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University

 

 

 

This is to certify that the

dissertation entitled

The Solubility of Nonpolar Gases in

Organic Liquids and Water
presented by

Richard Paul Kennan

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in PhYSiCS

 

 

 

2 g Major professor

Date March 9, 1990

 

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

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TO AVOID FINES return on at baton one due.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

#6

 

 

 

 

 

 

 

 

MSU I: An Alfirmotivo ActloNEquol Opportunity Institmion

The Solubility of Nonpolar Gases in

Organic Liquids and Water.

BY

Richard Paul Kennan

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR 0F PHILOSPHY

Department of Physics and Astronomy

1990

In

@059

ABSTRACT
SOLUBILITY OF NONPOLAR GASES IN ORGANIC LIQUIDS AND WATER.
By

Richard Paul Kennan

We have measured the Ostwald solubility L(T), as a
function of temperature in the approximate range
10.0-50.0°C, for 133Xe gas in 45 organic solvents, viz., 16
alkanes, 13 alkanals, 6 carboxylic acids, 4 alkanals, 3
cyclaalkanes, and 3 perfluaroalkanes. From our data for
each salute-solvent system we determine the following

. . . . . *
thermodynamic functions of solution: chemical potential Ana:

-* -¥ * -*

-RTOnL, enthalpy Ah2 , and entropy Asa, where Auz= Ahz-
-¥

TAsz, all based on the number density scale. The average

observed entropy of salvation of Xe is A§:= -4.1 :t 0.5
cal/moi K, remarkably independent of solvent. The results
are analyzed with scaled-particle theory from which we
obtained the effective hard core diameters a‘, and the
cavity energies gcnand enthalpies he" for all the solvents
at 25°C. Thermodynamic perturbation theory is used to find
the total enthalpy of salvation for the Xe-alkane systems.
We generalize the analysis to evaluate the salvation
enthalpy for all of the noble gases in the alkanes. We
discuss the role of configuratianal entropy, as well as
molecular dynamics approaches to calculation of free
energies of salvation. Finally the results are examined

empirically and values are given for the contribution to

chemical potential, enthalpy, and entropy of salvation, of
the six functional groups: CH2 (linear molecules), CH3, OH,
COOH, CHO, and CH2(cyclamalecules).

We have also measured the pressure dependence of the
Ostwald (L) and male-fraction (x2) solubilities for the
nonpolar gases N2, Ar, Kr, and Xe in water at 25.0°C. The
pressure ranges studied for each gas were approximately:
N2(44-116 atm), Ar(22-101 atm), Kr(33-81 atm), and Xe(5-48
atm). For N2, Ar, and Kr we see clear deviations from
Henry’s Law, f2=knx2. The data are analyzed in terms of the
Kirkwaod-Buff solution theory. The role of solvent -induced
(hydrophobic) interactions shall be discussed” For the
Kr-water system we shall compare our experimental results to
recent computer simulation predictions. We also use
statistical mechanics arguments to introduce a new
solubility parameter which is appropriate to high pressure
solubility measurements. Extensions of our analysis to

other gas-liquid data is discussed.

To Mom and Dad

iv

ACKNOWLEDGEMENTS

The experiments described in this thesis were suggested
by Professor Gerald L. Pollack. I cannot overemphasize his
contribution to this work through his guidance,
encouragement, and friendship. I would also like to thank
Professors Jules Kovacs, Aaron Galonsky, Dan Stump, and
Jerry Cowen for serving on my guidance committee.

I would like to thank Jeff Himm and Gary Holm for their
assistance in the laboratory in the early and late stages of
this research project, and Charles Scripter for all of his
help with computers. Many of the figures in this thesis were
produced with the generous aid of John and Linda Kennan.
Thanks are also due to Peggy Andrews for her general
assistance.

It would have been impossible to complete this work
without the help of my friends both in the department and
outside it. For moral support above and beyond the call of
duty, I wish to express my deepest gratitude to Carrie
Huang, Joel Gales, and Diandra Leslie-Pelecky. On the home
front I would like to thank Richard and Nancy Olsen-Harbich
for their kind friendship over the years. I also would like

to thank the Physics Department's brilliant dramatic

ensemble, the N.R.F.T.C. Players.

Most of all I want to thank my family, especially my
parents for whom I have seldom done my filial duty.
The research reported here has been supported under the

Navel Medical Research and Development Command.

vi

Table of Contents

Chapter Page
1. General Introduction and Overview ................ 1
2. Introduction: Solubility of Xenon in 45 Organic

Solvents ........................................ 3

3. Theoretical Background .......... . ................. 6
3.1 Phase Equilibria ....................... 6

3.2 Two Component Systems ...... . ........... 8

3.3 Ostwald Solubility ..................... 9

3.4 Salvation Thermodynamics .............. 10

3.5 Physical Interpretation...... ...... ...13

3.6 Other Concentration Scales ............ 17

4. Experimental.......... .......................... .19
4.1 Outline of Method ..................... 19

4.2 Radioactive Tracers ....... .. ......... .22

4.3 Solution Components ..... . ............. 23

4.4 The Experimental Apparatus............27

4.5 Volume Determination ................ ..28

4.6 Data Acquisition............ ..... .....33

4.7 Experimental Procedure ................ 36

5. Resu1ts..00000000000000.0.000...0.0.0.00000000000'040
5.1 Thermodynamic Analysis. ............... 40

5.2 Entropy of Salvation .................. 49

5.3 Enthalpy of Salvation.... ..... . ..... ..54

5.4 Temperature Dependence of L and Au....56

5.5 Total Entropy and Enthalpy............57

5.6 Male Fraction Scale ................... 63

6. Theoretical Analysis. ................ ...........65
6.1 The Excess Chemical Potential ..... ....65

6.2 Distribution Functions................70

6.3 The Van der Waals Picture of Liguids..80

*

6.4 Determination of Aug..................87

6.5 The Scaled Particle Theory ............ 93

6.6 Application to Real Liquids..........101

6.7 Evaluation of the Cavity Terms.......106

6.8 The Interaction Term.................134

6.9 Comparison with Experimental Data....146

7. Empirical Analysis and Predictions.. ..... .......159
8. Conclusions.. ................................... 171

vii

9. Introduction: Solubility of Inert Cases in

Water at Elevated Pressure ....... . ........... .178
10. Experimental ................................... 181
11. Results ........................................ 188
12. Pressure and Concentration Dependence .......... 194
13. Data Analysis .................................. 201
14. Number Density Scale ........................... 211
15. Conclusions .................................... 219
Appendix
A. Derivation of Equation (6.78) ................... 222
8. Computer Program to Evaluate ULM ........ . ....... 225
List of References .............................. 229

viii

List of Figures

Figure 1. Schematic of two component solution ............. 11

Figure 2. Three phase system of xenon gas in
equilibrium with octane and water at 20 OC ................. 11

Figure 3. Schematic description of the salvation
process. First the center of mass of the solute is
placed at fixed position R0, and then the particle

is released. The contributions to the chemical

potential are indicated next to each arrow ................ 16
Figure 4. Idealized solubility apparatus.... .......... ....21
Figure 5. Experimental apparatus.......... ............ ....31
Figure 6. Schematic of detection electronics ...... . ....... 32

Figure 7. Normalized counting rate vs time for a

typical experiment (Figure 2 of reference 18). The

letter A indicates the time the valve was opened,

and the letter B indicates the time a run might

end, about 8 hours after equilibrium has been reached ..... 39

Figure 8. L(T) for Xe in six representative organic

solvents ............................................ - ...... 48
t

Figure 9. Excess chemical potential, A“; (T) for

six representative organic solvents (Figure 1 of

reference 17).. ................. .. ......... . .............. 50

Figure 10. Radial distribution function, g(r),
for a typical gas, liquid, and solid..... ................. 75

Figure 11. Lennard-Jones potential, U(r). Lower
diagram shows repulisive portion of U(r) as
determined in WCA theory ................ . ................. 82

Figure 12. Radial distribution functions for
(i)Lennard-Jones fluid g(r), (ii) for repulsive
portion of L—J potential g°(r), (iii) for hard

spheres gh.(r) ....... .... ................................. 83

ix

Figure 13. Definition of a cavity in the SPT .............. 94

Figure 14. gcav(R12)/kT Xi cavity radius R12 ............. 100

Figure 15 Effective hard sphere diameter for
45 organic solvents 1; number of carbon atoms ............ 111

Figure 16. (a) gcav for Xe in 45 organic solvents

at 25.0°C. (b) corresponding excess chemical

1:
potential, Au: = —RT£aL .................................. 115
Figure 17. Molecular diameter ii polarizability
for several inert gas solutes. Solid line shows
extrapolation for noble gases ............................ 116

Figure 18. Excess chemical potential 11 solute
polarizability for noble gases in Tetradecane,

Decane, and Hexane. Filled in points along ordinate

are the SPT predicted values for a hard sphere

solute of 2.58A .......................................... 117

Figure 19. Excess chemical potential is solute
polarizability for noble gases in polar solvents

methanol, and ethanol. Filled in points along

ordinate are the SPT predicted values for a hard

sphere solute of 2.58A .................................. 118
Figure 20. (a) hbav for Xe in 45 organic solvents
11 number of carbon atoms.(b) corresponding

*
experimental excess enthalpy, Ah2 ....................... 123

Figure 21. (a) scav for Xe in 45 organic solvents
1; number of carbon atoms. (b) corresponding

*
experimental excess entropy, Asz ......................... 124

Figure 22. Heat of vaporization for 45 organic
solvents at 25° C ys_number of carbon atoms ............... 125

Figure 23. Number Density for 45 organic solvents
at 25° C _1 number of carbon atoms. ....................... 126

Figure 24. Thegmal Expansivity for 45 organic
solvents at 25 C is number of carbon atoms ............... 127

Figure 25. Cavity free energy, gc , g§,T for Xe

0V
in 6 representative organic solvents ..................... 133

Figure 26. Lennard-Jones potential with hard
sphere cutoff ............................................ 138

Figure 27. Verlet-Weiss and Percus Yevick pair
distribution functions for hard sphere fluid.
Packing fraction = 0.5............... .................... 139

Figure 28. Calculated and experimental excess
enthalpies of salvation for Xe in alkanes ................ 147

Figure 29. Calculated and experimental excess

enthalpies of salvation for He, Ne, Ar, Kr, and

Xe in (a) octane, (b) tetradecane. ........... . ........... 156
t

Figure 30. Excess chemical potential Apz=-RT¢nL

21 number of CH2 groups at 25°C ........... . ............. 160

Figure 31. Experimental data for Xe in 37 solvents.
Excess chemical potential divided by solvent number
density is number of CH2 groups at 25°C.. ... ............. 163

Figure 32. Experimental data for Xe in 37 solvents.
Excess enthalpy divided by solvent number
density y§_number of CH2 groups ......................... 164

Figure 33. Experimental data for Xe in 37 solvents.
*
TAs2 divided by solvent number density is
number of CH2 groups ............ . ...... . ................. 165

Figure 34. giM/p‘ y; number of CH2 groups for

Xe in 37 solvents......... ....... .... .................... 170
Figure 35. Schematic of experimental setup

(a) high pressure equilibration system (b)

analysis system..................... ...... ..............182

Figure 36. Solute fugacity y§,mole- -fraction
solubility at 25°C.... ..... . ......... . .................. 192

Figure 37. Henry' 3 constant y§,mole- -fraction
solubility at 25° C............................ .......... .193

xi

Figure 38. &n(fz/xz) - Pvzo/RT y§_mole—fraction
solubility at 25°C ....................................... 203

Figure 39. 6n(f2/x2) — Pvzo/RT y§_mole—fraction
solubility for He and N2 at 25°C for various

values of Va0 ............................................ 210

xii

List of Tables

Table I. Solubility data for experiments with

“BXe in 45 organic solvents. First row gives
Ostwald solubility and the second row is the

*
excess chemical potential, Apz= -RT¢nL (cal/mol) ........

Table II. Experimental data for excess enthalpy,
* *
Aha, and entropy A52: for 1“Xe in 45 organic

solvents .................................... . ...........

Table III Enthalpy and entropy of solution. The
second and third columns are the total entropy
and enthalpy of solution. The fourth and fifth
columns are the entropy and enthalpy evaluated

on the mole fraction scale ..............................

Table IV. Heat of vaporization, number density

and thermal expansivity for 45 organic solvents .........

Table V. SPT calculations for 45 organic solvents.
The second column is the effective hard sphere
radius. The third column is the packing fraction.

The fourth column is cavity free energy for 1E'Xe ..... ...

Table VI. Cavity enthalpy and entropy ...................

Table VII. Parameters to evaluate interaction
energy. Second column is hard sphere cutoff.
Third column is the energy integral using

the Verlet-Weiss distribution function. Fourth
column is the resulting Lennard—Jones energy

parameter for the pure organic solvent ...... .............

Table VIII. Predicted enthalpy of salvation

for “Xe in alkanes. ......... ..

Table IX. Excess chemical potential, enthalpy,
and entropy for He, Ne, Ar and Kr in the alkanes,
including hexane, octane, nonane, dodecane, and

tetradecane ........................ ............ .........

xiii

..42

..51

..60

.108

.112

.128

Table X. g“ , h , s , and um. for He,

V GOV CGV
Ne, Ar, and Kr in the alkanes,inc1uding pentane
through hexadecane ....................................... 152

Table XI. Empirical energy parameters for
Xe solubility in 37 organic solvents ..................... 166

Table XII Predicted y: expޣimental values
of Ostwald solubility for Xe in selected
organic solvents ......................................... 166

Table XIII. Experimental results for solubility
of N2, Ar, Kr, and Xe in water at 25.0°C. The

second column is the solute partial pressure. The

third column is the solute fugacity. The fourth

and fifth columns are the mole-fraction and Ostwald
solubilities respectively. The sixth column is

the new solubility parameter r. The seventh column

is the excess chemical potential of the solute in

the liquid. The eighth column is the experimental

average value of partial molar volume....... ............. 191

Table XIV. Results of Kirkwood-Buff analysis. The
third column gives the partial molar volume of the
solute used in Eq.(12.l4). The fourth and fifth
columns give the results for the KB integral Ga:

and the log of ghe infinite dilution limit of Henry's
constant, cha )....... ........................ . ........ 204

xiv

1 .Introduction

The phase equilbrium problem is ubiquitous to modern (or
any) life. Whether it is the delivery of oxygen from the
lung to the blood stream, or an industrial extraction
process to remove unwanted hydrocarbons from natural gas, or
the removal of a nasty stain from your best sweater; we are
. constantly confronted with the problem of how a substance is
transferred from one medium to another. In general, when two
phases are brought into contact there is an exchange of
matter until the concentration in each each phase attains a
constant value. To understand the physics that determines
this equilibrium state is a difficult, but important task.

The goals of the research outlined in this thesis are to
understand the solubility of gases in liquids and to predict
solubilities in general gas-liquid systems. In pursuit of
this goal we have measured the solubility of simple gases in
various organic solvents and water. The resulting body of
data has been analyzed using current molecular theories
based on thermodynamics and statistical mechanics. By
studying gas solubility one may be able to build a
theoretical and empirical framework that can serve as a
basis for more complicated phase equilibrium systems.

The thesis is divided into two parts. The first section

1

details experiments involving solubility measurements of the

133Xe in several homologous series of organic

radioisotope
solvents. The solvents are chosen to range from simple

nonpolar liquids to more complicated hydrogen bonding

liquids. The second section covers experiments on the
pressure dependence of gas solubility in water. The solute
gases are: N2 , Ar, Kr, and Xe. These gases are meant to

represent a series of prototypical nonreactive solutes. The
experimental results are applicable to problems of medical,
industrial, and. environmental interest. Specific examples
will be sited in the introduction to each section.

Gas solubility can also serve as a 'probe in
understanding intermolecular interactions. Due to the
advances in computational techniques it has become possible

1 ’2 The resulting

to accurately model pure liquids.
thermodynamic properties obtained are sensitive to the
intermolecular potentials chosen. Optimized potential
functions are determined by varying parameters until the
best results are obtained.3 By extending these ’computer
experiments’ to dilute liquid mixtures it may be possible to
understand the various factors which influence the
potentials and place them on a more sound physical basis.

Accurate experimental data (n1 well chosen solute-solvent

systems would be useful towards this end.

2.1ntroduction: Solubility of Xenon in 45 Organic Solvents.

This section of my thesis describes experiments which
measure the Ostwald solubility (L) as a function of
temperature of’ 133Xe in 45 organic solvents including;
alkanes, alkanols, cycloalhanes, alkanals, carboxylic acids,
and perfluoroalkanes.

A great deal of progress has been made in understanding
the physics of simple liquid-gas systems such as the
solubility of an inert gas in its own liquid, or the
solubility of one inert gas in the liquid phase of another

4 ’ 5 These experiments

inert gas, e.g., Aer, KroXe, etc.
are prototypical in the sense that they are the simplest
gas-liquid systems that one can study.

A logical next step is to look at simple solutes in more
complicated solvents '(the case of simple solvents with
complex solutes isn’t experimentally accessible: Just try
dissolving hexane in liquid nitrogen). To this end we have
chosen the inert gas xenon in the several homologous series
mentioned above.

Some of the reasons why Xe was chosen as a solute are:

All the inert gas elements are monatomic and do not

«basically interact with the solvents under conditions of
' 3

these experiments. Much is known about interactions and
properties of these elements. Xenon has a commercially

33 . .
1 Xe, whose concentrations in the

available radioisotope,
gas phase can be easily measured.

Liquids are often considered in two groups: water6 (and
aqueous solutions) and other liquids (mainly organic). We
have selected 45 solvents so as to bridge these groups.
Most of our solvents are nonpolar molecules, some are medium
chain length molecules with a polar head group, and CHBOH
and HCOOH are small polar' molecules for' which hydrogen
bonding is important. The obvious motivation for looking at
homologous series is that one can try to spot trends that
can be generalized to other systems.

Practical aspects of this work stem from biological and
industrial applications» IXenon. has several applications
which are dependent on its solubility and diffusion.
Because Xe is highly soluble in fats and relatively

1'33Xe is widely

insoluble in aqueous solutions, the isotope
used in nuclear’ medicine to study cerebral blood flow,
pulmonary function etc. A further application of Xe is as
an inhalational anesthetic,7 a property associated with Xe
solubility in lipids of cell membranes.8 Since the
mechanism of general anesthesia is not understood some
workers in the field believe that Xe is the prototype
anesthetic to study.9 Also, solubility properties of Xe

under pressure are useful for studying decompression

sickness and inert gas narcosis, two problems of deep sea

diving.10 Finally, there are environmental and safety

1”Xe and other

11,12

questions associated with the emission of
radioactive inert gases from nuclear reactors.

There is presently a great deal of interest in
developing synthetic blood substitutes (now euphemistically
referred to as oxygen carriers in the industry).
Perfluorocarbons and related compounds form the basis of
many' new blood substitute candidates because they carry
oxygen efficiently and do not induce an immune system
response.13 Thus understanding gas solubility in simple
perfluorocarbons and their analogous hydrocarbons would be
quite useful.

While a first principles understanding is beyond the
scope of this work, we hope to develop empirical and
analytic techniques that will allow the prediction of Xe
solubility from knowledge of bulk properties of the solute
and solvent. We aim ultimately to generalize, at least
qualitatively, the solubility parameters obtained for Xe to

solubility of other inert gases.

3 .Theoretical Background

3.1 Phase Equilibria
Chemical potential (u) is a fundamental quantity in the
determination of phase equilibria in multicomponent

systems The most useful definitions for our purposes are:14

a G ' a A
u‘ 3 —— I . 3 (301)
a ntlnr,n' a'ni'ny,n'

where G is the Gibb’s free energy of the system, A is the

 

Helmholtz free energy. n‘ is the number of molecules of type
i,‘T is the absolute temperature, P is the pressure, V is
the volume and N’ represents all other molecules in the

i‘h type.

system with the exception of the
Suppose we have a two phase system, for example a liquid
in contact with its own vapor, the condition for

equilibrium at constant T and P is:
”('(TJ’) = u9(T.P) - (3.2)

We shall use the subscript L to denote the liquid phase and
q to denote the gas phase. For low pressures we assume the
gas phase is ideal, ignoring internal degrees of freedom we

have:

p9 = -kT m (pg A3) = -kT On (p9 A3 / kT) , (3.3)
where p9 is the number density of the gas (molecules/unit
volume), It is Boltzmann’s constant and A is called the
thermal wavelength for a particle of mass m; i.e.
A=h/(2nka)1/z-

The chemical potential for the liquid can likewise be

evaluated:
3 *

where pt denotes the number density of the liquid and Ap*is
called the excess chemical potential. It is the
contribution to the free energy from intermolecular
interactions in the liquid. A more detailed derivation of
the above equations is reserved until chapter 6, but for now
we shall content ourselves with these results. Equating.
(3.3) and (3.4) we find:

*
= -kT s 305
Au On (p, / pg) ( )

Thus by knowing the vapor pressure and density of the liquid
we can determine its excess chemical potential. Typically,
far from the triple point,a liquid is about 1000 times as
dense as its vapor. This would lead to an excess chemical

potential of about ~17 kJ/mole at room temperature. Although

8

this is a reflection of the strong binding energy in the
liquid we must keep in mind that there are also entropic
contributions to the chemical potential. These can play a

large role as we shall see later.

3.2 Two Component Systems

The relations ¢derived in the previous section can
easily be generalized to two component gas-liquid systems.
We now imagine a liquid solvent in equilibrium with a
gaseous solute (Fig. 1). The condition for equilibrium is
that the chemical potential of the dissolved solute equals

the chemical potential of the solute in the gas:
u; = H? . (3.6)

Throughout this thesis the subscript 1 and 2will refer to
the solvent and solute respectively, and superscripts 9 and
C will refer ,respectively, to the gas and liquid, according
to the usual conventions.

The statistical mechanics of solute -solvent mixtures

15,16

starts with a standard partition function from which

one may obtain by standard techniques the following chemical
potential for a single solute molecule in the liquid

solvent14’l7:

u; = -kT a. < exp(-B° / kT ) > + kT on p; A3 .(3.7)

9

The key quantity in Eq.(3.7), Bo, is the binding energy of a
single solute molecule to a fixed configuration of the
solute-solvent system. The ensemble average of the
exponential in Eq.(3.7) is taken over all such
configurations. This term is also referred to as the excess
chemical potential for the dissolved solute, All;- The
second term in (3.7) is equivalent to the chemical potential
of an ideal gas at density pf (the number density of solute
molecules dissolved in the solvent).

For the chemical potential of a solute molecule in the

gas (assumed ideal) one has:

pg = kT a. (93 A3) . (3.8)
in which pg is the number density of the gaseous solute

molecules. Equating (3.7) and (3.8) we find:
* _ L 9
M2 - ‘kT 1CD ( pz / 92 ) o , (3.9)

3.3 Ostwald Solubility

In the experiments described in this thesis we directly
measured the Ostwald solubility as a function of temperature
L(T) of xenon gas in various organic solvents. Ostwald
solubility is an intuitive as well as a theoretically
significant measure of solubility. It is defined as the
(equilibrium) ratio of the (volume) concentration of

dissolved gas molecules in the liquid solvent to their

10

concentration in the gas phase. If p; , pg are the number

densities of solute 2 in the liquid and gas phases

respectively, then:
L = pg / pg . (3.10)

At 20°C the Ostwald solubility of Xe is about 0.12 in water6

and about 4.4 in n-octane,18’19

a common nonpolar solvent
(see Fig. 2).

One can immediately see the importance of the Ostwald
solubility by substituting Eq. (3.10) into (3.9), which

gives:
Au; = -kT Ln (L) . (3.11)

The Ostwald solubility provides a direct measurement of the

excess chemical potential.

3.4 Solvation Thermodynamics
We start this section by reviewing standard
thermodynamics. The Gibbs free energy for a system can be

expressed in terms of its state variables aszo:

G(T,P,N) = E + PV - TS = H - TS , (3.11s)

l'

G(T,P,N) = Znipi , (3.11b)

 

 

m

(9000

 

Figure 1. Schematic of two

m‘9g95

C)
808%)!

 

 

 

[Xe] =1

gas

 

[Xe}=4

octane

 

[Xe}=0.1

 

water

 

 

11

. solute

o solvent

component solution.

[1 = concentration
(dimensionless)

T=20 ‘2:

Figure 2. Three phase system of xenon gas in
equilibrium with octane and water at 20°C.

12

where E is the energy of the system, S is the entropy, H is
the enthalpy (H=E+PV), and n1 is the number of molecules of
type i (of which there are r different species) whose
chemical potential is pi. The differential is:

r

dG = -SdT + VdP + Z “idni . (3.12)
-1

The most important information we shall be concerned with is

the first derivative of the free energy, i.e:

as
s: -[———] , (3.13a)
p

we may then evaluate H in terms of G and its temperature
derivative by using Eq.(3.11a). The partial molar enthalpy
and entropy (h, and s respectively) can be found by

differentiating (3.11b):

[——-—a u‘] h T (3 13b)
8. = _ , o = u. + S. ’ O
1 a T 1 1 1
where °
r
S = nsSI H = “oh. 0 (3.13C)
.1 1 1 , .1 1 1

Generalizing these, we may find the partial molar enthalpy
and entropy of the solvation process by taking the

temperature> derivative' of the «excess chemical ‘potential,

Aug:

13

 

6 Au
*_- 2 *_ It *
As2 - [ ) , Ah2 - Apz + TAs2 . (3.14)
6 T
It *
We call Ah2 and A32 , respectively, the excess partial molar

enthalpy and entropy. One may also find the Helmholtz free
energy of salvation and the internal energy of solvation.
However these differ from G, and H by a factor of Pv2 (where
vzis the molar volume of the solute in the solvent) which is
usually negligible [see section 3.21 of ref. 14 for a

complete discussion].

3.5 Physical Interpretation of the Salvation Process

In section 3.2 we introduced the concept of the excess
chemical potential, ALI;- I shall now present a physical
interpretation of this quantity developed by Ben-Naim.21 For
simplicity we consider a two component system at temperature
T, and pressure P with N1 and N2 representing the number of
molecules of solvent and solute respectively, Because the

Gibbs free energy is an extensive quantity the mathematical

derivative in equation (3.1) can be replaced by:
“2 = G(T,P,N1,N2+1) - G(T,P,N1,N2) . (3.15)

This statement is valid for a macroscopic system, where the
addition of one molecule may be viewed as an infinitesimal

change in the variable ii In order to interpret various

2.
contributions to the chemical potential Ben-Naim introduced

14

the concept of the pseudo-chemical potential. It is defined
as:
”Escudo: G(T,P,N

N +1; fio) - G(T,P,N1,N2) . (3.16)

1’2

This corresponds to the chemical potential for placing the
solute atom at a fixed position within the solven. We assume
the solvent is homogeneous and macroscopic so the actual
position no is irrelevant as long as it is within the bulk
of the solvent. By explicitly solving both (3.15) and

(3.16) from partition functions one can show21:

pg = "gum” + kT bu (pg/13) . (3.17)

Therefore it is immediately apparent that the
pseudo-chemical potential is equal to the excess chemical
potential, A“; defined in Eq. (3.7). We may interpret
equation (3.17) as follows. The full chemical potential can
be viewed as a two step process for adding an extra
particle to the solvent. First, we place the molecule at a
fixed position. The change in free energy for this is
“3"“d° (= Au; ). Next, we release the constraint imposed
on the fixed position; this leads to an additional change in
free energy, kam (pzAa). For classical systems 92A3<<1,
thus the free energy resulting from the release of the
particle is always negative. This quantity is referred to

21

as the liberation free energy. An explanation of the

15

various factors that make up the liberation free energy
are as follows. When the particle is released it acquires a
translational kinetic energy which leads to a free energy of
kT bn(A3). Also, when the particle is released it may now
wander throughout the volume of the solvent, which gives
rise to a free energy -kT On(V). Finally, once the particle
is released it is indistinguishable from the other N2
particles in the solvent. This adds a free energy
contribution of kT (11(N2). Putting all of these together
forms the liberation free energy. The entire discussion is
outlined in Figure 3.

The important property of equation (3.17) is that it is
generalized for any kind of molecule, whether it be atomic
argon or a complex protein, all that is required is that
classical statistical mechanics be obeyed. Of course there
still is the problem of developing a solid theoretical
calculation of Au; , however we now at least have some feel

for what it is we are dealing with.

3.6 Other Concentration Scales

The Ostwald solubility, which has dominated much of our
previous discussion, is based on the number density scale.
This means that it is dependent on the number density of the
solute in both the liquid and gas phase. However, there are
many other measures of gas solubility. The most popular
alternative is the mole fraction scale. The mole fraction

solubility (xi) is defined as the eqilibrium ratio of the

16

 

 

 

®
® @

 

 

 

 

 

kT in( p2 Ag)

 

 

 

 

Figure 3. Schematic description of the salvation
process. First the center of mass of the solute is
placed at fixed position 30, and then the particle

is released. The contributions to the chemical
potential are indicated next to each arrow.

17

number of moles of dissolved solute divided by the total

number of moles of solution:

x. = ——————' . (3.18)

For a two Component solution this reduces to:

= N2 / (N1 + N (3.19)

2) 9
where N2 and N1 are the number of moles of solute and
solvent respectively. Note that unlike the Ostwald
solubility this makes no direct reference to the gas phase
concentration. If the gas phase is ideal we can relate x2

to the Ostwald solubility, L19:

-1
._. 1[ R_T_ . 1] , (3.20)
L

where P2 is the pressure of the solute gas and 171 is the
molar volume of the solvent.
For a dilute solution the chemical potential for

component i may be written as 22:
“i = p: + kT &n(xz) . (3.21)

The quantity u: in equation (3.21) is known as the standard

state of the dissolved solute. It is hypothetical in the

18

sense that we cannot give a physical interprtation of it.
For example, one might say that it is the chemical potential
of the pure solute (i.e. x2=1), however equation (3.21) only
holds in the case of very dilute solutions. Despite this
flaw the mole fraction scale is still useful and has many

20’22 Its strength lies in the fact that it can

applications.
be used to describe liquid mixtures and electrolyte
solutions, where knowledge of the vapor phase is often hard
to obtain.

There are of course many other concentration scales that

20 but only the number density scale avoids

are available,
the problem of defining standard states and allows a strict
physical interpretation of the associated chemical

potential.

4. Experimental

4.1 Outline of Method

For the discussion in this section. please refer to
Figure 4. Here we have a two compartment chamber separated
by a valve. The upper chamber contains the solute gas while
the lower chamber contains the liquid solvent and a-stirring
device. An accurate pressure gauge is connected to the gas
volume which will allow us to determine the gas density at
any time through its equation of state. The gas volume in

the upper chamber is denoted V“’, the liquid volume is Vt ,

and the gas space in the lower chamber is V‘z’

8
system is immersed in a constant temperature bath. The

. The entire

(1)
V'
8

amount of gas. This is determined by measuring the initial

procedure of the experiment is to fill with a known
pressure, determine the gas density from it, and multiplying

the density by the initial volume:

(1)
Nuu41.1:"hnieun ' Vg ' (4'1)
We then open the valve and start the stirrer. The pressure
will drop until equilibrium is reached and the solvent is
saturated. We now measure the final pressure and determine

the final density, (For this idealized system we

pfinel.
19

20

neglect the vapor pressure of the solvent . In a real
experiment we must take the solvent vapor pressure into
account.) 'The final number’ of solute particles can. be

expressed as:

_ gas (v(1)+ V(2)) + dissolved: V

Nfinel- final)‘ 8 8 pfinel AC ’ (4.2)

where the second density is that of the solute which is in
the liquid. We now can introduce the Ostwald solubility,

which for this system is obviously:

dissolved

 

final
L = 9.. . (4.3)
pflnel
We now substitute L * p9" for p“"°”°d in equation
final final
(4.2). Since the system is closed the number of solute

molecules is conserved, thus we can equate (4.2) with (4.1)

and solve for L:

(I) (‘1 (2)

v v 4 v

L = a -3— - -3——-3- , (4.4)
"’4 V:

where a is the ratio of initial to final gas density,
(pi/pf).

To briefly summarize, we have outlined a simple
technique in which L is obtained by recording the decrease

in pressure during equilibration.

21

Pressure Gauge

 

valve

(1)

 

 

 

 

 

Figure 4.

Idealized solubility apparatus.

22

4.2 Radioactive Tracers

There are several problems which come up when actually
measuring the Ostwald solubility as described in the
previous section. First of all, one must be sure to measure
the pressure drop associated with the solute gas. Two
contributions of background gases are, (a) the vapor
pressure of the solvent, and (b) any other gases dissolved
in the liquid. For dilute solutions the vapor pressure of
the solvent does not change significantly from its pure
value,15 so it can be corrected for. The second problem is
overcome by carefully degassing the solvent when loading it
into the apparatus. This can lead to fairly elaborate and
time consuming setup procedures.

Another problem is the necessity of noninvasive pressure
measurement. In other words, we do not want to change the
system by measuring its pressure. Many of the more accurate
pressure gauges and sensors are able to detect changes in
pressure through moving parts such as diaphragms or
Bourdon tubes.23 A result of this is often a small change
in gas volume which must be accounted for due to our need to
know absolute differences in mass (density*volume).

The most difficult problem to overcome is sensitivity
over a wide range of gas concentration. As mentioned
earlier, gas solubility in various solvents can vary by more

6’24 The best reasonably priced

than factors of one hundred.
transducers and gauges have an uncertainty of 0.05 percent

of the full scale reading. Therefore a 0.05% error at one

23

pressure could become a 5% error on another. To get around
this one must use a series of pressure sensors for the range
appropriate to the system being studied. A related
difficulty is that for systems where the Ostwald solubility
is low, the observed change in pressure will be small
compared to the total pressure (or even the vapor pressure).
This can be overcome by using large liquid volumes, however
for some expensive solvents this is not a feasible
alternative.

By using radioactive tracers one can avoid all of these
problems. Since the tracer is the only gas which is
observed, background gases can be ignored. Furthermore, the
tracer can be monitored by external detectors, which are
truly noninvasive. And finally, since radioactive decay has
an uncertainty of (NcmmflJ)-o.5 one can get desired accuracy
over a wide range by simply waiting for enough counts.
Since the goal of this experiment is to look at gas

solubility over a wide range of solvents, the advantages of

using radioactive tracers is apparent.

4.3 Solution Components

The solute gas chosen for our experiments is the
radioisotope Xenon-133. Xenon is one of the noble gases.
Its atomic radius is 2.23A, which qualifies it as the
largest and most polarizable of the 5 inert gases with
stable isotopes, viz.,He, Ne, Ar, Kr, and Xe. The next noble

gas in this series is radon which has no stable isotopes.

24

235U fission and is readily

Xenon-133 is a byproduct of
available commercially. It is unstable and decays with a
half life of 5.245 days.25 Because of its high solubility
in fats and low solubility in aqueous solutions, 133Xe is
used extensively in nuclear medicine to study cerebral blood
flow pulmonary function, etc.18 The decay process for’133Xe
is as follows: the isotope first decays by beta emission to

an excited state of 133C

3, this nuclear excited state then
decays with a half life of 6.3 * 10"9 sec by emiting an 81
keV gamma ray. The beta rays are rapidly attenuated, but
the gamma ray intensity can readilybe quantitatively

133

measured to determine xe concentration.

Xenon-133 was purchased from the Hedi-Physics Company
(Plainfield, N.J.) in 20 millicurie (mCi) aliquots (this is
a recommended human dosage for cerebral blood flow studies).

A typical amount of ‘33

Xe used during a run was of the order
of 100 pCi. In practice, one aliquot usually supplied enough
xenon for a month (a: 10 runs). The corresponding partial
pressure of the tracer is appoximately' 1‘ picoatmosphere

(10"12

atm), therefore» it’s safe to say that our results
correspond to the limit of infinite dilution. The tracer was
usually mixed with air at 1 atm. In order to make sure the
air did not effect our results control experiments were done
in which' 1”Xe was mixed with naturally occuring
nonradioactive xenon at a total pressure of 1 atm. No

difference was observed. The solvents studied were an

extension of previous work involving n-alkanes and

25

n-alkanols26 (the prefix n- means that the molecule is a
straight chain). They include cycloalkanes, alkanals,
carboxylic acids, and perfluoroalkanes.

N-alkanes can be described as straight chains of singly

bonded carbon atoms which are saturated with hydrogen:
H H H
C H 9 H-C-C-eeeoe-C-H e (405)
n 2n+2 H H H

They are composed of n22 acyl groups (CH2) and two methyl
groups (CH3). We studied the alkanes ranging from pentane
(C5) to eicosane (020). The alkanes were purchased from
Humphrey Chemical Co. (New Haven, Conn.), and were all at
least 99% pure.

N-alkanols differ from n-alkanes in the addition of a

hydroxyl (OH) group to a terminal carbon:

, H
"' eeeee " C - 0H s (406)
H“ -

C

nH2n+l 1 2

H H

OH 9 H - C - C

H H

Ethanol was obtained from Aaper Alcohol and Chemical Co,

while the other alkanols were from Aldrich Chemical

(Milwaukee, Wis.). Purities were: 99.9% (methanol), 99+%

(propanol and butanol), 99% (pentanol, octanol, decanol, and

undecanol), 98% (hexanol, heptanol, and dodecanol), 97%
(nonanol and tetradecanol), and 200 proof ethanol.

Carboxylic acids are formed by replacing the terminal

hydrogen with a carboxyl group (COOH):

H H H
on H2n+lcmn ’ H - 1.01,- :2- eeeee "" g“- COOH e (4e?)

Alkanals, also known as aldehydes, have a terminal CHO group
(CnH2n+1CHO)' Cycloalkanes (CnHzn) are similiar to alkanes
except that the carbon atoms form a closed ring; as a
consequence they have no methyl groups. These solvents were
also obtained from Aldrich Chemical. Their purities were
the highest reasonably available: formic acid (95%-99%,
remainder water), acetic acid (>99%), propanoic acid (>99%),
n-butanoic acid acid (>99%), n-pentanoic acid (>99%),
n-heptanoic acid (>99%), propanal (>99%), n-butanal (>99%),
n-pentanal (99%) , n-heptanal (95%), cyclopentane (78%;
99.6% saturated 05 hydocarbons), cyclohexane (>99%), and
cyclooctane (>99%).

Perfluoroalkanes are alkanes with fluorine substituted
for hydrogen (CnF2n+2)° They were obtained from SCM
Specialty Chemicals (Gainesville, FL). Their purities were
also the highest reasonably available: perfluorohexane
(99%, of which 85% is n-CGFIO),

mixed isomers), and perfluorooctane(90% mixed isomers).

perfluoroheptane (97%-99%,

Other perfluoroalkanes were either prohibitively expensive

or not available.

4.4 The Experimental Apparatus
A diagram of the apparatus is shown in Figure 5. The

design is similiar to that used in previous work,26 except

27

the upper and lower portions are now held together by an
indium sealed brass flange instead of a threaded brass
connection. The apparatus can be disassembled at the flange
for cleaning and loading. The upper portion of the
apparatus consists of two valves and a brass gas volume.
The lower portion is a pyrex flask which is joined to a
glass-to-metal seal. The brass flange is soldered to this
seal. A small matching groove is cut on both faces of the
flange and is filled with 0.040" diameter indium wire. The
top and bottom are connected by six screws which press the
indium to form‘s reliable seal. The indium can be reused
many times by reforming the wire in a hydraulic press.

The lower volume contains a glass encased stir bar in
order to mix the solvent. Most commercial stir bars are
coated with Teflon because it is non-reactive. We found it
necessary to remove this coating because xenon is very
soluble in Teflon. The stir bar is then encased in glass to
protect it from potentially corrosive solvents.

A ball valve separates the upper and lower volume. The
advantage to using a ball valve is that its volume is well
defined. In other words, once the valve is opened the
volume that is exposed is that of the hole in the ball
regardless of how much you open it. The second valve, a
Hoke valve, is for loading the gas. This does not have any
critical volume requirements because it remains sealed
throughout the entire run.

During a run the apparatus is immersed in a fluid bath

28

in order to regulate temperature. The bath was controlled
to $0.1”: by a Lauda/Brinkman K-2/RD circulator. The bath
fluid was a 4:1 mixture of water:ethylene glycol in order to
prevent freeze-up of the circulator cooling coils.

We used Pb sheilding a 1 cm to reduce background
radiation and to isolate the volume Vé1)from the rest of the
apparatus. This is important because we want the detector
to measure the concentration in the gas phase only. Any
scattered gamma emissions from 133Xe in the liquid which
reach the detector will lead to errors. One cm of Pb
6 27

attenuates BlkeV gamma rays by a factor of 10 We

checked the shielding by putting ‘33

Xe in V with the
rest
apparatus in the bath and the shielding in place. No counts

were observed above background.

4.5 Volume Determination

One of the° most important criteria for determing
solubility by the method outlined in section 4.1 is the
accurate determination of volumes.‘ The relevant volumes for
the apparatus are indicated in Figure 5. Of these seven
volumes five are the same for every run, namely, V‘1’,V

g rest

(2)
, V;b, Vfib , and V.. . The remaining two , V8 and Vt , are

determined at the start of each new run. Each volume is

described below:

(1)
V
8

= the rest of the gas volume ,

= initial gas volume ,

(2)
V
8

29

= total volume excluding V(1) ,
rest 8

Vt = volume of the liquid ,

Vhb = volume of hole in ball valve ,
V“ = volume of stir bar ,
V = volume below the ball valve .

They are related by:

= V + V (408)

rest ee hb

(2) ’
v8 — rest. vsb VC . (4.9)
To determine V" (ee stands for everything else), we
weigh the apparatus before and after filling V“ with water
at a known temperature. We make a buoyancy correction for
the weight of air displaced by the water. If mob. is the

observed mass, then the true mass,muu.,is:

p

sir,T
mum: mob. / [1 - Fla—7;] , (4.10)

2

where p 'r is the density of x at temperature T. One can

X,

then determine V" by:

Vee= mtrue / pl! 0,1' . (4.11)

The volume of the hole in the ball was calculated from

measurements made on the disassembled valve. We can then

30

find V}..t from (4.8). The volume of the glass encased stir
bar, V.b, was found by measuring the water it displaced in a

graduated cylinder.

Once V"... was known , we were able to find v; 1 ) by
performing a dilution run. This consists of loading 1331Xe
in Véi) with V}..t evacuated. Once we determine the

concentration of gas in Vé"

, c°(normalized to correct for
radioactive decay), we open the main valve and allow the gas
to expand into Véz’. After several hours the system
equilibrates and we note the final normalized concentration,
ct. Since the amount of xenon is conserved (subject to

radioactive decay) we have:

(i) (1)
cov8 - f(V8 + Vr.‘t) , (4.12)
which in turn gives:
V“’ = V s a / ( 1- a ) , '(4.13)

8 rest

31

 

 

 

 

 

 

 

 

 

 

 

 

 

; Nal a
I Detector ]
I I
5 5
. . I I
Pb Shielding ’ g
\ I 1
I I
I- a
I I
5 5
I I
I I
Valve a v1' V1
.- g '
Pb Shielding /
Valve
H L—I ‘
' brass flange
glass-metal
seal "—’ 2
V
,_. 9
l Viiq ' l
stir5ar

Figure 5. Experimental apparatus.

 

 

32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High Volltage
Power Supply
0 d Signal Single
etector an . —. a Amplifier Channel
Photomultiplier P" Amp fl
Pre- Amp Analyzer
Power
sand
. AOOSec ._.—4
_ . Timer _ Counter
Tngger
Connections 4 Counting
200 Sec __ EM”
‘ ' Com uter
“m" Timing Pulse p
CRT Printer

Figure 6. Schematic of

 

 

 

 

 

 

detection electronics.

33

where a = (C0 / of). For the apparatus used in these
measurements Vuus 250cm3 , and Véus 50cm3.
4.6 Data Acquisition

The electronics are shown in Figure (6). Gamma ray
intensity is monitored by a well -counter NaI(T1)

scintillation crystal connected to a photomultiplier
(Harshaw Chemical Company, type 7SF8). A high voltage power
supply (Power Designs, Inc., model 1570) supplies 1100 volts
for the photomultiplier via the preamp (Ortec model 276).
Signals from the photomultiplier go through the base
preamplifier and a Nimbin mounted amplifier (Ortec Model
485), then through a single channel analyzer (SCA, Ortec
Model 406A), and finally into a microcomputer counter (Micro
Development Tool 1000). The SCA is set with a suitably wide

window (10-90keV) so as to permit passage of ‘33

Xe signals
while keeping the system stable and the background low. The
window was set by feeding signals of known amplitudes into
the amplifier and observing the output on a multichannel
analyzer (Northern Electronics model NS633). The resulting

133Xe spectrum . Two t imers

output was then compared to a
(Ortec Model 719) are supplied a 0.1 sec timing pulse by the
computer.

The computer is programmed to take data and print out
the results. It counts the number of decays in a 400 sec

interval and then waits an additional 200 sec before

counting again (the time intervals are somewhat arbitrary

34

and are based on historical reasons). A result is printed
every ten minutes with averages printed every hour.

Two corrections are made to the raw data. Because the
electronics are not perfect, a pulse into the detector is
not recorded if it follows an earlier pulse too closely,
i.e, the detector has a ’dead time’ The dead time can be
measured by sending signals of known frequency into the
electronics and observing the output. In this way, we
determined the dead time for our system to be about t a:
3.33xlo—6 sec.

In order to calculate the dead time correction we must
exploit the random nature of the radioactive decay process.
The probability of k events in a time interval, given a mean
rate of u events in that interval, is described by a Poisson

distributionza:

k

u
p(k) = . (4.14)
kleu

 

We now take the dead time, t, to be our unit of time. Thus
p becomes the mean number of events in time t. The
probability that a single signal occurs within time t of an

earlier signal is given by:
9(1) = u e'" . (4.15)

Since p is small (typically of order 10-3) we can expand the

exponential. To first order this reduces to:

35

P(l) u . (4.16)

The actual counting rate is then approximated by:

Nact = Nobs(1 + P(1)) = Nob.“ + u) , (4.17)
where Now. is the observed number of counts over period T.
We approximate u by using the observed counting rate

multiplied by the dead time:

u = ( Nob./ T ) * t . (4.18)

Plugging into (4.17) we have:

Naet = Nob.( 1 + ( Nob./ T ) * t ) . (4.19)
We have programmed the computer to make this correction
using the parameters : T=400sec, and r=3.33x10-63ec.

The second correction we make is to subtract off
background counts. We measure background radiation over
periods of several hours with the evacuated apparatus and
shielding in place. Background is subtracted after the dead
time correction is made. Typical background rates are about
1500 counts/400 sec. During an actual run counting rates
vary from 250,000 counts/400 sec (at the beginning) to

50,000 counts/400 sec (at the end).

36

4.7 Experimental Procedure

I shall now outline the experimental procedure to

33 . .
1 Xe in an arbitrary

determine the Ostwald solubility of
solvent. Referring to Figure (5) may be useful.

The solvents are generally used as purchased, we do not
degas them. Control experiments were done on degassed
solvents and have shown the dissolved gases do not effect

26

xenon solubility ( if we equilibrated the solvent with 1

atm of air, the gas concentration would amount to about 0.01
mole/liter).

With the ball valve closed and the apparatus

(1)
v
8

loading valve (Hoke bellows valve). A trace amount of

disassembled, air is evacuated from volume through the

1’33Xe is allowed to expand into V”’, and then air is let in

8
to bring the total pressure to 1 atm. The loading valve is
then closed.
The glass portion of the apparatus is loaded with about
200 ml of solvent, and is weighed (we make the usual
buoyancy correction).' Using known densitieszg’30 we
calculate the liquid volume. The perfluoroalkanes and

cyclopentane were mixed isomers, so we measured the density

ourselves using a standard volumetric method. This allows us

(2)
8

the stirring bar, the apparatus is assembled (see Section

to calculate gas volume V using (4.9). After putting in

4.5). The apparatus is then mounted rigidly in the

temperature bath so the detector geometry does not change in

37

the course of the run.

With the main valve closed, we start the computer and
begin taking data. During a run the initial counting rate,
ci, is measured hourly for six to eight hours. Each value

is corrected for decay, and the mean value is calculated.
(1)
g 9
by a steady counting rate, we open the main valve and start

Once the gas is uniformly distributed in V as indicated
the stirrer. The solubility can then be calculated from a

generalization of equation (4.4):

V(1)+ V(2)

Vr

 

 

where 1 is the decay constant for 133Xe (l=0.9177x10-4min-1)
and At is the time elapsed between measurement of c1 and
cf. We calculate the Ostwald solubility each hour until L
reaches a constant value. Time to equilibrium is typically
12 hours. We wait an additional 8 to 12 hours to insure
equilibrium has been reached(see Figure 7).
In order to obtain L(T) in these experiments one needs
to know the densities o(T) at each temperature at which L(T)
is measured. The idea is that after charging the apparatus
with solute and solvent and then measuring L at some initial
temperature, say 25°C, one can measure L at a different
temperature by just changing the temperature and waiting for
the new solute-solvent equilibrium. Once the bath

temperature is changed we recalculate equation (4.20) using

38

the new liquid and gas volumes, VC and V’Z). This procedure

8
works because the masses of solute and solvent are fixed in
the sealed apparatus. Thus, in a typical determination of
L(T) we measured solubility sequentially at temperatures of
20,30,40,50,10 and, finally, again 20°C (the temperatures we
cycle through depends on the solvent of course). We went

though a cycle like this two times independently for each

solvent.

39

 

 

5C) . I [if I I {rfrr I i [i I rl i I r [r‘
a—40u '-
g A. _
c .

3 .
>20 . -
b O
E . "
a: .
h, . _
in —-
.3 . _

7 '. ‘-
se' - -
.< 5 "
a)4 -, ..
'3 B
X3. l-
3... i

o ’ f 1‘ 6 3 ' ‘4

Time (hours)

Figure 7. Normalized counting rate vs time for a
typical experiment (Figure 2 of reference 18). The
letter A indicates the time the valve was opened,
and the letter B indicates the time a run might
end, about 8 hours after equilibrium has been reached.

5. Results

5.1 Thermodynamic Analysis

Table I gives the results of our experiments for 45
solvents. These include: 16 alkanes, 13 alkanols, 6
carboxylic acids, 4 alkanals, 3 cycloalkanes, and 3
perfluoroalkanes. The first row for each solvent gives the
measured value of L(T) at each temperature. The second row
gives the corresponding chemical potential, Au; (cal/mole),

which is calculated from L(T) by equation (3.11):

Au;=-RTan .

The data given at each temperature are each averages of at
least 2 separate runs. Typical uncertainties for 6L/L are
about 0.015. The uncertainty 6(Au;) = -RT(6L/L) varies from
£6 cal/mole at 278.15K to :10 cal/mole at 323.15K.

Figure 8 shows the Ostwald solubility plotted as a
function of temperature (C) for six representative solvents,
one for each homologous series. They are: n-hexane,
n-hexanol, cyclohexane, n-heptanoic acid, n-heptanal, and
n-perfluorohexane. Each of these solvents has a similiar
backbone with different terminal groups attached (or

hydrogens replaced by fluorines). Figure 9 is a plot of the

40

41

corresponding chemical potential versus temperature. For
the solvents illustrated the largest solubility occurs in
hexane, this corresponds to the most negative free energy,
while the lowest solubility occurs in perfluorohexane, which
has the highest free energy. From these figures we can see
the general tendency in the Xewmu- organicwlvmt system
for L(T) to decrease with temperature, while ApZCT)
increases with temperature.

From equation (3.11) it is clear that the sign of Au; is
negative if L is greater than one, as is the case for 44 of
these solvents (HCOOH excluded). For all such solvents a
positive free energy is required to remove a solute molecule
from a fixed position in the solvent to a fixed position in
the gas. Conversely for those in which L is less than one, a
positive free energy is required to place a solute at fixed
position in the liquid.

Over the temperature range we used, the experimental
data for Ap;(T) can be fitted well by a straight line of the
form. Ap;(T)=a+bT. Since the partial molar entropy is
proportional to the temperature derivative of the chemical

*
potential, 3: = -(a"2/6T)p , one can “has.“

-RT on (L) = Au;(T) = All: - ME: , (5.1)

42

gable I. Solubility data for experiments with
Xe in 45 organic solvents.

the Ostwald solubility,

L(T),

First row gives
and the second

a):
row gives the excess chemical potential A“: =

~RT¢nL. Temperature is in degrees celsius.

 

 

 

 

 

 

 

 

 

 

 

 

Temperature 10 20 30 40 50
alkanes
n-carbon
5 L(T) I 6.41 5.48
Aule) I -1045 -991
6 L(T) I 5.91 5.07 4.55
Au:(T) I -999 -945 -913
7 L(T) I 5.41 4.67 4.13 3.75 3.37
t
AMI(T) I -950 -896 -854 -822 -780
8 L(T) I 4.99 '~4.36 3.90 3.47 3.31
a
Au.(T) I -904 -856 ~820 -774 -769
9 L(T) I 4.70 4.14 3.70 3.32 2.99
s
Aule) I -a71 -828 -788 -747 -703
10 L(T) I 4.42 3.92 3.52 3.14 2.84
t
Auzfl‘) I -836 -796 -758 -712 -670
ll L(T) I 4.16 3.72 3.35 3.00 2.71
8
Aung) I -805 -765 ~728 -684 -640
12 L(T) I 4.03 3.59 3.22 2.90 2.64
a
AM:(T) I -784 -744 -704 -662 -623
13 L(T) I 3.88 3.44 3.09 2.80 2.53
I
Au.(T) I ~763 -720 -679 -641 -596
14 L(T) I 3.76 3.35 3.02 2.72 2.49
a
Au.(T) I -745 -704 -666 -623 -586
15 L(T) I 3.24 2.92 2.64 2.41
c
Au'lT) I -685 -645 -604 ~565

43

 

 

 

 

Table cont.....

16 L(T) 3.14 2.35 2.57 2.35
Aule) -667 -631 -537 -549
17 L(T) 2.75 2.51 2.30
Au:(T) -612 -573 -53s
13 L(T) 2.71 2.47 2.25
Au:(T) -601 -563 -521
19 L(T) 2.42 2.21

.
Au'(T) -550 -509
20 L(T) 2.36 2.17

.
““2‘7’ -534 -493

 

Table I cont.....

44

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Temperature 10 20 30~ 40 50
alkanols
n-carbon
1 L(T) - 2.46 2.20 1.93 1.79
Au:(T) . -507.4 -460.4 -411.5 -362.3
2 L(T) - 2.79 2.47 2.22 ' 2.02 1.35
Au;(r) . -577.9 -527.9 -431.5 -436.0 -395.4
3 L(T) - 3.02 2.65 2.33 2.16 1.93
Au:(T) . -621.3 -567.7 -521.3 -432.2 -439.0
4 L(T) - 3.04 2.63 2.40 2.17 1.93
*
AnalT) - -625.3 -574.7 -527.4 -432.2 -439.0
'5 L(T) - 2.97 2.62 2.36 2.13 1.95
3
Aule) . -613.3 -561.3 -516.5 -467.0 -419.6
6 3(3) . 2.97 2.62 2.34 2.12 '1.92
.
. Auz(T) . -611.9 -559.1 -512.7 -467.0 -419.6
7 3(7) - 2.91 2.57 2.31 2.09 1.91
Ap;(T) - -601.6 -594.4 -503.3 -457.5 -413.5
3 L(T) . 2.36 2.52 2.25 2.05 1.33
Ap;(T) a -590.7 -537.3 -439.6 -445.3 -404.4
9 L(T) - 2.79 2.49 2.24
Au;(T) a -577.5 -530.5 -434.5
10 L(T) . 2.74 2.43 2.20 2.00 1.33
AuZlT) . -567.3 -513.4 -475.5 -432.0 -333.4
11 L(T) . 2.34 2.11 1.92 1.76
Au:lT) - -496.2 -449.3 -406.3 -363.3
12 3(7) - 2.12 1.94 1.73
Au:(r) - -453.5 -410.3 -369.6
14 L(T) - 1.91 1.72
Au;(T) - -404.0 -364.3

 

Table I cont.....

45

 

 

 

 

 

 

 

 

 

 

 

Temperature 10. 20 30 40 50
carboxylic acids
n-carbon
1 3(7) = 0.474 0.456 0.437 0.420 0.414
Au;(T) - 420.1 453.1 493.7 539.3 566.3
2 3(7) . 1.71 1.53 1.47 1.37
Au;(7) - -312.0 -275.2 -239.7 ~204.3
3 3(7) - 2.97 2.66 2.41 2.21 2.04
i
332(7) - -611.7 -570.0 -530.0 -493.2 -456.6
4 3(7) - 3.22 2.39 2.61 2.37 2.16
0
Aule) a -653.7 -613.9 -577.0 -537.5 -494.3
5 ' 3(7) - 3.20 2.34 2.56 2.31 2.11
I
Aule) . -653.9 -607.4 -567.2 -521.3 -473.6
7 3(7) . 3.12 2.73 2.50 2.27 2.07
R
“"2‘T’ - -639.5 -594.9 -553.0 -511.0 -463.1
alkanals .
3 3(7) 2 3.09 2.30 2.55
fl
332(7) - -634.9 -600.0 -564.4,
4 3(7) - 3.47 3.12 2.33 2.56 2.40
*
332(7) - ~700.0 -662.5 -625.6 -535.7 -560.9
5 3(7) - 3.50 3.15 2.34 2.59 2.40
Au;lT) - -705.2 -663.3 -623.4 -592.4 -560.0
7 3(7) . 3.32 2.93 2.70 2.47 2.26
I
332(7) . -675.7 -636.3 -593.4 -561.9 -524.2

 

46

Table 1. cont....

 

 

 

 

 

 

 

 

Temperature 5 10 15 20 25
n-carbon
perfluoroalkanes
6 L(T) I 2.48 - 2.39 2.30 2.20 2.11
Aule) I -501.8 -489.8 -477.9 ~459.3 -442.4
7 L(T) I 2.23 2.16 2.09 . 2.01 1.95
Aule) I -443.3 -433.8 ~423.2 -406.7 -396.3
8 L(T) I 2.15 2.06 2.00 1.92 1.85
*
Aule) I —422.1 -408.0 -396.3 -379.7 -365.8
Temperature 30 40 50
n-carbon
perfluoroalkane
7 L(T) I .l.87 1.76 1.67
e
Ap2(T) I -378.0 -350.4 -327.8
8 L(T) I 1.79 1.67 1.59

a
AM2(T) I -349.4 -318.0 -296.6

 

47

Table I cont.....

 

 

 

 

 

 

 

Temperature 5 10 15 20 25

n-carbon

cycloalkanes

5 L(T) I 6.88 6.49 6.04 5.75 5.33
Ap:(T) - -1066 -1053 -1030 -1019 -991.3

6 L(T) I 5.28 5.00 4.70
Aule) I -952.2 -937.3 -916.8

8 L(T) I 4.26 4.02 3.79
Au:(7i . -329.4 -309.9 -739.7

Temperature 30 40 50

n-carbon

cycloakanes

5 3(7) - 5.04
Ap;(T) a -974.3

 

 

6 3(7) . 4.46 3.93 3.53
‘R
032(7) = -9oo.1 -359.7 -313.5
3 3(7) - 3.53 3.21 2.91
Ap;(T) . -763.3‘ -725.3 -636.2

 

 

Ostwald Solubility, L

48

 

 

 

 

I I I I @ T 1 I I FT T l I I l I I I [ I I fit
: O Alkanes :
_ X Alkanols o 9 3
To Acids ‘j
o
- 0 Cyclo 8 I
: >< Perfluoro :
—+ Aldehydes o -—
I— -l
- 0
_. + ..l
.— D -4
._.- )1 + —-(
.— . D .4
_ X n a 1
4.
— X -(
_ X X x D J»
L— X n .ifl
x
l— .7
.4
l l l L l ' l l J I i L L l I L i L l I l i l i
O 10 20 30 4O 50

Temperature, °C

Figure 8. L(T) for Xe in six representative organic
solvents.

49

i.e., the slope of Au;(T) is the negative average entropy
of solvation (AEZ) over the temperature interval, and the
intercept at T=0 (T is the absolute temperature) is the
average enthalpy of solvation (Ahz). One might at first
think that this interpretation ignores the effect of a
strong temperature dependence of the enthalpy having a
strong temperature dependence which could then be seen in

a:
Au2(T), however this is not correct since31

(6H/6T)p" : T(aS/6T)p" . (5.2)
Any temperature dependent contribution from the enthalpy to
the free energy is cancelled by a temperature dependent
contribution from the entropy (provided we take the
derivatives under the proper conditions). This does not
mean that aAh:/8T is meaningless, in fact this quantity is

14 This quantity turns

the specific heat of solvation, Ac:.
out to be very small, but it may be of interest in more
detailed studies in the future. The tabulated values of

Ah; and 3;: for the 45 solvents studied can be found in

Table II.

5.2 Entropy of Solvation
In viewing Figure 9 one can see the slopes of the
straight line fits are all quite similiar. This means that

the solvation entropies all fall close together. This

Chemical Potential. Apao' (cal/moi)

50

 

 

 

 

 

-4£"3 r'Ii l ’ Izii" ii’i i‘ i ii I ‘i r*ii
l r l l l «
- 3“
500 b «c 3“ 0%
_. -— ,0 ‘°
_ ‘9 (jfififi
-600 —- o _,
_ «63,903 :
—700 -- "j
-300 —— -
—900 *-
-1000 -—
_fi11c"3L- 1 1 LJ lllllLII Ill [I I (I LUlJ_L-
0 10 20 30 40 50

Temperature. C

t
Figure 9. Excess chemical potential, A“2(T) for

six representative organic solvents (Figure 1 of
reference 17).

51

Table II. Exggrimental data for excess enthalpy and
entropy for Xe in 45 organic solvents.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

alkanes

n-C Afi:(;:t) A§:(%§%x
S ~2582 -5.429
6 -2264 -4.482
7 -2109 -4.121
8 -2109 -4.260
9 -2044 -4.l45
10 -2007 -4.128
11 ~1958 -4.069
12 -1922 ~4.017
13 -1924 -4.104
14 -1879 -4.005
15 -1869 -4.038
16 ~1819 -3.928
17 -1769 -3.820
18 ~1807 ”’ ”.3393—
19' -l762 ~3.875
20 -1727 -3.808

L. -.

cycloalkanes

n-C 36333;) A§;(§§{K‘
5 -2120 -3.77
6 -2065 -3.85
3 -2022 -4.14'

 

 

52

Table II. cont .....

 

 

 

 

 

 

 

 

 

 

 

 

 

alkanols

n-C Ah:(§§%) A§:(%§a‘
1 -1330 -4.34
2 -1369 -4.57
3 -1910 -4.57
4 -1943 -4.66
5 -1924 -4.64
6 -1959 -4.77
7 —1922 -4.67
3 -1900 -4.64
9 ‘ -1390 -4.65
10 -1326 -4.45
11 ~1790 -4.41
12 -1720 —4.20
14 -1630 -3.90

 

 

 

 

 

Table II. cont....

53

carboxylic acids

 

 

 

 

 

 

 

46:42:) :<—-
1 -640.2 -3.75

’-2 -1366 -3.60
3 -1702 -3.86

__; -1319 -4.10
5 -1335 -4.35
7 .134, -4.26

 

 

 

 

 

alkanals (aldehydes)

 

 

 

 

 

 

 

 

 

 

 

 

n-c Alibi-2t) 3%.:
3 -1632 -3.52
4 ~1710 -3.57
5 -1743 -3.67
7 -l742 -3.77
perfluoroalkanes
n-c 9532-3?) 653.53%;
6 —1335 -2.99
7 ~1188 -2.67
8 -1221 -2.87

 

 

 

 

 

54

observation is further supported on Table II which indicates
-*
that the values of Asz are all negative and commensurate in

magnitude. The average value of the salvation entropy over

all our solvents is, (A§:)‘v. -4.1t0.5 cal/mol K .

The entropy is associated with the process of taking the
solute from fixed position in the gas and placing it at
fixed position in the solvent. The negative entropy can be
thought of as solvent ordering during this process. We can
conclude from our data that the amount of ordering is more
or less independent of the solvent used. This isn’t too
surprising since our solvents were generally large floppy
molecules compared to the solute. We shall see later that
the salvation entropy is much more dependent on the solute
particle. It is interesting to compare our results to Xe
solubility in water. For the Xe-oHZO system, the
corresponding A52, averaged from 10-40°C is about -18
cal/mol K , calculated from the data of Benson and Xrause.32
This is indicative of a high degree of solvent ordering
upon hydration. This ordering can be associated with the
high polarizability of Xe, hydrogen bonding, dipole-dipole
interactions, and the small relative size of H20 molecules.9
These «combined effects can lead to ice-like structures

9,32

surrounding the solute , and thus decreases the entropy

of the liquid.

5.3 Enthalpy (Energy) of Salvation

Before analyzing the enthalpy a brief reminder, for the

55

systems we are looking at the excess enthalpy of salvation
is essentially the same as the excess energy of salvation
since PV is small in a liquid. If we assume the gas phase
is ideal then the salvation enthalpy, Ahz, can be thought of
as the energy required to take the solute from infinity and
put it at a fixed position in the liquid.

One can see from Figure 9 and Table II that for all the
solvents, except HCOOH (formic acid), the salvation process
is enthalpically dominated, i.e. AfileA§:> 1. For these
solvents (excluding HCOOH) we find AhZ/TA§:=1.510.15, also
remarkably constant. The average value of enthalpy for the
same solvents is A52: -1840 :1: 250 cal/mole. Values for
enthalpy are spread more widely than for entropy as one can
see from Figure 9. The alkanes and cycloalkanes tended to
have a large negative enthalpy, while the perfluoroalkanes
and the small polar molecules (such as acetic and formic
acids) have enthalpies of (comparatively’ small. magnitude.
All values of enthalpy were negative. This demonstrates a
net attraction between the solute and the solvent.

Solubility of Xe and other noninteracting gases in water
is entropically dominated. Using the XeIHZO data from
Benson and Krauseaz we find that 36:: -4050 cal/mole,
averaged from 10-40°c, which leads to Afi:/TA§: = 0.75. Even
though the enthalpic contribution is larger in water than in
the organics it is overwhelmed by the entropy caused by
solvent organization. It shouldn’t really be surprising that

the enthalpy in water is high since water has a permanent

56

dipole) moment, but the interactions between. other water
molecules are also very strong. This competition is
reflected in the salvation entropy and keeps the solubility
of nonpolar gases in water low. A simple way to think of it
is that water likes the solute, but it likes itself a lot
better and rearganizes to keep as many hydrogen bonds intact

as possible.

5.4 Temperature Dependence of L(T) and Ap;(T)

It is clear from our earlier discussions that the
temperature dependence of the chemical potential is
determined by the entropy. Thus we attribute the positive
slope of the curves of Figure 9 to a negative entropy. We
now ask: What determines the temperature dependence of the
solubility itself, L(T). To illustrate this clearly we

start with the now familiar relation (equation 5.1):
* -* -*
Au2(T) = -RT LniL) = Ah - TAs2 .

Solving for L(T) we have:

 

 

- 35* 35*
3(7) = exp [ 2 + 2 ) . (5.3)
37 R

Assuming A32 and A3: are essentially constant in the
temperature range of interest we differentiate (5.3) with

repect to temperature:

57

-*
3n
LL: 1,—22 . (5.4)
dT R T
33

One can also show:

 

-*
-Ah2 (T-To) )

L(T) = L(To) exp [ . (5.5)

R T T
a

where T is the temperature of interest and To is a chosen
reference temperature. This leads us to the surprising
' result (at least for me it was) that the temperature
dependence of the Ostwald solubility is due to the enthalpy,
while the temperature dependence of the chemical potential
is due to the entropy. We conclude that the experimentally
observed decrease in gas solubility with with increasing
temperature (see Figure 8) is associated with negative

enthalpies of salvation.

5.5 Total Entropy and Enthalpy

Up to this point we have focused on the entropy and
enthalpy associated with the excess chemical potential, 50;.
This is the chemical potential for the fictional process of
placing the solute at fixed postion in the solvent. Its
value in constructing a physical model of salvation has been
discussed earlier. We would now like to look at the total
entropy and enthalpy for the entire (real) salvation

process. Our starting point is the chemical potential for

58

the solute and the equlibrium condition (equations 3.6-3.8):

p9 = kT On (pg A3)

2
u: = kT On (p; A3) + An;
9 _ L
“2"“2 ‘

Taking the temperature derivative at constant pressure and

composition we calculate the total entropy:

tot_ _ _§_ 4 _ 9 _ 3 _ 9
As2 - 8T [(12 112 ); [s2 s2) , (5.6)
where:
8p“
39 = -k on (p9 33) - kT -1— ——2 (5.7)
2 2
pg a'r P
4
6p
3: = -k Ln (9% A3) - kT[ —l— -—E

] +33: - (5.3)
P

For fixed particle number we can write the derivatives of
density as negative derivatives of volume. This allows us to
substitute the isobaric expansivity, a = (1/V)(aV/8T)P into
(5.7) and (5.8). For dilute solutions we assume the
expansivity of the liquid mixture is the same as that of the
pure liquid, at . The gas expansivity, 019 , is easily

derived from the ideal gas equation of state:

59

a l / T . (5.9)

Putting all this together, the relationship between the

a
total entropy, As;°t , and the salvation entropy, As2 is

33‘“ = 33* - k can.) - k + kTa

2 2 C . (5.10)

We can now find the total enthalpy through equations (2.6)
and (2.13b):

Aug“ = 0 = 311;“ - 733;“ . (5.11)

Therefore:

31);“ = TAs: - 1:73:1(3) - kT + szoi£ , (5.12)

which is equivalent to:

2

tot _ * _
Ah.2 - Ah2 + kT kT at . (5.13)

tot

’°’and As2

Table III shows Ah2 for the 45 solvents used.

Table III.

60

Enthalpy and entropy of solution.

The second and third columns are the total

entropy and enthalpy of solution.
and fifth columns are the entropy and enthalpy

The fourth

evaluated on the mole-fraction scale.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

alkanes
ze— e3—
5 —2899 -9.72 —2818 -16.90
6 -2615 -8.77 -2691 -16.37
7 ~2478 -8.31 -2415 -15.36
8 -2493 -8.36 -2429 ~15.34
9 -2440 -8.18 —2378 -15.09
10 -2410 —8.08 -2351 -14.93
11 -2379 —7.98 -2308 -14.74
12 —2348 -7.88 -2278 -14.56
13 -2350 -7.88 -2278 -14.51
14 -2307 -7.74 —2239 -14.30
15 ~2301 —7.72 —2233 -14.23
16 -2253 -7.76 -2188 -l4.00
17 -2206 -7.40 -2154 -13.68
18 -2246 —7.54 -2191 -13.91
‘cycloalkanes
9:633 93:23.3 A5323) 3%..)
5 -2478 ~8.31 —2385 -15.68
6 —2437 ~8.17 -2263 -15.34
8 -2440 -8.18 ~24l7 -1S.84

 

 

 

 

 

 

 

 

61

Table III cont...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

alkanols
“72%) A‘s';<:.:t.n A6:<:::> Aé:<:.::.(
1 -2263 -7.59 -2250 -18.82
2 —2268 -7.61 —2257 —17.87
3 -2325 -7.80 -2300 -17.39
4 ~2372 -7.96 -2345 ~17.l3
5 -2361 -7.92 -2331 -l6.79
6 -2397 —8.04 -2364 -l6.64
7 -2359 -7.91 -2317 -16.26
8 -2342 -7.86 -2304 -16.05
9 -2343 -7.86 -2310 -15.90
10 -2272 -7.62 -2234 -15.50
11 -2239 -7.51 -2210 -15.35
carboxylic acids
n-C Arlyfi) A§;(;—:{-K) Ah:(;::) ":(Egt—K
1 -1052 -3.53 —1051 -17.99
2 -1780 -S.97 —l780 -l7.02
3 -2101 -7.05 -2088 -16.67
4 ~2223 -7.46 -2204 -16.49
5 -2301 —7.72 -2287 -16.48
7 -2276 -7.63 -2239 -15.84

 

 

 

 

 

 

62

Table III cont...

 

  

 

  

 

 
  

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

alkanals (aldehydes)
-t cal. -!. col. -0 cat -0 col.
n-C Ahz(;3T) A52(mouc) Ah2(;3T)‘Asz(moLK)
3 -1966 -6.60 -l946 -l6.13
4 -2071 -6.95 -2039 —15.83
5 -2115 -7.09 -2076 -15.61
7 -2152 -7.22 “2137 -15.38
perfluoroalkanes
-t. cal. -t cal. -0 col -0 col.
n-C Ah2(;:T) 2(moLK) Ah2(;3T)‘Asz(moLK)
6 -l632 -5.48 -1606 -l3.5
7 -1515 -5.08 -l487 -l3.0
8 -1565 -5.25 -1536 -13.1

 

 

 

 

 

 

63

35,36(we

Expansivity data was obtained from standard sources
measured the expansivity far the perfluoroalkanes and
cyclopentane using our temperature dependent density data,
p(T) ). Although the salvation enthalpy, Ah: is not
accessible to direct measurement, the total salvation
enthaply, Ahgu, is. This is the heat of solution at
constant pressure. The heat of solution has been measured
for various gases in water using calarimetric techniques,
and general agreement is found with solubility derived
34

values. No such measurements have been made to date on

our system.

5.6 Male Fraction Scale

The standard approach to solubility is to calculate
chemical potential on the mole fraction scale Au; (see
discussion in section 3.6) :

o-—

The corresponding enthalpy (A52) and entropy (A3:) are

obtained by writing An; = Ah:

- TAEZ and proceeding in a
manner analogous to what we have used. The tabulated
enthalpies and entropies can be found in Table III. Note
that the enthalpy an the sale fraction scale (A52) is very
close to the total enthalpy of salvation (Ah;°t). This fact

has strengthened confidence in the mole fraction scale, but

also note the entr0py (AE2) is not the same as the total

64

-*
°‘) or the entropy of salvation (Asz). In fact

entropy (As;
the mole fraction entropy has no straightforward

interpretation (and,in my opinion, is often misused).

6. Theoretical Analysis

6.1 The Excess Chemical Potential, Au;.

Now that we have experimentally determined the excess
chemical potential, also referred to as the free energy of
salvation, our task is to predict it from as close to first
principles as possible. This goal will constitute the bulk
of this chapter.

The chemical potential is derived from the partition
function. In order to keep things clear we will first work
out the case for a single component, N particle, atomic
liquid in the canonical ensemble (all results are
generalizable to the grand .canonical ensemble). The

classical partition function is:

Q(V.T) a —L—” I 6i" I d3" apt-BIG". i5")! . (6.1)
N! h

where l’is the hamiltonian of the system, ha' (h is Planck’s
constant) accounts for quantum corrections to the
differential volume elements in phase space, N! corrects for
the indistinguishability of the particles, 8 is the

Boltzmann factor (um, and (if-",3” is a paint in the an

66

dimensional phase space that describes the system.16 An

important simplification that arises in the classical

appoximation is the separation of potential and kinetic

terms to characterize the liquid state.35 This allows us to
write the Hamiltonian as:
2(3'.B" = K(B”) + U(B") . (6.2)

where K(p') is the kinetic energy of the classical degrees

I
of freedom, K(B')=£ (pf/2m), and 0(3') is the potential
is!

energy. We can now write the partition function as:

 

- 1 an an __ an
Q(V.T) - [m h" Jdr Idp expl B K(p )1] x

-l; I d?" expl-B U(f")] . (6.3)
V .
In writing equation 6.3 we have used the fact that [drul V"

= 1 (note that we now have two integrals over d3"). The

first bracketed term 'is the partition function for an N

particle ideal gas, Qua-1° The second term is referred to
as the configurational partition function , Qcon . 16
Evaluation of the ideal partition function is
straightforward:36

A-au

v” , (6.4)

 

ideal- N1

67

A denotes the de Broglie wavelength, A=(2uBh2/m)”2. The

Helmholtz free energy, A(N,V,T), is:

A = -kT {n.Q(V,T) . (6.5)

For our partition function this is equivalent to:

A=-kT{mQ -kTOnQ =A +AA , (6.6)
ides co

1 n ideal

is the free energy for an ideal gas and AA: is

I
the excess free energy. The corresponding chemical

where A
ideal

potentials are found by differentiating with respect to N,
it therefore becomes obvious that the chemical potential can

be expressed as:

+Ap , (6.7)

where u” = -kT 01(pzA3). To derive an expression for the

eel

excess chemical potential we use identity (3.15):

Au*= AA*(N+1,v,T) - AA*(N,v,T)

. (6.8)

 

v econmwm.) )

= kT 6n [
Q (N+1.V.T)
can

The ratio Qc"(N+1,V,T)/Qcon(N,V,T) is given by:

68

Iexpl-flul.1(;"1)l d;u+1
= q . . (6.9)
Qc°n(N.V.T) Iexpt-BUlh'H dr"

Q (N+1,V,T)
can

 

If the potential energy is pair decomposable we may write

the potential energy of the N+1 particle system as:

+N+1 _ 9N
U".1(r ) - Uu(r ) + a0 , (6.10)

where mo is the binding energy of the (N+1)'t particle with

all the others in configuration 3'. Equation (6.9) becomes:

1 exvl-Bool expl-BU'(?')J d?'*‘

= . (6.11)
Qc°n(N.V.T) jexpl-BU'(3')1 a?”

Q (N+1.V.T)
can

 

In a system with translational invariance, the point 1"”

may be taken as the origin 'for the remaining N position
vectors. 'This allows us to integrate over’ 3"‘, ‘which

yields a factor of V. Equation (6.11) reduces to:

Q nmfl’v’” v I ”pl-3%] eXPI-BU.(I-")l di’J'

Qc°n(N,V,T) Iexp[-BU.(r')] d?"

 

 

= V <exp(-pqo)> , (6.12)

where the angular brackets denote a canonical ensemble

average over the N particle system. Substitution of (6.12)

69

in (6.8) gives:

Au*= -kT 6n <exp(-p¢o)> . (6.13)

For a two component system ‘we can write the partition
function for a simple structureless solute (2) dissolved in

a fluid (1)

Q = Qid..l Qiczleal v-(N‘HNZ) x

Id?"‘j6§'zexp[-p( 01(3"‘) + uigf",?"2) + uzgf'zil. (6.14)

For dilute solutions the contribution from solute-solute

interactions, 022 is negligible.14

Following the same steps
as before one can show that the free energy for adding the

(Nz+1)th particle is:
Au; = - kT 6n<exp(-pso)> , (6.15)

where B0 is the binding energy of the added salute to a

fixed configuration of the solute-solvent system, i.e:

N291 +N2+i - N2 4N2
012 (r- ) — 012(r ) + Bo , (6.16)
and the ensemble average is taken over all coordinates of
the (N1+N2) particles.
Now we can appreciate an important simplification that

occurs by using an inert structurelesss solute like Xe as a

70

probe. If we used a more complicated molecule with internal
degrees of freedom (such as vibrational and rotational) it
is conceivable that the potential energy would be a function
of the solute conformation (shape), we would have to perform
a double ensemble average: one over all spatial
configurations of the (N1+N2) molecules and a second over
all possible configurations of the solute molecule itself,
i.e:

An; = -kT on < <exp(-BB°(P2))> > (6.17)

P i
2

P2 represents the internal coordinates of the solute

L4 A clear example where this would be necessary

molecule.
is the salvation of a polymer in water. By using an inert
gas probe we can focus on free energy changes due to the

liquid solvent and not have to worry about contributions

from the solute.

6.2 Distribution Functions

An alternative approach to describing the liquid state
that has proven to be quite powerful is the distribution
function method. I shall present a brief overview of the
technique, a more complete discussion is given in reference
36. As in the last section I shall work in the limit of an
atomic liquid to keep the development simple,, but the
results are readily generalizable.

The phase space probability distribution function in a

71

classical system is defined as:

expl-Bz(?'.§')l/ Idf'jdfi'eXPt-Bl(3'.3'). (6.18)

The probabilty of a state (3",3') is simply f(?",3")d§'d§".
We have shown earlier that the Hamiltonian factors into
kinetic and potential terms. Therefore we can write the
total phase space distribution function as a momentum
probability distribution, M3"). and a configurational
probability distribution, P(r'). The latter will be
important in this discussion. The configurational
probability distribution tells the probability for observing
the system at configuration space point '1'", it is defined

as:
“-11-, = expt-BUG'H / I dr'expl-BUH’J'H . (6.19)

Note the denominator is equal to (V'x Qcon).

We) can. determine distribution functions for' a small
number' of ‘particles by integrating over (all coordinates
except those pertaining to the particles of interest. The
joint probability distribution for. finding a labelled

particle 1 at 31 and a labelled particle 2 at 32 is:
(ZIN) 4 4 _ a 9 e an
F (r1,r2) .. I dra I dr‘....I dr. P(r ) . (6.20)

Since we cannot label identical particles, a more meaningful

72

quantity is the reduced probability distribution function,

(2")(ifi
1

p ,fz). It is defined as the probabilty distribution

for finding any pair of particles at fiand $2:

-) _ _ (21):)» ..

1,rz) - N(N 1) I’ (r‘,r2) . (6.21)
The factor of N(N-1) refers to the fact that there are N
ways of choosing the first particle and (N-l) ways of
choosing the second. Similiarly, the n-particle distribution

function is defined as:

I d?"'" expl-BU(§')]

 

p(n/N)(;n)= N!
(N-n)!

. (6.22)
I d?” expl-BU(?')]

For an homogeneous fluid, the single particle distribution

function is simply the bulk density:
p(1")(;1)= p = N/ v . (6.23)

In the special case of an ideal gas, U(f'")=0 and QC”. 1.

Thus the n-particle distribution function becomes:

N!

p‘"""('§n) = p" m '= p"(1 + O(n/N)) . (6.24)

For example the pair distribution function for an ideal gas

is:

73

3 ,3 ) = pz(l - l/N ) a p2 , (6.25)

(ZIN)(
1 2

p

where the last equality depends on N being large; which it
is for any macroscopic system. We now introduce yet another

.9

distribution function , g(f1,r2):

g(f .? ) = p p . (6.26)

which is the fractional deviation of the two particle
distribution from the ideal gas limit. It is essentially a
measure of how much the system deviates from complete
randomness.35 If the system is isotropic as well as
homogeneous, the pair distribution function “3132) is a

function only of the separation between the two particles,

r12: Ifi- 32| ; it is then referred to as the radial
distribution function and simply written as g(r).16 From
our definition of g(r) one may write the following:'

ps(r) = p(p(z’"’(0.i") / p 2) = p ‘2’") p . (6.27)

Since we have already seen 9”“)(31) =

p, equation (6.27)
can be interpreted as follows; pg(r) is equivalent to the
conditional probability density that a particle will be
found at 1" given that another is at the origin. This is
based on the well known theorem of statistics: If A and B

are random 'variables *with. joint probability' distribution

74

P(AnB), then the conditional probability distribution,

P(AIB), that A occurs if B also occurs 1837:

P(A|B) = P(AnB)/ P(B) , (6.28)

where P(B) is the probability distribution for B. Our
definition of pg(r) can also be stated as follows; pg(r)
gives the average density of particles a distance r from a
particle at the origin. The average number of particles in

volume element dV at distance r is thus:
dN(r) = pg(r)dV . (6.29)

Figure 10 shows g(r) for an atomic solid, liquid and gas.
The radial distribution function plays an important role
in the physics of liquids. There are two main reasons for
this. First, the radial distribution function is directly
measurable by scattering experiments (although it can be

difficult to interpret results for complex molecular

75

 

 

 

 

0 1 L L

0 20 30
Fun . .
cootdination she" Second COOl’dlllallOIl Sh!"

The radial distribution function for a simplefluid.

‘Y

 

 

 

 

l

4)..
'*-—-Som1

3i—
‘ Lhum
\\

2” r)

I ————— —— —— \ —

\
1 (I l l 1),,
0 la 20

Radial distribution functions for liqAUid and solid argon at the triple point
(oi-34 ).

Figure 10. Radial distribution function, g(r),
for a typical gas, liquid, and solid.

76

liquids). Secondly, if the particles interact through
pairwise additive forces, thermodynamic properties of the
fluid can be written in terms of integrals over g(r).

The two thermodynamic functions we shall require in our
analysis are the excess internal energy of salvation, unit,
and the excess free energy of salvation, Ad‘. I shall now
present brief explanations of how these quantities are
determined from g(r). Detailed derivations can be found in
reference 36.

The internal energy of an atomic fluid can be written

I!)
u
NH»
Cl

NkT + (6.30)

where the first term is the mean kinetic energy (same as for
an ideal gas), and the second term is the mean potential
energy. Using our definition of pg(r) we can easily find 6.
Consider a tagged particle at the origin. If we assume the
intermolecular interactions, u(r), are pairwise additive we
can express the potential energy between the tagged particle

and all other particles at distances between r and r+dr as:
d0 = u(r)dN(r) = u(r)pg(r)dV = u(r)pg(r)4sr2dr (6.31)
The total potential energy of the liquid is found by

integrating over r, and multiplying by N/Z. The factor of N

arises since any particle could be the tagged particle, and

77

the factor of two compensates for double counting of pair
interactions. The average potential energy per particle is

therefore:

Cl
ll
2|...

m
= 211p! u(r)g(r)r2dr . (6.31)
o

For the case of a dilute solution (we use the conventional
subscript notation in which 1 refers to the solvent and 2
refers to the salute), the excess internal energy (excluding

kinetic ener8Y) of the solute is:

z
duinr' u12(r12)p1g12(r12)4ur12 driz ’ (6'32)

In Eq. (6.32) the function g12(r12) is the solvent-solute
pair correlation function and p3g12(r) is the number density
of solvent molecules a distance r from the tagged solute.
The excess internal energy for this solute molecule is found
by integrating over r12

0.

uint- 4np1Io g12(r)u12(r)r2dr . (6.33)

This is sometimes called the excess internal energy of
salvation. Note that for a pure liquid, for which ughz}
goes into “hid“, the excess salvation energy is twice the
average excess energy per particle , 3. This is because the
average energy in equation (6.31) is the potential energy to

assemble. the whole liquid, while the salvation energy in

78

equation (6.33) is the potential energy to bring one extra
particle into a pre-assembled system.

In order to calculate the excess free energy of
salvation we must introduce the idea, due to Onsager38 and
Kirkwood,39 of a coupling parameter 5 which can vary from
{=0 to 5:1. We imagine the following process: we would like
to introduce a new particle into the system by turning on
its interaction with the other particles in the system.
When §=0 the new particle does not interact with the system

at all, and when §=1 we have the full intermolecular

interaction u12(r12). This can be expressed as:

u12(§,r12) = Eutzhtz) . (6.34)

To find the excess chemical potential we use equation (6.8):

Ap* = AA*(N+1,V,T) - AA*(N,V,T) .

Since the Helmholtz free energy is related to work in
thermodynamics ‘we can conclude that the excess chemical
potential is the isothermal, reversible work that has to be
done on the system against intermolecular forces in order to
add one more molecule to the system under conditions of
constant volume and temperature.40 In other words Au* is
the work done on the system in going from the initial state

with N molecules coupled with each other and 1 molecule not

79

coupled (5:0) to a final state with N+1 molecules fully
coupled (5:1). Following the same steps as before we choose
the new molecule to be at the origin. For an arbitrary
intermediate value of 5 we can write the radial distribution
function about the central molecule as g(r,§). The
potential energy of interaction of the central molecule is
§u(r). This means that u(r)d§ is the work that is done on
the system by this one interaction if E is increased by d5.
The work done on the system when 5 increases by d5, due to

all the molecules between r and r+dr is:
du(r,§) = u(r)d§ pg(r,§) 4nr2dr (6.33)

The total work, Ap*, is the integral of (6.33) over r and

over 5 from 5:0 to 5:1:
* 1 a 2
Au = 4np I I u(r)g(r,§)r drd§ . (6.34)
a a

For a two component system the salvation free energy is:
_ * i a 2
Apz = 4uptj I u12(r)g12(r,§)r drdE . (6.35)
o o

A generalized form of this statement is:

1
Au; = (New: . (6.36)
0

80

where U(E) is the spartially averaged coupling energy of the
salute to the rest of the system with the interaction

potential reduced from its full strength by a factor of 5.41

6.3 The Van der Waals Picture of Liquids

One of the fundamental problems in developing a theory
of liquids is that there is no idealized model comparable to
the ideal gas or the harmonic solid; both of which can be

treated exactly.31’42

These provide a reference system from
which one can base perturbation expansions or at least get
an intuitive feel for the physical situation, in order to
develop more sophisticated theories.

The van der Waals picture of liquids has helped a great
deal to overcome this problem. The basic idea is to look at
the different roles of strong short-ranged repulsive
intermolecular forces and longer-ranged attractive forces in
determining the structure and dynamics of a dense fluid.
Though the concept was first utilized by van der Waals in
his treatment of nonideal gases, there have been many other
contributors to our current understanding.43 The renaissance
of this idea was spurred by the discovery from computer
simulations that a system of infinitely hard spheres
(essentially billiard balls, albeit tiny ones) undergoes a
first order fluid-solid phase transition that can be related
to freezing and melting of real materials.44’45

The van der Waals picture asserts that the relative

arrangements and motions of molecules in a liquid are

81

determined ‘primarily by packing effects produced by the
short-ranged repulsive interactions. Attractive forces,
since they vary relatively slowly, play a minor role in
structure and to first order can be treated as a mean field
which exerts no intermolecular force but provides the
cohesive energy that holds the system together at fixed
density and temperature.

As an example we shall briefly look at a Lennard-Jones
fluid, which is defined by the intermolecular potential,

u(r) (see Figure 11):
u(r) = 4.:[w/r)12 - (a/r)6] , (6.37)

the collision diameter, a, is the separation of two
particles where u(r)=0, and e is the depth of the potential
well at the minimum in u(r). The properties of, the
Lennard-Jones .fluid are well known from computer
simulations,46 and it serves as an excellent model of atomic
liquids like argon.47 The repulsive (l/r)12 term arises
from electron cloud overlap and Pauli exclusion, and the
attractive (1/r)6 term accounts for induced dipole - induced

dipole interactions. To examine the role of repulsive and

Potential [ U(r) ]

Potential [ U(r)]

82

 

0.501 U“)

 

 

 

'8

 

 

 

0.0 1 .0 2.0 3.0
Intermolecular Distance (r)

Figure 11. Lennard-Jones potential, U(r). Lover
diagram shows repulisive portion of U(r) as
determined in WCA theory.

L00

 

4

0.50 3

1

A
q
V

O

 

000

 

4

~050‘

 

4.00 ‘

(

-L50‘

5U(r)

--------)-------.
,4
C

 

 

 

-200 fi- r I
00 L0 20 30

Intermolecular Dlstance (r)

i

g(r)

83

 

 

 

l k golf)

| —-- g}\9(r)

3““ I. 000 g(r)

2»- '
. O

1- l- '.-— .--- -. .. --
l . l

(3 __ "'
d - 2d r

Figure 12. Radial distribution functions for
(ilLennard-Jones fluid g(r), (ii) for repulsive
portion of L-J potential g°(r), (iii) for hard

spheres gh.(r).

84

attractive interactions in determining liquid structure
tests were done to compare g(r), the radial distribution
function for the full potential u(r), to go(r), the radial
distribution function that one would obtain from only the
replusive forces (at the same temperature and density). The

potential used to determine go(r) is (Figure 11):

uo(r) = u(r) + e rsro

u°(r) = 0 r>ro . (6.38)

In Eq.(6.38) ro is the point at which the minimum of u(r)

21/6

occurs; i.e. r = a. Figure 12 shows the strong

a
correlation between g(r) and g°(r) as determined from
computer simulations and analytic theory.48

To take this experiment one step further it was found
that there exists a hard sphere system for which the radial
distribution function, gh.(r), is closely related to the
actual distribution function, g(r) (Figure 12).49 The hard

sphere fluid is characterized by the sphere diameter, d,

which appears in the hard sphere potential (Figure 11):

ii I a rsd

u = 0 r>d . (6.39)

The effective hard sphere diameter is chosen to most closely
reproduce the structural features of the real fluid.

The fact that the attractive forces play a minor role in

85

structure was rationalized by Verlet.50 In order to

understand his argument a few new points must be introduced.

The local particle density is defined as:
M?) = 2M '1’- - ii) , (6.40)

where 8(x-xi) is the Kronecker delta function. The average
density at point f is simply <p(?)>, which for a homogeneous
system is the bulk density, p. The Fourier transform of

(6.40) is:

pk: I exp(-ik-f)p(f)df

= S: exp(-k-f'i) , (6-41)

is!

with an autocorrelation function defined as:

I

sat) = T‘ pup > . (6.42)

The function S(k) is called the static structure factor.

For a homogeneous fluid it can be related to the Fourier

transform of g(f) 35:

SHE) = 1 + p I exp(-ik-'f') 3mg): . (6.43)

S(k) can be experimentally determined from scattering

86

experiments on the liquid. One can then perform an inverse
Fourier tranform to determine g(f').16
One can show that the limit of S(k) for k40 is 35:
- _ 0
3(0) - kaxT - xT/ xT . (6.44)

where xr is the isothermal compressibility of the liquid:

- 1 23

IT " p [arr] 9 (6.45)
N r

and x: is the isothermal compressibility for an ideal gas:

x2 = 1/ka . (6.46)

If we apply an external field to the fluid with

potential (M3), one can show that this leads to a Fourier

transformed density response, 69* 35:
6pk = -S(R)¢(§)p/k7 , (6.47)
where (HE) is the Fourier transform of Ni"). Equation

(6.47) is a form of the more general fluctuation-dissipation
theorem.

Verlet’s arguement' can now be summarized. Equation
(6.47) shows that the structure factor determines the
density response of a system due to a weak, external field.

If the external potential is associated with the potential

87

of a test particle at the origin, the long-range part of
that potential gives rise to a long wavelength response in
density. In the long wavelength limit (k90), the response
is proportional to 8(0) by equation (6.47). By equation
(6.44) we see that 8(0) is proportional to the
compressibility» ‘Eor typical liquids, the compressibility
is very small ( xT/ x: m 0.02 near the triple point).

Therefore, the density change caused by a long wavelength

perturbation is not significant. This phenomenon is
referred to as ’repulsive-force screening’.51 At lower
densities, such as in the critical region, the

compressibility (and hence 8(0)) becomes large . ‘This can
lead to large density fluctuations. In this regime
attract ive forces become very important and the van der

Waals model is no longer valid.

6.4 Determination of Au; , ’Hard’ and ’Saft’ Contributions
The success of the van der Waals model in predicting
structure has lead to the development of thermodynamic
perturbation theories. The idea is to split the excess
chemical potential into two parts: a reference term due to
the ’hard’ repulsive interactions, and a perturbation term

due to the ’soft’ attractive interactions:

+ An . (6.48)

:7:
M2 = A“ soft

hard

To treat the ’soft’ part of the potential we perform energy

88

averages using the structural information obtained from the
’hard’ part of the potential (this is analogous to quantum
mechanical perturbation theory where we compute the
expectation value of the energy by employing the wave
function of the unperturbed hamiltonian). In this section we
shall investigate the validity of equation (6.48).

As we saw in the previous section the intermolecular

potential can be written as:
u(r) = uh(r) + u.(r) , (6.49)

where the subscripts h and 8 refer to the ’hard’ and ’soft’
parts of the potential. This simple division assumes the
solute-solvent pair potential is a function of separation r

only. The excess chemical potential is:
Au; = -kan<exp(-BBO)> = -kT6n<exp(-BBh - 88.)) , (6.50)

where Bh and B. are the hard and soft contributions to the
binding energy of the salute to all the solvent molecules in
a specific configuration. If Bh and B. were independent

random variables one could then write:

Au; = -kTOn[<exp(-BBh)><exp(-BB’)>]
= -kT!m<exP("BBh)> - kan<exp(-BB.)> . (6.51)

s
and therefore Auz could be split into two factors. The

89

point is that Bh and B. are generally not independent random

variables so equation (6.51) is invalid.

A proper interpretation can be obtained by writing the

average in equation (6.50) as 14:

<exp(-BBh - 3B.)>

I d?" exp[-pU(?F) -th] exp(-BB.)

 

I d?” expl-BU(;") ]

I dr'exp[-BU(;') -th] I df'expl-BU(?') -pah] exp(-BB.)

 

I df'exp[-BU(3') ] I df'exp[-BU(f') — BBh]
= <exp(-BBh)> <exp(-BB.)>h . (6.52)

where the subscript h on the second average refers to a
conditional average. This means the second average is taken
assuming that a hard solute particle already exists in the

liquid. The excess chemical potential is then:

Au; = -kT6n<exp(-8Bh)> -kT.€n.<exp(-BB.>h

= Au: 4» Apt/h . (6.53)

The first term, Au:, is the free energy to add a single,
hard solute molecule at a fixed position. The second term
but,“ is the conditional free energy to couple the soft part
of the solute-solvent interaction given that the hard part

of the potential has already been coupled. This is perhaps

90

better understood using the coupling parameter approach. As
we saw earlier the excess chemical potential can be found by
utilizing a coupling parameter to turn on the potential and
essentially grow the particle in the liquid. Since this
process is carried out reversibly it does not matter what
path we take to couple the particle. One possible way is to
turn on the hard part of the potential first and then turn
on the soft part, this is exactly what we have done in
equation (6.53). (As you may have guessed one could just as
well perform the calculation in the opposite order and
couple the attractive forces first and then calculate the.
conditional averages for the repulsive potential.)

The idea of breaking up the salvation process has been
utilized by many researchers in developing simple models to
predict solubility. The salvation process is normally
modeled by the following physical process: (1) One makes in
the solvent a cavity just large enough to fit a solute
molecule. The free energy associated with this process is
called gc". For this part of the process the solute is
considered to be a hard sphere. (2) One now allows the

solvent and salute to interact with the soft potential; the

52.53,

associated free energy is gint

Au; = g + g . (6.54)

From our previous discussion it is obvious that the cavity

*
free energy is equivalent to A"): and the interaction free

91

*

energy is equivalent to Autnv

Equation (6.54) provides the basis for most successful
analytic theories to date on solubility (the term successful
implies both accuracy and flexibility). We shall now adopt
a model, due to Reiss et al.54 and Pierroti,55 in order to
analyze our data. The primary assumption made by these
authors is: Since there is no analytic expression for gun,
in a real liquid one exploits the van der Waals model and
calculates gen for an effective hard sphere system whose
radii are chosen to best reproduce features of the pure
solute and solvent. Although our solute (Xe) qualifies as
spherical, the organic solvents certainly are not. One must
regard this appoximation as, in some sense, averaging over
spatial configurations of the solvent.

e
If we combine our experimental expression for Auz with

equation (6.54) we have:
Au; = mum. = g + g . (6.54a)

Equation (6.54a) can be interpreted in thermodynamic as

well as in statistical mechanical terms. If one calls v2

the partial molar volume of solute atoms in the solvent, v:

the molar volume of solute gas, and mg the volume fraction
of solution occupied by solute, then we have L = (adv: /v2).

Substituting for L in equation (6.54a) gives:

g + g + 127mg - n'ronz/v‘z') = 0 . (6.54b)

cev int

92

Equation (6.54b) has the following thermodynamic
interpretation 3: ‘The free energy for transferring at
equilibrium one mole of solute gas into the solution is the
zero sum of four terms. The first two terms are for the
physical process already described. The third term may be
interpreted as the free energy change associated with the
entropy of mixing -R6n¢é bentropy increase, free energy
decrease) in a real solution. Finally the fourth term is
the free energy required for isothermal and reversible
compression of the solute from its volume in the gas to itsv
volume in solution (entropy decrease, free energy increase).
The idea that entropy of mixing depends on volume fraction
in real solutions is supported by the work of

Flory,56Huggins,57 and Longuet-Higgins.58

93

6.5 The Scaled Particle Theory for Hard Spheres

We now must calculate the free energy for introducing a
hard sphere solute with diameter a2 into a hard sphere fluid
with solvent diameter a1. At present there exists no exact
theory even for this simple system , although it can be
solved numericallysg. However the scaled particle theory
(SPT) has proven to be a very good approximation. Details
of this theory are quite lengthy (see refs 60,61) so only a
brief outline will be given below.

The basic idea of the SPT is to calculate the reversible
work to produce a cavity at fixed position in a fluid of
hard spheres. A cavity is defined as a sphere of radius, r,
from which the centers of all fluid particles are excluded.
It is apparent from Figure 13 that a cavity of radius, r, in
a fluid of spheres of diameter, a , is produced by the

1

introduction of a hard spherical solute of diameter, a2,

such that:

r =-—1———£ . (6055)

The equivalence of these two processes allows us to conclude
that the free energy to create a cavity in the fluid is the
same as that to introduce an appropriately chosen hard

sphere solute.

94

 

 

 

Figure 13. Definition of a cavity in the SPT.

95

We can see from Figure 13 that for r=a1/2 the diameter
of the solute molecule goes to zero. Thus, a cavity with r
= a1/2 corresponds to a point solute. If the cavity is
smaller than a1/2 only one solvent center can occupy it at
any time. The corresponding probability of finding the

center of a molecule in the cavity is:
p (r) = 4 urap r s a /2 (6 56)
l 3 ’ 1 '

where g) is the number density concentration of solvent
molecules. The probability that the cavity is empty is

therefore:
_ _ _‘ _ 4 3

The reversible work theorem is a general theorem of
statistical mechanics which states that the probability of
observing a fluctuation in a system is equal to the
reversible work required to produce the fluctuation divided
by kT.62 Thus the probability of finding in the solvent a

cavity of radius r that is empty is:
p°(r) = exp(-W(r)/kT) . (6.58)
Since we imagine a process of empyting the cavity at

constant N,V,T, we may identify W(r) may with the change in

Helmholtz free energy for the process and write:

96

polr) = exp(-AA(r)/kT) - (6.59)
Substituting (6.57) in (6.59) we find:
AA(r) = -kan( 1 - g- ur3p) . r 5 81/2 (6.60)

In the limit of a macroscopic cavity the free energy is just
the thermodynamic work of compression:

AA(r) = PV = P g nr3 , r>>a1 (6.61)

where P is the macroscopic pressure in the liquid. We now
have two exact results for the cavity free energy. In order
to bridge the gap between these two limits we introduce a
new function, p°(r+dr), the probability that a cavity of

radius r+dr is found empty. This may be written as:
poir+drl = po(r)po(dr/r) . (6.62)

where p°(r) was defined earlier and po(dr/r) is the
conditional probability that a spherical shell of width dr
will be empty given that the sphere of radius r already is
empty.

If we expand p°(r+dr) to first order in dr, we obtain:

97

dpo(r)

 

po(r+dr) = po(r) + dr . (6.63)

dr

Equating (6.62) with (6.63) gives:

 

dpo(r)
Poirlpo(dr/r) = po(r) + dr
dr
dén Po(r)
Po(dr/r) - 1 = ‘———————— dr . (6.65)
dr

We now introduce the auxiliary function, G(r), defined by:
p4xrzG(r)dr = 1 - po(dr/r) . (6.66)
Substituting into (6.65) we have:

dén p°(r) 2
————————— = -p4nr G(r) , (6-67)
dr

which upon integration, yields:
l'
on p°(r) - 0n po(r=0) = -pI41rlzG(l)dl . (6.68)
0

Since po(0) = 1, equation (6.68) reduces to:

r 2
0n p°(r) = -pI4ul G(l)dl , (6.69)
O

98

or, using equation (6.59):
r 2
AA(r) = kTpI4xl cum). . (6.70)
0

The work required to create a cavity of radius, r, is a
scaling process, in that we build up the cavity from l=0 to
l=r. This is the same as building up a hard sphere particle
at a fixed» position in the fluid, hence the name Scaled
Particle Theory.

60

To find AA(r) Reiss gt, :5; used statistical

considerations to suggest a simple analytic form for G(r):

G(r) = A + (B/r) + (C/rz) . (6.71)

The coefficients A, B, C, are determined by using the

limiting forms of G(r):

G(r) = (1 - gnrap)‘1 r s a1/2

G(r) = P/kTp r>>a1 , (6.72)
along with the assumption that AA(r) and its first two
radial derivatives are continuous at aI/Z. The result is:

_ 2 a
AA(r) - K0 + Kir + Kzr + Kar ,

in which the coefficients are:

99

K. = kT{ -6n(1-y) + 3 [y/(1-y)]2} - (nPaia/G)

x1 = -(kT/a,){(6y/(1—y)1 + 13[y/(1-y)12} + «9.12
K2 = (kT/612)([12y/(1-y)1 + 13[y,(1_y,]2, - ZuPa1

_ 4
K3 -§7¢P e (6.73)

In Eq. (6.73), y=ua13p/6, is called the packing fraction of
the hard sphere solvent (it is the fraction of space
occupied by the fluid). The salvation free energy, AA(r),
is for a constant T,V,N system, however this is the same as
the Gibbs free energy in a T,P,N system provided the average
volume (V) in the latter is equal to the exact volume V in
the former system.14 We can therefore equate AA(r) with the
cavity energy, gc.v, defined in the previous section.

Figure 14 shows gen/kT as a function of the reduced cavity
radius R’= ant/a2 for a hard sphere solvent with packing
fraction y = 0.5. The open circles are those described by
equation (6.73) while the closed circles are described by
the exact relation (6.60). The cavity free energy increases

monotonically as a function of cavity radius and is always

positive.

GCAV (kcal/mol)

20

15

10

100

 

 

  
 

 

 

I I I I I I I I I I I I I I I I I

_l
‘1
y=0.5, al=5A _
— 0 SPT —'
>< Eq.(6.60) a
-— —1

lyvl XL k IXI I I I I I I I I I I I I I I . I

2 4 6 8 _ 10

R12 (angstroms)

Figure 14. gcavm‘zi/k’r 1:, cavity radius R“.

101
To summarize, the free energy required to create a
cavity large enough to accomodate a hard sphere of diameter
a2 in a hard sphere fluid with particle diameter a1 and
number density p1 is, by rearranging terms in Eq. (6.73):
- - 2 -
Sc“,- kT{[6y/(1 y)][2(r12/al) (rm/an]

+[18y2/(1-y)2][(r12/61)2 - (rm/a!) + 1/4 1 - mum}

I . (6.74)

O’HH

+uPa3[% (rm/a1)3 - 2(r12/a1) + (rizlail -

with r12: (a1+a2)/2 , and y=naaap1/6. ‘The total excess

chemical potential is defined by equation (6.54):

A"; = gcsv+ gint ° (6°54)

An aside for experts: The SPT also leads to an equation

of state for the hard sphere fluid which is exact up to the
third virial coefficient, which agrees over the entire range
of fluid density with the equations of state obtained
through machine computation, and which is identical to the
equation of state obtained by exact solution of the

Percus-Yevick equation.63’35

6.6 Application to Real Liquids
The SPT is extended to the study of real liquids by: (a)

using the actual liquid density of the solvent and (b) using

102

a simple model and the heat of vaporization to determine the
optimal hard sphere diameter a1 of the solvent. This model
essentially examines the salvation process for the solvent
in its own liquid. The excess chemical potential per male
for a single component gas-liquid system is given by
equation (3.5):

*
= -RT ,

where 06 and p9 are the molar densities of the liquid and

gas respectively. Assuming the gas phase is ideal, we have:
*
An = -RT 6n ( at RT / P9 ) , (6.75)

where P9 is the vapor pressure of the liquid. We now apply
our model to this system and equate the excess chemical
potential to that of cavity formation plus attractive
interaction:

*

Au=~RTaOn(szT/P9)=g +3 . (6.76)

cav int

It follows from equation (6.76) that:

Ah = h 4-)) As

csv int' cav int

(6.77)

H
m
+
W

-*
The excess molar enthalpy. Ah , can be related to the molar

heat of vaporization, AHv as follows:

103

-AH ( 1 - Pv / RT) = AE*+ RT - RTza
v 6 L
= h + h - RT + RTzd s -AH (6.78)
t C. v

cav in

where the heat of vaporization is defined as, AHV= (H "-
s

H ), a is the coefficient of thermal expansion for the
liquid it
pure liquid, and VI. is the molar volume of the liquid. We
assume ,as usual, that Pvt/RT << 1, so we can neglect it
(see appendix # for a detailed derivation of 6.78).

We shall now look at the interaction term. The molar
energy of vaporization of a normal liquid is approximately

given by:36’64

Q
AU;m -2nNApI u(r)g(r)r2dr , (6.79)
0

where u(r) is the pairwise additive intermolecular potential
, g(r) is the radial distribution function (see Eq.(6.31) ),
and NA is Avogadro’s number. On the other hand, the molar

energy of interaction can be derived from equation (6.33):

Q
“int: 4uprIau(r)g(r)r2dr . (6.80)

Hence, it follows that for such a liquid:

AU = -u / 2 . (6.81)

104

Now we invoke the thermodynamic relation:
H = E + Pv = K + U + pv , (6.82)

where E is the total internal energy, and K is the kinetic

energy. We may then reexpress the heat of vaporization as:

AH = H - “C = (Kq- Kl) + (Ug- Ut) + P(Vq- Vt)

a RT + AUv . (6.83)

We have assumed that V§>>Y£, K7“ KL’ and the gas is ideal.

Plugging (6.81) in (6.83) we find:

I1 = 2RT - ZAH . (6.84)
int v
Once again we use the fact that, in a liquid, the PV term is
negligible, hence u‘n{= h .

Finally, by introducing equation (6.84) into (6.78) we
get the following relation for the standard molar enthalpy
of vaporization:

AH =h +6:t RT2+RT . (6.85)

V CIV
This is a general equation which should be satisfied by any
theory which predicts cavity terms. Now we can use this to
determine the effective hard sphere diameter in the SPT.

The enthalpy of cavity formation is found using standard

105

thermodynamics: i.e. he": -T2(a(gc"/T)/8T)P 55:
hm: a, 672 (y/z) [(6/2) + (say/22) + 11 . (6.86)

where z=(1-y), y is the packing fraction, and “L is the
thermal expansivity of the liquid. We have also assumed that
(1/v)(av/6T) = -(l/y)(ay/8T). Putting this result in (6.85)

we find, finally:

2
AB = RT + a, RT2[ (115%) . (6.87)
' (l-y)

The SPT is related to real fluids by using the heat of
vaporization, density, and thermal expansivity to find an
effective hard sphere diameter for the fluid. It is by no
means a first principles theory since it uses the
experimentally measured values for these variables (there is
no theory to date that can predict the density of even
atomic liquids let alone molecular liquids). However, the
SPT is consistent with our current goal of being able to
predict solubility by'knowing only bulk properties of the
solute and solvent. Other techniques exist to determine the
effective hard sphere diameter of the solvent, a‘, but they
require a detailed knowledge of the intermolecular potential
and lots of computer simulation (or very simpLe molecular

43,49

structure). This option was not presently available to

us.

106

6.7 Evaluation of the Cavity Term:

Table IV lists the heat of vaporization, number density,
and thermal expansivity for each of the solvents studied.
Since our experiments were generally done in a range of
5-50°C we have chosen an intermediate temperature of 25.0°C
to evaluate a‘. In theory, a1 should be temperature
independent. In order to evaluate the number density we

require the mass density29’30’65

and the molecular weight,
i.e.; p'(mol/cm3) = pu(g/cm3)/M.W.(g/mol).

Table V and Figure 15 show the calculatated values of a1
for the solvents at 25.0°C, we have not included nonadecane,
eicosane, dodecanol, and tetradecanol because they are
normally solids at this temperature. The values of ai were
calculated from solving equation (6.87) for y and using the
known density to solve for a1 by the relation;
a1=(6y/xp1)“3. ’The points on Figure 15 vary monotonically
in each homologous series with number of carbons but they,
bear no simple relation to actual chain lenghts. In
alkanes, for example, the C-C bond length is about 1.5A. Of
the hydrocarbon solvents we studied, the more polar
molecules tend to have a larger effective radius than their
nonpolar analogues. One can obtain values of a1 other ways
besides the one we use here. For example, one can use gas
viscosity,66’67 or second virial coefficient data.68’69

However, for all but a few of the short alkanes (and for

those) molecules the agreement with our 'values of a1 is

107

good55) the results are limited by lack of available data.
In order to evaluate go" we use Eq. (6.74). For the
cavity radius we have rm= (a1+ a2)/ 2. For Xe we took the

hard core molecular diameter to be a2= 3.973A; this is the
potential parameter a (or 21161.0) in the Lennard-Jones
(6-12) potential for Xe.73

The calculated values of go" are in Table V and Figure
16a. All the values of go" are positive and range from 2.5
to 9.4 kcal/mole. This shows that for a hard sphere fluid
it always takes a positive amount of work to make a cavity.
The fact that go" is positive means that its contribution
will always tend to lower solubility (L=e-Au/"), therefore,
systems with low solubility are dominated by repulsive
interactions (as intuitively expected). In contrast, the
experimental quantities Au; for these solvents, with the
exception of formic acid, are negative and in the range from
about -0.4 to -l.0 kcal/mole. Since, by equation (6.54),
Au; is just the sum of gen and 81:". these results imply

that g1M must be negative and its magnitude must be a few

to several kcal/mole. This isn’t surpring since one would

Table IV. Heat of Vaporization,

 

108

number

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

density, and thermal expansivity for
organic solvents at 25.0°C.
alkanes
n-C AHV(::f:) pntxtogéclattso-gc)
5 6.39 5.19 1.558
6 7.54 4.5788 1.368
7 8.74 4.0852 1.266
8 9.92 3.6851 1.183
9 11.10 3.3535 1.114
10 12.28 3.0740 1.071
11 13.46 2.8399 0.9704
I—12 14.64 2.6378 0.9417
13 15.83 2.4613 0.9432
14 17.01 2.3083 0.9300
15 18.20 2.1706 0.9097
16 19.22 2.0524 0.8986
17 20.6 1.9409 0.879
18 21.7 '1.8434 0.8661
cycloalkanes
n-C AHV(::::) pntxiogécIat(10-3C)
5 6.85 6.1644 1.3298
6 7.90 5.5400 1.2475
8 10.36 4.4702 0.98805

 

 

 

 

 

 

 

109

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table IV. cont .....
alkanols

n-C AHV(::?:) pn(x10§écIa£<io_3C>
1 8.94 14.79 1.189
2 10.18 10.267 1.093
3 11.51 8.0178 1.006
4 12.50 6.5509 0.9265
5 -13.61 5.5443 0.8804
6 15.00 4.8128 0.8720
7 16.20 4.2446 0.8828
8 17.00 3.8043 0.8491
9 18.60 3.4444 0.7875
10 19.82 3.1453 0.8283
11 21.00 2.8988 0.8105

carboxylic acids

n-C AHV(::::) pn<x1036c4atrio—3c>
1 11.03 15.891 1.0249
2 12.49 10.475 1.0114
3 13.7 8.0341 1.0974
4 15.20 6.5163 1.0693
5 16.56 5.5117 0.99941
7 18.1 4.2244 0.91035

 

 

 

 

 

 

 

110

Table IV. cont .....

alkanals (aldehydes)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n-C AH (kcal) p (xiogécIa (io-3c>
v mole n 3
3 7.09 8.2083 1.4619
4 8.05 6.6543 1.3084
5 9.17 5.6298 1.2474
7 11.40 4.2907 1.0333
perfluoroalkanes
n-C AH (kcal) p (xioiéCIG (iO-3C)
v mole n 5
6 7.606 2.9844 1.6698
7 8.69 2.6978 1.5047

 

 

 

 

 

 

8 9.77 2.4216 1.4119

 

Hard Sphere Diameter, a1 (A)

111

 

 

 

 

10 ’- I I r I I I I I f I I I I I I I l T I
r _
9 __ .2
.. o _
_ o _
_ o a
_ o -
— O —-(
8 c o -
_. )1 O ..
.— X )1 O —(
__ X __
7 _ )1 O ..
- X E o O Alkanes ‘
I- -(
- 31 5 ° )1 Alkanols 7
6 — a O , —
- Cl Ac1ds ~
- E 6 <> 9
.— .l
_ B + o 0 Cyclo 4
5 — X Perfluoro —
r- .g + '7
- + Aldehydes 7
4 fl 1 I l I l L I l I l I i I I I L 1 d
O 5 10 15

Number of Carbon Atoms

Figure 15 Effective hard sphere diameter for
45 organic solvents 1; number of carbon atoms.

112

Table V. SPT calculations for organic
solvents. The second column is the effective
hard sphere radius. The third column is

the packing fraction. TEEI£ourth is the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

cavity free energy for Xe.
alkanes
n-C a‘(A) y gégei/mou
5 5.476 0.44614 3052
I 6- 5.877 0.48656 I 3396
7 6.226 0.51611 3645
8 6.543 0.54055 3861
9 6.838 0.56143 4052
10 7.108- 0.57791 4191
11 7.386 0.59925 4463
12 7.623 0.61176 4577
13 7.836 0.62006 4613
14 8.044 0.62914 4682
15 8.251 0.63838 4765
16 8.436 0.64516 4805
17 8.635 0.65423 4907
18 8.813 0.66075 4958
cycloalkanes

n-C a1(A) y gcéeal/mol)
5 5.286 0.47681 3766
6 5.582 0.50450 4019
8 6.233 0.56678 4775

 

 

 

 

 

 

113

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table V. cont....
alkanols
n-C a1(A) y gcé$°"m°"
1 4.082 0.52668 7366
2 4.686 0.55305 6854
3 5.156 0.57554 6703
4 5.580 0.59588 6667
5 5.948 0.61083 6595
6 6.274 0.62238 6508
7 6.567 0.62929 6327
8 6.843 0.63822 6274
9 7.134 0.65467 6521
10 7.358 0.65612 6279
11 7.591 0.66382 6289
carboxylic acids
n-C a‘(A) y gcésaL/moi)
1 4.092 0.57015 9429
2 4.746 0.58623 8143
3 5.189 0.58790 7131
4 5.610 0.60244 6872
5 5.986 0.61891 6854
7 6.606 0.63774 6601

 

 

 

 

 

 

 

114

Table V. cont....

alkanals (aldehydes)

 

 

 

 

 

 

 

 

 

 

 

 

 

n-C a1(A) y gcéeal/mol>
3 4.780 .46945 4210
4 5.239 .50109 4337
5 5.623 .52417 4414
7 .6.342 .57316 4821
perfluoroalkanes
n-C a‘(A) y gcésai/moi>
6 6.662 0.46209 2520
7 7.045 0.49389 2740
8 7.417 0.51733 2876

 

 

 

 

 

 

115

 

 

 

 

 

 

 

 

 

10000 i T r T I 7 I I l I 7 fl 1 1 I
‘3 1
L T = 25.00 C O Alkanes ‘
8000 .. C) ‘1 Alkanols «
)—
- C1 Acids _
A __ )1 -l
0) C1 0 Cyclo 4
3 h K 11 El] 0 .4
8 ~ :1 11 g r! X Perfluoro'
)1 I! I! -(
E 6000 r t Aldehydes -—
3 L -(
z ' + o O C' '3 d
. o _
QB ‘ + + t O O O 0 q
4000 L— o O o 0 _
o
b- 0 d
O
o .
h K x d
L X .
2000 ' . . - 1- 1 . 1 | . .. . I . . ..
O 5 10 15 20
Number of Carbon Atoms
500 ,_ a II I 1 I l 1 I I l I I I l I 7 I 1 . 4
Z i
250 :— O Alkanes -j
“' K Alkanols 7
2 ” C1 Acids .
E Q t 0 Cyclo 1‘
> t X Perfluoro <
(0 " .7
3 -250 :- 0 f Aldehydes J
- - -(
IN _ x x -(
1 )— x X n -(
Q -500 "" ’2‘ xx 1‘ x “
.- E n n x X -(
.. c] C] E] ' O -l
.. + + O O O .4
_ o o o 4
-750 ;- o o O o 1
- 0 .
.. 9 0 ..
- 2L 1 l L l I l l L l 1 I l l l L
-1000 . 9 L
0 5 10 15 20

Number of Carbon Atoms
Figure 16. (a) g“W for Xe in 45 organic solvents
at 25.0°C. (b) corresponding excess chemical
potential, (in: = -RT&nL..

Molecular Diameter 02 (A)

116

 

 

 

 

 

1"— _
O- LiLlllLll 111 1 l 1 ' Lll'ill"‘.ll]
0 l 2 - 3 4 5 6

Polarizabiiity a (10'24 cma/molecule)

Figure 17. Molecular diameter ¥§_polarizability
for several inert gas solutes. Solid line shows

extrapolation for noble gases.

-—R'I‘ [u(L) = Ape” (kJ / mole)

117

 

 

O Tetradecane -*

U Decane

0 Hexane

0‘

O

I
Ln

 

'1,[ ..l I
111' "lll‘III

 

fl

C1

_.I

d
.1
d
_1

q

 

 

O 1 2 :3 4

Polarizability a (10'24 c-ma/molecule)

Figure 18. Excess chemical potential 11 solute
polarizability for noble gases in Tetradecane,
Decane, and Hexane. Filled in points along ordinate
are the SPT predicted values for a hard sphere
solute of 2.58A.

-—R‘I‘ ln(L) = Ape” (cal / mole)

4000 . ei,, , r, r
k t l ' T T ’ ' 7

t. T = ZSIP C ‘
3000 — __‘
'_' 1

U Methanol ~

2000 0 Ethanol —

118

 

 

 

 

 

 

-24

Polarizability a (10 cma/molecule)

Figure 19. Excess chemical potential 1; solute

polarizability for noble gases in polar solvents

methanol, and ethanol. Filled in points along

ordinate are the SPT predicted values for a hard
sphere solute of 2.58A.

119

intuitively expect the attractive interaction to lower the
free energy of solvation and increase the solubility.
Figure 16b shows the experimental values of Au; for Xe in
the alkanes; it is encouraging to note that the SPT has much
of the same systematic behavior.

The physical significance of gc.v is supported by

extrapolation of measured solubilities of inert gases in

55’66’67 In this technique the excess chemical

solvents.
potential is plotted versus solute polarizability, up, in a
single solvent at fixed temperature. It has been shown that
extrapolation of this data to zero polarizabilty is
equivalent to finding the free energy required to introduce
a hard sphere of diameter 2.58A into the real solvent.55’72
The hard core diameter is determined by extrapolating a plot
of solute diameter versus polarizabilty. Such a curve is
shown in Figure 17 for the inert gases He through. Xe.
Figure 18 shows a plot of -RTlml. versus up for the inert

gases in n-hexane. Extrapolating to ap=0 is equivalent to

writing:

* .
Auz (ap= 0, a2- 2.58A,T) - gc.v(a1,az-2.58A,91,T) . (6.88)

This value can then be compared to the SPT prediction of
equation (6.74) for a solute of diameter 2.5&A. As Figure
18 shows, the predicted gcw agrees to within 2% of the

extrapolated value. Pierotti has used this test on a wide

120

variety of non-polar solvents with good results.55 ‘We

performed this test for some of our solvents in order to see
if the SPT was applicable. Even for alkanes as long as
n-Cunao the calculated gc" agrees with the extrapolated
value to within 5x ( we couldn’t test most of our solvents
in this manner due to lack of experimental data). The case
is different for the polar solvents. Figures 19a and 19b
show the extrapolated and SPT results for methanol and
ethanol. Clearly for these solvents there is a discrepancy.
This can be traced back to our derivation of the effective

hard sphere diameter from the solvent heat of vaporization.

We recall that Eq. (6.79):
0
AUva -2nNApI'u(r)g(r)r3dr , (6.79)
o .

holds for normal liquids in which there are only two -body
radial forces. For polar liquids one must account for
many-body correlations. This renders equation (6.79), and

therefore equation (6.87):

2
AB = RT + a:z RT2[ 13131-3?) , (6.87)
' (l-y)

invalid. We can also see that for methanol and ethanol the
descrepancy between the two results decreases as the solvent
chain length increases. Although we can’t test it at this

time; we are confident that for the longer acids, alkanals,

121

and alkanals the SPT works much better. It should also be
mentioned that for the highly polar solvents the effective
hard sphere diameter evaluated from solubility data together
with the SPT agrees very well with other experiments; this
involves using the SPT relation for gc‘v to evaluate a1 from

the extrapolated result.72’73

In other words we use a plot
like Figure 19 to find the free energy to dissolve a hard
sphere of 2.58A. The SPT is then applied to this result to
give the effective radius of the solvent a1. This has
proven to be a powerful method for estimating the molecular
diameter of a solvent. Such a technique has even been
applied successfully to water, which is a very complex
11gu1d.7°’73

From the calculated values of gc"(T), we obtain the

enthalpy of cavity formation by the Gibbs-Helmholtz

equation:

h = -'r"’[a(gc"/ T)/ 8T] . (6.89)

OIV P

In this calculation .we again neglect the temperature

54,55

dependence of a‘. Using our previous expression for

g (T), we find:
IV

C

36y
(1-y)

 

he": 31.2% Téi [13; [A] + lel + 1] .

where we have used the notation,

122
[A] = [ 2(r12/ a1)2 - (r,2/ a1) 1 .and

[B] = [ (r12/ a1)2 - (r12/ a!) + 1/4 1 . (6.90)

This expression reduces to equation (6.86) in the limit
a =a (i.e. r /a =1). Figure 20a shows h (25°C) versus
1 2 12 1 cav

for our solvents. For comparison Figure 20b shows
carbon
-1:

the experimental excess enthalpy of salvation Ahz for these
solvents.

One may obtain the entropies of cavity formation,
s (25°C), from g and h by the thermodynamic relation
ClV CIV 08V
s = (h ' 8 )/ T. The 3 values thus obtained are
CIV CIV CIV 0"
shown on Figure 21a. However, they appear to be unreliable.
They do not have the systematic dependence with series that
one might expect, nor do they correlate with our
experimental results (Figure 21b). In order to check a
possible cause for this we have plotted in Figures 22-24,
respectively, the heats of vaporization (ARV), number
density (pa), and thermal expansivity (5.2) versus carbon
number for our solvents. Of these three input parameters in

the SPT only the thermal expansivity seems to vary roughly

in some of the homologous series. Perhaps better

he“, (cal/moi)

AHz'. (cal/moi)

123

 

 

 

 

 

 

 

 

10000 I . . . eI . ,
" i
__ O Alkanes :
' ’1 Alkanols #
—- Cl _.
8000 __ U 0 Cl Acids 4
- D 0 Cyclo i
_ x n n x K :x n :x Perfluoro .
6000 — t Aldehydes q
a -l
- O o .4
.. O O
_ t * ‘ 0 O o O 0 j
o
4000 - 0 0 ~—
... o O '1
- x .l
- X X .4
L- I -4
2000 #4 l l l l l J 1 J 1 I. l L 9 l
0 5 10 lo 20
Number of Carbon Atoms
L fir I r T I Y I f I T 1' d
t O Alkanes *
r K Alkanols ‘
- 1000 - o Ac1ds —
L ocydo ~
” x x X Perfluoro ‘
t x t Aldehydes *
-1500r- —*
- # .
" O + 1
9 e
o
P ‘1 n g x z 1“ O O O "
-2ooo+— C" o O o —.
l- 0 O O 4
I o i
- — —1
2500 l L f£ 1 l J; l L 1 l l :l_ I ‘ L l l
0 5 10 15 20

Figure 20. (a) h
CQV

Number of Carbon Atoms

for Xe in 45 organic solvents

11 number of carbon atoms.(b) corresponding

1:
experimental excess enthalpy, Ahz .

ASZ', (cal/moi K°)

124

 

 

 

 

 

 

 

 

4 I I I I I I I I I I I I T I I I I I I q
_ 0 Alkanes :
.— D .I
3 -— a x Alkanols —
: O U Acids :
6‘ - .
a: 2 — 0 Cyclo —
s 2 *
E : xx 0 + x D X Perfluoro 1
§ 1 _0 X x + Aldehydes —
3 : =1 + 9 o o o .
n .
3 " O O K O O “
8 O _ )1 o I! O o O _
U) h + n O "
I- )1 ' O '1
— K n -1
_1I-— X ——
: o
_2 l l l l J l l l l I 1 l l l I L l l l
O 5 10 15 20
Number of Carbon Atoms
_2_5 I I I I I I I I I I I I I I I I I I
I x 3
I x :
-3.5 :- D + + _q
-0 + .
_ o + .
__ c: . 0 ° .
-4 0 _ O O O O O —
I 0 o .
.. D o x x ‘
-4.5 :" n n x x O Alkanes i
.. F x " x I! Alkanols ‘
: x 0 Acids :
-50 -- 0 Cycle m
: X Perfluoro :
~ 1 + Alldehydes ~
P
_5.5 L l l L ¢ 1 l 1 l l l 1 1 l l l l
O 5 10 15 20

Number of Carbon Atoms

Figure 21. (a) 3mV for Xe in 45 organic solvents
11 number of carbon atoms. (b) corresponding
*

experimental excess entropy, A3,-

Heat of Vaporization, AHV (keel/mole)

25

20

15

10

125

 

 

+ .

8
l

O
x
C]
o
x
+

l

 

 

5

10

O -4

O --l

O i

o s

O .1

Alkanes ‘

Alkanols ‘

Acids -

Cyclo 1

Perfluoro -*

Aldehydes ‘
l I l 1 l l

1 5 20

Number of Carbon Atoms

Figure 22. Heat of vaporization for 45 organic
solvents at 25 C 11 number of carbon atoms.

Density, ,0n (102‘ molecules/CHIS)

126

 

 

 

 

Li I I fr I fiI I I I r I I I r I
15 "-11 —'
' T = 25.0° C O Alkanes ~
“' ’1 Alkanols «
' U Acids a
' g 0 Cyclo 4
10 '— X Perfluoro ‘
- t Aldehydes ‘
.. a .
.. a i
_ 0 p
o
5 — 5 5 o -—
.. 5 1
5 6
8
I— X X g .I
X ‘3 O O O
.— O O O -I
b ' , . . L I L . 1 1 L . 4
O J l l l l J
0 5 10 15 20

Number of Carbon Atoms

Figurg 23. Number Density for 45 organic solvents
at 25 C 11 number of carbon atoms.

'l'hermal Expansivity, (XI (l/Co)

2.00

1.75

1.50

1.25

1.00

127

 

 

 

 

.— I I I I I I I I ‘1
" O Alkanes ._‘11
" x K Alkanols 1
l- ...
- c1 Acids 4
_ o g
_ X '
_ ¢ 0 Cycio —_I
‘- 0 X X Perfluoro j
' o .
- ‘* O ‘- Aldehydes 1
I_ + o —n"
- O .
_ :1 Cl :1 + o .
- 11 x g O O . d
— x u 3 -I
_ z x _q
' :i
— 4 l I ' ' 1° l ' I 1 I l J1
0 5 10 15 ‘20

Number of Carbon Atoms

Figure 24. Thegmal Expansivity for 45 organic
solvents at 25 C 11 number of carbon atoms.

Table VI. Cavity enthalpy and entropy.

128

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

alkanes
5 3251 0.67
6 3482 0.29
7 3710 0.22
8 3900 0.13
9 4067 0.05
10 4224 0.11
11 4332 -0.44
12 4467 -0.37
13 4613 -0.002
14 4739 0.19
15 4850 0.28
16 4929 0.41
17 5067 0.54
18 5148 0.64
cycloalkanes

n-C cav(§£§) scav(S£%kl
5 3725 -0.14
6 3980 -0.13
8 4398 -1.26

 

 

 

 

 

129

Table VI cont....

 

 

 

 

 

 

 

 

 

 

 

alkanols
n-c ...(fi -
l 7760 1.32
2 7027 0.58
3 6675 -0.10
4 6440 -0.76
5 6291 -1.02
6 6336 -0.58
7 .6335 0.03
8 6190 -0.28
9 6289 -0.78
10 6366 0.29
11 6385 0.32

 

 

 

 

 

carboxylic acids

 

 

 

 

 

 

-C g, 3.9}. .
n cav mol. cav mall:
I 9730 1.01
2 8538 1.33
3 8046 3.07
4 7820 3.18
5 7620 2.57
7 7017 1.39

 

 

 

 

 

130

Table VI cont.....

alkanals (aldehydes)

 

 

 

 

 

 

 

 

 

 

 

 

-c 21'.) 32'. ‘
n cav mot cav moLK‘
3 4563 1.18
4 4510 0.58
5 4611 0.66
7 4722 -0.33
perfluoroalkanes
_C 2}. L“. .
n cav moi. cav mot]:J
6 2893 1.25
7 3046 1.03
8 3166 0.97

 

 

 

 

 

131

measurements of these are required. In any case it appears
as though the son’s are approximately centered on zero,
whereas the total excess entropy, As: , is about -4 cal/mol
it. We conclude that the SPT alone does not adequatly
predict the salvation entropies. This is in sharp contrast
to previous workers, who used the mole fraction scale to
analyze their results.55’67’7o’71’74

A surprising result of these SPT calculations is that
the cavity formation process is almost completely enthalpic
(energetic). Table VI clearly shows that the entropic term
is less than 10% of the enthalpic term (this is even more
enthalpic than our experimental data; recall Ah:/TA§:81.8).
This is in strong contrast to a real hard sphere fluid which
is completely controlled by entropic considerations. The

key difference between these two situations can be seen in

equation (6.90):

36y
(l-y)

 

hm: RTza‘c 1%; [If—y [A] + 2[13] + 1] . - (6.90)

If the fluid were truly hard spheres then the thermal
expansivity and hence the cavity enthalpy would be zero. By
putting in the real solvent behavior we are accounting for
the full interaction between solvent molecules. At first
this may seem improper, but if we write out explicitly the
expression for the excess chemical potential due to

a
repulsive interactions Aph from equations (6.52) and (6.53)

132

we find:

A“: = _kT m jd¥"exp[-BU(?~") -thl .
h fd?"exp[-BU(;')]

It is clear that in evaluating go" we should include the
full solvent-solvent interaction. It is only the
salute-solvent interaction which is broken up. Whether the
SPT properly accounts for this by using the real expansivity
requires further testing.

To complete our present discussion Figure 25 shows the
calculated values of gc“(T) for six typical solvents, one
from each homologous series. These results were obtained by
applying equations (6.74) and (6.87) at each temperature for
which the calculation is made; i.e., a1(T) values were
obtained from equation (6.87) for each temperature. The
AH"p(T) data for these calculations were obtained from

Watson’s relation and critical temperature (Tc) data 33:

0.38
AHV(T2) = AHV(T‘) [(1- T2/ Tc) / (1- T1/ T9] (6.91)

The calculated temperature dependence of gc"(T) on Figure
25 is in the opposite direction from the observed values on
Figure 9. However the slopes of the lines on Figure 25 are

not the entropies of cavity formation.

a... (cal/Incl)

Cavity Energy.

Figure 25. Cavity free energy, 9

133

CGV

in 6 representative organic solvents.

8000

7000

6000

5000

4000

3000

2000

1211' £01:

 

 

 

 

 

 

 

 

 

 

I II I II [ill I II III I‘II I II I II
"" u-I
:__ n-c,H,coon __':
_"__ tremor!

.. ..J
— a
- X~ n-C3H7CH0 -
P -
— n u d
I- n x a: .-
' “‘CJHI ‘
- d
: NH n-CJ' 1‘ :
_1_ll I [II [I I II I II I 11.1 In I ll

10 20 30 4O 50

. 0

Temperature. C

Xe

134

6.8 The Interaction Term

After the cavity is formed in the hard sphere solvent,
one introduces a solute molecule into the cavity and lets it
interact with the solvent molecules; the free energy
associated with this step is glut. The natural next step is
to calculate g1M for Xe in our solvents.

This proves to be very difficult even for our simplest
solvents. Since zinc is a Gibbs free energy one may write

it in terms of enthalpic and entropic contributions as:

glut: hint- Tslnt m uint - Tsint ' (6°92)
This means that the two principal quantities which determine
giM are the potential energy of interaction, ulnt’ and the
entropy of interaction, sint. Both of these are difficult
to calculate.

Solvent reordering, caused by the solute-solvent
interaction, is probably the origin of most of the entropy
of interaction and one therefore expects that sun} 0.54
Neff and McQuarrie75 include a term in their expression for
gun which may account for this entropy; it gives the
contribution to the free energy (due to changes in
solvent-solvent structure when a solute is introduced. This
term [equation (27) of Ref. 74] depends on a derivative of
the solvent-solvent distribution function, g11(r), which is
hard to calculate.

55,71

neglect sint and make the

Other workers

135

approximation that gins uint. I believe that sun cannot
be neglected, but rather may be commensurate with the
entropy of salvation As:. This contention is supported by
consideration of the entropy of interaction in pure solvents
for which it can sometimes be estimated. For example, Reiss
at al.54 have calculated the heat of vaporization for liquid
Ar at its boiling point using the SPT and thermodynamics.
From their results [equations (4.3)-(4.11) and Table II of
Ref. 54] one finds sin£= -4.6 cal/mol K.

We have made a similiar calculation for other pure
liquids and find sing: -10 cal/mol K for CCl‘ at 25°C, and
sun: -12 cal/mol K for n--CGHH at 25°C. The entropy of

interaction is found by combining equations (2.5), (6.54),

(6.84), and (6.92):

*

A" = -RTm(pAC/pq) = gcav + uint - Tsint
= g + 2(RT - AH ) - Ts
cav v int
sint= [RT&n(pz/ag) + gcav+ 2(RT - AHV)] / T . (6.93)

These sint’s must properly be compared to the salvation
entropies for the pure solvent, A§*,which one finds as
follows: for the pure solvent we plot the excess chemical
potential, Ana“, versus T along the gas-liquid coexistence

line. The temperature derivative is then written as:

(Inf/aw); (emf/aw), + (adu*/8P)T(dP/GT)O . (6.94)

 

136

where the subscript c means the derivative along the
coexistence line. We then utilize the thermodynamic

identity; vi: Ufifi/BP)T , and equation (6.94) reduces to:
a s *
(dAu /dT)c= -As + Av (dP/dT)c . (6.95)

In Eq. (6.95) Av* is the molar volume of salvation, which is
approximately equal to the molar volume of the pure
liquid.16 Values we obtained for the corresponding entropy
defined by equation (6.95) are As*(CCl‘) = -9.1 cal/mol K,
and As*(n-CSH“) = -10.6 cal/mol K., both at 25°C. For
these two solvents we calculated scfl’s of about 2 cal/mol
K. Thus for pure solvents it appears that s1M makes an
important contribution to the total entropy.

The interaction energy for a single solute molecule with

the solvent may be written as:

...

a . 9
uint- pLI g12(r) u12(r)dr , . (6.96)
where the terms are as defined previously. Evaluation of
equation (6.96) is difficult for nonspherical solvents. An
approximate method which assumes spherical symmetry has been

55’71 In. this technique a

used by’ Pierotti and. others.
Lennard-Jones interaction potential u12(r) is assumed for
u12(;) and the solvent distribution is taken to be uniform,
i.e., 812:1’ outside the cavity radius r12. The integration

is over the solute volume, i.e., from r12 to infinity.This

137

is equivalent to an effective Lennard Jones potential with a
hard sphere cuttoff at 012: (01+az)/2, as illustrated in
Figure 26 (this is a more reasonable cutoff than ro because
the actual potential is much steeper for r<o12 and is thus a
better approximation to a hard sphere. Such a perturbation
scheme was first suggested by Barker and Hendersen76).
Lennard-Jones potential parameters are usually chosen to fit
either solubility data or gas phase properties of the pure
solvent and pure solute. However the potentials associated
with the solvent are not consistent with the properties of
bulk solvent liquids.

I shall now outline a model which is more consistent
with our present goal of estimating solubility from
knowledge of bulk properties of the solute and solvent.
This model is presently designed for non-polar solvents, and
will be used to analyze the Xeealkane system. The procedure
is as follows:

1) We treat the molecules effectively as spheres with a
radial Lennard-Jones interaction between molecular centers
and a hard sphere cutoff.(For polar solvents we would have
to include dipole terms in the interaction).

2) The solvent structure is determined solely by the

hard sphere packing (van der Waals picture). Therefore we

U(r) / e

138

 

 

I I I I I I I I I I I I

l
I
I
l
I‘- hard sphere cutoff —
I _ 7
I
I
I
I
|
l

 

 

 

 

 

Figure 26. Lennard-Jones potential with hard
sphere cutoff.

 

900

139

 

 

8IIIIIIIIIIIIIIIIIIIIIIII
L m
)—- "1

61.— _
.. . a
_ < — Verlet—Welss
L. _

4 — O Percus Yeviok —-

 

 

 

 

 

0 0.5 1 1.5 2 2.5 3
x=r/a1

Figure 27. Verlet-Weiss and Percus Yevick pair
distribution functions for hard sphere fluid.
Packing fraction = 0.5.

140

write equation (6.96) as:

m
_ he 2
“int- 4np1Ir g12(r) u12(r) r dr , (6.97)

where: 3:; is 'the radial distribution for a hard sphere
reference system, ro is the cavity radius (a1+ az)/2.

3) Use the SPT to determine the effective hard core
diameter of the solvent.

4) Use known Lennard-Jones potential parameters for the

solute.77’78’79

5) Apply standard mixing rules for the solute-solvent

interaction:35
_ 12_ s
“12- 4€12[ (Caz/r) (aiz/r) )
_ 1/2
512- (£152)
_ _ SPT
012- (01+ 02)/ 2 - (a1 + a2)/ 2 . (6.98)

6) To determine the Lennard-Jones energy parameter for
the solvent, 61, we apply this model to the pure solvent and

use equation (6.33):

o
- _ = ho 2
uint- ZRT ZAHv 4np1Ir 311(r) u11(r) r dr

0

u11(r) = 4‘1[ (at/rliz- (oi/r)6]

_ as”

o 1-- a . (6.99)

141

The primary difference between this model and previous
models is that we use the actual hard sphere distribution
function (Pierotti assumed. 812: l), and 'we evaluate the
Lennard-Jones energy parameter for the solvent, 61. from
heat of vaporization data (as opposed to using solubility
data to determine £1).

The hard sphere distribution function for a l-component
system of hard spheres is well known from computer
simulations and is uniquely determined by the density and
diameter of the fluid molecules (when applying this to real
fluids temperature only enters indirectly in that the
density is temprature dependent). We shall employ the
Verlet and Weiss 81expression for g(r). This is a
semi-empirical modification of the Percus-Yevick analytic
solution for g(r) which reproduces numerical results very

80’81 All of their results are in terms of the

accurately.
dimensionless variable x defined by; x=(r/a). Figure 27
shows the Verlet-Weiss pair distribution functions for a
hard sphere fluid with a packing fraction of 0.5. Near x=1
there is significant local structure, i.e., gv'(l)m5.8.
Since the potential drops of rapidly (r's) it is very
important to properly account for nearest neighbor effects.
This is why the assumption of a uniform fluid distribution
outside the cavity will probably greatly underestimate the
interaction energy. It is also clear from Figure 27 that

the fluid correlations fall off rapidly and by the time one

reaches x=3 the fluid is essentially randomized (no long

142

range order). We shall therefore adopt the following

convention for g(x), i.e.:

g(x) = 0 x<1
g(x) = g(x) lsx<3
verlet-weiss
g(x) = 1 x23 . (6.100)

The interaction energy for the pure solvent can then be

written as:

3

um=16n,p,af [Ix-1832:) [(1/x)‘°-(1/x)‘] dx

0
+I [(1/x)‘°-(1/x)‘)dx] . (6.101)

x-3

Evaluating the second integral we find:

uint(slksnes)=16fl€1p18: [ (INTI) - (1/81) ] , (6.102)

where INTl is the first bracketed term in (6.101). A
computer program was written to evaluate INTl (listing in

82 We

appendix: ##) using the standard. trapazoidal rule.
generally found good convergence in the integral by using
500 or more steps. Table VII lists the values of INTI for
the alkanes. We used the bulk number densities and SPT

packing fractions from Tables IV and V. Having evaluated

the integral we can apply equations (6.99) and (6.102) to

143

find the Lennard-Jones energy parameter, 81’ for the pure

solvent:

These are tabulated in column 4 of Table VII; column 4 lists
the values of 61 one would obtain by assuming g = 1. We can
now use these values to determine the interaction energy for

Xeealkanes. The Lennard Jones potential parameters for Xe

are 77; eZ/k= 221°K, and az=a2=3.973A. To evaluate the
solute-solvent pair correlation function, g:;, we use the
single fluid approximation35 which states that for a

multicomponent system 3 scales with a , i.e.:
~ H” H”
8pv(r/auu) = 3x (r/ax) , (6.104)
for all u,v. For our case this reduces to the statement:

gnu”) = g(x) . (6.105)

where xa2= r/at2 . Therefore we may write equation (6.97)

 

144

Table VII. Parameters to evaluate interaction
energy. Second column is the hard sphere cutoff.
Third column is the energy integral using the
Verlet-Weiss distribution function. Fourth
column is the resulting Lennard-Jones energy
parameter for the pure organic solvent.

 

 

 

 

 

 

 

 

 

 

 

 

alkanes

n-C atéA) INTl 1/k
5 4.7245 0.2731 477
6 4.9250 0.2722 526
7 5.0995 0.2703 585
8 5.2580 0.2677 648
9 5.4055 0.2648 708
10 5.6795 0.2620 773
11 5.6795 0.2576 834
12 5.7980 0.2546 921
13 5.9045 0.2524 973
14 6.0085 0.2498 1044
15 6.1120 0.2470 1115
16 6.2045 0.2448 1177

 

 

 

 

 

 

145

Tableas VIII. Predicted enthalpyC of solvation
for aXe in n-alkanes at 25. .

 

 

 

 

 

 

 

 

 

 

 

 

alkanes

n-C int hcav hpred
5 -5069 3251 -1818
6 -5300 3482 -1818
7 -5503 3710 -1793
8 -5654 3900 -1754
9 -5800 4067 -1733
10 -5919 4224 ~1695
11 -6023 4332 -1691
12 -6185 4467 -1718
13 -6214 4613 -1601
14 -6296 4739 -1557
15 ~6373 4850 -1523
16 -6422 4929 -1493 _

 

 

 

 

 

 

146

nint(Xe->alkan00)=161¢€12p18:2 [(INTI) - (1/81) ],

where the bracketed terms are exactly the same as for the

. _ 1/2 _ . .
pure fluid, £12— (5182) , and 812' (a1+ a2)/2 . Combining

this result with equation (6.102) and (6.99) we find:

In

u_ t(Xe-’alkaneo) = (Ci/€2)°'5(a12/a1)3[2RT - ZAHV]. (6.106)
Table VIII shows the values of‘umt for Xe in the alkanes.

6.9 Comparison with Experimental Data
We may now combine our calculated values for he and

IV

lunt to find the total enthalpy (energy) of solvation.
Figure 28 shows a plot of the combined theoretical term, h.=
lunv+ u‘n‘, and the experimentally determined enthalpy, A52.
versus number of carbons, for Xe in alkanes ranging from
hexane (Cs) t0“ hexadecane (016). The theoretical and
experimental data both have the same systematic tendency;
i.e. the enthalpy becomes less negative for the longer chain
alkanes. According to our model this means that the solvent
size dependence, a1, in the cavity term slightly dominates
over the solvent energy dependence, evin the interaction
term. The general agreement between the two sets of data is
within 20%. This points out one of the biggest problems in

calculations of this sort, viz;to describe the total

salvation process we usually calculate

Enthalpy of Solvation (cal/mol)

147

 

 

 

 

 

O l I I I l I I I I I I I I
_500 T htheory=hcav+hint —:
_ htheory (8:1) fl
—1000 _ -—-‘
' o o o 0 ° 0 0 o o o o ‘
: htheory (g=gv.w.) :
-1500 A x x x X —
- x x x x I
Z x x X 0 O o o 0 i
—2000 _ O O O -—J
7' O 0 exp _
: 0 Z
—2500 _ l 1 1 1 1 I l 1 1 1 ' I 1 1 l 1
5 10 15 20
n—carbon

Figure 28. Calculated and experimental excess
enthalpies of salvation for Xe in alkanes.

148

two separate processes which each have characteristic

energies that are opposite in sign and of larger magnitude
than their sum. This puts an extra burden of accuracy in
the calculation of each term. For example the discrepancy
between h. and Ah: for decane is about 16% of the
experimental value which in turn is only 7% of he" or 5% of
uh“. To claim that a model as simple as ours could do
better than this would be an exaggeration. For completeness
Figure 28 also includes the theoretical predictions that one
would obtain by using a random fluid distribution, g11= g =

12

1, throughout the entire calculation of 11 Agreement

int
obtained in this case is not as good as that obtained with
the hard sphere distribution of Verlet and Weiss.81

We expect that equation (6.106) should not just be
applicable to Xe, but should hold for any non-polar solute
with a well defined diameter, a2. In order to test this
hypothesis we have looked at solubility data for the inert
gases He, Ne, Ar, and Kr in several alkanes. Solubility
data was obtained from reference (24). The original data
were quoted on the mole fraction scale. In order to convert

data to the number density (Ostwald solubilty) scale we

inverted equation (3.20):
L(T) = RT / [Pv1(T)[(1/x2(T)) - 11] , (6.107)

where v1 is the molar volume of the solvent and P is the

pressure of the gas (all measurements were at 1 atm). Once

149

the temperature dependent solubilities were calculated, we
could easily find the excess chemical potential, enthalpy,
and entropy using standard thermodynamics . Table IX gives

4: -* -*
the values of Apz, Ahz’ and Asz for the inert gases in the

solvents hexane, octane, nonane, decane, dodecane, and

tetradecane. Figures 29a and bshow the predicted and

experimental enthalpies verses 6.01““ for the inert gases

in octane and tetradecane, respectively. The solute
77,78,79

parameters are shown below.

 

 

solute ezlk (oK) az= 82(A)
He 6703_-___—276§——__-
Ne 34.9 2.78
Ar 119.3 3.448
Kr 172.7 3.59

 

Table X lists all the relevant theoretical quantities that
we can calculate for the alkanes; i.e. g , ii , s , and
GOV CIV GOV

uint'

Once again our simple theory seems to capture some-of
the general systematics but is not sufficiently accurate to
make reliable predictions. A particular flaw can be seen in
the predicted values of enthalpy for the less polarizable
solutes like Ar, Ne, and He. Experimental data indicates

that for these solutes the enthalpy increases with solvent

size, however the theory predicts just the opposite. This

150

Table IX. Excess chemical potential, enthalpy
and entropy for He, Ne, Ar, and Kr in the alkanes,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

including: hexane, octane, nonane, dodecane,and
tetradecane.
Helium (20 C)
n-C A633) Ah:(§%t-) :(;:{K)
6 1801 2271 1.602
8 1975 2312 1.150
9 2048 2715 2.274
10 2102 2260 0.537
12 2214 2153 -0.2076
14 2279 1840 -1.496
Neon (20 C)
M 'Mz‘i‘ii" Abbi-3:) Ashfia
6 1587. 1699 0.383
8 1732 2046 1.070
9 1816 1902 0.295
10 1857 1941 0.288
12 2014 2096 0.283
14 2048 1838 -0.715

 

 

 

 

 

 

151

Table IX. cont .....
Argon (20 C)

 

 

 

 

 

 

n-c 41.3%) Ah:(§§-{-) 432133;“)
6 436 -297 -2.501

a 594 301 -o.999

9 631 43.9 -2.002
10 676 53.4 -2.123
12 756 276 -1.636
14 811 79.0 —2.495

 

 

 

 

 

 

 

Krypton (20 C)

 

 

 

 

 

 

 

n-C Au: (:3) A1133) As: (fix)
6 -168 -787 -2.11
a -49.3 -816 -2.62
9 2.86 -716 -2.46
10 44.5 -767 -2.77
12 110 -594 -2.40
14 146 -850 -3.41

 

 

 

 

 

 

152

 

 

 

 

 

 

 

 

 

 

 

 

Table X. 9““. cav, sec“. and “1m for He
Ne, Ar, and Kr in the alkanes, including
pentane through hexadecane
Helium (25 C)
M 9...<.%E> ...fi) ...—.9... 2%)
5 1766 1744 -0.07 -529
6 1965 1857 -0.36 -564
7 2109 . 1971 -0.47 -595
m8 -w-~;;55-« 2065 -0.57 -620
9 2345 2147 -0.66 -644
10 2428 2227 -0.67 -664
11 2580 2273 -1.03 -682
12 2647 2340 -1.03 -706
13 2672 2417 -0.78 -714
14 2715 2482 -0.78 -729
15 2765 2538 -0.76 -742
16 2791 2579 -0.71 -752

 

 

 

 

 

 

 

153

 

 

 

 

 

 

 

 

 

 

 

 

Table X. cont .....
Neon (25 C)
5 1891 1888 -0.01 -1344
6 2103 2011 -0.31 -1430
7 2258 2136 -0.41 -1506
8 2392 2239 -0.51 -1566
9 2510 2330 -0.61 -1622
10 2598 2416 -0.61 -1671
11 2762 2469 -0.98 -1716
12 2834 2542 -0.98 -1774
13 2860 2625 -0.79 -1794
14 2905 2696 -0.70 -1828
15 2958- 2757 -0.67 -1861
16 2986 2802 -0.62 -1885

 

 

 

 

 

 

 

\‘

154

 

 

 

 

 

 

 

 

 

 

 

 

Table X. cont...
Argon (25 C)

"-0 9... SEE-D ..Jfi’ ... 373%; ...(EZi’
5 2504 2603 0.33 -3137
6 2766 2762 -1.05 -3304
7 2990 2961 -0.10 -3450
6 ' 3167 3110 -0.19 —3563
9 3323 3240 -0.28 -3670
10 3439 3363 -0.25 -3760
11 3659 3445 -0.72 -3840
12 3753 3550 -0.66 -3955
13 3784 3666 -o.40 -3963
14 3842 3766 ~0.26 ~4046
15 3911 3853 -0.19 -4105
16 3945 3915 -0.10 -4144

 

 

 

 

 

 

 

155

 

 

 

 

 

 

 

 

 

 

 

 

 

Table X. cont ......
Krypton (25 C)

n-C gcav 51%) cos/(2%) scav 76:31:) 1751(3)
5 2646 2771 0.42 -3958
6 2945. 2964 0.06 -4160
7 3161 3154 -0.02 -4337
8 3347 3314 -0.11 -4472
9 3512 3454 -0.20 -4601
10 3634 3586 -0.16 -4708
11 3868 3674 -0.65 -4803
12 3967 3787 -0.60 -4943
13 4000 3911 -0.30 -4976
14 4060 4017 —0.14 -5050
15 4133 4111 -0.07 -5121
16 - 4£;;—~—-4177 -0.03 -5167

 

 

 

 

 

 

 

Ahz'. (cal/moi)

(cal/mol)

I
2 0

Ah

 

 

 

 

 

 

 

 

:X. . I 1 r I . I r -
2000 F“ x
E0 OCTANE
1000 _ X experiment —
: o O theory ‘
1
- 4
_ X J
O "" —(
r J
: O '1
L . i
-1000-— -d
I- O -4
Z 1
-2000 — 4
L 1 i I v I l l L L l l l 41 Li 1 ‘ d
0 50 100 150 200 250
E solute/l k (CK)
" . fi fi T I Yfi ' ( Y i
2000 g x _‘1
Z TETRADECANE I
1000 )— O X experiment J
: . O theory :
0 :— " i
I O i
. . 1
—1000 — o —J(
_. 4f
' 7
I 1
-2000 _' "‘2
L 7
(.4 ' LQ { L l L J l l l l l l L i l l 1 A 4
0 50 100 150 200 250

E solute/k (CK)

Figure 29. Calculated and experimental excess
enthalpies of salvation for He, Ne, Ar, Kr, and
Xe in (a) octane, (b) tetradecane.

157

seems to indicate that for these salutes we are

overemphasizing the attractive contribution to h . A
I

possible expanation for this lies in our use of the single

fluid approximation for g i.e. g11(r/a) = g12(r/a12). It

12‘
has been shown that this approximation gets worse as the
solute and solvent become more disparate in size. The most
prominent effect is that the main peak in the correlation
function tends to become smaller as the ratio of salute to
solvent size decreases.83 An interesting feature in Figure
29 is that the enthalpy increases as the solute becomes

less polarizable, and crosses over to a positive value at

argon. If we recall equation (5.4):

A)?"
9.1: = 1,..32 , (5.4)
dT a T

we see that for the less polarizable gases the Ostwald
solubility actually increases with temperature. This hs
apposite to our usual experience (when you heat a liquid you
drive it towards the vapor phase!), but is easily understood
from our model, i.e. as the solute becomes less polarizable
it behaves more like a hard sphere, and the SPT shows that
it always takes a positive energy to make a cavity. This
positive energy then leads to our observed temperature
dependence via (5.4).

This seems about as far as we can take a simple model
for explaining gas solubility. The usual recourse at this

point is to introduce solubility parameters that are

158

characteristic of the solute-solvent combination. 33 ’ 84

Although this leads to very accuarte predictions over a wide
range of temperature and solute concentration we shall leave

it to the chemical engineers.

7. Empirical Analysis and Predictions

Although analytic techniques give only limited
understanding of these solubility data, as discussed in the
previous section, ’empirical techniques may be applied
usefully to them as we demonstrate now.

The idea underlying our empirical analysis is to
construct the thermodynamic quantities Aug, Ahz, and A3; for
each solvent as the sum of contributions from the functional
groups which make up the solvent molecules. For our 45
solvents we shall attempt to decompose Au; into Xe
interactions with the functional groups: CHz (in linear
molecules), C83, 08, C008, CEO, and CH2 (in cyclomolecules).

Figure 30 is a plot of the excess chemical for 40
solvents verses number of C32 groups at 25°C. At first
glance this does not look like a promising candidate for
separation into functional group contributions. One might
hope that the addition of a Cflz to any molecule might bring
about the same change in excess chemical potential in each
homologous series, but this is clearly not the case since
the slopes for each series are so different. In fact for
the polar molecules Au; turns upward as we approach zero 082
groups, while for the nonpolar alkanes and cycloalkanes the

chemical potential becomes more negative (this holds for the

159

Apz', (cal/Incl)

-400

—600

—800

-1000

160

 

 

 

 

 

0

Number of CH2 groups

E I I -0 Alkanes :
_ ’1 Alkanols 1
— Cl Acids _
— <> Cyclo —
_ + Aldehydes ‘
5' ..J
5 10 15 20

Figure 30. Excess chemical potential Au:=-RT£nL
2; number of C14z groups at 25°C.

161

perfluoroalkanes as well). In order to get some insight
into what is happening we must look at our expression for

*
Apz; i.e. equation (6.35):

2

a ’ °°
Aug = I; Io4npigaz(r,l)u12(r)r dr d1 . (6.35)

As one increases the number of CH2 groups there are several
parts of Eq. (6.35) that vary; the solvent number density ,
p1 (as illustrated on Table IV and Figure 23), the pair
correlation function 812’ and the intermolecular potential
u12(r). There is a great deal of evidence from computer
simulations that the pair potential can be broken into group
contributions. If we assume this is true and that the
correlation function changes slowly, we may be able to
appropriately scale our data.

Figure 31 is a plot of the data, for 37 solvents,.fram
which one may obtain group contributions for Aug. The
ordinate is Aug/p1 at 25°C and the abcissa is the number of
CH2 groups in the solvent molecule.

The four sets of points on Figure 31 which give data for
alkanes, alkanals, carboxylic acids, and alkanals, can be
fitted to straight lines which have approximately the same
slopes. This makes empirical analysis possible. One
therefore takes the weighted average slope of these four
lines to be the contribution to Aug/p1 of each CH2 group in
a linear molecule. The analysis omits solvents which have

one or zero CH2 groups, but the data for these are included

162

on Figure 31. This quantity, called 6“
(cal/mol)/(mol/liter), is given at the upper left of Table
IX. In like manner the average slope of the curve for
cycloalkanes on Figure 31 gives the corresponding 6 for an
additional CH2 group in a cyclomolecule. The
perfluoroalkanes’ data are not good enough to be analyzed
this way because the samples were mixed isomers. If one
extrapolates on Figure 31 the points for alkanes to the
limit of zero CH2 groups one gets as intercept the
contributions to Aug/p1 of the two CH3 groups; i.e.
2€P(CH3)’ at the ends of these molecules. The intercept on
the ordinate axis obtained by extrapolation of the alkanol
data is €H(CH3) + eu(OH), from which one may obtain £p(OH)
by subtracting off eu(CH3) previously obtained. In similiar
fashion we obtained 6u(COOH) and 6u(CHO). Values for all
six of these en’s are shown in Table XI. The formula by
which one reconstructs an estimated Au; at 25°C is, say, for

an alkanol with, say, m CH2 groups:
* .
Ap2(est) - p1[ eu(CH3') + meu(CH2,lin) + eu(OH) ] , (7.1)

from which L(25°C) may be obtained from Ap;(est)= -RT5n L.
Figure 32 shows values for average enthalpy obtained
from our solubility measurements. The ordinate is MEI/p1

and the abscissa is, as in Figure 31, the number of CH2

Au,” / ,01 (cal/mol)/(mol/liter)

163

 

 

 

 

 

 

O ‘ I I’ I F’I LI I I I I I I I I I I
f O X alkanes 4
- I O O alkanals ‘
-50 — I Q I acids
- I . . A alkanals 4
' O cycloalkanes -
Z 5 . e j
-100 —- f o ' 0 ——
.. x o 3
L. X ..
- x ..
.. X ..
-150 — x x -
_ .X _
x-
I- x 4—
)— X X -.
I. X X "‘
"ZCN)‘_' ‘x "'
.- l l L I l l L .1 l l l l l L l L I d
0 6 12 _ 18

Number of CH2 groups

Figure 31. Experimental data for Xe in 37 solvents.
Excess chemical potential divided by golveht number
density y; number of CH2 groups at 25 C.

A1730" / p, (cal/mol)/(mol/liter)

164

 

-100

 

 

 

_ ‘1 ' ' ' ' lz'. T I I r” * I I I I I
- l . l I
- . X alkanes :
_200-_ 2 e alkanols .7
_ ' O O s acids —'
.. . A alkanals :
- . O O cycloalkanes_
43007 X X x e ._-
.. . -(
.. x e -
.. X a -
-400— X _"
-. x .-
- x _
.. x :
-500H— x —~
- >< :
"6‘H3- I I I I I I I I I I 1, l I I I IX I 1
0 6 12

Number of CH2 groups

Figure 32. Experimental data for Xe in 37 solvents.
Excess enthalpy divided by solvent number
density 11 number of CH: groups .

...-0
CD

T115?" / p1 (cal/ mol)/ (mol/ liter)

165

 

 

 

 

0 L- I r I I I I I I I r I I I I I I
0 X alkanes -
I 0 O alkanols ..
-100 . . I acids —n
" ‘ a A alkanals -
- I O c cloalkanes-
_ ' A . y 3
_. x x Q J
-200 — Q __
- I
' I
.. Q ..
_ C ..
-300 I:- X x —‘
.. X ..
_ x >< 4
-400 I- l l l l l l L l l l l I l l l >l< l d
6 12 18

Number of CH; groups

Figure 33. Experimental data for Xe in 37 solvents.
*
TAsz divided by solvent number density 21

number of CH2 groups.

 

in

166

Table XI. Empirical Energy parameters for Xe solubility
37 organic solvents.

 

Group Energy contribution (25°C)

(cal/mol)[(mol[liter)

‘p ‘H ‘76
CH2 (lin) -7.45 -28.02 -20.36
CH3 -47.67 -99.01 -50.21
0“ +18.41 + 4.64 -14.89
COOH +10.16 - 9.57 -20.86
CHO + 7.22 + 3.02 .- 5.32
CHZ (cycl) - 2.98 -22.09 -19.11

 

 

 

 

solvents at 25.0°C.

 

Table XII. Predicted igugxperimental values of
Ostwald solubility for

Xe in selected organic

 

Predicted Experimental
Solvent L (25°C) L (25°C)
n-ClZHZ6 3.50 3.39
n-C7H1308 2.42 2.44
n-CachOOH 2.52 2.70
n-CAH9CHO 2.69 2.99

 

 

 

167

groups in the solvent molecule. For these quantities, as on
Figure 31, the curve for alkanes, alkanals, carboylic acids,
and. alkanals, are approximately' parallel straight lines.
One obtains the enthalpic contribution eh for each of the
six component functional groups by a similiar analysis as
for 6“. The values of Ch obtained in this way are given in
the middle column of Table XI. The formula by which one
constructs AB: is, say for the same alkanol as in equation

(7.1):
Ah:(est) = p1[ eh(CH3) + meh(CH2,lin) + eh(OH) ] . (7.2)

The A52 values predicted by equation (7.2) are averages over
the experimental temperature intervals; nominally they
correspond to about 25°C.

Figure 33 shows values for the average entropy obtained
from our solubility measurements. The ordinate is TA§:/p1.
It has been multiplied by the average temperature T=298.15 K
so that the ordinate’s dimensions will be
(cal/mol)/(mol/liter) the same as on Figures 31 and 32. The
componental entropy contributions 5T8 are shown on table V
for the six functional groups. Values of 518 were obtained
by applying to the data on Figure 33 the same technique as
before. The formula by which one constructs A3: from these

componential contributions is:

A§:(est) = pal T296[€7§CH3) + meT§CH2,lin) + £1§OH)] , (7.3)

 

168

for the same solvent as in equations (7.1) and (7.2).
Finally, one may estimate the temperature dependence of
*
Auz, over the temperature intervals near those for which the

original data apply, from:
It -* -*
Apz(T,est.) = Ah2(est.) - TAsz(est.) . (7.4)

In applying this equation, one uses equations (7.2) and
(7.3) along with eh and 818 values in Table XI, and one
obtains an estimate of solubility L(T) for Xe is whichever
solvent that has been reconstructed.

As a test of the ability of equations like (7.1) to
predict Ostwald solubility, we show on Table XII a
comparison. between. predicted. and, experimental values for
five solvents, each chosen from about the middle of' its
series. The predictions are obtained from L(25°C)
=exp[-Au;(est/RT298)]. The predicted values of L(25°C) are
within 12% of the experimental values for all of our
solvents with two or more CH2 groups. The probable
explanation for deviations at low carbon number is that the
solvents are becoming more ordered and the correlation
function begins to change rapidly at short chain length.

The above test is largely one of consistency rather than
predictability. But if suitable data were available, one
could test whether our 6" values could be used to predict Xe

solubility in, say, dicarboxylic acids, polyalcohols, and

169

other solvents .

Finally, Figure 34 shows a plot of (Au; - 3mm)“;1 =

shat/p1 verses number of CH2 groups. An extrapolation of
the cycloalkane data to zero CI-i2 groups gives 81m: 0 , as
one would expect. We note for completeness that the

straight line fits are somewhat better than those in Figure

31.

 

 

gm/ ,0, (cal/ mol) / (mol/ liter)

170

 

 

 

 

0 ‘7 I I I I I I I I I I I I I I I I
‘ X alkanes -
—400 ‘ x . O alkanois
. A x . a acids -
' a . x A alkanals -
- O cycloalkanes -
__ _. e A x ’ _.
800 _ I . x _
.. O x

_ ' e x _
4—1200'—‘ ~ O X -—
- . x a
.. . X _
— x -(
- 1600 —- x “
_ x -
x _

_2000 I I I I I l I I I I I I I I I I
0 6 12 18

Number of' CH2 groups

Figure 34. giM/p‘, a number of CH2 groups for
Xe in 37 solvents.

 

8. Conclusions

In conclusion we discuss briefly how this work can be
used to predict and understand solubility data and what
further experimental and theoretical work would be useful
towards these ends.

The empirical results associated with Table XI and
Figures 31.32.33 are suggestive but need testing on other
systems before they can be- relied on for predictions. Some
other C-, H-, and O-containing solvent functional groups for
which the values of e", eh. and 61,8 would be interesting
are, e.g. phenyl and its derivatives, carbonyl,alkoxy, etc.

If solubility predictions from these prove to be robust one
could extend the technique to find contributions of groups
containing other elements, say halogens, N, 8, etc.

A parallel. direction for empirical development is
solubility of other inert gases and, beyond that, gases that
are isaelectronic, or nearly so, to inert gases, etc. We
anticipate that the en’s, en’s, and crs’s empirically found
for different gases with respect to a single solvent
functional group would be simply related to the

Lennard-Jones potential parameters of the solutes.71
Finally there is some indication that the mixed
experiaental-theoretical term (-RT£nL - g°")/ p1, which

according to equation (6.54) equals gun/p , is meaningful

i
for espirical and analytic purposes. As more data become
171

 

 

172

available, its use for quantitative predictions can be
explored.

There are several interesting reasons 'why' analytical
approaches to these data are difficult. In the following
paragraphs I would like to summarize some of our
observations on this topic.

The quantities which currently lend themselves to
theoretical analysis are the free energy of cavity
formation, g , and its associated enthalpy, h , and

CIV OIV
entropy, s , and the energy of interaction u .
cav . int

Scaled particle theory provides a formal procedure by
which one may calculate g?.v. There are several alternative
theories which predict cavity free energies most notable of

85 This

which is the surface tension theory of Sinanoglu.
approach requires a calculaton of the surface energy
associated with cavity formation in a solvent. The surface
tension’s contribution is modified to take into account the
microscopic cavity size. Recent reviews have shown that
this theory tends to predict cavity energies which are too
high, and is inconsistent with pure liquid heat of

72

vaporization data. In earlier papers we tested it on the

perfluoralkanes and found cavity energies to be 2.5 times
higher than the SPT prediction.65 The general consensus is
that Sinanoglu’s theory can be used to get rough
approximations of the cavity terms but SPT is much preferred
when the parameters necessary to apply the later are known

or can be evaluated.72

173

Boublik gt 9;. have developed a version of the SPT for
nonspherical hard. molecules; such. as spherocylinders
(cylinders with spherical endcaps) and other convex

86 At first this may seem like a better alternative

bodies.
to describing solubility in alkanes, unfortunatly, this
theory cannot yet account for mixtures of particles of
different shapes.14 More work in this direction may prove
to be useful.

Although scaled particle theory may be well suited to
predict cavity energies there are restrictions to its
application. The solute should be as hard and spherical as
possible and must not have strong directional interactions

with the solvent molecules. An example of an inappropriate

(but important)solute would be a protein, for which a cavity

would be an ill defined quantity. The solvent can be any
real liquid. However,the effective solvent diameter,' a1,
must be determined with great accuracy. For nonpolar

solvents one can evaluate a1 from heat of vaporization data
through equation (6.87). For polar solvents this equation
is invalid and one usually has to use solubility data to
evaluate a1.71 I

The fact that the SPT works at all is actually quite
amazing and. is ‘probably the result. of’ many' errors that
cancel each other out. This means that we should be
cautious when looking at higher order terms; i.e., the

enthalpy and the entropy. We have demonstrated that the SPT

does not predict the entropy of salvation. This is contrary

174

to previous thought. The reason this has gone unnoticed in

the past is because most workers choose the mole fraction

55,74

scale. The chemical potential on the mole fraction

scale is determined from [equation (8) from ref.55]:

*
Apz = -RT5nx2 Auz + RT6n(RT/v1) , (8.1)
where v1 is the molar volume of the solvent. To find the
corresponding entropy we take the derivative with respect to
temperature:

-#
As2 = As2 - R + RTcIc - ROn(RT/v1) . (8.2)

The extra terms in equation (8.2) are usually much larger
than A5: but they have no physcial interpretation. For
example, for Xe in hexane we have A§:= -4.4 cal/mol X, but
A32: -15.8 cal/moi K. This excess pseudo-entropy serves to
mask the deficiencies in sc ; i.e., instead of predicting

IV

As = 3391': 0.29 cal/mol K, we would say As: s8". spun“ =
-ll.1 cal/moi K, which doesn’t look so bad when compared to
Asa. This does not necessarily mean that the SPT is wrong;
it is possible that the rest of the entropy could be in the
interaction term, g .
int

To settle these many questions one must turn to computer

simulations. Using equation (6.36) we obtain :

1
mum. = (0(5) dg . (8.3)
O

175

Swope and Andersen41 have used molecular dynamics to
evaluate (8.3) for inert gases in water. The idea is to run
your simulation and gradually couple a solute particle into
the solvent. This is done by turning on the solute-solvent
potential in a stepwise manner as suggested by Eq. (8.3) and
calculating the coupling energy, U(g), at each value of 5
until the particle is fully coupled. One can then integrate
the smoothed results to obtain the excess chemical
potential. Once the particle is fully coupled it is
possible to calculate the entropic contribution to the free

energy by simply subtracting off the energetic part; i.e.
a ‘ -:I:
Apz = I 0(5) 65 m U(g=1) - 7432 . (8.4)
o

Jorgensenl has developed interaction site potentials
which predict the properties of pure alkanes and pure
alkanals. These are potentials in which the molecule is
represented by a set of discreet interaction sites that are
commonly, but not invariably, located at the sites of the
atomic nuclei. For the alkane potentials Jorgensen has used
the various functional groups as the interaction sites.
These have also been used to predict the solubility of
various alkanes and alkanols in water.87 While this data is
certainly valuable one must note that solubility in water is
a much more complicated process than in nonpolar solvents

due to the complex structure of liquid water. In principle

176

it is possible to do the same calculation for inert gases in
alkanes and alkanols and maybe other solvents.

By' carefully choosing the way the solute - solvent
potential is coupled it may be possible to gain insight into
how repulsive and attractive interactions affect the
salvation entropy. This in turn would clarify the
deficiencies in analytic models and help us to correct them.

To complete our experiment it would be interesting to
look at Xe solubility in the lighter alkanes, i.e.; ethane,
propane, butane and methane (despite the fact that methane’s
boiling point is lower than xenon’s at 1 atm. The trick is
that by using a tracer the solute gas is so dilute that it
will not condense until a significantly lower temperature77
). These are simpler solvents and thus may help bridge the
gap in our understanding of solubility. Also, our
experimental technique offers a unique advantage over other
cryogenic experiments in that we can easily measure

89’90 This

solubility in the limit of infinite dilution.
makes the subsequent analysis much easier since one can
neglect solute-solute interactions.

Our technique may also prove to be useful in analyzing
solubility near the solvent’s critical point (we again have
an advantage over other methods since we can ignore solvent
vapor pressure effects). A great deal of recent work has
been along the lines of gas solubility near the critical

temperature of watergl; performing complementary experiments

on non-polar and polar systems such as ours could prove to

 

177

be very enlightening.

9.Introduction: Solubility of Nonpolar Gases in Water at

Elevated Pressure.

This section of my thesis reports solubility
measurements of the. gases nitrogen, argon, krypton, and
xenon in water. For each of these gases we measured, at
25.0°C, the dependence of gas solubility on partial pressure
of the solute. The pressure ranges studied were
approximately: 44-116 atm for N2, 22-101 atm for Ar, 33-81
atm for Kr, and 5-48 atm for Xe. The data are analyzed in
terms of molecular theories based on thermodynamics, and
statistical mechanics. These experiments are an extension
of previous work on Xe solubility in organic liquids‘ and
aqueous solutions.”’92

The study of gas solubility in liquids is an. old and
well developed subject in chemistry and physics. For dilute
solutions at sufficiently low pressure, the mole fraction

93

solubility, x2, is well described by Henry’s Law :

P = k x (9.1)

where P2 is the partial pressure of the solute gas and ha is
a constant of proportionality. The so called Henry’s

constant, k”, is characteristic of the solute-solvent system

178

179

at a given temperature. At higher pressures Eq.(9.1) must
be modified to account for nonideality of the vapor.
Lewis94 introduced a new form of Henry’s Law in which the
solute partial pressure is replaced by its corresponding
fugacity; f2=kflx2. At still higher pressures the measured
solubility fails to obey this relation as well. It shall be
shown that by measuring deviations from Henry’s Law one can
obtain information on the salvation process.

The importance of understanding the effects of pressure
on gas solubility comes from its many applications. At an
industrial level, extraction and separation processes are
often performed at elevated pressure. In biology and
medicine, interactions of inert gases with living organisms
are also important. For example, the gas mixtures breathed
at high pressure in deep sea diving consist mainly of inert
gases. Interactions between these gases and the diver are
responsible for decompression sickness and inert gas

narcosis.10’95

Solubility limited phenomena are also
relevant in marine biology. Several species of deep water
fish have swim bladders in which partial pressures of
nitrogen up to 10 atmospheres have been observed.96 In
order to determine the mechanisms used by the fish to
inflate its swim bladder, it is important to know the
concentrations of dissolved gases available at a given
depth. Such variation is generally determined by the

hydrostatic pressure.

Another motivation for this research is to look at the

180

role of hydrophobic effects on gas solubility. Hydrophobic
interactions are important for understanding the stability
of biologically important macromolecular structures.

Hydrophobic effects are thought to play a key role in

97,98

protein conformation and enzyme specificity. Several

experimental studies have investigated the behavior of

simple polar and nonpolar organic solutes in water.99’100

Inert gases would in some sense be the prototypical
hydrophobic solutes because their interactions with water
are very weak compared to those of water with itself.

The inert gas elements have been studied extensively as

77,78

prototype solids, liquids, and gases, so many of their

interactions are well known. The low pressure solubility of

these gases in water have also been determined

6’32 With the advent of improved analytic73’101

41,102

accurately.
and computational techniques it has become possible to
predict these solubilities with good accuracy. By providing
data in different ranges of temperature and pressure one can

test the robustness of these theoretical predictions and

make appropriate refinements.

10. Experimental

The two most common units of gas solubility are the
Ostwald (L) and mole-fraction (x2) solubilities. They are

defined as:

 

I.) 92 n2
3 fl and X2 3 W e (10 a 1)
pz 1 Z

Ostwald solubility is the ratio, at equilibrium, of the
concentration of dissolved gas molecules in the liquid
solvent to their concentration in the gas phase. The mole
-fraction solubility is the equilibrium fraction of solute
molecules in the solvent. In recent literature measurements
in terms of the Ostwald solubility are characterized as
being on the ’number density-scale’.14

Figure 35shows a schematic of the experimental
apparatus. The Figure is divided into two parts to show
each step in the measurement process. First the liquid is
saturated with the solute gas at high pressure as depicted
in 35a. Once equilibrium is attained the liquid is
isolated and the sample cell is transferred to the analysis

system, 35b, where the amount of dissolved solute in the

pressurized liquid is determined. Together, this
181

182

To Vacuum Pump

 

 

 

 

 

 

 

 

 

v...mooooooooooooooooooom

(b)

 

Analyzer

 

1 Pressure
Transducer

 

 

 

 

    

[lflll

 

 

 

 

 

 

 

 

 

hmm

3

Figure 35. Schematic of experimental setup
(a) high pressure equilibration system (b)
analysis system.

 

183

constitutes a modified Van Slyke method 103 in which the

sample volume is the entire liquid volume. Our technique is
different from Van Slyke’s in that it allows us to measure
directly the mole -fraction solubility, Ostwald solubility,
and partial molar volume.

We briefly outline some experimental details below. The
sample cell is a 100cm3 stainless steel volume with a ball
valve to isolate the liquid and allow for easy transfer
between the equilibration and analysis stages of the
experiment. The solute gases we used were obtained
commercially. Their purities were: N2(99.99%),
Ar(99.998%), Kr(99.997%), and Xe(99.996%), (N2, Kr, Xe from
Linde Gas Products, Ar from Matheson Gas Products).

At the beginning of the experiment we load the sample
cell with twice distilled water which has been degassed
using standard methods. The water is loaded under vacuum to
prevent air packets from forming. The cell is weighed and
the mass of the water, after making proper .buoyancy
corrections for the weight of displaced air, is denoted mo.
To help prevent gas spaces from developing in the cell upon
liquid compression we add some extra water above the ball
valve. The mass of the additional water is Am+. The cell
is then connected to gas volume V8 and pressurized. Mixing
is achieved with a specially designed magnetic stirrer. Gas
pressure is monitored with an accurate capacitive transducer
(Setra Systems), and. a pressure gauge (Heise Inc.) for

calibration. The sample cell and gas volume are submerged

184

throughout the entire experiment in a constant temperature
water bath at 25.0:0.1°c.

After some time, typically 2-3 days depending on
stirring speed and solubility, the pressure readings reach a
steady state. To be sure that equilibrium is attained we
monitor the pressure for an extra 8-12 hours. At this point
we record the final pressure of the solute gas, Peq’ and the
ball valve on the sample cell is closed. The gas volume is
then slowly depressurized. Excess liquid above the ball
valve is collected and weighed. This mass is denoted Am-.

The water remaining in the sample cell is now:

In = m + Am - Am . (10.3)

The analysis system as illustrated in Fig. 35b consists
of a detachable gas volume, V8a , with a pressure transducer
and mercury manometer. The gas volume is variable in the
sense that different sized cells can be attached to the side
arm of the analyzer. We choose the gas volume such that the
pressure of the evolved solute will be within the optimum
range of our sensing devices. This pressure can be
estimated. via Henry’s Law with. well known low pressure

6’32 For the solute gases and pressure

solubility data.
ranges in this experiment we used two analyzer volumes,
namely: slBOcc (N2 and Xe) and s360cc(Ar, Kr, and Xe). To

measure the amount of solute in, the saturated liquid we

connect the sample cell to the analyzer and evacuate the

185

upper volume. The main valve is then opened thus releasing
the dissolved gas into V8a shown in Fig. 35b. With stirring
a steady value of pressure is usually reached within 2-6
hours. The final pressure of the solute gas in the
analyzer, Pa’ is found after correcting for the vapor
pressure of the water.79

To calculate the mole-fraction and Ostwald solubilities

we first must find the density of the solute gas, 928(P), at

Pa and Peq' Accurate virial expansions were used in the

appropriate pressure ranges for N2 , 104 Ar , 105 Kr , 106 and
Xe . 107 One can then find the number of moles of gaseous
solute in the analyzer to be:
- 8 a (2)
n2 - p2 (Pa) [Vg + V8 ] , (10.4)

where V;Z)is the extra. gas volume that arises from the

decompression and outgassing of the pressurized liquid.
Assuming the low pressure liquid density is that of pure

water we have:

(2)_ _' a
v8 "V6.14 In"20 /p"20(25.0 C) , (10.5)

where V is the volume of the sample cell and m is
cell H20

defined in Eq.(10.3). To find the total number of moles of
solute we must also account for any remaining in the liquid.

Adding this correction we find for the total amount of

solute in the sample cell at Peq

186

n2,total= 928(P8)[ [vga+ V;2)] + Lo(mlIZO/p"2°) ] , (10.6)

where L° is the low pressure limit of the Ostwald solubility

6,32

at 25.0°C. The number of moles of water in the sample

cell, n1, is the mass of the water divided by the molecular

weight, i.e., n1 = IIIn 0(g)/18.01534(g/mol). One can then
2

easily calculate the mole fraction solubility from
Eq.(10.2). At the equilibrium pressure Peq the liquid
volume is fixed to be that of the sample cell, therefore the
number density of solute molecules in the liquid is p21:
Dividing this quantity by the gas number

n / V .
2,total cell

density at peq one obtains the Ostwald solubility, L, as

given in Eq.(10.2). One may also use this technique to

obtain a rough estimate of the solute molar volume by:

325: [Vc.”- [mazo/pnzo(25 C°,1'atm)]exp(-x(P.q- Pa)]/ n2,

(10.7)

where x is the isothermal compressibility of pure water,108

x = 4.571310“5 atm-1, and n2 is given by Eq.(4). Essentially
this expression means that if we take Va”, the known
volume of the pressurized liquid, and subtract off a term
due to liquid compression then the remaining volume, which
is in the square brackets of Eq.(10.7) is due to the solute

molar expansion. Dividing this by the amount of solute in

the liquid gives the molar expansion produced per mole of

187

solute added, i.e., 62. Since our solutions are very
dilute, typically xzarIO-‘, we assume in this development
that the density is the same as for pure water. This
assumption is also supported by the fact that our results
agree fairly well with literature values.109’110

We stress that the strength of this technique is that by
fixing both the sample volume and mass, one may easily find
the Ostwald solubility and the molar volume. Other methods
usually sample only a known mass of saturated liquid and
therefore are limited in that they measure only mole
fraction directly. Separate measurements are then needed to
calculate the solubility on the number density (Ostwald)
scale. It has been argued that the number density scale has
certain advantages over the mole fraction scale in analyzing

99 The

results from a statistical mechanics perspective.
primary weakness of our current technique is that the
measurement takes a relatively long time. A better
alternative wouLd be to use small samples of fixed volume

111,103 This

from a large reservoir of equilibrated liquid.
is a more traditional approach with the important addition

that we know the sample volume at Peq.

11. Results

The experimental results we obtained are shown in Table
XIII. The first column gives the pressure at which the
measurement was made, the second column gives the
corresponding fugacity. The third and fourth columns
tabulate solubilities on the mole fraction and number
density (Ostwald) scale respectively. We also include the
average value obtained for partial molar ‘volume of the
solute. The fifth and sixth columns give the data in terms
of a new solubility parameter which will be discussed below.
Judging from reproducibility and by comparison with other
datas’112 a conservative estimate of our fractional
uncertainty in solubility is about 1-1.5%. The primary
sources of error are pressure measurement(:2.0 psi for high
pressure, $0.5 Torr for low pressure), volume measurement
(t0.l% reproducibility), and microbubble formation in the
pressurized liquid. The first two terms lead to an
uncertainty of less than 0.2% so it seems that the last term
is the main contributor.

Fugacitiea were calculated from volumetric properties of

the real gas via20

P
f = p exp[ 'fi; I [v(p’) - 3%? ldp’ ] . (11.1)

188

189

The integral in Eq. (11.1) measures the cumulative deviation
of the real gas volume, v(p’), from that of the
corresponding ideal gas at pressure p’. To obtain v(p’) we

used the virial expansions referred to in the previous
104-107

 

section. The fugacity is related to the chemical
potential per mole of the gas by 126:
12 A3
p29: Run[ J , (11.2)
k T

where f2 is in atmospheres and k=R/NA= cma-atm/molecule-°K
and A=h/(2nka)1/2is the thermal wavelength of the solute.16
Figure 36 shows a plot of the equilibrium fugacity
versus mole-fraction solubility for our data. Also included
is data for N2 in water at 25.0°C from Ref.(112). As one
can see, the agreement with our measurements is good. The
curves in Fig{ 36 have the common feature that the
mole—fraction solubility increases approximately linearly
with fugacity. Table XIII shows that the Ostwald solubility
tends to decrease with f2. If the system was in complete
accord with Henry’s Law we would expect the quantity ku=flx2
to be constant for each gas. Figures 37(a-d) show such
plots with respect to solute concentration. For N2, Ar, and
Kr there is a clear increase in k" by as much as 7-10% in
the concentration range studied. For xenon there appears to

be very little variation in kn' A problem that we

encountered with xenon was that above a pressure of 17

 

 

190

atmospheres the solubilty decreased rapidly and the molar
volume increased to about 125cm3/mole. The data point from
Table XIII at 23.33 atm seems to be anomalous in the sense
that the solubility has decreased sharply ,compared with
lower pressures, but we did not see a large molar expansion.
We associate this effect with the onset of clathrate
formation.113 This speculation was confirmed in separate
experiments with xenon-water mixtures at Izq>30 atm.
Instead of subjecting the equilibrated sample to analysis we
quickly depressurized the cell and pour the liquid into a
beakeru Several ice-like crystals, as large as 5mm in
diameter, were observed. These melted away within minutes.
We have included on Table XIII the solubility measured
during the clathrate phase but we are not sure that the
results are reliable since it is known that this is a glassy
state which may require very long equilibration times.113
It might be worthwhile to repeat our low pressure xenon
measurements because the pressure ranges we explored were at
the lower end of our sensing accuracy. Our technique using

radioactive tracers which would be useful toward this end.18

191

Table XIII. Experimental results for splubility
of N2, Ar, Kr, and Xe in water at 25.0 C. The

second column is the solute partial pressure. The
third column is the solute fugacity. The fourth

and fifth columns are the mole-fraction and Ostwald -

solubilities respectively. The sixth column is

the new solubility parameter r. The seventh column
is the excess chemical potential of the solute in

the liquid. The eighth column is the experimental

average value of partial molar volume.

 

 

 

 

Solute P(atm) ftiatm) x'I-lo‘) I. r 781524151) V'ica'laoi) 1‘:
6' 44.55 44.25 4.902 0.01465’ 0.01501 2466 g
50.46 50.10 5.462 0.01466 0.01464 2495 ,
63.66 63.17 6.609 0.01446 0.01463 2503
77.26 76.63 6.100 0.01423 0.014355 2514 31 z 2 ‘ ’
93.30 92.54 9.562 0.01397 0.014045 2527 ‘“‘
102.19 101.36 10.32 0.01360 0.01363 2536
115.60 114.99 11.46 0.01360 0.01356 2546
Ar 22.505 22.190 5 426 0.0322 0.0331 2019
25.50 25.01 6.136 0.0321 0.0331 2019
36.12 37.24 6.993 0.0313 0.0327 2026
44.36 43.20 10.36 0.0309 0.0325 2030
55.95 54.11 12.61 0.0301 0.0321 2036 ,3 2 1
66.66 66.15 15.41 0.0292 0.0316 2047
71.67 66.73 16.06 0.0293 0.0317 2044
79.19 75.65 17.43 0.0267 0.0313 2053
67.90 63.61 19.06 0.0262 0.0309 2059
101.27 95.70 21.59 0.0276 0.0306 2065
x: 33.30 31.01 13.45 0.0506 0.0566 1679
40.64 37.26 15.91 0.0465 0.0579 1666
41.16 37.69 16.22 0.0461 0.0563 1663
52.14 46.64 20.03 0.0463 0.0562 1665 3° 3 2
56.66 51.93 21.53 0.04345 0.05625 1705
69.73 60.06 25.13 0.0417 0.0567 1700
60.56 67.91 27.67 0.0391 0.0556 1712
60.66 67.99 27.79 0.0369 0.0554 1714
x0 4.791 4.669 3.592 0.0994 0.1041 1340
6.344 6.103 4.716 0.0962 0.1046 1336
6.151 7.600 5.979 0.0950 0.1036 1342
9.415 6.947 6.930 0.0950 0.1046 1336 ,7 g 2
11.25 10.56 6.069 0.0917 0.1034 1345
12.10 11.33 6.750 0.0915 0.1045 1336
14.69 13.55 10.36 0.0660 0.1034 1345
15.74 14.44 11.035 0 0669 0 1034 1345 --_-
x. 23.33 20.46 13.60 0.066 0.090 1426 44
32.76 27.19 17.46 0.057 0 0664 1451 117
33.29 27.52 17.62 0.057 0.0659 1454 125
47.69 35.99 21.17 0.040 0.079 1506 131

192

 

 

 

 

120 ' i ' i I T V I l I r 7 —.
'_' o 4
- bfitrogen «
100 - 9 _.
.- 0 .1
_ o .
so ' ° :
,\ - O o Argon _:
g ' 0° :7 ~
d ' O 4
v 60 — a -—
r? ‘ 9 o ' C ‘
3 :_ O o a 7
go 4:0 _ 0 cm Krypton 1
2. : a 7
- 1? 00 I
20 :— ‘(enon 1
_ xx XX ‘ . .J
.. xx .
x x

O t. ' l ' l L 1 ' ' l ' ' I ' q

0 0.001 0.002 0.003

Mole Fraction. x2

Figure 36. Solutg fugacity u mole-fraction
solubility at 25 C.

193

 

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s constant 2:, mole-fraction

I
°c.

Figure 37. Henry
solubility at 25

12. Pressure and Concentration Dependence

Deviations in Henry’s Law, such as those shown by our
data, are best analyzed in terms of the solute chemical
potential. In our experiments the vapor pressure of the
solvent, P(azo,zs°c) a 23.76 torr, was small compared to the
solute gas pressures and can be ignored. The chemical

potential per mole of gaseous solute can then be written as:
O fz(T,P)
2 2 p¢

where 142’”) is _the standard chemical potential of the gas
at temperature T and pressure P’. The choice of reference
pressure is such that the gas behave ideally at P’. We
shall follow the usual convention and set P0 equal to 1

20,22

atm. This expression for 142' is identical to Eq.(12.l)

but will prove to be' easier to work with in subsequent
discussions.

The chemical potential of the solute in the liquid, "21’
is completely described by the variables T,P, and x2. At

constant 'I‘ one may express the differential change in ”2‘ as
114

194

 

 

- auz d auz
- x2 + dP . (12.2)
PT 36 ‘I'

To understand how "21 depends on solute concentration we

use the Kirkwood-Buff (KB) solution theory.115

This is an
exact formal treatment in which the solution properties are
obtained from integrals over 'pair~ correlation functions.

For two components the KB integral is defined as:
m
G = I [g (r) - 1]4ur2dr , (12.3)
ii 0 ii

where g‘j(r) is the orientationally averaged pair
correlation function between species i and j, and depends
only on intermolecular separation r. To relate these

quantities to pZ'CT,P,x2) we use Eq. (22) of Ref. (115):

 

 

1
1 [ a "2 J = 1 + 91(2612 Ga: 622) ,
R T 0 x2 P,T x2 1 + p1x2(611+ G22- 2G12)
(12.4)
where 91 is the molar concentration of the solvent. For

dilute solutions Eq. (12.4) can be expressed in terms of the

infinite dilution limits of the molar volumes and solvent

isothermal compressibility, x1 26:

196

 

 

l 6 p i 1 1
__ [ 2 J = —— + O [ -G + V O - 2V 0+ RTX O]
R T a x 9,7 x v 22 1 2 1
2 2 1
+ 0(x2)2 . (12.5)

Throughout this thesis we use the superscript o to refer to
the infinite dilution limit. Integrating Eq. (12.5) and

keeping terms to first order in x2 yields 116:

1 _ o 2
112 (x2.T.P) - 112 (T.P) 7‘ RTlmxz + XZE(T) + 0(x2) . (12.6)

where

O

_ _ _ o
g(T)-RT( 622+ v1 2v2

+ RTx1°)/v1° , (12.661)

and p20(T,P) is a constant of integration.

The KB integral. 622, plays an important role in
determining the concentration dependence of the chemical
potential. It is a measure of correlations among solute

particles in solution. This can be illustrated by

multiplying 622 by the solute concentration pz:

0
_ _ 2
p2622 - Io pz[ g22(r) 1 ] 4nr dr . (12.7)

This quantity reflects the total average excess (or

deficiency) of solute molecules surrounding a fixed solute

197

with respect to a uniform distribution.99 For example, if
622 is positive the solute molecules are more correlated
than if they were distributed randomly. This sort of
solvent induced clustering in aqueous solutions is commonly
referred to as a hydrophobic interaction.97 An exaggerated
example of such a situation would be micelle formation in
detergent solutions.117 A driving force in this example is
the fact that the detergent has both polar and nonpolar
regions which compete to minimize the free energy. The
resulting conformation is that solute molecules cluster
together with polar regions in contact with the water and
the nonpolar regions in contact with each other. This sort
of mechanism plays a vital role in biological process such

118 It is

as membrane formation and pmotein conformation.
not obvious that such mechanisms should come in to play for
nonpolar gases, particularly those we have studied, since
these solutes have little structure, i.e., the solutes are
mono- or di- atomic and cannot be divided into polar and
nonpolar regions. It has been shown however that solvent
induced interactions can lead to clustering in hard sphere
mixtures where there are no attractive interactions at
an.119

A more familiar occurence of 622 is through its relation
to the osmotic pressure of a solution. By analogy to the
virial expansion for real gases one can show that the
osmotic pressure ([1) of a two component solution obeys an

equation of the form 22:

 

 

 

198

n/RT = p2 + B p + B p + ... . (12.8)

N*

The second osmotic virial coefficient of the dissolved

solute is 126:

32 = -TG . (1249)

*
The value of 82 is frequently used in statistical theories
of polymer configurations as a measure of the interactions
between segments along the polymer chain.120

The pressure dependence of the chemical potential is

described by the well known thermodynamic relation 20:
6112l
[__) . .2 , (.2...)
BP 71 ‘1'

i!

where v2 is the molar volume of the solute in the solvent.

Applying this to Eq. (12.6) allows us to evaluate pzeas:

P ,
)1 =1 v (p') dP’ + C(T) , (12.11)
0 2

where C is an integration constant which depends only on
temperature. If the solution is well below the critical
temperature of the solvent, the molar volume is weakly

dependent on pressure and can be taken outside the

199

integral.121 We then substitute its value at infinite

dilution, v20. Putting this result in Eq. (12.6) gives:

0

- 2
p - C(T) + v2 p + 31‘0an + ngvr) + 0(x2) . (12.12)

The equilibrium condition between the vapor and the

liquid is “29 = 02'. Using Eq.(12.l) for p29 and Eq.(12.12)

for pal gives:

f (T9?)
02°(T) + R'wn[ z—— ]

x P¢
2

= C(T) + P v20 + ngw) + 0(x2)2.

(12.13)

In the limit of low pressure and infinite dilution the
bracketed term on the left hand side of Eq.(12.13) reduces
to Henry’s constant kuo, so that one may evaluate C(T) -

15¢(T) = RTbn(k"°). This leads to the desired result:

f2(T,P) ] o
x P¢
2

Run[ = Monwf) + Pvz + ngvr) + 0(x22) .

(12.14)

This expression was previously derived in Ref. (116), but we
have filled in some of the steps. The observed deviations
from a constant kno in (f/xz) can be attributed to increased

Pv work in the liquid, and possible solute-solute

200

interactions. In the limit x2§(T)-40 we obtain a well known

result in chemical engineering:

 

00(1‘ /x P") =bn(k °) + 2 (12.15)
2 2 H RT

Equation (12.15) is known as the Krichevsky-Kasarnovsky

112’121 For the gas-liquid systems in this paper

equation.
(Pv‘o) is typically 20f100 times greater than (szT).
Therefore, by Eq. (12.14), we will only be sensitive to g(T)
if G22 is significantly larger than the molar volumes, i.e.,

g(T) a: XZRT ( C - Gzzlvao)’ where C is of order (-3) for

typical solute molar volumes and solvent compressibilities.

13. Data Analysis

Equation (12.14) can be used to obtain information on
the pressure and concentration dependence of gas solubility.
Figure 38 shows a plot of the quantity [cnlfz/sz¢) -

PvzolRT] m x for each of our solutes. A linear

2
regression fit to the data gives for the ordinate axis
intercept the value of mug) and the slope determines
E(T)/RT. From g(r) and Eq. (12.6a) one may then find 622.

The calculated slope was very sensitive to the choice of
molar volume so that reliable values of Ge: are difficult to
determine. To illustrate this point, Table XIV lists values
of 622 and 6n(ku°) obtained for each solute at for different
assuemd values of v 2°. The values of v 2° are taken from
Ref. 109, in which standard literature values are compiled,
along with the authors’ own direct measurements. The values
of Véo given in Table XIV represent the range of generally
acceptable data. A common way to obtain the partial molar
volume of gases in liquids is to use solubility data in
.conJunction with Eq.(12.15). This obviously is not
appropriate to our analysis, which relies on Eq.(12.14),
since the resulting molar volume would give zero for £(T).
o

When possible we have used values of v2 which were

determined directly. and have indicated when otherwise. For

201

 

202

xenon the only direct measurement of molar volume is by

Biggerstaff gt __1,. at elevated pressure.110 The data on

Figure 38 were fitted using the following intermediate
122

values of molar volume; 32°(N2) = 34.2cm3/mol, 32°(Ar)

32cm3/mol, 32°(Kr) = 32cm3/mol, and 32°(Xe, P<17atm)

43.50m3/mol (the average nitrogen value corresponds to a

measured value in Ref 122). The corresponding values of
respectively.

On Figure 38 we have shown on the ordinate axis using
filled symbols the experimental values of (11(kno) for each
solute at 25.0°C as given in reference 6, viz., kfl°(N2)=
85251, kH°(Ar)=39746, ka°(Kr) = 22252, kn°(Xe) = 12885. The
intercepts of our graphs are much less dependent on choice
of molar volume than the slopes and agree with the known

results very well. Thus, the values of k"0 that one obtains

from our plot are: k"°(N2) 84669, k"°(Ar) = 39875 ,

12925. We also found that

k"°(Kr) = 22261, and ka°(Xe)
the xenon data at higher pressures could be fit to the same
curve as the pre-clathrate data, provided we used our
experimental value of v2°(Xe,P>20atm) a 125cm3.

It is interesting to note what happens when Va0 is

treated as a free parameter which can be varied to obtain

the best straight line fits by chi squared minimization.

Ina/x...) — Pvz/RT

203

 

 

 

 

12.0 _ ‘ T T T ' ' i I I I 1 r
115 '—
i 89—6-6—99-9 Nitrogen
11.0
% —9-0——-0——-Oo+—o—o
10.5

Argon

 

 

 

 

10.0 _ 3 $——o—°——a—a
L. i~ Krypton
9-5 W
)- .. .
: Xenon '
9.0 L ; L ' l L l L i l i L l
0 0.001 0.00 0.003

Mole Fraction. x2

Figure.36. dnifz/xz) - PVéo/RT x§,mole-fraction

solubility at 25°C.

 

204

Table XIV. Results of Kirkwood-Buff analysis. The
third column gives the partial molar volumeaof the
solute used in Eq.(12.14). The fourth and fifth
columns give the results for the KB integral Ga:

and the log of ghe infinite dilution limit of Henry's
constant, £n(k" ).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solute T(°C) vaoicm’/mol) 621(cm3/mol) (u(kaoi
N. 25.0 33‘ —236 11.346
35.7‘ -27 11.347
hr 25.0 32"' +169 10.5935
Kr 25.0 33‘ +24 10.010
31' +69 10.011

x: 25.0 41‘ +196 9.4666
46’ +240 9.4669

He 0.0 15.5 -157 11.791
23.7 +927 11.601

29.7 +1720 11.609

He 25.0 ~ 15.5 -22 11.674
23.7 +996 11.661

29.7 +1745 11.666

N: 25.0 33- -4.5 11.360
35.7 +345 11.369

143 50.0 33 -92 11.604
35.7 +330 11.611

Hz 0.0 20 -16.5 10.953
25.2 +262 10.956

26.7 +343 10.960

6: 25.0 20 +7.0 11.170
25.2 +314 11.173

26.7 +403 11.174

11' 50.0 20 +3.0 11.242
25.2 +305 11.245

26.7 +392 11.245

ca‘ 25.0 34.5 -13 10.566
37.4 +223 10.596

036‘ 37.6 51 +125 10.595

53 +470 10.609

1h

205

In every case the optimum fit leads to |GZZ|<50cm3/mole.

However the resulting value of the ’effective molar volume’,

vz'”, is often not in agreement with experiment, i.e., we
obtain by this procedure v2“! ( N2 ) =36¢malmol ,
vz'”(Ar)=26cm3/mol , v2°"(Kr)=28cm3/mol,
andvz'”(Xe)=10cm3/mol. The value for Xe is certainly

unphysical since Xe is known to have the largest molar
volume of the four gases studied.

Table XIV also shows the results obtained by application
of this analysis to data of other worker on high pressure

gas solubility in water. These data appear below the double

horizontal line on Table XIV. The systems include
nu25-100061m;0°c,25°o),123 “2(25-10008tl; 0°o,25°o,50°0)
112 112 124
, N2(25-10003tm; 25°C,50°C), c114 (0-6000p81; 25°C),

125

Czi-i‘5 (0-10,000psi; 37.8°C). For H2 and N2(50°C) the gas

phase fugacities used were those of Demming and Shupe.127
For the remaining gases the fugacities were calculated in
the manner described earlier; the equation of state data we
used to calculate f2 is taken from references 128, 104, 129,
and 130 for He, N2, 011‘, and C2116, respectively. Although
ethane isn’t a spherical solute we include it because of its
similarity to Xe in aqueous solutions; i.e. both gases have
comparable solubilities at 1 atm .and both form aqueous

clathrates at 25.0°c.6’ 1139 131

The analysis uses all the
published solubility data except the first point of the
methane data (P=34lpsia) for which x2 seemed anomalously

high. Once again we see that the calculated values of 622

 

206

are strongly dependent on the choice of molar volume. Molar
volume data are taken from Ref.(109). For He there is no
direct measurement of molar volume. Of the three quoted

values shown in Table XIV, two are obtained from solubility

0 132,96

measurements (v2 = 15 . 5cm3/mol , v2°=29 . 7cm3/mol) and

the other is an estimate from the pure liquid triple point
density.116 These data, also, have the general feature that

one can choose an effective molar volume which leads to the

best fit of the data. The effective molar volumes that one

obtains are: vz'”(i-Ie)=l5.5cm3/mol, v2°"(H2)= 20cm3/mol,
V2.tt ( N2) = 33cm3/mol , v2." ( CH4) = 34cm3/mol , and
vz'"(C2H6)=50cm3/mol. All are close to the experimental

results with the exception of He and H2.

Watanabe and Andersen116

have performed molecular
dynamics simulations of the Kr-water system. Their results
indicated that the Kr solutes tended to undergo ’hydrophobic
repulsion’ in solution. By integrating the solute-solute
correlation function, as evaluated from ,computer
simulations,they found. that. Gé2(Kr) = -1004A3/molecule =
-604cm3/mol. This means the solutes tended to avoid one
another in solution. In the appendix of that paper they
include an analysis of solubility measurements which is
similar to ours. They find a definite correlation between
maxi) for a given solute and its corresponding second
osmotic virial coefficient, B: of Eq. (12.9). The general

tendency was that as max) increased, B: decreased. By

extrapolating experimental results they postulated that

207

622(Xe)a -7750m3/mol, 622(Kr)u -362cm3/mol, 622(Ar)a
+16cm3/mol, and G22(Ne)u +821cm3/mol. They assert that the
less soluble gases (large kuo) are less polarizable and tend
to demonstrate hydrophobic clustering. The more polarizable
gases such as Kr and Xe are relatively hydrophilic and
prefer to be surrounded by water. We see no systematic
variation of this type. We believe that the systematic
variation observed in reference 116 for He, N2, H2 and CH4
is attributable to choice of molar volume, and selection of
data (for example, they neglect all nitrogen data above
x2=0.003 at 25°C and above x2=0.004 at 50°C). Our results
do not rule out such behavior, but within the experimental
uncertainty of v2O we cannot support such findings. It is
necessary to point out that the Kr solutions of Watanabe and

116 computer simulation were 5‘10 times more

Andersen’s
concentrated than the solutions we made. It is possible
that at this higher concentration one starts to see
clathrate behavior for krypton such as ‘we observed for

133 which form around the solute

xenon. The ice-like cages
in the clathrate phase could provide a mechanism for
’hydrophobic repulsion’ between solutes.

Our results are more consistent with the theoretical
calculations of Pratt and Chandler.101 They developed a
theory of the hydrophobic effect which is based on an
integral equation for the relevant pair correlation

functions of the aqueous solution. The theory is more

accurate than previous analytic attempts because it utilizes

208

the experimentally determined oxygen-oxygen correlation
function of pure water as an input parameter. Table V of
Ref. 101 tabulates the calculated values of second osmotic
virial coefficient, B:, for hard spheres in water in and a
hard sphere solvent. Although Pratt and Chandler’s model
predicted some hydrophobic clustering of the solutes, the
resulting values of 622 only ranged from -15 to 96
(cmalmol). SinCe a hard sphere is in some sense the
limiting case of an inert gas solute with zero
polarizability it might be reasonable to expect that 6:;
would be the largest possible value for the inert gases.
Unfortunately no attempt was made by the authors to
calculate G22 for more polarizable gaseous solutes.

To briefly summarize, we found that the solubility data
could be accurately fit by taking account only pressure
effects on the solute chemical potential. The contribution
of solute-solute interactions expected at high
concentrations, is uncertain since the partial molar volumes
v2o are not known accurately enough.

There is evidence, however, that solute concentration
effects may play a role in some aqueous systems. Figure 39
shows solubility data on the systems He-HéO and Nz-HZO, both
at 25.0°C. On Fig. 39 the quantities plotted are the same
as those on Fig 38, i.e. {51(f2/x2P4’) - PVZO/RT w x2.
We show curves representing each value of molar volume
listed on Table XIV. Although the best fit for He is

obtained with v20 = 15.5cm3, this seems too low in relation

to

other

values

209

of

the

noble

gases.

109

P“

1n (f/xz) — v2° P / RT

12.0

11.8

11.6

11.4

11.2

11.0

210

 

 

Y \I L V V l
h .‘ ' ' r 1‘ w 4 ,‘
., .
_ u .e He vz" = 15 ems/mole 7
3 a
O

4
h ’ He v; = 23 ems/mole -
- -(

— I .

—(

' He v3” - 29 ems/mole ‘

+N3 Vg° - 33 Ema/mole

   

—

I—

 

 

lllllllllllllllllllilllllllll

 

0 0.001 0.002 0.003 0.004 0.005 0.006

Mole Fraction, X2

Figure 39. anifz/xzi - PVéo/RT z§,mole-fraction

solubility for He and N2 at 25°C for various
values of vzo.

14. Number Density Scale

So far we have analyzed the high pressure results in
terms of the mole fraction scale. In this chapter we
consider the thermodynamics of solvation on the number
density scale. This scale along with its many advantages
has been put forth by Ben-Naim.134

To illustrate how this applies to our high pressure
solubility data we write the chemical potential (in the

T,P,N ensemble) for the solute in the gas and liquid phases:

142' = RT40n(ngA3) + 12:9 (14.1a)
(12' = maupz’fi) + 11;" . (14.1b)

where A = h/(2xka)“z 1.8 the thermal wavelength of the

14 The first term on the right hand sides

solute particle.
of Eqs.(l4.la) and (l4.lb) represents the free energy per
mole of an ideal gas at the specified solute density, and
temperature. The second term, 11:, represents the excess
free energy per mole (over the ideal case) due to
intermolecular interactions in the gas and liquid phases.
Formally u: should also include contributions from the
internal partition function of the solute. For simple

solutes like the inert gases these contributions are mostly

electronic and we assume as usual that they do not change

211

212

when the solute goes into the solvent. The excess chemical
potential in phase i, p:i, (i= 1 or g) can be interpreted as
avagadros number multiplied by the free energy to take an
extra solute from infinity and place it at fixed positions
in phase i of the assembled system. (see discussion in
Chapter 6 for details.) By equating 1121 with 1129 we find
the equilibrium condition in terms of the Ostwald

solubility, L, to be

It It It
Au=p’-pg=-R'ronL, (14.2)
2 2 2
where L = (pal/p29). The key quantity that one wishes to
understand in developing a theory of solvation is the excess
chemical potential in the liquid, p:l. When the gas phase

s
is at low density it can be treated as ideal, i.e., p29: 0.

l (see

For this case Eq.(l4.2) gives a direct measure of p:
Eq.(3.11)). When the gas is at a higher density one must
account for its nonideality. For the nonideal case we use
Eq. (11.2) to find the chemical potential in terms of the

solute fugacity. Using the equilibrium condition we now

find an expression for p21:

*1 92 RT
(.12 = -m~m[ —] = «7011(3) . (14.3)
f

This introduces the new dimensionless solubility parameter,
1 = (pleT / f2). In the low pressure limit 1 reduces to L.

Equation (14.3) also holds when the solvent vapor pressure

213

isn’t negligible, as would apply near the critcal
temperature of the solvent. In that situation Eq.(ll.1) no
longer gives the solute fugacity correctly. Rather one must
then employ standard mixing rules in the gas phase to find
f .121,l31
2
Because our experimental technique allows for' direct
measurement of pa1 we can easily evaluate L and 1. Columns
5 and 6 of Table 13 lists the results of our solubility
measurements in terms of the parameter 1. Also included are
the values of ”:1 as given by Eq. (14.3).
The statistical mechanics of solute-solvent mixtures

135,16

starts with a standard partition function from which,

as we have shown in chapter VI, one may obtain the excess
chemical potential for a single solute molecule in the

liquid solvent14:

*1

"2

= -kTOn<exp(-Bo/kT)> . (14.4)

The key quantitiy in Eq.(14.4), 30’ is the binding energy of
a single solute to a fixed configuration of the solute
solvent system. The ensemble average of the exponential is
taken over all configurations. In practice Eq.(14.4) is
difficult to evaluate unless one performs some sort of

weighted average which neglects improbable

102,136 1

*
configurations. Another way of writing p2 is to use

a coupling parameter approach:

214

*1_ 1
pa - Io 0(1)01 , (14.5)

where the averaged interaction potential between the solute
and the liquid is a function of the coupling parameter,

1.38,39,40

Normally one ignores solute-solute interactions
with this technique. If solute-solute interactions become
important the resulting chemical potential will be a
function of the solute concentration in the liquid, i.e.,
“:1: F(pzl) 'The solute density can then be determined
through Eq.(l4.3).

The pressure and concentration dependence of u:l can be
evaluated in an manner analogous to our earlier derivation.

Starting from Eq. (22) of Ref (115) one can easily show for

dilute solutions:

-Rron(1) = -RT0n(L°) + Pv2° + pz’ RT( -022 - v20 +RTx° )

+0(p2‘)2 (14.6)

where Lois the infinite dilution limit of the Ostwald
solubility,and -RT6n(Lb) is the excess chemical potential in
the same limit.

The number density scale also offers advantages in
studying the temperature dependence of the salvation
process. The excess molar enthalpy and entropy in the

liquid are:

 

€3p*l
*
6T 9,. 2

where pzl is given by Eq. (14.3).

The excess free energy of solution is commonly described
on the mole fraction scale by Henry’s constant via; 1.1;” =
RTbn(k") , where k": (fa/x2). In this case the term excess
refers to a hypothetical standard state22 and ideal
solution.22 Henry’s constant is related to the solubility

parameter 1 by:

7=(p,‘+ (02’Hi'l‘lkll (14.88)

4: pilRT/k" , when pzl<<p11 . (14.8b)

The corresponding entropy and enthalpy evaluated on the
mole fraction scale includes terms which depend on the
thermal expansivity of the liquid. These terms are
irrelevant in the description of the salvation process on a
statistical mechanical level. A more complete discussion of
this topic is found in chapter III and Ref. (134).

In the limit that solute-solute interactions are
negligible a useful expression for the solvation entropy at
a given pressure P may be obtained by applying Eq.(l4.7) to

Eq. (14.6). The result is:

 

a‘v
3*‘(p) = 3*”°’ - p[ 2 ] , (14.9)
2 Z P n

216

t
where 821(0)

is the entropy associated with the infinitly
dilute solution at low pressure. This result seems
reasonable since changes in molar volume would be closely
tied to solvent ordering around the solute.

An application of the new solubility parameter 1 is in
the case of gas solubility at temperatures near the solvent
critical point. This subject has become a source of great
interest because of its applications towards industrial
power131 and the study of geological processe8137. For
dilute solutions approaching the solvent critical
temperature the gas pressure is dominated by the solvent
vapor. It has been experimentally observed, for water and
nonaqueous solvents, that Henry’s constant ka passes through
a minimum and then declines as the temperature is raised

from the triple point to the critical point.131 In a recent

article Japas and Sengers91

have noted that for x2<<1 a
linear relation exists between the solution density 91 and
the quantity A=T0n(k"/f:’), where f1*is the fugacity of the
pure solvent vapor at temperature T.91 For nonpolar gases
in water this correlation exists over a range of more than
150 K below the steam critical point of 647 K (see figures
2-6 of ref 91). We can understand this result by using Eq.

(14.8b) to convert from Henry’s constant to 1. We can then

solve for A in terms of 1:

A = Nana/21*) m Tbn(p1lRT/fi*12) = TOnHI/yz), (14.10)

217

- l _ i
where 11- (p1 RT/fi) and 12 - (p2 RT/fz). We have also
assumed that for dilute solutions the solvent fugacity f1 is

*
equal to that of the pure solvent f1 . This is probably
4

justified since x2810- . Multiplying Eq. (14.10) by R
gives:
RA a 121mm!) + RTanz) = 11’,"- u? . (14.11)

Therefore the quantity A is proportional to the difference
in excess chemical potentials between the solvent and the
solute, both evaluated in the liquid phase. In chapter 6 we
derived expressions for the excess chemical potential using
a coupling parameter approach. Applying this to the solute

and solvent we find:

1 m
*1_ 1 9 9 3+ 1
p2 - 4npi‘fo Io u12(r) g12(r,l) d r d1 + O(p2 )

m pilllNT-12] . (14.128)
“*1: 471131119 11 (i‘) g (1" 5) d3? d5 + O(p l)
1 1 o o 11 11 ’ 2

01 pilfINT-lll , (14.12b)

where 1 and E are the solute-solvent and solvent-solvent
coupling parameters respectively, and [INT-12] and [INT-ll]
represents 471 times the integrals appearing in 14.12a and

14.12b. Using this result Eq (14.11) can be written as:

 

218

A a p1'([INT-11] - [INT-12] ]/R . (14.13)

For dilute solutions the solvent density p l is

1

approximately equal to the total solution density 91' It is
then apparent that the obseverd linear relation between A
and p1 is a consequence of Eq (14.13). This tells us that
as we approach the critical temperature of the solvent the
integrals [INT-11] and [INT-12] either become weakly
dependent on the solution density or linearly dependent on

the solution density. Why this happens at high temperature

is unknown and merits further study.

15. Conclusions

In conclusion we discuss briefly how this work can be
used to predict and understand solubility data and what
further experiments would be useful towards these ends.

We have found that the solubility of simple nonpolar
gases in water can be adequately predicted as a function of
pressure using Eq.(12.L4). Although salute-salute effects
may play some role in the salvation process the change in
chemical potential with pressure is primarily due to the
increased Pv work to place the solute in the solvent. The
possible exceptions to this are He, H2, and perhaps Xe. The
first two cases were discussed earlier.We include Xe in this
list because our experimental technique was not as sensitive
as we would have liked it to be in the pre-clathrate range.
The other reason to study Xe is because it is the mast
polarizable inert gas (excluding radon). If the salvation
properties of He and Xe were known with sufficient
accuracy3one could set limits on the range of expected
hydrophobic behavior of the inert gases.

It is known that gas solubility in non-aqueous liquids
can be dependent on concentration. Fbr example, Hz solute
in N33 solvent shows large deviations from Eq. (12.15) at

121,138

mole fraction concentrations near 10 percent. To

219

220

account for these effects one can use thermodynamic

121

arguments based on a two suffix Margules equation (a

semi-empirical form of the excess free energy)to show that:

u(f/xzp") = an(kfl°) + , (15.1)

 

where x1 is the mole fraction concentration of the solvent
and A is a constant. This equation is known as the
Kritchevsky-Ilinskaya equation.139 It is often used in
chemical engineering to model solutions of light gases such
as He or H2 in liquid solvents where the solubility is
appreciable.121 It is only valid if the partial molar
volume of the solute is independent of pressure and
composition over the range studied. If this assumption
holds the constant A can be determined by fitting solubility
data. Equation (15.1) can be understood at a statistical
mechanical level. If the solution is sufficiently dilute we
may substitute (x12-1)s -2xz . By comparison with Eq.
(12.14) the coefficient A is identified as - +§(T). This
allows for convenient transformation of data that has been

21 Further experiments

compiled in engineering journals.1
with nonpolar liquids will be useful for comparison with
aqueous solutions.

We have also argued that the number density scale

provides a better framework to analyze solubility data. For

dilute systems like those discussed in this paper it is

221

fairly straightforward to convert from the mole fraction to
the number density scale by Eq. (14.8b). However for more
concentrated solutions one would need a separate measurement
of the total solution density in order to extract the
relevant thermodynamic properties, particularly the enthalpy
and entropy. Our experimental technique automatically
measures the saturated solution density by sampling a fixed
volume of liquid. We feel that present experimental

methodslll’125

could easily be modified to include density
measurement. This may be especially valuable in the study
of gas-liquid mixtures near the solvent critical point,

where the solute molar volumes tend to increase drastically.
110,131,140

APPENDIX

Appendix A: Derivation of Equation (6.78).

For a liquid in equilibrium with its own vapor can

rewrite Eq. (6.76) as:

RTOMPq/RT) - 1!th - go" - 81114 = 0 , (A.1)
where we have assumed the vapor phase to be ideal. We now
take the total derivative of A.l with respect to temperature

along the liquid-gas coexistence curve.

Mung/133) - 6.011% + 11'1- §, [07199] - 111- %, [441117]

-.. 4. [+4],- 4. (a... + 4.1-0 .

The subscript c refers to the derivative along the
coexistence curve. We note that this along the coexistence
curve the chemical potential in the gas and the liquid are
always equal, which is why Eq.(A.2) is zero. The total
derivative of the chemical potential with respect to

temperature can be written as:

(44); [21.42420 .

222

 

223

Using standard thermodynamics Eq. (A.3) becomes:

[31): .. . V131). . ....)

Multiplying Eq. (A.2) by T and using identity (A.4) we have:

2 d 2 d
RTbMPg/RT) - RTlmp‘c + RT 3? [W9] - RT 3‘? [MT]

c C

“23'4"” ]c+ Tscv

+ Tsim- T(vc"+ vmt)[§%) = 0. (A.5)

Subtracting (A.5) from (A.l) yields an expression for the

enthalpies h .and h :
cav int

_ 2 d _ 2 d 2 d
hcfl+ h1M RT a-T-[bnp‘c]c RT aT[lmRT]c + RT “[m‘i].
- 7'va 3714:0119] = 0 , (A.6)
9 c
where V; = vc"+ V1114. is the molar volume of the liquid

solvent. The Clausius-Clapyeron equation allows us to write
the dervatives of the vapor pressure in terms of the heat of

vaporization (Hv) via:
d[0nP]-H/RT‘°- (A 7)
3T 9 - v ° '
c

Putting this result into Eq.(A.6) and evaluating the other

temperature derivatives in terms of the gas and liquid

224

expansivities we find the desired result:

h +11 -RT+RT201

cav int ‘C

+ HV[ 1 + Pvt/RT ] = 0 . (A.8)

no nnnnnnnnnnnnn fifififififi

nnn

fififi n O fiflfi

Appendix 3: Computer Program to Evaluate 01m

THIS PROGRAM CALCILATES M PAIR CORREJTIM
FWCTION FOR A HARD SPHERE FLUID. USING TIE
SBAI-EH’IRICAL APPROXIHATION BY

VERLET ET AL WTLIIED IN

’EQJILIMIW TI‘EDRY 0F 51101.5 LIMDS

PHYS REV A, VII. 5, W 2, 1972

THEIR 7664441006 MODIFIES THE PERCUS YEVICR 504.071044 60R

HARD SPHERES, so THAT 17 ms MICAL 5114.4.A71044 0A7A.

70 00 74415 44E ~50 44:57 use 7146 PERcus-vmcx

501071044 BASED 044 1745 72644414105 0147114460 144

‘ANALYTICAL REPRESENTATION 04: 7746 96146145 YEVICK HARD SPi-ERE RADIAL
01571410471044 WIOR' BY 9.11. 5141774 A443 0.4606150!

4404. PHYS.,1970, V0119, 440 3, 411-416

74415 VERst 0417 EVALUATE 7606 LP 70 34153.0 AF7ER mm It
TAKE 0(X/R)-l.0

666666666ALSO HAS BEEN WIFIED T0 EVALUATE TIIE BERRY w INTERACT!”

Uint FRN HICH E WILL DETEflIK TIE Lacuna-304:5 PM PM
TEE ME LINID .

O....O............OOCOOOOOOOOOOOOOOO
vARIAaLes ARE DESCRIBE 144 544174445 PAPER
MICIT 604019444 (6)
01449451044 CT(3) ,65(3),6s1(3).652(9),653(3)
017464451044 61(3).611(3)
014467451044 6A0(3),660(3).661(3)
011467451044 662(3),663(6)
014464451044 60(2)
REAL R.E,I,7441R0.ALPHA,VP,744,6E7A,VERL
REAL 00,01,02.03,m,0707,440u47,6044,07071
REAL INTERVAL,4I’0INT,UTOT,SW,LPREV,INTEGRAL

FIRST E EVALUA7E 7446 714466 ZERO’S 04= 5(6)
Mismmnemmmecma
S(t)s((l-N)662)6t663) 6 (6604604066662) 406646644641) 0 (126446042640)

TYPE 6, 'PACRINO FRACTIM RI’
ACCEPT 6,14 .

......‘OOOOCOOCOOOOOOOCO0....
CMERT REAL PACKIM FRACTIM T0 P.Y. EQUIVALENT PACKING FRACTIM
SEE VERLET pg 461
44-44- 44.44/16)
F IS TIE WIN old I” VERTLET’S Pm
6044-.75.44.4441-.7117444-.114.44.44)/(1-44)..4

......OOOOOO.....OOOOOOOOOOO
VARIM ALOEMAIC WWII”
SEE SMITH st sl pp 412-413

F834 (36") 446R

THIRD-i .0/3.0

ALPHA-(1626 (Rm/F) 662) 66 . 6

YP-(l 6ALPHA) 66 (THIRD)

444—1+(.6.(1-ALH4A)«(7441R0))

TYPE 6,”

225

n nnnnn

10

2226

TYPE 6 F

TYPE 6 :ALPHA

TYPE 6 YP

TYPE 6,YM
CJ=CEXP(CHPLX(0.00000,2.09439510))
CJ=EXP(26PI6i/3)

TYPE 6 CJ

CJ2=CJ6CJ

CJ _18CEXP(CMPLX(0. 00000, ~2. 09430510))

CJ _2-CEXP(CNPLX(0. 00000, -4. 10879021))

CJ l:(CJ)66-l CJ _2:(CJ)66-2

TYPE . ,CJ2

TYPE 6 ,CJ 2

CXO=(CMPLX(YP, 0. 0000)6CJ0)6 (CMPLX(YN, 0. 0000)6CJ0)
CXls(CMPLX(YP, 0. 0000)6CJ1)66(CMPLX(YH, 0. 0000)6CJ_1)
CX2:(CMPLX(YP, 0. 0000)6CJ2)6 (CMPLX(YU, 0. 0000)6CJ_ 2)
TYPE 6 CXO

TYPE 6 :CXI

TYPE 6,CX2

type 6,third

BETA=(26N6F)66(third)

CT(l)s(-26N6BETA6CX0)/(l-N)
CT(2)-(-26N68ETA6CX1)/(l-N)
CT(3)a(-26N68ETA6CX2)/(l-N)

'TYPE 6,’t0, t1, t2 8’

TYPE 6,CT(i),CT(2),CT(3)

THESE ARE THE THREE R0075 0F 5(4)

4404 446 SHALL EVALUATE THE OTHER VARA6LEs

446 USE 7446 SAME NOTATION A5 5441714 A447 4464436R50N EXCEPT
MMWMBMMMEMWJeMWMMfl
A150 CHANGE summons 1-0 To 2 66604465 44.1 70 3

FIRST RE SHALL MAKE SURE THAT OUR ROOTS GIVE S(t)-0
DU 10 K21, 3

038(1. 0-N)662.0

0286.06N6(l.0-N)

01818.06N6N

00:12. 06N6(l62. 06H)
CS(K)2036CT(K)6CT(K)6CT(K)6026CT(K)6CT(K)6016CT(K)-00
TYPE 6 ,CS(K )

CONTINUE

NOR EVALUATE OTHER CONSTANTS

DO 100 Ksl,3
CL(K)I(l.06(N/2))6CT(K)6(2.06N)61.0
CL1(K)31.06(N/2)

CSl(K)83.O6036CT(K)6CT(K)62.06026CT(K)601
C$2(K)s6.06036CT(K)62.0602

C53(K)86.0603

CAO(K)=CT(K)6CL(K)/CSI(K)
CBO(K)I(-12.06N6CL(K))/(CSI(K)6C$l(K))

(if)

4545 45 45

227

C81(K)=(i.0-(CT(K)6CS2(K)/CSI(K)))6CL(K)+(2.06CT(K)6CL1(K))
C82(K)=CT(K)6CL(K)

6044714406
0.44. NOW RE'vE DEFINED ALL THE FRICKIN’ VARIABLES LETs P100 THEM
IN AND EVALUATE 0n(x) n: 1 70 2

TYPE 6,’HON LARGE IS THE INTERVAL BETWEEN POINTS'
ACCEPT 6 , INTERVAL

TYPE 6,’XR IS EVALUATED STARTING PRO“ 1.0’
XR:1.0

NPOINTSs2/INTERVAL

TYPE 0,'NPOINT$I',NPOINT5

00 2000 KOUNT81,NPOINTS-l

CX01TOT=0.0

CX02TOT=0.0

OPEN(1,FILE-’PERYEV.DAT’,STATUSI’NEN’)

DO 1000 Kal,3
CXGlaCA0(K)6CEXP(CT(K)6(XR-1))
CX01T0T8CXGITOT6CXOI

6x02a660(R).(661(R)+662(R).(XR-2.0))+CEXP(67(R).(xR-2.0))
6x02707-6x02707.6x02

CONTINUE
TYPE 6,CX01TOT
TYPE 6,CX02TOT

C0(i)=CX01TOT/XR

- C0(2)8CX02TOT[XR

46 ARE ALMOST 00446, 4404 A115 446 0075 TA 00 Is EVALUATE cm
566 60 (1) I44 PAPER

IF (XR.GE.3.0) GTOT-REAL(1.0)

IF (XR.LT.3.0) 0TOT=REAL(CG 1)+60(2))
IF (XR.LT.2.0) 0T0T6REAL(C0 1))

IF (XR.LT.1.0) 0TOT=REAL(0.0)

IF (xR.60.1.0) 07071-0707

VERL-24.0.Pon.(XR-1)/(R.0ToTI)
0T0T=0T0T+(F0N/XR)6COS(VERL)6EXP(-VERL)

TYPE 6,'0T0Ta’,0T0T,'XR-',XR

570R6 0ATA IN FILE FOR 70P DRAWER PERYEV.DAT
WRITE(1,6) xR,070T

....OOOOOOOOOIOOOOOOOOOO

EVALUATE THE POTENTIAL ENERGY

1999

2000

228

UTUT=GTOTt((XRtt-‘)-(XRtt-10))
IF (MARKER .EQ. 0) THEN
MARKER=1
UPREVsUTOT
GO TO 1999
ELSE
SUM=SUM0UTOT+UPREV
UPREV=UTOT
END IF

XR=XR+INTERVAL

CONTINUE

INTEGRALalNTERVALoSUN/Z
TYPE 0,’INTEGRAL=’,INTEGRAL

CL05E(1)

STOP
END

 

 

9.

10.

ll.

12.

13.

14.

15.

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