𝑗≠1 },𝜌𝑜 ,𝜎𝐴 𝑇,𝑉,{𝑛 𝑗 },𝜎𝐴
𝜕(𝐴𝑎𝑠𝑠𝑜𝑐 𝜌) 𝜕𝜎Γ−𝐴
+( ) ( )
𝜕𝜎Γ−𝐴 𝑇,𝑉,{𝑛
𝜕𝜌1 𝑇,𝑉
𝑗 },𝜌𝑜
But chemical reaction equilibrium requires that the derivative with respect to 𝜌𝑜 and the
derivative with respect to 𝜎Γ−𝐴 = 𝑥1 𝜌𝑋 𝐴 to be zero because they involve reacting species. These
derivatives provide relations for the chemical potentials of sites and monomer as influenced by the
apparent density. Though the component apparent density is determined by composition,
temperature and pressure, the minimization of the Helmholtz energy at a certain apparent density
occurs when the derivatives with respect to reacting species 𝜌𝑜 and 𝜎𝐴 are zero. Thus, the only
nonzero term is first term on the right-most side
𝜇1𝑎𝑠𝑠𝑜𝑐 1 𝜕(𝐴𝑎𝑠𝑠𝑜𝑐 𝜌)
= ln 𝜑̂1𝑎𝑠𝑠𝑜𝑐 = ( )
𝑅𝑇 𝑅𝑇 𝜕𝜌1 𝑇,𝑉,{𝑛 𝑗≠1 }
Eq. 0-11
1 𝜕(𝐴𝑎𝑠𝑠𝑜𝑐 𝜌)
= ( )
𝑅𝑇 𝜕𝜌1 𝑇,𝑉,{𝑛𝑗≠1 },𝜌𝑜 ,𝜎𝐴
222
APPENDIX E: Key Material Balance Equations
The chemical equilibria balance equations are obtained by the derivatives of Eq. D-3 with
respect to 𝜎Γ−𝐴 and 𝜌𝑜 as provided by Marshall and Chapman112 as equation (17) and (18) in that
work. Recognizing the density of free acceptor sites hosted by component (1) is 𝜎Γ−𝐴 = 𝑥1 𝜌𝑋 𝐴 ,
then
𝑥1 𝜌𝑋 𝐴 𝑥1 𝜌𝑋 𝐴 Δ2
−1 = Eq. E-1
𝜌𝑜 1 + (Δ2 − Δ𝑁 )𝜌𝑜
2
𝑥1 𝜌 𝑥1 𝜌𝑋 𝐴 (Δ2 − Δ𝑁 )(𝑥1 𝜌𝑋 𝐴 )2 Δ2
=( ) − Eq. E-2
𝜌𝑜 𝜌𝑜 (1 + (Δ2 − Δ𝑁 )𝜌𝑜 )2
Eq. E-2 provides a material balance. Eq. E-1 can be rearranged to identify the contributions of 𝛼
(monomer) and 𝛽 hydroxyls to the free site density (note 𝜌𝛼 = 𝜌𝑜 and for the 2B bonding scheme
𝜌𝛽 = 𝜌𝛾 )
𝐴
𝑥1 𝜌𝑋 𝐴 𝛥2 𝜌𝑜
𝑥1 𝜌𝑋 = 𝜌𝑜 + = 𝜌𝛼 + 𝜌𝛽 ;
1 + (Δ2 − Δ𝑁 )𝜌𝑜
Eq. E-3
𝐴
𝑥1 𝜌𝑋 𝛥2 𝜌𝑜
𝜌𝛽 =
1 + (Δ2 − Δ𝑁 )𝜌𝑜
An alternative arrangement of Eq. E-1 is found by combining the terms with 𝑥1 𝜌𝑋 𝐴 using a
common denominator and simplifying to give
𝑥1 𝜌𝑋 𝐴 𝜌𝑜
= Eq. E-4
1 + (Δ2 − Δ𝑁 )𝜌𝑜 1 − Δ𝑁 𝜌𝑜
Noting the appearance of the left side of Eq. E-4 we can write Eq. E-1 as
𝑥1 𝜌𝑋 𝐴 Δ2 𝜌𝑜
=1+ Eq. E-5
𝜌𝑜 1 − Δ𝑁 𝜌𝑜
223
Squaring Eq. E-5 replaces the first term of Eq. E-2 and squaring Eq. E-4 to insert into the second
term of Eq. E-2 results in equation (20) of Marshall and Chapman112 which is
2Δ2 𝜌𝑜2 Δ2 Δ𝑁 𝜌𝑜3
𝑥1 𝜌 = 𝜌𝑜 + + Eq. E-6
1 − Δ𝑁 𝜌𝑜 (1 − Δ𝑁 𝜌𝑜 )2
The monomer can be determined by recognizing that Eq. E-6 is a cubic in 𝜌𝑜 . Instead, we present
an iterative method that is solved simultaneously with 𝑋 𝐴 . The fraction of sites (of a single type,
i.e. acceptor or donor sites) bound is found by difference between Eq. E-6 and Eq. E-5 multiplied
by 𝜌𝑜 to give
Δ2 𝜌𝑜2
𝑥1 𝜌(1 − 𝑋 𝐴 ) = Eq. E-7
(1 − Δ𝑁 𝜌𝑜 )2
Inserting the square of Eq. C-14 into the right-side results in a key equality
𝑥1 𝜌Δ2 (𝑋 𝐴 )2
(1 − 𝑋 𝐴 ) = Eq. E-8
(1 + (Δ2 − Δ𝑁 )𝜌𝑜 )2
Defining for convenience
𝑠 = 1 + (Δ2 − Δ𝑁 )𝜌𝑜 Eq. E-9
we can rearrange Eq. E-8 to the more useful form
𝑠2 1
𝑋𝐴 = 2 𝐷
= Eq. E-10
𝑠 + 𝑥1 𝜌Δ2 𝑋 1 + 𝑥1 𝜌Δ2 𝑋𝐷 /𝑠 2
where we have now indicated the distinction of donors and acceptors. The donors and acceptor
can be exchanged in the equation (𝑋 𝐷 = 𝑋 𝐴 ) since they are equal for the system type considered
here, and the similarity of Eq. E-10 with the 2B-TPT-1 is obvious where 𝑠 = 1. A final key
expression results from solving the following quadratic obtained by rearranging Eq. E-5
(Δ2 − Δ𝑁 )𝜌𝑜2 + (1 + 𝑥1 𝜌Δ𝑁 𝑋 𝐴 )𝜌𝑜 − 𝑥1 𝜌𝑋 𝐴 = 0 Eq. E-11
224
and solving for the physically meaningful root gives
2𝑥1 𝜌𝑋 𝐴
𝜌𝑜 = Eq. E-12
1 + 𝑥1 𝜌Δ𝑁 𝑋𝐴 + √(1 + 𝑥1 𝜌Δ𝑁 𝑋𝐴 )2 + 4(Δ2 − Δ𝑁 )𝑥1 𝜌𝑋𝐴
This monomer density is determined by solving Eq. E-12 simultaneously with Eq. E-10 using
successive substitution. A trial value of 𝑋 𝐴 is used to generate 𝜌𝑜 in Eq. E-12, which is used to
calculate 𝑠 and 𝑋 𝐴 from Eq. E-10 and the iteration is continued until convergence is obtained.
225
APPENDIX F: Excess Helmholtz Energy
The excess Helmholtz energy can be obtained by
𝑎𝑠𝑠𝑜𝑐
𝐴𝐸 𝐴𝑎𝑠𝑠𝑜𝑐 𝐴𝑎𝑠𝑠𝑜𝑐
𝑝𝑢𝑟𝑒 1 𝐴𝑎𝑠𝑠𝑜𝑐
𝑝𝑢𝑟𝑒 2
( ) = − 𝑥1 − 𝑥2 Eq. F-1
𝑅𝑇 𝑅𝑇 𝑅𝑇 𝑅𝑇
Combining Eq. D-6, Eq. D-7, and Eq. E-1
2
𝐴𝑎𝑠𝑠𝑜𝑐 𝜌𝑜 2𝜎Γ−𝐴 𝜎Γ−𝐴 𝜎Γ−𝐴 𝜎Γ−𝐴
= 𝑥1 ln − + + 𝑥1 − ( − 1) Eq. F-2
𝑅𝑇 𝜌1 𝜌 𝜌𝑜 𝜌 𝜌 𝜌𝑜
𝐴𝑎𝑠𝑠𝑜𝑐 𝜌𝑜 𝜎𝛤−𝐴 𝜌𝑜
= 𝑥1 ln − + 𝑥1 = 𝑥1 ln + 𝑥1 (1 − 𝑋 𝐴 ) Eq. F-3
𝑅𝑇 𝜌1 𝜌 𝜌1
Note by Eq. E-12 that 𝜌𝑜 /𝜌1 is always finite as 𝜌1 approaches zero with a limiting value of unity.
𝜌𝑜 2𝑋 𝐴
= Eq. F-4
𝜌1 1 + 𝜌1 Δ𝑁 𝑋𝐴 + √(1 + 𝜌1 Δ𝑁 𝑋𝐴 )2 + 4(Δ2 − Δ𝑁 )𝜌1 𝑋𝐴
Thus
𝑎𝑠𝑠𝑜𝑐
𝐴𝐸 𝜌𝑜 𝜌𝑜,𝑝𝑢𝑟𝑒 1 𝐴
( ) = 𝑥1 ln + 𝑥1 (1 − 𝑋 𝐴 ) − 𝑥1 ln − 𝑥1 (1 − 𝑋𝑝𝑢𝑟𝑒 1) Eq. F-5
𝑅𝑇 𝜌1 𝜌𝑝𝑢𝑟𝑒 1
𝑎𝑠𝑠𝑜𝑐
𝐴𝐸 𝜌𝑜 𝜌𝑝𝑢𝑟𝑒 1 𝐴 𝐴
( ) = 𝑥1 ln ( ) + 𝑥1 (𝑋𝑝𝑢𝑟𝑒 1−𝑋 ) Eq. F-6
𝑅𝑇 𝜌1 𝜌𝑜,𝑝𝑢𝑟𝑒 1
226
APPENDIX G: Activity Coefficients
The derivation here is modified from the original publication where some of the steps were
restricted to 𝑉 𝐸 = 0 and 𝑉𝑘 = 𝑉𝑘 . As shown by Bala et al.,130 using a standard state as the pure
species at the same T and P as the mixture, the liquid phase activity coefficients are given by
1 𝜕𝐴𝑎𝑠𝑠𝑜𝑐
ln 𝛾𝑘𝑎𝑠𝑠𝑜𝑐 = ( ) | − 𝐴𝑎𝑠𝑠𝑜𝑐 |𝑝𝑢𝑟𝑒 𝑘
𝑅𝑇 𝜕𝑛𝑘 𝑇,𝑃,{𝑛
𝑗≠𝑘 }
assoc
𝑚𝑖𝑥 Eq. G-1
𝑃 𝜕𝑉 𝑃assoc
+ ( ) − 𝑉
𝑅𝑇 𝜕𝑛𝑘 𝑇,𝑃,𝑛 𝑅𝑇 𝑘
{𝑗≠𝑘}
1 𝜕𝐴𝑎𝑠𝑠𝑜𝑐 𝑉𝑘
( ) = ln 𝜑̂𝑘𝑎𝑠𝑠𝑜𝑐 − 𝑍 𝑎𝑠𝑠𝑜𝑐 Eq. G-2
𝑅𝑇 𝜕𝑛𝑘 𝑇,𝑃,{𝑛 𝑉
𝑗≠𝑘 }
Note that 𝑃𝑎𝑠𝑠𝑜𝑐 in the last term of Eq. G-1 uses the system contribution because the standard state
is at the system pressure. Combining these two equations and recognizing that the partial molar
volume terms cancel
𝑃assoc
ln 𝛾𝑘𝑎𝑠𝑠𝑜𝑐 = ln 𝜑̂𝑘𝑎𝑠𝑠𝑜𝑐 − 𝐴𝑎𝑠𝑠𝑜𝑐 |𝑝𝑢𝑟𝑒 𝑘 − 𝑉 Eq. G-3
𝑅𝑇 𝑘
Using Eq. 0-11 and applying to Eq. D-3 and recognizing that 𝑔11 is the radial distribution function
for component 1 association
𝜌𝑜 𝜕(Δ𝑐 (𝑜) /(𝑁𝐴 𝑉)
ln 𝜑̂𝑘𝑎𝑠𝑠𝑜𝑐 = ln −( ) Eq. G-4
𝜌1 𝜕𝜌1 𝑇,𝑉,{𝑛𝑗≠1 },𝜌𝑜 ,𝜎𝐴
227
𝜕(Δ𝑐 (𝑜) /(𝑁𝐴 𝑉)
( )
𝜕𝜌1 𝑇,𝑉,𝑛 𝑗≠1 ,𝜌𝑜 ,𝜎𝐴
2 𝜕 ln 𝑔11
𝜎Γ−𝐴 Δ2 ( )
𝜕𝜌1 𝑇,𝑉,{𝑛
𝑗≠1 } Eq. G-5
=
1 + (Δ2 − Δ𝑁 )𝜌𝑜
2 𝜕 ln 𝑔11
𝜎Γ−𝐴 Δ2 (Δ2 − Δ𝑁 )𝜌𝑜 ( )
𝜕𝜌1 𝑇,𝑉,{𝑛
𝑗≠1 }
−
(1 + (Δ2 − Δ𝑁 )𝜌𝑜 )2
Eq. E-8 can be inserted yielding
𝜕(Δ𝑐 (𝑜) /(𝑁𝐴 𝑉) 𝜕 ln 𝑔11
( ) = 𝑥1 (1 − 𝑋 𝐴 )𝜌 ( ) Eq. G-6
𝜕𝜌1 𝜕𝜌1 𝑇,𝑉,{𝑛
𝑇,𝑉,{𝑛𝑗≠1} ,𝜌𝑜 ,𝜎𝐴 𝑗≠1 }
𝜌𝑜 𝜕 ln 𝑔11
ln 𝜑̂1𝑎𝑠𝑠𝑜𝑐 = ln − 𝑥1 (1 − 𝑋 𝐴 )𝜌 ( ) Eq. G-7
𝜌1 𝜕𝜌1 𝑇,𝑉,{𝑛
𝑗≠1 }
By inspection
𝜕 ln 𝑔11
ln 𝜑̂2𝑎𝑠𝑠𝑜𝑐 = −𝑥1 (1 − 𝑋 𝐴 )𝜌 ( ) Eq. G-8
𝜕𝜌2 𝑇,𝑉,{𝑛
𝑗≠2 }
The compressibility factor contribution is
1 𝜕(𝐴𝑎𝑠𝑠𝑜𝑐 )
𝑍 𝑎𝑠𝑠𝑜𝑐 = 𝜌( ) Eq. G-9
𝑅𝑇 𝜕𝜌 𝑇,{𝑛 }
𝑖
The contribution to 𝑍 𝑎𝑠𝑠𝑜𝑐 from the first four terms of 𝐴𝑎𝑠𝑠𝑜𝑐 /(𝑅𝑇) in Eq. D-7, is:
2
𝜕 𝜌𝑜 2𝜎Γ−𝐴 𝜎Γ−𝐴
𝜌 (𝑥1 ln − + + 𝑥1 )
𝜕𝜌 𝑥1 𝜌 𝜌 𝜌𝜌𝑜 𝑇,{𝑛 }
𝑖
Eq. G-10
2
1 2𝜎Γ−𝐴 𝜎Γ−𝐴 2𝜎Γ−𝐴 𝜎Γ−𝐴 𝜎Γ−𝐴
= 𝜌 (−𝑥1 ( ) + 2
− 2
) = (−𝑥1 + − )
𝜌 𝜌 𝜌𝑜 𝜌 𝜌 𝜌 𝜌𝑜
228
where the values of 𝜎Γ−𝐴 and 𝜌0 are considered constant for the differentiation since it is evaluated
at a stationary point. Inserting Eq. E-1 and the first equality of Eq. E-3
𝜎Γ−𝐴 𝜎Γ−𝐴 Δ2
= −𝑥1 + 2𝑥1 𝑋 𝐴 − (1 + ) Eq. G-11
𝜌 1 + (Δ2 − Δ𝑁 )𝜌𝑜
Recognizing Eq. E-8
2
𝐴 𝐴
1 𝜎Γ−𝐴 Δ𝑁
= −𝑥1 + 2𝑥1 𝑋 − 𝑥1 𝑋 − ( )
𝜌 1 + (Δ2 − Δ𝑁 )𝜌𝑜 Eq. G-12
𝐴) 𝐴 )(1
= −𝑥1 (1 − 𝑋 − 𝑥1 (1 − 𝑋 + (Δ2 − Δ𝑁 )𝜌𝑜 )
The derivative of the first four terms contributing to 𝑍 𝑎𝑠𝑠𝑜𝑐 becomes
= −2𝑥1 (1 − 𝑋 𝐴 ) − 𝑥1 (1 − 𝑋 𝐴 )(Δ2 − Δ𝑁 )𝜌𝑜 Eq. G-13
The derivative we need for the last term is
𝜕(Δ𝑐 (o) /𝑁𝐴 𝑉𝜌) 2
𝜎Γ−𝐴 Δ2
𝜌( ) = 𝜌 (− 2
𝜕𝜌 𝑇,{𝑛 }
𝜌 (1 + (Δ2 − Δ𝑁 )𝜌𝑜 )
𝑖
2
𝜎Γ−𝐴 Δ2 𝜕 ln 𝑔11
+ ( ) Eq. G-14
𝜌(1 + (Δ2 − Δ𝑁 )𝜌𝑜 ) 𝜕𝜌 𝑇,{𝑛 }
𝑖
2
𝜎Γ−𝐴 Δ2 (Δ2 − Δ𝑁 )𝜌𝑜 𝜕 ln 𝑔11
− ( ) )
𝜌(1 + (Δ2 − Δ𝑁 )𝜌𝑜 )2 𝜕𝜌 𝑇,{𝑛 }
𝑖
Recognizing Eq. E-8
𝜕 ln 𝑔11
= − (𝑥1 (1 − 𝑋 𝐴 )(1 + (Δ2 − Δ𝑁 )𝜌𝑜 ) − 𝑥1 (1 − 𝑋 𝐴 )𝜌 ( ) ) Eq. G-15
𝜕𝜌 𝑇,{𝑛 }
𝑖
𝜕 ln 𝑔11
= −𝑥1 (1 − 𝑋 𝐴 ) (1 + (Δ2 − Δ𝑁 )𝜌𝑜 − ( ) ) Eq. G-16
𝜕 ln 𝜌 𝑇,{𝑛 }
𝑖
Thus, the last contribution to 𝑍 𝑎𝑠𝑠𝑜𝑐 is
229
𝜕(Δ𝑐 (o) /𝑁𝐴 𝑉𝜌)
−𝜌 ( )
𝜕𝜌 𝑇,{𝑛 }
𝑖 Eq. G-17
𝐴)
𝜕 ln 𝑔11
= 𝑥1 (1 − 𝑋 (1 + (Δ2 − Δ𝑁 )𝜌𝑜 − ( ) )
𝜕 ln 𝜌 𝑇,{𝑛 }
𝑖
Combining Eq. G-13 and Eq. G-17, we find
𝜕 ln 𝑔11
𝑍 𝑎𝑠𝑠𝑜𝑐 = −𝑥1 (1 − 𝑋 𝐴 ) (1 + ( ) ) Eq. G-18
𝜕 ln 𝜌 𝑇,{𝑛 }
𝑖
And thus, because the standard state is at the same pressure as the mixture, the mixture pressure is
used expressed using the mixture 𝑍 𝑎𝑠𝑠𝑜𝑐 ,
𝑃assoc 𝑉𝑘 𝑉𝑘 𝜕 ln 𝑔11
− 𝑉𝑘 = −𝑍 𝑎𝑠𝑠𝑜𝑐 = 𝑥1 (1 − 𝑋 𝐴 ) (1 + ( ) ) Eq. G-19
𝑅𝑇 𝑉 𝑉 𝜕 ln 𝜌 𝑇,{𝑛 }
𝑖
The activity coefficient for the associating component is obtained by inserting Eq. F-3 evaluated
at purity, Eq. G-7, Eq. G-19.
𝜌𝑜
ln 𝛾1𝑎𝑠𝑠𝑜𝑐 = ln +
𝑥1 𝜌
𝑉1 𝜕 ln 𝑔11 𝜕 ln 𝑔11
+ 𝑥1 (1 − 𝑋 𝐴 ) ( (1 + ( ) ) −𝜌( ) )
𝑉 𝜕 ln 𝜌 𝑇,{𝑛 } 𝜕𝜌1 𝑇,𝑉,𝑛 Eq. G-20
𝑖 2
𝜌𝑜,𝑝𝑢𝑟𝑒 1 𝐴
− ln − (1 − 𝑋𝑝𝑢𝑟𝑒 1)
𝜌𝑝𝑢𝑟𝑒 1
For the non associating component, the fugacity coefficient is given by Eq. G-8, where the third
and fourth terms of Eq. G-19 drop out, and the expression is
𝑉2 𝜕 ln 𝑔11 𝜕 ln 𝑔11
ln 𝛾2𝑎𝑠𝑠𝑜𝑐 = 𝑥1 (1 − 𝑋 𝐴 ) ( (1 + ( ) )− 𝜌( ) ) Eq. G-21
𝑉 𝜕 ln 𝜌 𝑇,{𝑛 } 𝜕𝜌2 𝑇,𝑉,𝑛
𝑖 1
230
APPENDIX H: Regression Flow Diagram
, 2 ,
, 1
,
= , ,
, ,
2
, 1
,
,
,
= /( 1 , , , )
,
, 2 ,
231
APPENDIX I: Individual Isotherm Regression (Stage-1)
Figure I-1: Stage 1 regression of methanol in cyclohexane (left) and n-hexane (right).
Table I-1: Constants resulting from RTPT Stage-1 regression where Δ2 , Δ𝑁 and 𝜖𝐵𝐿 are
adjusted for each isotherm for methanol in cyclohexane (left) and n-hexane (right). Goodness
of fit (R2) is presented for each temperature.
methanol in cyclohexane methanol in n-hexane
Units T1 T2 T3 Units T1 T2 T3
T [K] 283.15 298.25 313.25 T [K] 283.35 298.25 313.25
𝚫𝟐 [cm3] 1946 289.9 50.00 𝚫𝟐 [cm3] 1175 942.8 270.2
3
𝚫𝑵 [cm ] 26605 16930 9095 𝚫𝑵 [cm3] 22801 14408 8175
[dm3/ [dm3/
𝝐𝑩𝑳 1804.9 1901.7 1608.8 𝝐𝑩𝑳 1489.2 1528.7 1431.5
mol-cm2] mol-cm2]
R2 --- 0.9917 0.9594 0.9527 R2 --- 0.9252 0.9308 0.9626
232
Figure I-2: Stage 1 regression of ethanol in cyclohexane (left) and n-hexane (right).
Table I-2: Constants resulting from RTPT Stage-1 regression where Δ2 , Δ𝑁 and 𝜖𝐵𝐿 are
adjusted for each isotherm for ethanol in cyclohexane (left) and n-hexane (right). Goodness of
fit (R2) is presented for each temperature.
ethanol in cyclohexane ethanol in n-hexane
Units T1 T2 T3 Units T1 T2 T3
T [K] 283.15 298.15 313.15 T [K] 283.35 298.15 313.15
𝚫𝟐 [cm3] 329.8 938.1 326.8 𝚫𝟐 [cm3] 1188 1095 454.5
3
𝚫𝑵 [cm ] 25772 15332 8257 𝚫𝑵 [cm3] 25814 16238 10069
[dm3/ [dm3/
𝝐𝑩𝑳 1471.5 1426.3 1478.9 𝝐𝑩𝑳 1498.6 1389.5 1430.3
mol-cm2] mol-cm2]
R2 --- 0.9164 0.9928 0.9941 R2 --- 0.9781 0.9907 0.9890
233
Figure I-3: Stage 1 regression of 1-propanol in n-hexane (left) and 2-propanol in n-hexane
(right).
Table I-3: Constants resulting from RTPT Stage-1 regression where Δ2 , Δ𝑁 and 𝜖𝐵𝐿 are
adjusted for each isotherm for 1-propanol in n-hexane (left) and 2-propanol in n-hexane
(right). Goodness of fit (R2) is presented for each temperature.
1-propanol in n-hexane 2-propanol in n-hexane
Units T1 T2 T3 Units T1 T2 T3
T [K] 283.25 298.05 313.15 T [K] 283.35 298.15 313.25
3
𝚫𝟐 [cm ] 1498 699.7 369.5 𝚫𝟐 [cm3] 1645 970.1 558.7
3
𝚫𝑵 [cm ] 22451 12603 7156 𝚫𝑵 [cm3] 22832 11972 6236
3 3
[dm /mol- [dm /mol-
𝝐𝑩𝑳 1308.1 1288.9 1306.1 𝝐𝑩𝑳 1225.4 1127.8 1062.0
cm2] cm2]
R2 --- 0.9957 0.9853 0.9960 R2 --- 0.9838 0.9986 0.9988
234
Figure I-4: Stage 1 regression of 1-butanol in cyclohexane (left) and n-hexane (right).
Table I-4: Constants resulting from RTPT Stage-1 regression where Δ2 , Δ𝑁 and 𝜖𝐵𝐿 are
adjusted for each isotherm for 1-butanol in cyclohexane (left) and n-hexane (right). Goodness
of fit (R2) is presented for each temperature.
1-butanol in cyclohexane 1-butanol in n-hexane
Units T1 T2 T3 Units T1 T2 T3
T [K] 283.25 298.15 313.25 T [K] 283.45 298.15 313.25
3
𝚫𝟐 [cm ] 1674 539.6 362.7 𝚫𝟐 [cm3] 1415.5 181.6 463.6
𝚫𝑵 [cm3] 23471 12271 8026 𝚫𝑵 [cm3] 24139 11841 8408
3 3
[dm /mol- [dm /mol-
𝝐𝑩𝑳 1515.3 1564.1 1479.6 𝝐𝑩𝑳 1474.8 1295.6 1376.7
cm2] cm2]
R2 --- 0.9922 0.9975 0.9978 R2 --- 0.9818 0.8260 0.9961
235
Figure I-5: Stage 1 regression of 1-pentanol in n-hexane (left) and phenol in n-hexane (right).
Table I-5: Constants resulting from RTPT Stage-1 regression where Δ2 , Δ𝑁 and 𝜖𝐵𝐿 are
adjusted for each isotherm for 1-pentanol in n-hexane (left) and phenol in n-hexane (right).
Goodness of fit (R2) is presented for each temperature.
1-pentanol in n-hexane phenol in n-hexane
Units T1 T2 T3 Units T1 T2 T3
T [K] 283.25 298.25 313.15 T [K] 283.25 298.15 313.25
3
𝚫𝟐 [cm ] 900.6 3268 955.0 𝚫𝟐 [cm3] 2353 1334 878.8
𝚫𝑵 [cm3] 22165 19703 9612 𝚫𝑵 [cm3] 19409 10924 7356
3 3
[dm /mol- [dm /mol-
𝝐𝑩𝑳 1296.3 1597.6 1434.0 𝝐𝑩𝑳 4021.0 3763.9 3793.1
cm2] cm2]
R2 --- 0.9841 0.9824 0.9843 R2 --- 0.9981 0.9990 0.9962
236
Figure I-6: Stage 1 regression of 1-hexanol in cyclohexane (left) and n-hexane (right).
Table I-6: Constants resulting from RTPT Stage-1 regression where Δ2 , Δ𝑁 and 𝜖𝐵𝐿 are
adjusted for each isotherm for 1-hexanol in cyclohexane (left) and n-hexane (right). Goodness
of fit (R2) is presented for each temperature.
1-hexanol in cyclohexane 1-hexanol in n-hexane
Units T1 T2 T3 Units T1 T2 T3
T [K] 283.25 298.25 313.35 T [K] 283.35 298.25 313.35
3
𝚫𝟐 [cm ] 850.6 875.3 745.1 𝚫𝟐 [cm3] 1094 1048 553.6
𝚫𝑵 [cm3] 20555 14275 8813 𝚫𝑵 [cm3] 25632 15065 9129
3 3
[dm /mol- [dm /mol-
𝝐𝑩𝑳 1341.2 1477.0 1444.7 𝝐𝑩𝑳 1477.7 1410.5 1416.7
cm2] cm2]
R2 --- 0.9959 0.9976 0.9978 R2 --- 0.9921 0.9951 0.9974
237
APPENDIX J: Scaling Parameters for Ethanol and 1-Butanol
The following table provides parameters for the scaling function described in APPENDIX A:
Detailed Summary of Attenuation Coefficient Function. Note that the values for 1-butanol are
unchanged from the previous work.122
Table J-1: Scaling constant parameters for ethanol and 1-butanol.
parameter units ethanol 1-butanol
𝑓 cm·dm2/mol
𝑚1 111.82007 121.51223
𝑓 cm·dm2/mol
𝑚2 -125.11856 -130.83618
𝑓 cm·dm2/mol
𝑚3 0 0
𝑏1 dm2/mol -407576.02 -443631.92
𝑏2 dm2/mol 471668.30 477672.97
𝑏3 dm2/mol 29583.780 29639.512
𝑓 cm-1
𝜈̃𝑅 3408.48207 3424.3849
𝑓 cm-1
𝜈̃𝐵 3644.92767 3650.9240
𝑓 dm2/mol
𝜖𝑅 29583.77998 29639.512
𝑓 dm2/mol
𝜖𝐵 0.04596044 5.140849E-3
𝜖3645 dm2/mol 489.22132 819.56086
Δ𝑓 cm-1 8.2589 8.3506
238
APPENDIX K: Parity Plots
Figure K-1: Parity plot for solutions of 1-butanol + cyclohexane at 30°C, 40 °C, 50 °C, and 60
°C.
Figure K-2: Parity plot for solutions of ethanol + cyclohexane at 30°C, 40 °C, 50 °C, and 60
°C.
239
APPENDIX L: Hydroxyl Populations
This section provides the hydrogen bond populations at 40 °C and 60 °C as evaluated using area
of the fitted peaks and the fitted attenuation coefficient.
Figure L-1: Hydroxyl populations for binary solutions of ethanol + cyclohexane at 40 °C (left)
and 60 °C (right).
Figure L-2: Hydroxyl populations for binary solutions of 1-butanol + cyclohexane at 40 °C
(left) and 60 °C (right).
240
APPENDIX M: Hydroxyl Fractions
This section provides supplemental plots at 40 °C and 60 °C for the fraction of alcohol in each
hydrogen bond moiety compared to the RTPT model.
Figure M-1: Fraction of alcohol in specific hydroxyl configurations for binary solutions of
ethanol + cyclohexane at 40 °C (left) and 60 °C (right) overlayed with the RTPT model
prediction (--). Predicted monomer fraction is depicted in black.
Figure M-2: Fraction of alcohol in specific hydroxyl configurations for binary solutions of 1-
butanol + cyclohexane at 40 °C (left) and 60 °C (right) overlayed with the RTPT model
prediction (--). Predicted monomer fraction is depicted in black.
241
APPENDIX N: Limiting Activity Coefficient Regressions
Figure N-1: Cyclohexane dilute in methanol (upper-left), ethanol (upper-right), 1-butanol
(middle-left), 1-pentanol (middle-right). Experimental data is from Lazzaroni et al.139
242
Figure N-2: Cyclohexane dilute in 1-hexanol. Experimental data is from Lazzaroni et al.139
Figure N-3: 2-propanol dilute in cyclohexane (left) and cyclohexane dilute in 2-propanol
(right). Experimental data is from Lazzaroni et al.139
243
APPENDIX O: Mapping RTPT onto TPT-1
Figure O-1: Mapping of RTPT predictions onto TPT-1 for 1-propanol (upper left), 1-butanol
(upper right), 1-pentanol (lower left), and 1-hexanol (lower right) in cyclohexane.
244
Figure O-2: Mapping of RTPT predictions onto TPT-1 for 2-propanol (upper left), 2-butanol
(upper right), i-butanol (lower left), and t-butanol (lower right) in cyclohexane.
245
APPENDIX P: Phase Equilibria and Excess Enthalpy
Figure P-1: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)184 and 𝐻 𝐸
(right)185 for ethanol in cyclohexane.
Figure P-2: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)186 and 𝐻 𝐸
(right)174 for 1-pentanol in cyclohexane.
246
Figure P-3: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)137 and 𝐻 𝐸
(right)187 for 1-hexanol in cyclohexane.
Figure P-4: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)188–190 and
𝐻 𝐸 (right)191 for 2-propanol in cyclohexane.
247
Figure P-5: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)192 and 𝐻 𝐸
(right)178 for 2-butanol in cyclohexane.
Figure P-6: RTPT (-) and TPT-1 (--) modeling of experimental phase behavior (left)193,194 and
𝐻 𝐸 (right)195 for i-butanol in cyclohexane.
248
APPENDIX Q: Contributions to the Excess Enthalpy at 318.15 K
Figure Q-1: Contributions to 𝐻 𝐸 for methanol (upper left), ethanol (upper right), 1-propanol
(lower left), and 2-butanol (lower right) in cyclohexane at 318.15 K.
249
Figure Q-2: Contributions to 𝐻 𝐸 for i-butanol (left) and t-butanol (right) in cyclohexane at
318.15 K.
250
APPENDIX R: Attenuation Function Parameters
Regression of the attenuation function resulted in the parameter values presented in Table
R-1. The variables correspond to the equations provided in APPENDIX A: Detailed Summary of
Attenuation Coefficient Function. Parameters are provided with many significant figures because
the equations require mathematical differences in large numbers.
Table R-1: Molar attenuation coefficient parameters for measured alcohol systems.
𝒇 𝒇 𝒇 𝒇 𝒇
𝝂̃𝑹 𝝐𝑹 𝝂̃𝑩 𝝐𝑩 𝒎𝟏 𝚫𝒇
[cm-1] [dm2/mol] [cm-1] [dm2/mol] [cm·dm2/mol] [cm-1]
methanol 3528.257 27288.09 3614.807 0.043244 25.88124 39.20276
ethanol 3451.392 29125.72 3617.187 0.046733 18.93279 45.65274
1-propanol 3483.178 27228.19 3631.943 0.043689 17.96059 40.53972
1-butanol 3457.282 29236.2 3620.377 0.046913 19.29024 52.26124
1-pentanol 3455.176 27149.68 3623.741 0.043562 17.81814 55.22821
1-hexanol 3445.965 28504.22 3619.338 0.045737 18.67722 55.48942
i-butanol 3415.853 31797.42 3613.372 0.050383 21.52852 61.2826
2-propanol 3386.512 29347.7 3615.519 0.047091 18.92334 50.1112
2-butanol 3384.383 28389.56 3609.366 0.045554 17.84182 60.24368
t-butanol 3376.146 27617.83 3600.002 0.044309 24.39824 50.73637
251
APPENDIX S: Tabulated XA Values
Table S-1: Tabulated XA values for methanol (left) and ethanol (right).
methanol XA ethanol XA
x 30 ℃ 40 ℃ 50 ℃ 60 ℃ x 30 ℃ 40 ℃ 50 ℃ 60 ℃
2.50E-03 --- 0.991 0.980 0.982 1.07E-03 1.000 1.000 1.000 1.000
4.97E-03 0.909 --- --- --- 4.25E-03 0.967 0.979 0.976 0.990
1.00E-02 0.683 0.819 0.910 0.965 6.37E-03 0.915 0.977 0.993 0.988
2.00E-02 0.459 0.573 0.678 0.833 1.27E-02 0.754 0.875 0.916 0.950
3.00E-02 0.328 0.465 0.623 0.735 2.53E-02 0.542 0.655 0.778 0.849
4.02E-02 0.307 0.421 0.492 0.607 3.15E-02 0.474 0.580 0.682 0.773
4.96E-02 0.257 0.363 0.449 0.527 4.85E-02 0.342 0.439 0.527 0.622
6.02E-02 0.215 0.297 --- 0.512 6.40E-02 0.284 0.369 0.442 0.532
7.06E-02 0.205 0.286 0.349 0.437 9.42E-02 0.208 --- 0.348 0.413
7.97E-02 0.188 0.289 0.346 0.418 1.50E-01 0.155 0.197 0.255 ---
8.98E-02 0.173 0.188 0.299 0.365 2.02E-01 0.111 0.146 0.196 ---
9.99E-02 0.165 0.179 0.291 0.342 3.03E-01 0.083 0.128 0.164 ---
252
Table S-2: Tabulated XA values for 1-propanol (left) and 1-butanol (right).
1-propanol XA 1-butanol XA
x 30 ℃ 40 ℃ 50 ℃ 60 ℃ x 30 ℃ 40 ℃ 50 ℃ 60 ℃
9.98E-04 1.000 1.000 1.000 0.992 9.48E-04 0.944 0.949 0.958 0.950
2.46E-03 0.962 0.971 1.023 0.998 1.18E-03 0.982 0.962 0.967 0.963
4.98E-03 0.950 0.968 1.052 0.997 5.08E-03 0.952 0.979 0.990 0.973
9.95E-03 0.795 0.905 0.941 0.950 5.80E-03 0.916 0.960 0.983 0.992
2.01E-02 0.565 0.688 0.777 0.860 5.92E-03 0.933 0.974 0.983 1.005
3.01E-02 0.447 0.553 0.644 --- 1.00E-02 0.788 0.889 0.942 0.937
4.00E-02 0.400 0.508 0.602 --- 1.12E-02 0.744 0.857 0.930 0.956
4.99E-02 0.309 0.404 0.512 --- 1.12E-02 0.755 0.851 0.923 0.961
5.96E-02 0.280 0.372 0.465 --- 2.18E-02 0.549 0.645 0.794 0.873
7.01E-02 0.240 0.318 0.408 --- 2.72E-02 0.467 0.601 0.714 0.816
8.04E-02 0.212 0.289 0.366 --- 5.04E-02 --- --- --- 0.584
9.00E-02 0.195 0.268 0.334 --- 6.80E-02 0.251 0.326 0.413 0.490
9.96E-02 0.184 0.244 0.313 --- 8.18E-02 0.217 0.285 0.361 0.442
1.52E-01 0.140 0.192 0.240 --- 1.00E-01 0.188 --- 0.324 0.394
2.03E-01 0.112 0.153 0.198 --- 1.50E-01 0.153 0.203 0.251 0.317
2.53E-01 0.095 0.133 0.168 --- 1.50E-01 0.149 0.196 0.251 0.314
3.02E-01 0.082 0.119 0.152 --- 2.00E-01 0.127 0.168 0.213 0.264
2.00E-01 0.125 0.164 0.212 0.262
253
Table S-3: Tabulated XA values for 1-pentanol (left) and 1-hexanol (right).
1-pentanol XA 1-hexanol XA
x 30 ℃ 40 ℃ 50 ℃ 60 ℃ x 30 ℃ 40 ℃ 50 ℃ 60 ℃
1.00E-03 1.000 1.000 1.000 1.000 0.001 1.000 1.000 1.000 0.999
2.50E-03 0.989 0.986 1.000 1.000 0.003 0.976 0.977 0.978 0.978
5.00E-03 0.929 0.972 1.013 0.980 0.005 0.938 0.967 0.971 0.971
1.00E-02 0.762 0.870 0.919 0.957 0.010 0.742 0.904 0.927 0.942
2.00E-02 0.527 0.648 0.766 --- 0.020 --- 0.643 0.752 0.832
3.00E-02 0.395 0.511 0.627 --- 0.030 0.403 0.511 0.643 0.740
4.00E-02 0.325 0.422 0.557 0.671 0.040 0.331 0.438 0.542 0.636
5.00E-02 0.268 0.368 0.481 0.579 0.050 0.284 0.385 0.474 0.569
6.00E-02 0.259 0.322 0.411 0.531 0.060 0.250 0.330 0.421 0.505
7.00E-02 0.212 0.282 0.369 0.476 0.070 0.222 0.299 0.392 0.471
8.00E-02 0.194 0.249 0.337 0.430 0.080 0.205 0.277 0.358 0.425
9.00E-02 0.169 0.228 0.299 0.391 0.090 0.192 0.253 0.333 0.389
1.00E-01 0.153 0.212 0.279 0.356 0.099 0.183 0.240 0.313 0.372
1.50E-01 0.111 0.152 0.213 0.276 0.151 0.154 0.199 0.247 0.301
2.00E-01 0.085 0.119 0.168 0.220 0.201 0.126 0.165 0.207 0.252
2.50E-01 0.078 0.105 0.142 0.182 0.252 --- --- --- 0.217
3.00E-01 --- 0.088 --- 0.166
254
Table S-4: Tabulated XA values for 2-propanol (left) and 2-butanol (right).
2-propanol XA 2-butanol XA
x 30 ℃ 40 ℃ 50 ℃ 60 ℃ x 30 ℃ 40 ℃ 50 ℃ 60 ℃
9.98E-04 1.000 1.000 1.000 0.992 1.00E-03 1.000 1.000 1.000 1.000
2.46E-03 0.962 0.971 1.023 0.998 2.47E-03 0.989 0.978 0.983 0.983
4.98E-03 0.950 0.968 1.052 0.997 5.00E-03 --- 0.986 0.987 0.970
9.95E-03 0.795 0.905 0.941 0.950 9.99E-03 0.915 --- --- ---
2.01E-02 0.565 0.688 0.777 0.860 2.00E-02 0.639 0.755 0.839 0.875
3.01E-02 0.447 0.553 0.644 --- 2.99E-02 0.517 0.629 0.782 0.796
4.00E-02 0.400 0.508 0.602 --- 3.98E-02 0.434 0.561 0.653 0.744
4.99E-02 0.309 0.404 0.512 --- 5.02E-02 0.370 0.467 0.570 0.646
5.96E-02 0.280 0.372 0.465 --- 9.97E-02 0.227 0.299 0.372 0.458
7.01E-02 0.240 0.318 0.408 --- 1.50E-01 0.175 0.223 0.270 0.348
8.04E-02 0.212 0.289 0.366 --- 2.00E-01 0.144 0.157 0.189 0.244
9.00E-02 0.195 0.268 0.334 --- 2.49E-01 0.143 0.160 0.201 0.253
9.96E-02 0.184 0.244 0.313 ---
1.52E-01 0.140 0.192 0.240 ---
2.03E-01 0.112 0.153 0.198 ---
2.53E-01 0.095 0.133 0.168 ---
3.02E-01 0.082 0.119 0.152 ---
255
Table S-5: Tabulated XA values for i-butanol (left) and t-butanol (right).
i-butanol XA t-butanol XA
x 30 ℃ 40 ℃ 50 ℃ 60 ℃ x 30 ℃ 40 ℃ 50 ℃ 60 ℃
9.94E-04 1.000 1.000 1.000 1.000 4.90E-04 1.000 1.000 1.000 1.000
2.46E-03 0.980 0.973 0.976 0.986 1.02E-03 1.000 1.000 1.000 1.000
4.91E-03 0.954 0.971 0.966 0.994 2.49E-03 0.993 0.991 0.994 1.000
9.93E-03 1.577 0.841 0.898 1.004 5.00E-03 0.970 0.982 0.991 0.996
1.98E-02 0.580 0.643 0.795 0.928 1.00E-02 0.850 0.922 0.963 0.972
3.03E-02 0.448 0.549 0.665 0.761 2.00E-02 0.657 0.758 0.862 0.907
3.96E-02 0.380 0.468 0.577 0.669 3.01E-02 0.543 0.647 0.768 0.833
5.05E-02 0.329 0.404 0.502 0.611 3.99E-02 0.469 0.587 0.687 0.746
6.03E-02 0.284 0.351 0.444 0.555 4.98E-02 0.416 0.527 0.619 0.706
7.98E-02 0.235 0.295 0.390 0.473 5.98E-02 0.369 0.482 0.587 0.659
1.00E-01 0.206 0.254 0.341 0.415 6.99E-02 0.343 0.425 0.538 0.625
1.49E-01 0.157 0.200 0.261 0.319 7.99E-02 0.307 0.401 0.496 0.586
2.02E-01 0.141 0.164 0.225 0.254 8.99E-02 0.284 0.388 0.465 0.554
2.48E-01 0.128 0.147 0.193 0.239 9.99E-02 0.271 0.356 0.436 0.520
1.50E-01 0.216 0.291 0.360 0.426
1.99E-01 0.187 0.245 0.308 0.381
2.49E-01 0.155 0.206 0.268 0.330
2.99E-01 0.132 0.195 0.244 0.301
256
APPENDIX T: Excess Volume Comparison
Table T-1: Excess molar volumes for selected binary systems at 298.15 K.
Solute methanol methanol 1-propanol 1-hexanol
Solvent cyclohexane cyclohexane n-hexane n-hexane
T [K] 298.15 298.15 298.15 298.15
x1 0.96005 0.0286 0.4988 0.5053
V1 [cm3/mol] 40.733 40.733 75.146 125.305
V2 [cm3/mol] 108.757 108.757 131.578 131.578
Vmixture
43.623 106.932 103.598 127.974
[cm3/mol]
Videal [cm3/mol] 43.450 106.812 103.430 128.408
VE [cm3/mol] 0.173 0.121 0.169 -0.433
VE/V (in %) 0.398 0.113 0.163 -0.338
257
APPENDIX U: Calculated Thermal Expansivities of Perdeutero Compounds
Table U-1: Calculated thermal expansivities of perdeutero compounds using finite differences (Part I).
cyclohexane-d12 toluene-d8 benzene-d6 tetrahydrofuran-d8 pydridine-d5 chloroform-d
T/[K] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1]
278.15 1.0574 1.1864 1.0118 1.2350
283.15 1.1905 1.0618 1.2029 1.1981 1.0148 1.2371
288.15 1.2023 1.0628 1.2066 1.2133 1.0139 1.2481
293.15 1.2158 1.0618 1.2123 1.2263 1.0178 1.2638
298.15 1.2267 1.0838 1.2155 1.2491 1.0265 1.2904
303.15 1.2502 1.1110 1.2460 1.2821 1.0401 1.3165
308.15 1.2575 1.1034 1.2502 1.2842 1.0372 1.3187
313.15 1.2626 1.1018 1.2420 1.2860 1.0385 1.3226
318.15 1.2948 1.1208 1.2683 1.3173 1.0597 1.3522
323.15 1.3192 1.1530 1.2946 1.3391 1.0715 1.3820
328.15 1.3375 1.1602 1.3146 1.3591 1.0817 1.3905
333.15 1.3542 1.1596 1.3247 1.3772 1.0952
338.15 1.3820 1.1726 1.3328 1.1066
343.15 1.4081 1.1809 1.3464 1.1100
348.15 1.2181 1.3690 1.1250
353.15 1.2270 1.1356
358.15 1.2238 1.1263
363.15 1.2411 1.1498
368.15 1.1828
258
Table U-2: Calculated thermal expansivities of perdeutero compounds using finite differences (Part II).
dimethyl
acetone-d6 dichloromethane-d2 ethanol-d5 acetonitrile-d3 methanol-d4
sulfoxide-d6
T/[K] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1] 𝛼𝑃 /[10-3 K-1]
278.15 1.3820 1.3380 1.0490 1.3313 1.1740
283.15 1.3961 1.3456 1.0675 1.3464 1.1778
288.15 1.4229 1.3576 1.0823 1.3662 1.1862
293.15 1.4312 1.3818 1.0883 1.3812 1.1996
298.15 1.4523 1.4214 1.1042 1.3940 1.2056 0.9215
303.15 1.4788 1.4223 1.1324 1.4199 1.2257 0.9402
308.15 1.4919 1.4254 1.1325 1.4297 1.2295 0.9301
313.15 1.5286 1.4582 1.1283 1.4376 1.2375 0.9168
318.15 1.5536 1.1617 1.4640 1.2694 0.9365
323.15 1.5829 1.2036 1.4968 1.2992 0.9501
328.15 1.6130 1.2284 1.5293 1.3198 0.9555
333.15 1.2409 1.5431 1.3386 0.9614
338.15 1.2730 1.5549 1.3642 0.9638
343.15 1.3033 0.9611
348.15 1.3401 0.9770
353.15 1.3716 0.9785
358.15 0.9671
363.15 0.9679
368.15 0.9711
259
APPENDIX V: Effect of Method for Determining Isobaric Thermal
Expansivity
Figure V-1: Comparison of isobaric thermal expansivity calculation by two different means for
acetone-d6 (upper left), acetonitrile-d3 (upper right), benzene-d6 (lower left), and chloroform-
d (lower right). Data points were calculated using the finite difference method and then fitted
to a polynomial expression which is depicted as a solid line. The dashed line was calculated by
differentiating the polynomial expression that was fit to the density values. Differences are
most evident at the low and high temperature endpoints.
260
Figure V-2: Comparison of isobaric thermal expansivity calculation by two different means for
cyclohexane-d12 (upper left), dichloromethane-d2 (upper right), dimethyl sulfoxide-d6 (lower
left), and ethanol-d6 (lower right). Data points were calculated using the finite difference
method and then fitted to a polynomial expression which is depicted as a solid line. The
dashed line was calculated by differentiating the polynomial expression that was fit to the
density values. Differences are most evident at the low and high temperature endpoints.
261
Figure V-3: Comparison of isobaric thermal expansivity calculation by two different means for
methanol-d4 (upper left), pyridine-d5 (upper right), tetrahydrofuran-d8 (lower left), and
toluene-d8 (lower right). Data points were calculated using the finite difference method and
then fitted to a polynomial expression which is depicted as a solid line. The dashed line was
calculated by differentiating the polynomial expression that was fit to the density values.
Differences are most evident at the low and high temperature endpoints.
262
APPENDIX W: Protiated Molar Density Regression Coefficients
Table W-1: Protiated molar density regression coefficients used in molar
volume comparison. Regressed using NIST accepted density values.
mol 1
𝜌[ 3
] = 𝐶1 𝑇 2 + 𝐶2 𝑇 + 𝐶3
cm
C1 C2 C3
[mol/cm3K2] [mol/cm3K] [mol/cm3] R2
/10-8 /10-5
ethanol -2.1814 -0.54912 0.020622 0.99909
chloroform -1.2972 -0.84098 0.016049 0.99533
pyridine -0.30447 -1.0862 0.015873 0.99941
toluene -0.56179 -0.66921 0.011852 0.99953
dichloromethane -3.2153 -0.27394 0.019173 0.99458
benzene -0.64921 -0.97283 0.014661 0.99939
cyclohexane -0.52955 -0.80478 0.012106 0.99960
acetone -0.53025 -1.6907 0.019025 0.99739
methanol -1.2493 -2.1937 0.032202 0.99914
acetonitrile -1.6125 -1.6737 0.025343 0.99948
dimethyl 0.25777 -1.4442 0.018098 0.99921
sulfoxide
tetrahydrofuran -0.26914 -1.3568 0.016517 0.99913
*underbar indicates the last significant digit
263
APPENDIX X: Uncertainty Analysis Equations
Calculation of Standard Uncertainty in Densimeter A-value
𝐴(𝜏𝑎𝑖𝑟 , 𝜏𝐻2𝑂 , 𝜌𝑎𝑖𝑟 , 𝜌𝐻2𝑂 ) Eq. X-1
𝜕𝐴 2 𝜕𝐴 2 𝜕𝐴 2
𝑢2 (𝐴) = ( ) 𝑢(𝜏𝑎𝑖𝑟 )2 + ( ) 𝑢(𝜏𝐻2𝑂 )2 + ( ) 𝑢(𝜌𝑎𝑖𝑟 )2
𝜕𝜏𝑎𝑖𝑟 𝜕𝜏𝐻2𝑂 𝜕𝜌𝑎𝑖𝑟
Eq. X-2
𝜕𝐴 2
+( ) 𝑢(𝜌𝐻2𝑂 )2
𝜕𝜌𝐻2𝑂
The period of oscillation for the air and water calibrants are 𝜏𝑎𝑖𝑟 and 𝜏𝐻2𝑂 , respectively.
The uncertainty in this value is the same for both calibrant fluids; 0.0001 s. The tabulated density
values for air and water were assumed to be uncertain in the last decimal place; 1E-6 g/cm3 for air
and 1E-5 g/cm3 for water. The partial derivatives were evaluated using the following expressions.
𝜕𝐴 2𝜏𝑎𝑖𝑟
| | = |− | Eq. X-3
𝜕𝜏𝑎𝑖𝑟 𝜌𝐻2𝑂 − 𝜌𝑎𝑖𝑟
𝜕𝐴 2𝜏𝐻2𝑂
| |=| | Eq. X-4
𝜕𝜏𝐻2𝑂 𝜌𝐻2𝑂 − 𝜌𝑎𝑖𝑟
2 2
𝜕𝐴 𝜏𝐻2𝑂 − 𝜏𝑎𝑖𝑟
| |=| | Eq. X-5
𝜕𝜌𝑎𝑖𝑟 (𝜌𝐻2𝑂 − 𝜌𝑎𝑖𝑟 )2
2 2
𝜕𝐴 𝜏𝑎𝑖𝑟 − 𝜏𝐻2𝑂
| |=| | Eq. X-6
𝜕𝜌𝐻2𝑂 (𝜌𝐻2𝑂 − 𝜌𝑎𝑖𝑟 )2
Calculation of Standard Uncertainty in Densimeter B-value
𝐵(𝜏𝑎𝑖𝑟 , 𝜌𝑎𝑖𝑟 , 𝐴) Eq. X-7
2 (𝐵)
𝜕𝐵 2 2
𝜕𝐵 2 2
𝜕𝐵 2
𝑢 =( ) 𝑢(𝜏𝑎𝑖𝑟 ) + ( ) 𝑢(𝐴) + ( ) 𝑢(𝜌𝑎𝑖𝑟 )2 Eq. X-8
𝜕𝜏𝑎𝑖𝑟 𝜕𝐴 𝜕𝜌𝑎𝑖𝑟
264
The uncertainty of 𝐵 was then calculated analagously to 𝐴 using equations which follow.
𝜕𝐵
| | = |2𝜏𝑎𝑖𝑟 | Eq. X-9
𝜕𝜏𝑎𝑖𝑟
𝜕𝐵
| | = |−𝜌𝑎𝑖𝑟 | Eq. X-10
𝜕𝐴
𝜕𝐵
| | = |−𝐴| Eq. X-11
𝜕𝜌𝑎𝑖𝑟
Calculation of Combined Uncertainty for Density Measurement.
𝜌(𝑇, 𝜏𝑠 , 𝐴, 𝐵, 𝑃𝑢𝑟𝑖𝑡𝑦 (𝑥), 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐷𝑒𝑢𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 (𝑥𝐷 )) Eq. X-12
2
𝜕𝜌 2 2 𝜕𝜌 2 2 𝜕𝜌 2 2 𝜕𝜌 2 2
𝑢 (𝜌) = ( ) 𝑢 (𝑇) + ( ) 𝑢 (𝜏) + ( ) 𝑢 (𝐴) + ( ) 𝑢 (𝐵)
𝜕𝑇 𝜕𝜏𝑠 𝜕𝐴 𝜕𝐵
2 2 Eq. X-13
𝜕𝜌 𝜕𝜌
+ ( ) 𝑢2 (𝑥) + ( ) 𝑢2 (𝑥𝐷 )
𝜕𝑥 𝜕𝑥𝐷
𝑀𝐷 and 𝜌𝐷 denote the molecular weight and mass density of the deuterated compound, and 𝑀𝐻
denotes molecular weight of the protiated form. The period of oscillation for the sample is 𝜏𝑠 . For
samples where the purity was unavailable, 99% was assumed. It was assumed that the molecular
mass of the impurity was 12 g/mol less than the analyte. The effect of percent deuteration was
estimated as one minus the ratio of the molecular mass of the completely protiated compound
relative to the molecular mass of the completely deuterated compound. This difference was then
multiplied by the measured density of the deuterated sample. A detailed comprehensive error
analysis for each compound as well as the air and water calibrants can be found in the tables within
APPENDIX Y: Uncertainty Analysis Data.
𝜕𝜌
| | = |𝜌𝛼𝑃 | Eq. X-14
𝜕𝑇
265
𝜕𝜌 2𝜏
| |=| | Eq. X-15
𝜕𝜏 𝐴
𝜕𝜌 𝜏2 − 𝐵
| |=| | Eq. X-16
𝜕𝐴 𝐴2
𝜕𝜌 1
| |=| | Eq. X-17
𝜕𝐵 𝐴
𝜕𝜌 𝑀𝑖𝑚𝑝𝑢𝑟𝑖𝑡𝑦
| | = |(1 − ) 𝜌𝐷 | Eq. X-18
𝜕𝑥 𝑀𝐷
𝜕𝜌 𝑀𝐻
| | = |(1 − )𝜌 | Eq. X-19
𝜕𝑥𝐷 𝑀𝐷 𝐷
266
APPENDIX Y: Uncertainty Analysis Data
Table Y-1: Densimeter calibration constants.
Temperature Temperature τ(Air) τ(Water) ρ(Air) ρ(Water) A B
3 3 2 3
[°C] [K] [s] [s] [g/cm ] [g/cm ] [s cm /g] [s2]
5.00 278.15 5.9054 7.4642 0.001227 1.000183 20.86233 34.84814
10.00 283.15 5.9036 7.4615 0.001206 0.999785 20.85112 34.82735
15.00 288.15 5.9018 7.4585 0.001185 0.999123 20.84084 34.80667
20.00 293.15 5.9001 7.4551 0.001164 0.998208 20.82892 34.78693
25.00 298.15 5.8983 7.4514 0.001145 0.997052 20.81863 34.76611
30.00 303.15 5.8966 7.4474 0.001125 0.995667 20.80744 34.74649
35.00 308.15 5.8949 7.4433 0.001107 0.994064 20.79934 34.72683
40.00 313.15 5.8932 7.4386 0.001089 0.992256 20.78656 34.70717
45.00 318.15 5.8916 7.4338 0.001072 0.990255 20.77576 34.6881
50.00 323.15 5.8899 7.4288 0.001055 0.988071 20.76578 34.66901
55.00 328.15 5.8883 7.4237 0.001037 0.985716 20.75726 34.65055
60.00 333.15 5.8867 7.4183 0.001021 0.983203 20.74762 34.63205
65.00 338.15 5.8852 7.4128 0.001006 0.980544 20.73839 34.61471
70.00 343.15 5.8838 7.4071 0.000992 0.977749 20.7278 34.59855
75.00 348.15 5.8825 7.4013 0.000976 0.974831 20.7171 34.58358
80.00 353.15 5.8813 7.3955 0.000963 0.971801 20.70725 34.57009
85.00 358.15 5.8802 7.3895 0.00095 0.968671 20.69553 34.55709
90.00 363.15 5.8792 7.3833 0.000937 0.965454 20.6821 34.5455
95.00 368.15 5.8781 7.3770 0.000924 0.96216 20.6688 34.53343
267
Table Y-2: Error analysis for densimeter constants.
u(τAir) u(τH2O) u(ρAir) u(ρH2O)
0.0001 0.0001 0.000001 0.00001
Temperature Temperature ∂A/∂τ ∂A/∂τ ∂A/∂ρ ∂A/∂ρ ∂B/∂τ ∂B/∂A ∂B/∂ρ
u(A) u(B)
[°C] [K] [Air] [H2O] [Air] [H2O] [Air] [Air]
5.00 278.15 -11.8232 14.94401 20.88414 -20.8841 0.001917 11.8108 -0.00123 -20.8623 0.001181
10.00 283.15 -11.824 14.94423 20.88078 -20.8808 0.001917 11.8072 -0.00121 -20.8511 0.001181
15.00 288.15 -11.828 14.94782 20.8839 -20.8839 0.001918 11.80362 -0.00118 -20.8408 0.001181
20.00 293.15 -11.8352 14.95442 20.8907 -20.8907 0.001919 11.8002 -0.00116 -20.8289 0.00118
25.00 298.15 -11.8451 14.96405 20.9042 -20.9042 0.00192 11.7966 -0.00114 -20.8186 0.00118
30.00 303.15 -11.8579 14.97654 20.92164 -20.9216 0.001922 11.7932 -0.00112 -20.8074 0.00118
35.00 308.15 -11.8734 14.99218 20.94685 -20.9469 0.001924 11.7898 -0.00111 -20.7993 0.001179
40.00 313.15 -11.8914 15.00977 20.97179 -20.9718 0.001927 11.7864 -0.00109 -20.7866 0.001179
45.00 318.15 -11.912 15.03018 21.00295 -21.0029 0.001929 11.7831 -0.00107 -20.7758 0.001178
50.00 323.15 -11.9348 15.05306 21.03896 -21.039 0.001933 11.7798 -0.00106 -20.7658 0.001178
55.00 328.15 -11.9598 15.07841 21.08022 -21.0802 0.001936 11.7766 -0.00104 -20.7573 0.001178
60.00 333.15 -11.987 15.10575 21.12401 -21.124 0.00194 11.7734 -0.00102 -20.7476 0.001178
65.00 338.15 -12.0163 15.13531 21.17161 -21.1716 0.001944 11.7704 -0.00101 -20.7384 0.001177
70.00 343.15 -12.0476 15.16672 21.22103 -21.221 0.001949 11.7676 -0.00099 -20.7278 0.001177
75.00 348.15 -12.0809 15.20001 21.2733 -21.2733 0.001953 11.765 -0.00098 -20.7171 0.001177
80.00 353.15 -12.116 15.2353 21.32926 -21.3293 0.001958 11.76266 -0.00096 -20.7073 0.001176
85.00 358.15 -12.1527 15.27189 21.38584 -21.3858 0.001964 11.7604 -0.00095 -20.6955 0.001176
90.00 363.15 -12.191 15.30984 21.44296 -21.443 0.001969 11.75838 -0.00094 -20.6821 0.001176
95.00 368.15 -12.2304 15.34899 21.50232 -21.5023 0.001974 11.75628 -0.00092 -20.6688 0.001176
268
Table Y-3: Error analysis for acetone-d6 (purity of 99% was assumed).
acetone-d6 64.1161 [g/mol]
purity 0.991
xD 0.9996
MH 58.0791 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 52.1161 [g/mol] 0.05 0.0001 0.01 0.0001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
278.15 7.3277 0.903401 1.3820E-03 0.001249 0.702482 0.043303 0.047933 0.169081 0.085062 0.001696419 0.003325
283.15 7.3167 0.897158 1.3961E-03 0.001253 0.701804 0.043027 0.047959 0.167913 0.084474 0.001684753 0.003302
288.15 7.3057 0.890875 1.4229E-03 0.001268 0.701095 0.042747 0.047983 0.166737 0.083882 0.001673033 0.003279
293.15 7.2945 0.884482 1.4312E-03 0.001266 0.70042 0.042464 0.04801 0.16554 0.08328 0.001661078 0.003256
298.15 7.2835 0.878216 1.4523E-03 0.001275 0.69971 0.042184 0.048034 0.164367 0.08269 0.001649384 0.003233
303.15 7.2722 0.871727 1.4788E-03 0.001289 0.699 0.041895 0.04806 0.163153 0.082079 0.001637281 0.003209
308.15 7.2612 0.865325 1.4919E-03 0.001291 0.698214 0.041603 0.048078 0.161955 0.081477 0.001625319 0.003186
313.15 7.24976 0.858817 1.5286E-03 0.001313 0.697543 0.041316 0.048108 0.160737 0.080864 0.0016132 0.003162
318.15 7.23831 0.852197 1.5536E-03 0.001324 0.696804 0.041019 0.048133 0.159498 0.080241 0.001600853 0.003138
323.15 7.2269 0.845577 1.5829E-03 0.001338 0.696039 0.04072 0.048156 0.158259 0.079617 0.001588515 0.003113
328.15 7.2154 0.838812 1.6130E-03 0.001353 0.695217 0.040411 0.048176 0.156993 0.07898 0.001575906 0.003089
MEAN 0.003208
1
estimate
269
Table Y-4: Error analysis for acetonitrile-d3 (purity of 99% was assumed).
acetonitrile-d3 44.0704 [g/mol]
purity 0.991
xD 0.995
MH 41.0519 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 32.0704 [g/mol] 0.05 0.0001 0.01 0.001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
278.15 7.2612 0.856898 1.3313E-03 0.001141 0.696106 0.041074 0.047933 0.233326 0.058691 0.002337746 0.004582
283.15 7.25091 0.851194 1.3464E-03 0.001146 0.695494 0.040822 0.047959 0.231773 0.058301 0.002322224 0.004552
288.15 7.2406 0.845437 1.3662E-03 0.001155 0.694847 0.040566 0.047983 0.230205 0.057906 0.002306563 0.004521
293.15 7.2302 0.839643 1.3812E-03 0.00116 0.694246 0.040311 0.04801 0.228628 0.057509 0.002290797 0.00449
298.15 7.2198 0.83384 1.3940E-03 0.001162 0.69359 0.040053 0.048034 0.227048 0.057112 0.002275002 0.004459
303.15 7.2094 0.828019 1.4199E-03 0.001176 0.692964 0.039794 0.04806 0.225463 0.056713 0.002259176 0.004428
308.15 7.199 0.822082 1.4297E-03 0.001175 0.692233 0.039524 0.048078 0.223846 0.056307 0.002243016 0.004396
313.15 7.1885 0.816266 1.4376E-03 0.001173 0.691649 0.039269 0.048108 0.222262 0.055908 0.002227184 0.004365
318.15 7.178 0.810348 1.4640E-03 0.001186 0.690998 0.039004 0.048133 0.220651 0.055503 0.002211094 0.004334
323.15 7.1675 0.804402 1.4968E-03 0.001204 0.690318 0.038737 0.048156 0.219032 0.055096 0.002194938 0.004302
328.15 7.1569 0.798307 1.5293E-03 0.001221 0.68958 0.038459 0.048176 0.217372 0.054678 0.002178372 0.00427
333.15 7.1462 0.792193 1.5431E-03 0.001222 0.688869 0.038182 0.048198 0.215708 0.054259 0.002161738 0.004237
338.15 7.1356 0.786082 1.5549E-03 0.001222 0.688154 0.037905 0.04822 0.214044 0.053841 0.002145108 0.004204
MEAN 0.004395
1
estimate
270
Table Y-5: Error analysis for benzene-d6.
benzene-d6 84.1488 [g/mol]
purity 1
xD 0.9995
MH 78.1118 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 72.1488 [g/mol] 0.05 0.0001 0.0001 0.0001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
283.15 7.4069 0.960851 1.2029E-03 0.001156 0.710456 0.046082 0.047959 0.137022 0.068933 0.000140124 0.000275
288.15 7.3967 0.955072 1.2066E-03 0.001152 0.709827 0.045827 0.047983 0.136198 0.068519 0.000139724 0.000274
293.15 7.3865 0.949327 1.2123E-03 0.001151 0.709254 0.045577 0.04801 0.135378 0.068107 0.000139387 0.000273
298.15 7.3763 0.943563 1.2155E-03 0.001147 0.708625 0.045323 0.048034 0.134556 0.067693 0.000139001 0.000272
303.15 7.3662 0.937857 1.2460E-03 0.001169 0.708035 0.045073 0.04806 0.133743 0.067284 0.000139168 0.000273
308.15 7.3559 0.931877 1.2502E-03 0.001165 0.70732 0.044803 0.048078 0.13289 0.066855 0.000138788 0.000272
313.15 7.345735 0.926207 1.2420E-03 0.00115 0.706777 0.044558 0.048108 0.132081 0.066448 0.000138228 0.000271
318.15 7.3355 0.920374 1.2683E-03 0.001167 0.70616 0.0443 0.048133 0.131249 0.066029 0.00013832 0.000271
323.15 7.3253 0.914534 1.2946E-03 0.001184 0.705516 0.04404 0.048156 0.130417 0.06561 0.000138414 0.000271
328.15 7.315 0.908534 1.3146E-03 0.001194 0.704814 0.043769 0.048176 0.129561 0.06518 0.000138369 0.000271
333.15 7.3047 0.90259 1.3247E-03 0.001196 0.704148 0.043503 0.048198 0.128713 0.064754 0.000138145 0.000271
338.15 7.2944 0.896577 1.3328E-03 0.001195 0.703468 0.043233 0.04822 0.127856 0.064322 0.000137879 0.00027
343.15 7.2842 0.890641 1.3464E-03 0.001199 0.702844 0.042968 0.048244 0.127009 0.063896 0.000137738 0.00027
348.15 7.2739 0.884585 1.3690E-03 0.001211 0.702212 0.042698 0.048269 0.126146 0.063462 0.000137765 0.00027
MEAN 0.000272
271
Table Y-6: Error analysis for chloroform-d.
chloroform-d 120.3840 [g/mol]
purity 1
xD 0.9982
MH 119.378 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 108.384 [g/mol] 0.05 0.0001 0.0001 0.0001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
278.15 8.1702 1.531084 1.2350E-03 0.001891 0.783249 0.073303 0.047933 0.15262 0.012795 0.000195606 0.000383
283.15 8.1559 1.521629 1.2371E-03 0.001882 0.782299 0.072892 0.047959 0.151678 0.012716 0.000194798 0.000382
288.15 8.1416 1.512259 1.2481E-03 0.001887 0.781312 0.072475 0.047983 0.150744 0.012637 0.000194331 0.000381
293.15 8.1272 1.502753 1.2638E-03 0.001899 0.780376 0.072064 0.04801 0.149796 0.012558 0.000194063 0.00038
298.15 8.1128 1.493265 1.2904E-03 0.001927 0.779379 0.071644 0.048034 0.14885 0.012479 0.000194199 0.000381
303.15 8.09829 1.483483 1.3165E-03 0.001953 0.778403 0.071223 0.04806 0.147875 0.012397 0.000194327 0.000381
308.15 8.08338 1.473734 1.3187E-03 0.001943 0.777273 0.070766 0.048078 0.146903 0.012315 0.000193527 0.000379
313.15 8.068755 1.464048 1.3226E-03 0.001936 0.776343 0.070352 0.048108 0.145938 0.012234 0.000192874 0.000378
318.15 8.0542 1.45437 1.3522E-03 0.001967 0.775346 0.069925 0.048133 0.144973 0.012154 0.000193162 0.000379
323.15 8.0394 1.444381 1.3820E-03 0.001996 0.774293 0.069484 0.048156 0.143977 0.01207 0.000193436 0.000379
328.15 8.0245 1.434408 1.3905E-03 0.001995 0.773175 0.069029 0.048176 0.142983 0.011987 0.000192905 0.000378
MEAN 0.00038
272
Table Y-7: Error analysis for cyclohexane-d12.
cyclohexane-d12 96.2334 [g/mol]
purity 0.99
xD 0.996
MH 84.1595 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 84.2334 [g/mol] 0.05 0.0001 0.01 0.001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
283.15 7.3253 0.903197 1.1905E-03 0.001075 0.702629 0.043316 0.047959 0.112626 0.113319 0.001139838 0.002234
288.15 7.3156 0.897821 1.2023E-03 0.001079 0.702045 0.04308 0.047983 0.111955 0.112645 0.001133122 0.002221
293.15 7.3058 0.892403 1.2158E-03 0.001085 0.701505 0.042844 0.04801 0.11128 0.111965 0.00112636 0.002208
298.15 7.296 0.88697 1.2267E-03 0.001088 0.700911 0.042605 0.048034 0.110602 0.111284 0.001119575 0.002194
303.15 7.2862 0.881522 1.2502E-03 0.001102 0.700345 0.042366 0.04806 0.109923 0.1106 0.0011128 0.002181
308.15 7.2764 0.875949 1.2575E-03 0.001102 0.699676 0.042114 0.048078 0.109228 0.109901 0.001105832 0.002167
313.15 7.2665 0.870507 1.2626E-03 0.001099 0.699154 0.041878 0.048108 0.108549 0.109218 0.001099027 0.002154
318.15 7.2566 0.864958 1.2948E-03 0.00112 0.698564 0.041633 0.048133 0.107857 0.108522 0.001092147 0.002141
323.15 7.2466 0.859308 1.3192E-03 0.001134 0.697937 0.041381 0.048156 0.107153 0.107813 0.001085126 0.002127
328.15 7.23667 0.853621 1.3375E-03 0.001142 0.697266 0.041124 0.048176 0.106444 0.107099 0.001078045 0.002113
333.15 7.2266 0.84789 1.3542E-03 0.001148 0.69662 0.040867 0.048198 0.105729 0.10638 0.001070906 0.002099
338.15 7.2166 0.842139 1.3820E-03 0.001164 0.695965 0.040608 0.04822 0.105012 0.105659 0.001063768 0.002085
343.15 7.2064 0.836251 1.4081E-03 0.001178 0.695337 0.040344 0.048244 0.104278 0.10492 0.001056457 0.002071
MEAN 0.002153
273
Table Y-8: Error analysis for dichloromethane-d2.
dichloromethane-d2 86.945 [g/mol]
purity 1
xD 0.99965
MH 84.933 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 74.945 [g/mol] 0.05 0.0001 0.0001 0.00001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
278.15 7.98399 1.38682 1.3380E-03 0.001856 0.765398 0.066391 0.047933 0.191407 0.032092 0.000185035 0.000363
283.15 7.96968 1.377542 1.3456E-03 0.001854 0.764437 0.065985 0.047959 0.190126 0.031878 0.000184406 0.000361
288.15 7.9553 1.368283 1.3576E-03 0.001858 0.763434 0.065571 0.047983 0.188848 0.031664 0.000183934 0.000361
293.15 7.9409 1.358965 1.3818E-03 0.001878 0.762488 0.065164 0.04801 0.187562 0.031448 0.000183911 0.00036
298.15 7.9263 1.349505 1.4214E-03 0.001918 0.761462 0.064742 0.048034 0.186256 0.031229 0.000184411 0.000361
303.15 7.9116 1.339782 1.4223E-03 0.001906 0.760459 0.064319 0.04806 0.184914 0.031004 0.000183563 0.00036
308.15 7.897 1.330448 1.4254E-03 0.001896 0.759351 0.063881 0.048078 0.183626 0.030788 0.000182797 0.000358
313.15 7.8822 1.320817 1.4582E-03 0.001926 0.758394 0.063465 0.048108 0.182297 0.030565 0.000183092 0.000359
MEAN 0.00036
274
Table Y-9: Error analysis for dimethyl sulfoxide-d6.
dimethyl
84.170 [g/mol]
sulfoxide-d6
purity 0.9996
xD 0.9993
MH 78.133 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 72.17 [g/mol] 0.05 0.0001 0.0001 0.0001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
298.15 7.7078 1.183751 9.2150E-04 0.001091 0.740471 0.05686 0.048034 0.168766 0.084903 0.000154741 0.000303
303.15 7.6983 1.178296 9.4020E-04 0.001108 0.739956 0.056629 0.04806 0.167988 0.084512 0.000154771 0.000303
308.15 7.6888 1.172672 9.3010E-04 0.001091 0.739331 0.05638 0.048078 0.167186 0.084109 0.00015418 0.000302
313.15 7.6794 1.167389 9.1680E-04 0.00107 0.738881 0.056161 0.048108 0.166433 0.08373 0.000153598 0.000301
318.15 7.67 1.16197 9.3650E-04 0.001088 0.738361 0.055929 0.048133 0.16566 0.083341 0.000153682 0.000301
323.15 7.6606 1.156507 9.5010E-04 0.001099 0.73781 0.055693 0.048156 0.164882 0.082949 0.000153645 0.000301
328.15 7.65126 1.150982 9.5550E-04 0.0011 0.737213 0.05545 0.048176 0.164094 0.082553 0.000153435 0.000301
333.15 7.6419 1.145509 9.6140E-04 0.001101 0.736653 0.055212 0.048198 0.163314 0.08216 0.000153255 0.0003
338.15 7.63255 1.139969 9.6380E-04 0.001099 0.736079 0.054969 0.04822 0.162524 0.081763 0.000153006 0.0003
343.15 7.6233 1.134522 9.6110E-04 0.00109 0.735563 0.054734 0.048244 0.161747 0.081372 0.000152679 0.000299
348.15 7.6141 1.129064 9.7700E-04 0.001103 0.735055 0.054499 0.048269 0.160969 0.080981 0.000152737 0.000299
353.15 7.6049 1.123491 9.7850E-04 0.001099 0.734516 0.054256 0.048292 0.160175 0.080581 0.000152496 0.000299
358.15 7.5958 1.118071 9.6710E-04 0.001081 0.734052 0.054025 0.04832 0.159402 0.080192 0.000152028 0.000298
363.15 7.5867 1.112678 9.6790E-04 0.001077 0.733649 0.053799 0.048351 0.158633 0.079806 0.000151824 0.000298
368.15 7.5776 1.107301 9.7110E-04 0.001075 0.73324 0.053574 0.048382 0.157866 0.07942 0.000151673 0.000297
MEAN 0.0003
275
Table Y-10: Error analysis for ethanol-d6.
ethanol-d6 52.1054 [g/mol]
purity 0.99
xD 0.997
MH 46.0684 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 40.1054 [g/mol] 0.05 0.0001 0.01 0.001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
278.15 7.3371 0.910008 1.0490E-03 0.000955 0.703383 0.04362 0.047933 0.209577 0.105435 0.00210257 0.004121
283.15 7.3282 0.905235 1.0675E-03 0.000966 0.702907 0.043414 0.047959 0.208478 0.104882 0.002091581 0.004099
288.15 7.319195 0.900345 1.0823E-03 0.000974 0.70239 0.043201 0.047983 0.207352 0.104315 0.00208032 0.004077
293.15 7.3102 0.89549 1.0883E-03 0.000975 0.701928 0.042993 0.04801 0.206234 0.103753 0.002069133 0.004055
298.15 7.301175 0.890599 1.1042E-03 0.000983 0.701408 0.042779 0.048034 0.205107 0.103186 0.002057871 0.004033
303.15 7.2921 0.885656 1.1324E-03 0.001003 0.700913 0.042564 0.04806 0.203969 0.102613 0.002046506 0.004011
308.15 7.283 0.880569 1.1325E-03 0.000997 0.700311 0.042336 0.048078 0.202797 0.102024 0.002034777 0.003988
313.15 7.2739 0.875684 1.1283E-03 0.000988 0.699866 0.042127 0.048108 0.201672 0.101458 0.002023511 0.003966
318.15 7.2648 0.870689 1.1617E-03 0.001011 0.699354 0.041909 0.048133 0.200522 0.100879 0.002012034 0.003944
323.15 7.255565 0.865569 1.2036E-03 0.001042 0.6988 0.041682 0.048156 0.199343 0.100286 0.002000277 0.003921
328.15 7.2462 0.860271 1.2284E-03 0.001057 0.698185 0.041444 0.048176 0.198122 0.099672 0.001988092 0.003897
333.15 7.2368 0.855001 1.2409E-03 0.001061 0.697603 0.04121 0.048198 0.196909 0.099061 0.001975958 0.003873
338.15 7.2274 0.849661 1.2730E-03 0.001082 0.697007 0.04097 0.04822 0.195679 0.098443 0.001963687 0.003849
343.15 7.2178 0.844185 1.3033E-03 0.0011 0.696437 0.040727 0.048244 0.194418 0.097808 0.0019511 0.003824
348.15 7.2082 0.838658 1.3401E-03 0.001124 0.69587 0.040481 0.048269 0.193145 0.097168 0.001938407 0.003799
353.15 7.19848 0.832946 1.3716E-03 0.001142 0.695262 0.040225 0.048292 0.191829 0.096506 0.001925279 0.003774
MEAN 0.003952
276
Table Y-11: Error analysis for methanol-d4.
methanol-d4 36.0665 [g/mol]
purity 0.999
xD 0.99981
MH 32.0419 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 24.0665 [g/mol] 0.05 0.0001 0.001 0.00001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
278.15 7.3354 0.908813 1.1740E-03 0.001067 0.70322 0.043562 0.047933 0.302379 0.101413 0.000330765 0.000648
283.15 7.3257 0.903478 1.1778E-03 0.001064 0.702667 0.04333 0.047959 0.300604 0.100818 0.000328999 0.000645
288.15 7.3161 0.898172 1.1862E-03 0.001065 0.702093 0.043097 0.047983 0.298839 0.100225 0.000327279 0.000641
293.15 7.3064 0.892824 1.1996E-03 0.001071 0.701563 0.042865 0.04801 0.297059 0.099629 0.00032559 0.000638
298.15 7.2967 0.887461 1.2056E-03 0.00107 0.700978 0.042628 0.048034 0.295275 0.09903 0.000323843 0.000635
303.15 7.28706 0.882124 1.2257E-03 0.001081 0.700428 0.042395 0.04806 0.293499 0.098435 0.000322215 0.000632
308.15 7.2774 0.876649 1.2295E-03 0.001078 0.699772 0.042148 0.048078 0.291678 0.097824 0.000320418 0.000628
313.15 7.2677 0.871346 1.2375E-03 0.001078 0.699269 0.041919 0.048108 0.289913 0.097232 0.000318724 0.000625
318.15 7.2579 0.865866 1.2694E-03 0.001099 0.698689 0.041677 0.048133 0.28809 0.096621 0.000317147 0.000622
323.15 7.2481 0.860355 1.2992E-03 0.001118 0.698081 0.041431 0.048156 0.286256 0.096006 0.000315548 0.000618
328.15 7.2382 0.854688 1.3198E-03 0.001128 0.697414 0.041175 0.048176 0.284371 0.095373 0.000313828 0.000615
333.15 7.2283 0.849075 1.3386E-03 0.001137 0.696784 0.040924 0.048198 0.282503 0.094747 0.000312119 0.000612
338.15 7.2183 0.843322 1.3642E-03 0.00115 0.696129 0.040665 0.04822 0.280589 0.094105 0.000310417 0.000608
MEAN 0.000628
277
Table Y-12: Error analysis for pyridine-d5 (purity of 99% was assumed).
pyridine-d5 84.1307 [g/mol]
purity 0.991
xD 0.99
MH 79.0999 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 72.1307 [g/mol] 0.05 0.0001 0.01 0.01 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
278.15 7.5522 1.063524 1.0118E-03 0.001076 0.724004 0.050978 0.047933 0.151696 0.063596 0.001651213 0.003236
283.15 7.5426 1.058143 1.0148E-03 0.001074 0.723472 0.050748 0.047959 0.150928 0.063274 0.001642891 0.00322
288.15 7.5331 1.052785 1.0139E-03 0.001067 0.722917 0.050515 0.047983 0.150164 0.062954 0.0016346 0.003204
293.15 7.5236 1.047468 1.0178E-03 0.001066 0.722418 0.050289 0.04801 0.149406 0.062636 0.001626381 0.003188
298.15 7.5141 1.042124 1.0265E-03 0.00107 0.721863 0.050057 0.048034 0.148644 0.062316 0.00161813 0.003172
303.15 7.5046 1.03677 1.0401E-03 0.001078 0.721338 0.049827 0.04806 0.14788 0.061996 0.001609874 0.003155
308.15 7.4952 1.03134 1.0372E-03 0.00107 0.720715 0.049585 0.048078 0.147105 0.061671 0.00160147 0.003139
313.15 7.4857 1.026073 1.0385E-03 0.001066 0.720244 0.049362 0.048108 0.146354 0.061357 0.001593331 0.003123
318.15 7.4762 1.020683 1.0597E-03 0.001082 0.719704 0.049129 0.048133 0.145585 0.061034 0.001585035 0.003107
323.15 7.4667 1.015257 1.0715E-03 0.001088 0.719135 0.048891 0.048156 0.144811 0.06071 0.001576666 0.00309
328.15 7.4573 1.009804 1.0817E-03 0.001092 0.718525 0.048648 0.048176 0.144034 0.060384 0.001568256 0.003074
333.15 7.44779 1.004334 1.0952E-03 0.0011 0.717942 0.048407 0.048198 0.143253 0.060057 0.001559824 0.003057
338.15 7.4383 0.998805 1.1066E-03 0.001105 0.717346 0.048162 0.04822 0.142465 0.059726 0.001551299 0.003041
343.15 7.4288 0.993281 1.1100E-03 0.001103 0.716796 0.04792 0.048244 0.141677 0.059396 0.001542769 0.003024
348.15 7.4194 0.987779 1.1250E-03 0.001111 0.716259 0.047679 0.048269 0.140892 0.059067 0.001534295 0.003007
353.15 7.41 0.982168 1.1356E-03 0.001115 0.715691 0.047431 0.048292 0.140092 0.058731 0.001525645 0.00299
358.15 7.4006 0.976626 1.1263E-03 0.0011 0.715188 0.04719 0.04832 0.139301 0.0584 0.001517066 0.002973
363.15 7.3913 0.971169 1.1498E-03 0.001117 0.714753 0.046957 0.048351 0.138523 0.058073 0.001508681 0.002957
368.15 7.38162 0.965459 1.1828E-03 0.001142 0.714276 0.046711 0.048382 0.137708 0.057732 0.001499921 0.00294
MEAN 0.003089
1
estimate
278
Table Y-13: Error analysis for tetrahydrofuran-d8 (purity of 99% was assumed).
tetrahydrofuran-d8 80.155 [g/mol]
purity 0.991
xD 0.9982
MH 72.1057 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 68.155 [g/mol] 0.05 0.0001 0.01 0.0001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
278.15 7.4778 1.011463 1.1864E-03 0.0012 0.716871 0.048409 0.047933 0.151426 0.101573 0.001521068 0.002981
283.15 7.4673 1.005425 1.1981E-03 0.001205 0.716249 0.048148 0.047959 0.150522 0.100966 0.001512044 0.002964
288.15 7.4567 0.999353 1.2133E-03 0.001213 0.715585 0.047879 0.047983 0.149613 0.100357 0.001502978 0.002946
293.15 7.4461 0.993259 1.2263E-03 0.001218 0.714977 0.047615 0.04801 0.148701 0.099745 0.001493877 0.002928
298.15 7.435465 0.98713 1.2491E-03 0.001233 0.714309 0.047345 0.048034 0.147783 0.099129 0.001484744 0.00291
303.15 7.4248 0.98084 1.2821E-03 0.001258 0.713668 0.047075 0.04806 0.146841 0.098498 0.001475393 0.002892
308.15 7.4141 0.97475 1.2842E-03 0.001252 0.712917 0.04679 0.048078 0.14593 0.097886 0.001466276 0.002874
313.15 7.4033 0.968473 1.2860E-03 0.001245 0.712316 0.046523 0.048108 0.14499 0.097256 0.001456882 0.002855
318.15 7.392555 0.962201 1.3173E-03 0.001268 0.711652 0.046247 0.048133 0.144051 0.096626 0.001447558 0.002837
323.15 7.3817 0.955759 1.3391E-03 0.00128 0.710948 0.045964 0.048156 0.143087 0.095979 0.001437961 0.002818
328.15 7.37083 0.949366 1.3591E-03 0.00129 0.710193 0.045672 0.048176 0.14213 0.095337 0.001428432 0.0028
333.15 7.3599 0.942861 1.3772E-03 0.001299 0.709469 0.045384 0.048198 0.141156 0.094684 0.001418733 0.002781
MEAN 0.002882
1
estimate
279
Table Y-14: Error analysis for toluene-d8.
toluene-d8 100.1877 [g/mol]
purity 1
xD 0.9980
MH 92.1384 [g/mol] u(T) u(τ) u(x) u(xD) k
mass (im) 88.1877 [g/mol] 0.05 0.0001 0.0001 0.0001 1.96
Temperature Period Density ITE ∂ρ/∂T ∂ρ/∂τ ∂ρ/∂A ∂ρ/∂B ∂ρ/∂x ∂ρ/∂xD u(ρ) U(ρ)
[K] [τ] 3
[g/cm ] -1
[K ] [sample] [combined]
278.15 7.40808 0.960176 1.0574E-03 0.001015 0.710187 0.046024 0.047933 0.115005 0.077143 0.000137125 0.000269
283.15 7.3988 0.955099 1.0618E-03 0.001014 0.709679 0.045806 0.047959 0.114397 0.076735 0.000136808 0.000268
288.15 7.3896 0.950035 1.0628E-03 0.00101 0.709146 0.045585 0.047983 0.113791 0.076328 0.00013644 0.000267
293.15 7.3804 0.945002 1.0618E-03 0.001003 0.708668 0.04537 0.04801 0.113188 0.075924 0.000136062 0.000267
298.15 7.37127 0.940000 1.0838E-03 0.001019 0.708142 0.045152 0.048034 0.112589 0.075522 0.00013609 0.000267
303.15 7.3619 0.934814 1.1110E-03 0.001039 0.707622 0.044927 0.04806 0.111967 0.075105 0.000136212 0.000267
308.15 7.3527 0.929614 1.1034E-03 0.001026 0.707013 0.044694 0.048078 0.111345 0.074687 0.000135711 0.000266
313.15 7.3434 0.924557 1.1018E-03 0.001019 0.706553 0.044479 0.048108 0.110739 0.074281 0.000135362 0.000265
318.15 7.33416 0.919428 1.1208E-03 0.00103 0.706031 0.044255 0.048133 0.110125 0.073869 0.000135365 0.000265
323.15 7.3249 0.914252 1.1530E-03 0.001054 0.705478 0.044027 0.048156 0.109505 0.073453 0.000135601 0.000266
328.15 7.3155 0.908886 1.1602E-03 0.001054 0.704862 0.043786 0.048176 0.108862 0.073022 0.000135378 0.000265
333.15 7.306285 0.903706 1.1596E-03 0.001048 0.704301 0.043557 0.048198 0.108242 0.072606 0.000135046 0.000265
338.15 7.297 0.898406 1.1726E-03 0.001053 0.703719 0.043321 0.04822 0.107607 0.07218 0.000134947 0.000264
343.15 7.2878 0.893171 1.1809E-03 0.001055 0.703191 0.043091 0.048244 0.10698 0.071759 0.000134784 0.000264
348.15 7.27856 0.887859 1.2181E-03 0.001082 0.702662 0.042856 0.048269 0.106343 0.071333 0.000135127 0.000265
353.15 7.2692 0.882356 1.2270E-03 0.001083 0.702092 0.042611 0.048292 0.105684 0.07089 0.000134954 0.000265
358.15 7.26001 0.877032 1.2238E-03 0.001073 0.701602 0.042378 0.04832 0.105047 0.070463 0.000134597 0.000264
363.15 7.25069 0.871623 1.2411E-03 0.001082 0.701156 0.042144 0.048351 0.104399 0.070028 0.000134604 0.000264
MEAN 0.000266
280