TRACER AND MUTUAL DIFFUSION IN SEVERAL
[SSTHERMAL NON - IDEAL LIQUID
NON ~ ELECTROLYTE SYSTEMS

Thesis for the Degree of Ph. D.
MESHIGAN STATE UNWERSHY
C. MICHAEL KELLY]

1970

   
 
 

  

LIBRARY

Michigan Stats?
University P

{HR-“S

This is to certify that the
thesis entitled

TRACER AND MUTUAL DIFFUSION IN SEVERAL
' ISOTHERMAL NON-IDEAL LIQUID NON-ELECTROLYTE SYSTEMS

presented by

C . MICHAEL KELLY

has been accepted towards fulfillment
of the requirements for

Ph . D . CHEMICAL ENGINEERING

degree in

Major professor

Date February 25, 1970

 

0-169

ABSTRACT

TRACER AND MUTUAL DIFFUSION IN SEVERAL
ISOTHERMAL NON—IDEAL LIQUID
NON-ELECTROLYTE SYSTEMS

By

C. Michael Kelly

Hydrodynamic theory is used to develOp equatiOns
predicting the effect of intermolecular association upon
tracer and mutual diffusion. On the basis of simple
assumptions about the volume of associated complexes, it
is shown that Onsager's Reciprocal Relation should be
valid in certain associated systems.

An experimental study is made of tracer and mutual
diffusion in several systems. It is found that the
association characteristics of a given system may be
determined from plots of the tracer diffusivity-viscosity
product vs. composition.

It is further shown that several systems which are
non-associated, as can be seen from the D*n products, fail
to obey the Hartley-Crank equation. Possible reasons for

this failure are presented.

C. Michael Kelly

A study has been made of the method currently
employed for measuring ternary diffusivities. Weaknesses
in the current method are pointed out, and suggestions are
made for improvements. Within experimental precision,
however, ternary measurements support both the predictions
of hydrodynamic theory, and the Onsager Reciprocal

Relations.

TRACER AND MUTUAL DIFFUSION IN SEVERAL
ISOTHERMAL NON-IDEAL LIQUID

NON-ELECTROLYTE SYSTEMS
By

c. Michael Kelly

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Chemical Engineering

1970

r:
ézaOl-

7-/-7o

ACKNOWLEDGMENTS

The author wishes to express his appreciation to
Dr. Donald K. Anderson for his guidance and assistance
throughout the course of this study, and to the chemical
engineering faculty in general for their support and
encouragement.

The author is grateful to the National Science
Foundation for financial support during his graduate
studies, and to the Division of Engineering Research for
support of this work.

The patience and encouragement of the author‘s wife

Ruth is deeply appreciated.

ii

TABLE OF CONTENTS

Page
ACKNOWLEDGMENT. . . . . . . . . . . . ii
LIST OF TABLES. . . . . . . . . . . . V
LIST OF FIGURES . . . . . . . . . . . vi
INTRODUCTION . . . . . . . . . . . . I
BACKGROUND . . . . . . . . . . . . . 5
Hydrodynamic Flow Equations . . . . . . 5
Solution Thermodynamics . . . . . . . 9
Nonassociated Solutions . . . . . . ll
Associated Solutions . . . . . . . 15
THEORY . . . . . . . . . . . . . . 18
Solution Thermodynamics . . . . . . . 18
Nonassociated Solutions . . . . . . 18
Associated Solutions . . . . . . . 22
Tracer Diffusivities . . . . . . . . 26
Nonassociated Systems . . . . . . . 26
Associated Systems . . . . . . . . 28
Binary Mutual Diffusivities . . . . . . A0
Nonassociated Systems . . . . . . . 40
Associated Systems. . . . . . . A2
Ternary Mutual Diffusivities. . . . . . A5
Nonassociated Systems . . . . . . . 45
Associated Systems . . . . . . . . SA
EXPERIMENTAL . . . . . . . . . . . . 66
Tracer Diffusivities . . . . 66
Calculation of Tracer Diffusivities . . . 69
Mutual Diffusivities . . . . . 72
Analysis of Binary Mutual Difquion
Experiment. . . 7A
Calculation of Binary Mutual Diffusivities . 78
Analysis of Ternary Mutual Diffusion
Experiment. . . . . . . . . . 79

iii

Page

RESULTS AND DISCUSSION . . . . . . . . . 87
Tracer Diffusivities . . . . . . . . 88
Mutual Diffusivities . . 101
Error Analysis - Mutual and Tracer Difqui on 106
Ternary Diffusivities . . . . . . 109
Error Analysis - Ternary Diffusion. . . . 114

SUMMARY . . . . . . . . . . . . . . I24

FUTURE WORK. . . . . . . . . . . . . I26

APPENDICES . . . . . . . . . . . . . 129
A. Procedure for Tracer Diffusion

Experiment. . . . . . . . . . 130

Procedure for Mutual Diffusion

Experiment. . . . . . . . . . 138

C. Computer Programs . . . . . . . 147

D. Ternary Intermediate Data . . . . . 153

E. Experimental Results . . . . . . . 166

F. Thermodynamic Data. . . . . . . . 171

G. Nomenclature. . . . . . . . . . 17A
BIBLIOGRAPHY . . . . . . . . . . . . 179

iv

LIST OF TABLES
Table ' Page

1. Summary of Hydrodynamic Theory Predictions
of Tracer Diffusivities . . . . . . . 39

2. Predicted and Experimental Ternary Dif-
fusivities in the System Chloroform (C) —
Acetone (A) - Benzene (B) at 25°C. . . . 110

3. Reduced Sensitivity Coefficients, 8: in the
. » i

System Acetone (A) - Benzene (B)-
Chloroform (C) for all Choiced of
Coordinates . . . . . . . . . ', . 119

A. True Initial Times for Ternary Runs, as
Predicted by Equations (187) and (188) . . 122

LIST OF FIGURES

Figure Page

*
1. Din vs Mole Fraction for Associating

Components . . . . . . . . . . . 89

x
2. Din va Mole Fraction for Nonassociating

Components . . . . . . . . . . . 91

3. Mutual and Tracer Diffusivities for the
System 2-Butanone — Carbon Tetrachloride
at 25°C a o c o o o o o o o o o 914

A. Tracer Diffusivity - Viscosity Products for
the System 2—Butanone - Carbon
Tetrachloride at 25°C . . . . . . . 95

5. Mutual and Tracer Diffusivities in the
System p-benzoquinone (A) - benzene (B) at
25°C . . . . . . . . . . . . . 97

6. Mutual and Tracer Diffusivities in the
System Diethyl Ether - Carbon
Tetrachloride at 25°C . . . . . . . 99

7. Mutual and Tracer Diffusivities in the
System Chloroform (A) - Carbon
Tetrachloride (B) at 25°C . . . . . . 103

8. Mutual Diffusivities in the System Benzene
(A) - Chloroform (B) at 25°C . . . . . 105

x
9. Din vs Mole Fraction in the System Acetone
Chloroform at 25°C . . . . . . . . 111

x
10. D n vs Mole Fraction in the System Acetone
Benzene at 25°C . . . . . . . . . 112

vi

INTRODUCTION

Interest in molecular diffusion in liquids has
increased considerably in the past few years, from both
experimental and theoretical points of view. Several
techniques have been developed for measuring ordinary
(both binary and multi—component) and tracer diffusive
fluxes [22].. A large number of binary solutions, both
electrolytic and non-electrolytic, have been studied. In
the past few years a number of multi-component systems
have also been investigated. Although much work has been
done, when one considers the large number of simple
systems available it becomes apparent that the surface
has barely been scratched. Much more work needs to be
done before there can be a precise understanding of
molecular diffusion.

Theoretical efforts have centered on determining
the relationship between diffusive fluxes and the
physical and chemical properties of the system, such as
molecular weight, molecular size and shape, viscosity,
state variables and solution thermodynamics. There has
also been considerable interest in the relationship of
ordinary diffusion to tracer diffusion, both from a
predictive and a correlative standpoint. In multicomponent

l

systems, much emphasis has also been placed upon verify-
ing the theory of Onsager based upon the principles of
irreversible thermodynamics, particularly the Onsager
reciprocal relationships.

There have been two basic theoretical approaches
to the description of diffusion processes. One is based
upon modifications of the absolute reaction rate theory
of Eyring [16], and the other upon modifications of the
hydrodynamic flow model of Stokes [29]. This work will
follow the hydrodynamic approach.

According to hydrodynamic theory, transport of a
species through a solution in which there is a concentra-
tion gradient of that species takes place by means of
two processes. The first process is the flow of individual
molecules through the surrounding medium as a result of
a force acting upon those molecules. This has been
termed by Hartley and Crank 'intrinsic diffusion' [20].
The second process is the transport of molecules due to
flow of the medium. This flow occurs because of
hydrostatic pressure gradients which arise from the
differing volumes of the diffusing species. Hartley and
Crank termed this process bulk flow.

The first process can be characterized by a combina-
tion of chemical and physical properties of the diffusing
species which Hartley and Crank called the 'intrinsic

diffusivity.‘ This 'intrinsic diffusivity' is the

product of two terms, one involving the physical
properties of the diffusing species and the surrounding
medium, and the other involving solution thermodynamics.
The first term will be called here the "intrinsic
mobility" of the species, as suggested by Carman [6].

Equations have been developed relating diffusivities
to the intrinsic mobilities of diffusing species and
solution thermodynamics. These have been modified by
assuming that in some systems molecular interactions can
be characterized by a chemical association. In these
systems, a given stoichiometric component may undergo
intrinsic diffusion not only as monomers, but as dimers,
trimers, and other associated complexes as well.

In this work ordinary (binary mutual, and ternary)
diffusion has been studied by means of a Mach-Zehnder
interferometer [5], and tracer diffusion has been studied
by a capillary technique, for several systems of interest.
It is shown that the degree of associative behavior in a
given system can be determined from the tracer diffusi-
vities of the components. Equations are developed
relating association (as determined from tracer
diffusivities) to the intrinsic diffusion process, and
to solution thermodynamics, upon which ordinary diffusion
is highly dependent. These equations will be tested by

the diffusivity data previously mentioned.

It will be shown that hydrodynamic theory predicts
that the Onsager reciprocal relations are valid for non—
associated systems, and several specific types of
associated system. Experimental measurements made in the
ternary system acetone - benzene - chloroform agree with
Onsager's reciprocal relation within experimental
accuracy.

Unfortunately, precise experimental verification
by this method is quite difficult, for reasons which
will be discussed later. An analysis has been made of
the probable causes of low experimental precision in the
measurement of ternary diffusivities by this method.
Possible avenues of investigation will be suggested which
might lead to an improvement in the method. It is hoped
that future work along these lines will lead to a precise
experimental verification of Onsager's reciprocal

relation.

BACKGROUND

Hydrodynamic Flow Equations

 

Hydrodynamic theory states that a diffusing
molecule behaves like a particle undergoing viscous
flow through a continuous medium. The driving fOrce for
diffusion which causes this flow is generally agreed to
be the gradient of the chemical potential of the diffusing
species, acting in the direction opposite to the gradient

of the chemical potential:

Fid = - Vui (1)

Since there is assumed to be no acceleration, this must
be balanced by a drag force upon the molecule, due to
the viscosity of the medium.

Sutherland [29] and Einstein [10] independently
showed that the viscous drag force for a sphere flowing

through a continuous medium is given by

= - 2
Fsr 61rrsnvSm ( )

where rS is the radius of the sphere, n is the
viscosity of the medium, and vsm is the velocity of the

sphere with respect to the medium,-and the negative

Sign is because the drag force is in the direction
opposite to the flow.

If the molecule were truly a sphere diffusing
through a continuous medium, equations (1) and (2)

could be combined to obtain

dui
.. “dz = enrinvim (3)
dui
where Vui has been replaced by d§_’ denoting one-

dimensional diffusion. Multiplying by the concentra—
tion of the diffusing species, and solving for the flux
of that species gives

C. du.

=_ l l
i Vim Ci 6nrin dz (4)

 

m is the flux of species i with respect to the

where Ji
medium. However, most molecules are essentially non—
spherical, and unless the diffusing molecules are much
larger than the surrounding molecules the medium cannot
be considered continuous. Therefore, the radius of the
diffusing molecule rS will be replaced by an empirical
constant 2%, which will be called the 'friction
coefficient‘ of species i. This yields

Ci dui

Ji =-61-fi-a'E- (5)

The defining equation for the chemical potential of

species i is given by

0

pi = “i + R1 Ln ai (0)

Substituting into equation (5) gives, at constant T

and P,

 

Ji = - o.n EE- (7)

m Ci [aLnai Sci]
1 L. i T,P T,P

This is the expression for the flux of Species i due
solely to the random molecular motion of 1 molecules,
with respect to a coordinate system fixed in the sur-
rounding medium. However, this is not a directly
measurable quantity (although experiments can be conceived
which could measure this flux, they are beyond the
capabilities of current techniques). The flux which is
measured in most experiments is the flux of component i
with respect to a coordinate plane across which the
total volume flux is zero.

This flux may be obtained from equation (7) by
the following argument. First, the flux of component i
with respect to a coordinate system fixed in the
laboratory in an N-component system is given by the

expression

.m ' <8)

HMZ
<1
c_.
+
<

ll
<1

where vmc is the velocity of the medium with respect

to laboratory-fixed coordinates, and v is the

Vc
velocity of the volume-fixed coordinate system with
respect to laboratory—fixed coordinates. This can be
seen by making a balance of volumes of diffusing

species in the laboratory-fixed coordinate system. The

quantity

which has the units of velocity, represents the flux
of volume due to random molecular motion, relative to
the medium. To relieve hydrostatic pressure gradients,
the medium itself must flow, relative to fixed coordi-
nates, and this velocity is Vmc'
Unless the partial molar volumes of the diffusing
species are constant, the total volume changes as
diffusion proceeds. As a result, the volume-fixed
coordinate system acquires a velocity relative to fixed
coordinates, va. The total volume flux (which is a

velocity, and is represented by the left side of

equation (8)) with respect to fixed coordinates must

equal the velocity of the volume-fixed coordinate

system with respect to fixed coordinates. Equation

(8) can then be solved for the velocity of the medium:

i=1 1 i Vc (9)

Wirth [34] gives a detailed derivation of this

expression.

The measurable flux Jiv is given by

i (Vmc - VVc) (10)

Combining equations (7), (8) and (10) gives the

basic hydrodynamic flow equation:

v i
J = - ———
i din

 

Q)
A I:
3' p.
+
:3
},.1

’=1 0 Bz I (11)

Solution Thermodynamics

Historically, there have been two approaches to

describing non-ideal liquid solutions. It is agreed

that intermolecular forces lead to deviations from

Raoult's law. The differences between the two approaches

arise from different interpretations of these inter-

molecular interactions.

The more widely accepted approach, as originally

developed by van Laar [32] and van der Waals and their

10

followers, considers all intermolecular forces to be
general in character. They arise from such phenomena as
coulombic attraction and repulsion, dipole interaction
(both permanent and induced dipoles), and van der Waals
forces. This approach is called the "physical approach,"
and is readily applicable to most types of solutions.
The second approach, originally set forth by
Dolezalek [9], considers all deviations from
Raoult's law to arise from specific intermolecular
forces which lead to chemical bonds between molecules.
According to this theory, a solution of components A
and B consists of A monomers, B monomers, plus various
associated complexes such as A2, A3, A4,..., B2, B3,
B4,..., AB, A2B, A32, and so forth, depending upon
the specific interactions present. These individual
species are then assumed to obey Raoult's Law. The
proportions of the species present in solution are
determined by an equilibrium characterized by an

equilibrium constant, such as A + B ++ AB. Because of

KAB

this equilibrium assumption, this is known as the
"chemical model" of solution non-ideality.

Dolezalek originally proposed this model before
the nature of chemical bonding was well understood. He
was led by his model into some rather improbable

hypotheses. For example, he tried to describe the

ll

vapor-liquid equilibrium in the system nitrogen—argon
by postulating the dimerization of argon, a most unlikely
occurrence.

These two approaches are not mutually exclusive,
though for years there was rather heated debate
between the two schools. As our knowledge of chemical
bonding increased, it became apparent that some systems
really do associate in liquid solution. This is
especially true of molecules which are capable of
hydrogen bonding, such as water, alcohols, amines, etc.,
and of molecules which form charge-transfer complexes.
On the other hand, there are many non-ideal solutions
in which the formation of associated complexes is rather
unlikely. Further, there is no good reason to conclude
a priori that the various species present in an associated
solution should obey Raoult's Law, as Dolezalek assumed.
It would seem logical to try to combine the two

approaches.

Non—associated Solutions

 

Before considering associated solutions, it would
be well to look at non-associated solutions. Perhaps
the simplest method of describing activity data in
non-associated systems is to assume that the natural
logarithm of the activity coefficients can be expressed
as a power series expansion of the mole fractions of

the stoichiometric components. This is the approach

l2

taken by Margules in deriving the equations which bear
his name. The constants in the power series expansion
are restricted by the Gibbs-Duhem equation. Within
these restrictions, the values of the constants are
determined by fitting the series to experimental
thermodynamic data, generally vapor-liquid equilibria,
by means of least—squares analysis. The equation can
be made to fit experimental data to whatever degree of
accuracy desired by simply taking more and more terms
into the series expansion, though at the expense of
introducing more arbitrary constants. It can be extended
quite easily to multicomponent systems, and is not
restricted in its range of application except by the
number of terms in the series that one wishes to use [35].
Though this procedure is mathematically rigorous,
and useful for describing experimental data for use in
design calculations, it sheds very little light on the
true nature of interactions in liquid solutions. Van
Laar [32] proposed a far more restricted equation,
based upon theoretical considerations, for binary
systems. This equation, and modifications of it, have
been very successful in describing binary systems,
especially those for which the activity data are rather
symmetrical, and for which the molecular sizes and shapes
are not too different. It has several disadvantages,

though. It is not easily extended to multicomponent

13

systems, without introducing further assumptions and
more arbitrary constants, which tend to decrease the
physical meaning of the equation. Also, it cannot be
used reliably for those systems in which there is
considerable bonding—type molecular interaction, nor for
such systems as high-polymer solutions. Both the
Margules and van Laar equations, as well as several
modifications, are discussed in considerable detail by
Wohl [35].

Another approach which is based upon theoretical
thermodynamic considerations is that of Hildebrand and
Scott for regular solutions [25]. Since it forms the
basis for some later conclusions, it will be described
in a little more detail.

If a solution contains enough thermal energy, the
different intermolecular forces of the various com-
ponents will not be sufficient to cause any one molecule
to tend to aggregate with any particular type of
molecule, either like or unlike. The entropy of mixing
will then be the same as for an ideal solution. Such
a solution is termed 'regular', even though it is non-

ideal, and the partial molar entropy of mixing is given

by

AS = - R Ln x (12)

1A

By making three assumptions, Hildebrand and Scott
show that the heat of mixing in the binary regular

system of components i and j is

(13)

where oi is the volume fraction of component i

(neglecting expansion on mixing) and 61 is defined by

where EiV is the internal energy of vaporization. The
assumptions leading to this relationship are: (a)
the energy of interaction between two molecules depends
only upon the distance between them and their orienta-
tion, (b) the volume change of mixing at constant
pressure is zero, and (c) the mixing of molecules is
random. The third assumption is essentially the
definition of a regular solution. The first, although
not rigorously correct, has been the basis for most
successful attempts at modeling the liquid state. The
second can be eliminated by extensive modification, as
shown by Hildebrand and Scott [21], but will not be done
here.

For regular solutions, where the entropy of mixing

is ideal, the activity coefficients are given by

[>
El
H.

1

Ln yi = = if. 4). (c3 - s.)2 (114)

:30
*3
P
C4
L1.
P

This can be extended to multicomponent systems
quite easily. Under the same assumptions as before,

for a ternary system we find that

where

with 6 and ¢i as defined before. Detailed derivations

i
of equations (12) through (15) are given by Hildebrand

and Scott [21].

Associated Solutions

The intermolecular forces which define an
associated solution would seem to be precisely those
forces which disqualify that solution from being
considered a regular solution. In a regular solution,
the molecules mix as though they had no preference as to
the nature of their nearest neighbors. In associated
solution, on the other hand, any given molecule has a
distinct preference for another molecule as its nearest
neighbor, as expressed by the polymerization equilibrium.

For example, a molecule with a hydroxyl group will prefer

16

to have another molecule with a hydroxyl group as
nearest neighbor (rather than a saturated hydrocarbon,
say) due to its ability to form the hydrogen bond.

It might then be a good assumption that all those
forces which lead to a solution not being regular are
due to complex formation by chemical bonding. Certainly
chemical bonding would be the major contributor to non—
regularity in associated solutions. It would seem
logical then that the true species present mix to form a
regular solution.

The mole fractions of the true species are
determined by the equilibrium equation and the stoichio-
metric mole fractions. Consider for example a binary
system in which one component dimerizes as a regular
ternary system, consisting of monomers of each component
plus dimers. Equations (15) can then be used to predict
the activity data from knowledge of the equilibrium
constant K. Alternatively, the equilibrium constant can
be determined from activity data by adjusting it until
equations (15) give the best fit.

This procedure requires knowledge of the partial
molar volume and the molar energy of vaporization of the
dimer. This can be handled in either of two ways. These
quantities may be treated as adjustable parameters, in
which case equations (15) will be a three—parameter set

of equations for the binary system [13]. Otherwise,

l7

assumptions can be made about the values of these
parameters, or their relation to those parameters for

the monomers. Equations (15) will then be a one-parameter
set of equations for the activity coefficients in the
binary system. The latter method will be used later

for fitting activity data in both binary and ternary

associated systems.

THEORY

Solution Thermodynamics—-
Nonassociated Solutions

 

 

The Gibbs free energy of a system of N components

is given by

where n1 is the number of moles of i, G10 is the molar
free energy of pure component i, Xi the mole fraction
of i, and AG represents the difference in free energy
between one mole of real solution and one mole of an
ideal solution with the same composition.

The partial molar free energy of component i, the
chemical potential of i, is given by

:1 LG = “10 + RT Ln Y1 XI (17)

where “10 is a function of T and P only. Carrying out

the indicated differentiation on equation (16), and

equating to equation (17) gives

Ln yi - filr‘r‘ —°—— [AG 2 mil (18)

19

If a power series expansion is written for AG,
and the coefficients constrained by the Gibbs-Duhem
equation, after drOpping terms of higher order than X3

there results

+ X1X32al3 + X2X3 2a23

2 2 . 2
1 2 33112 1 2 3&122 + X1 X3 3a113 (l9)

2 2 2
113 2 3 3a223 + X2X3 3a233

X X

+ X1 2 3 6a123

This is the three-suffix Margules Equation for a ternary
system [35]. Binary systems may be treated as Special
cases, and the corresponding Margules Equation obtained
from equation (19) by simply setting X3 equal to zero.
Carrying out the differentiations in equation (18) gives

the activity coefficients:

2
= [- .. _.
Ln Yi 2X1X2 LA21 X1A21 X2A12J + X2 A12
. 2
_ _ ]
+ 2xlx3 [A31 x1113l X3A13] + x3 A13 (20)
+ (x2x3 - 2xlx2x3) [A21 + A13 + A32 - t]

2O

2
u — r _ _
n Y2 2X2X1 ~A12 X2A12 X1A21J + X1 A21
2
+ A _ _.
2x2xj [A32 X2A32 ‘ X3A23] + x3 A23 (21)
+ (xlx3 - 2xl x3) [A2l + Al3 + A32 0]
Ln Y = 2X X [A - X A - X ] + x 2 A
3 3 l ‘13 3 l3 1 31 l 31
2
+ 2X2X3 [A13 - X3A13 x1 31] + x1 A31 (22)
+ (xlx2 - 2xlx2x3) [A21 + A13 + A32 - C]
where the constants are defined by
2a12 + 3a122 = Ai2
2a12 + 3°112 ‘ A21
2&13 + 3a133 ‘ A13 (23)
2al3 + 3a113 = A31
2a23 + 3a233 = A23
2a23 + 3a223 = A32
3all2 + 3°i33 + 3a223 ' °°i23 = C
The binary analogues to equations (20), (21) and
(22) are
2
Ln yl = 2x lx2[A21 - xlA21 - X2Al2] + x2 Al2 (2A)
2
Ln Y2 = 2X2 x ltA12 — X2Al2 - xl A21] + x1 A21 (25)

21

Wohl [35] gives detailed derivations of equations
formally similar, and algebraically identical to
equations (20) through (25).

If isothermal ternary activity data are available
equations (20), (21) and (22) can be fit to the data by

a le st-squares technique, with seven adjustable

m

oarameters. This is an unreasonably large number, but
some may be specified by other means. If the ternary
data are available, then binary data for each of the

three pairs of components are almost sure to be available.
The binary data may be fit by a least-squares technique,
thus fixing six of the seven constants in equations (20)
through (23). The constant C may then be determined from
the ternary data.

In most cases of interest, ternary isothermal data
is not available, but these equations are still useful.
Wohl interprets the various constants in equation (19) in
terms of physical interactions between molecules. Thus
a12 represents the energy of interaction of molecules 1

and 2, a the interaction between two molecules of type

113
l and one of type 3, etc. From equation (23) we see that
under this interpretation C is a ternary interaction
parameter. Since in non-electrolyte solutions three-
molecule interactions are not so strong as two-molecule

interactions, C can probably be taken as zero. As long

as this holds, ternary activity data can be predicted

22

from data for the subsidiary binaries. This has been
done for the system acetone-benzene-chloroform. Data
for the system acetone—chloroform have been taken from
Hildebrand and Scott [21], for benzene-chloroform and
acetone—benzene from Timmermans [30]. These data, and
their approximations by equations (24) and (25), as well
A

as the least-squares parameters A 21,... are given

12’
in Tables F—l through F-A of Appendix F.

Solution Thermodynamics--
Associated Solutions

 

 

As discussed previously, an associated solution may
be thought of as a regular solution of the true species
present. By making approximations as to the values of‘V
and 6 for the associated complexes, the equilibrium
constants may be treated as adjustable parameters in
fitting the activity data. This will be done here for
a specific case, a binary system of components A and B,

where the association reaction
A + B ++ AB

occurs. To avoid later confusion, let us refer to the
stoichiometric components by letters A and B, and the
true species present by numbers 1, 2, and 12, where 1
refers to the monomer of component A, 2 to the monomer of

B and 12 to the dimer. The true equilibrium constant

23

for this reaction is given by Ka’ the ratio of

activities of species:

 

a
waif
l 2
Defining K and KY by
K = Xi2
X1X2
Y:
fi=_:‘_a.
Y Y1Y2
we see that
Ka = KK

Nikol'skii [27] has shown that the chemical
potential of a component in solution is equal to the

chemical potential of its monomer:

= X

AYA 1Y1

Hence, the activity coefficient of component A is

given by

The activity coefficient YA is a directly measurable

quantity, but Y1 is not.

(27)

(28)

(29)

(30)

(31)

(32)

24

For a regular solution, equation (15) holds:

 

V .
Ln Y1 = —l (51 —a>2 (15)
RT '
where 61 and 8 are defined by
~ s h
31"
51 = :__ (33)
V.
l
_ 3
5 = 2 n1 Si . (3“)
i=1
xii/’1.
4’1‘xV +xi7‘ +xV (35)

1 1 2 2 12 12

Similar equations give the values of Lny2 and LnY12°

Now, for a given X and XB’ the mole fractions of

A
the true Species depend only upon the value of K, and can
be determined from equations (28) and the stoichiometric

relationships

(36)

Let us make the following approximations:

25

a) V12 = V1 + V5 (37)
b) E¥2 = EX + E; (38)

Assumption a) is required for consistency of constant
molar volumes which will be assumed later, and which
Kett [23] determined to be a good assumption for the
associated system ether—chloroform-carbon tetrachloride.
Assumption b) is reasonable if the energy of the dimer—
ization bond is approximately the same in the vapor state
as in the liquid state, that is, if AE for the equilibrium
reaction is the same in both states. Note that AH will
probably be different, as there is a change in PV in the
vapor state.

Now equations (15) and (32) through (39) can be
combined to give ln YA in terms of measurable quantities

(V's, EV

'S, stoichiometric mole fractions, temperature)
and one adjustable parameter K.

It is now possible to determine the value of K from
experimental isothermal activity data, by a least-squares
technique. This has been done for the system ether-
chloroform from total-pressure data at 25°C from Kohnstamm
and van Dalfson [2A]. The results agree reasonably well

with the data of Guglielmo [30] at an apparently higher

unreported temperature for the vapor-liquid equilibrium.

26

This procedure can be easily extended to a ternary
system of A, B, and C where A and B dimerize as before,
and C is inert. The value of K found for the binary
dimerization equilibrium Should not change in the ternary
solution, provided the solvent C does not change the
mechanism of the reaction but merely dilutes the reactive
components. By making this reasonable assumption, it is
possible to predict the ternary activity data for such an
associating system from physical properties and activity
data for the associating binary pair of components.

These principles are easily carried over to other
forms of association.

Tracer Diffusivities-—
Nonassociated Systems

 

 

The tracer diffusivity is defined by a modified

version of Fick‘s Law:

 

(39)

where the superscript * designates the tagged molecules.-
Since the tagged molecules are considered to be
identical to the untagged molecules physically and
chemically, if there are no external pressure gradients
there will be no bulk flow.. For every molecule diffusing
in one direction, there will be another molecule diffusing

back in the other direction. Since these molecules have

27

the same volume, there will be no hydrostatic pressure
gradients, and therefore no bulk flow. In this case, the
velocity of the medium-fixed coordinate system and the
volume-fixed coordinate system will be zero (relative to
laboratory-fixed coordinates).

This means that for tracer diffusion equation (11)

becomes

 

 

 

 

C * d *
—'. ll-
J.V* = I.m* :: _ —_"'__. _I' ((40)
l l o.n dz
1
For tracer diffusion equation (17) becomes
* o. x x
= 7“ "7‘
hi hi + R1 Ln Yi Xi (A1)
*
Applying the chain rule gives the derivative of “i :
x x * Y x *
du du. dC d Ln . d Ln x. d C
= 1? d:=RT[‘—T“—‘l+ *lein)
°Z dci dci d0; Z

Since the solution is chemically uniform, it can be Shown

that

x
d Ln Yi
- 0
_____¥__

dCi

(43)

x
d Ln Xi

—"——-'¥'
d Ln Ci

28

Combining equations (A0), (A2) and (A3) gives

v, RT i (AA)

 

 

i din , (A5)

Carman [5] suggested that the combination of physical
properties on the right hand side of this equation be
termed the ‘intrinsic mobility' of species 1. This
designation will be adopted here. The important result
is that the tracer diffusivity of a non-associated com-
ponent is equal to the intrinsic mobility of that

component.

Tracer Diffusion-—Associated
[Systems
The most easily analyzed case of tracer diffusion

 

in an associated system is the binary system of A and B
where there is an association to form an AB dimer.
Denoting the species present as l, 2 and 12, the total

concentration of tagged component A is given by

C = C + C (A6)

The flux of A molecules in tracer diffusion is

29

 

V*-_ ‘ _TV* V*
JA ’ DA dz ‘ ”1 + J12 (“7)

Substituting equation (A0) for the fluxes gives

* *-

dC dC
l + 1 A2 ] (A8)

dz dz
°12

 

 

V* = _ ET _1
JA n Ecl

*
Differentiating equation (A6) with respect to CA gives

8C * “C *
1 ° 2 ’
A 8 CA

Substituting into equation (A8) gives

* ii- * it

so 1 acA RT 3012 A

*“ dz - o * dz

 

 

JV*=__Rr£.[l-

(50)
A °1n ac

2
A
If the physical and chemical properties of the
tagged molecules are the same as the untagged molecules,
we may assume that the distribution of tagged molecules

is the same in the two species as in the component:

c
____ = ___ = ___ (51)

This ratio is a constant, since it depends only upon the

*
proportion of CA and CA when the solution was made up.

Therefore,

.30

 

'X'
8C12 _ 12
- C (52)
acA A

Now for convenience, define the pseudo-mole fraction of

component 12 by
X12 = E“:‘C" (53)

Combining this with equation (52) and substituting into

equation (50) gives, after rearrangement of terms

0 *
V, _ RT - 1 1 1 X12 aCA _
J1 "—n'-3‘+<5‘-“3— 7—3? (5“)
1 12 1 A

Comparing this with the defining equation, Fick's Law,

leads to the desired result:

0
X
* r't
DA=5—;-£-C,—l-+(;L-3-l-) -——,l{21 (55)
2 12 1 A

The derivation for component B only requires renumbering
the species and components in the previous equation, so
that A becomes B, 1 becomes 2, and vice versa. Equation

(55) then becomes

0
x
* _ RT . 1 1 _g 12
DB-—HL3—+<5—-o -———X1 (56)
2 12 2 B

31

Wirth [3A] derived equivalent equations, and
carried the derivations out for two other simple systems,
a binary system where one component is inert and the other
forms a self-dimer, and a ternary system where two
components form a cross-dimer and the third component is
inert. These equations, along with those to be derived
in this work are given in Table l, on page .

Theoretically, equations corresponding to (5A) and
(55) can be developed for any associating system, provided
an equation can be written for each association
equilibrium. Practically, such equations become very
difficult to handle if there are more than 2 or 3 such
equilibria. Furthermore, Since a considerable part of
the value of these equations lies in their ability to
model the measured tracer diffusivity data, the number
of associations must be small, or there will be a large
number of adjustable parameters available to fit the data.
Also, whenever the equations become too complex, they
lose much of their physical meaning in the algebra.

We will now consider another simple system, and
develop equations predicting the tracer diffusivities.
Let components A, B and C be ternary system in which

there are two competing equilibria

A + B ++ AB

A+C++AC

32

Let the true species present be designated 1, 2, 3, l2 and
13 representing A, B, C monomers, AB and AC dimers
respectively. The relative concentrations at any compo-
sition can be determined from the equilibrium constants

K and K . The fluxes and concentrations are given by

l 2
the following:

 

8C
V * . V
JA x = _ DA "'5“: = J1 * + J1 V* + .113V* (57)
NC *
V * o B V V _
JB * = ' DB 32 = J2 * + J12 * (98)
8C
V * C V V
JC * = — DC -a-Z— = JB * + J13 * (59)
if 'X' * 9(-
CA - C1 + 012 + 013 (60)
* x * ,
C2 = C2 + C12 (01)
if * 9(-
CC = C3 + C13 (62)

Equations (58), (59), (61), and (62) are equivalent
to those for the binary system just considered. The
tracer diffusivities are analogous to those in equations

(5A) and (55):

X o
l 4) i2] 1(63)

2 °12 °2 B

 

33

a:
:I)
*3

1
l3

) i: 1 (6A)

+ (

_ 1
D - [3; o

 

 

 

O
:5

_I
C73

The equations for component A are slightly more

complex. From equation (57):

 

 

 

 

 

 

 

 

* x *
V* _ RT 3C1 RT 8C12 RT ac13 ,
JA — - [c n 82 + o n 32 + o n 32 J (05)
1 12 13
*
Differentiating equation (60) with respect to CA and
substituting into equation (65) gives
C n * x * x
V, _ RT 1 3 12 3"13 1 8C12 1 ac13 CA
JA - -7 5—(l-——-.xr - -———T)+C 0’ f] (66)
1 BOA BOA 12 BCA 13 BCA
* *
8C12 3C13
AS before, the derivatives ————¥ and ———;— can be written
8C BC
A A
0 C12
in terms of pseudo-mole fractions X12 = C + C “+ C and
A C
C
° 13 .
X = to give
13 CA + CB + CC
0 o . *
v, RT 1 1 1 X12 1 1 X13 °CA
JA - - —H E 3— ( o - 3—) X + (o - 3—0- X J 82
1 12 1 A 13 1 A

(67)

3A

From this it can easily be seen that

O O

x
1 12 1 1 13 2
+ (--—- -) + (— --) -——-] (08)
1 XA 013 O1 XA

 

 

which is the desired result. The generalization to a
larger number of competing equilibria is obvious. If
component A dimerizes with N other components, the

tracer diffusivity of A will be given by

 

. 0
DA = --]-:- C— + Z (5__ - a") 31(1 1 (69)
' 1 i=2 11 l A

This equation is general, and though it provides
physical insight into the effect of several association
reactions on tracer diffusion, it is probably not too
useful in fitting experimental data, since it allows N
adjustable parameters (the friction factors Gli) if the
equilibrium constants can be determined independently,
or N2 if they are also considered free. The physical
meaning of this equation is that the tracer diffusivity
of an associating component is equal to the intrinsic
mobility of that species, decreased by the difference
between the mobilities of the monomer and the associated
complex (corrected for the amount of association) for each

of the association reactions.

35

This same type of result can be obtained for the
slightly more complicated case in which one component
associates with itself. Consider the binary system in

which component A reacts to form a dimer, according to

*
and component B is inert. In this case, the flux of A

is given by

**
3011

11 RT
,2. 1 (70)

11n

 

 

 

where the dimer can carry either one or two tags, as

denoted by the number of asterisks. Stoichiometry Shows

* * 'X' **

= +
CA C1 + Cll 2Cll (71)

Proceeding as before, we write

 

3" * 3C * 3" ** BC *
v, RT ”1 RT 11 ”11 . A .
JA 1 { ___¥ [._——¥-+ 2-———1—u} -———(72
nol n no 32
3 11 3C GO
A A A
Differentiating (71) gives
3C * 3C * "C **
0
BOA BCA 30A

36

which, when substituted into (72) yields, after

rearrangement,

 

 

5(-
96*
V RT 1 1 1 ac11 1 1)°Cii 3 A
.1A * + - -—l;— + (3—— - g— h + 2(g- - g— x 1 «z
n 1 11 1 ac 11 1 ac °
A A
(7A)
Equation (7A) can be written
ac * ac **
V m, ..
JA * = - 5% 13$ + (g—i - 5;) (———£% + 2 -——i%—)J (75)
1 11 1 acA acA

The derivatives again depend only upon the equilibrium
constant K, although calculation of their values is some-

what involved. The tracer diffusivity, from equation (77)

is then
ac * ac ‘*
3%
DA =Eg-[5-l-+(5—l—-3-l-) <—1—}+2-—-i%—)1 <76)
1 11 l BCA 30A

Wirth [3A] gives a Slightly different derivation,
which leads to a formula which is less difficult to
evaluate:

* RT 1
-—-C

1

 

37

The form given by Wirth is more useful for trying to fit
data for which there is only one dimerization equilibrium.
It is diffucult to generalize, however, while the
derivation of generalized forms of equation (78) is
rather simple.

Consider the system where there are two components,

one of which undergoes two self—polymerization reactions,

A + A<H+A2

A2 + A*-->A3

and the second Component B is inert. The flux of tagged

A molecules is

 

 

 

 

* C * **
v, RT 1 3C1 1 3 11 8C11

JA = - fi—'£3_' 82 + o E 82 + 2 3z 3
1 11

3C * “C ** a" ***

-\ d U
+ O l (__%11_.. 2 __%ll__.+ 3 $11 . :)1 <78)
111 Z 02 Z

and the stoichiometric formula is

+ *** <79)

By the same process as before, this time leaving out the

intermediate steps,

 

“A * 3C *3!-
* 1 l 1 OK“

A ‘B'fla‘l' <.——-.—> .4...” 11>
1 11 1 3C GO
A A

8C * 3C ** 3" ***
1 1 'V
+ (O 1 _ 5-) ( 111 + 2 11 + 111 3 (80)
111 1 BOA BOA BOA

The generalization of this equation by this method to a
system where one molecule undergoes repeated Simple self—
polymerization reactions is straight-forward, but
notationally very difficult.

The effect of repeated polymerization on tracer
diffusivity is easily seen from equation (80), however.
The tracer diffusivity in this case is given by the
intrinsic mobility of the monomer, decreased by a
correction factor for each polymer. These correction
factors involve the intrinsic mobilities of each polymer,
the number of tags carried by each polymer, and the
amount of each polymer present (determined by the equilibrium
constants).

Once again this model is not too useful in fitting
tracer diffusion data unless the number of polymerizations
is small, and there is some information indicating that
there is no further polymerization beyond a certain point.
Of course, assumptions could be made to reduce the number
of adjustable parameters. For instance, it would seem

reasonable that for hydrogen bonding molecules, after the

TABLE l.-—Summary

39

of Hydrodynamic Theory Predictions of Tracer Diffusivities.

 

System and
Association

Predicted Tracer Diffusivities

 

A, a, c,....u

 

 

 

 

 

 

 

 

 

 

 

 

 

no association L)i noi i = A’ B’ C”"' N
O
n X
«a DA *_.[_1.(_l_-_1)}1(23
“’ n 01 012 01 A
A + B ++AB
( O
I." ‘ a
b, —; [—l + ( l - —l) A“ 1
-J O) 0.13 02 [LB
7 .1. O l" O
A, a, c L, 1% [EA + ’01 - EA) :- + (El - El) 33 j
“ 1 12 1 “A 13 1 ‘A
A + a ++AB
, .. O
A + C +*AC -A
H ~_i;_1.1_1-1>%23
”a ” °2 °12 OT *5
1( O
1.11. . __l_ _ .1613;
VG n ‘03 \013 01/ 2X“ 4
n c
A, a, L, A w ET . _1 + E+l ( 1 _ 1. X11 7
A + 5-6»: “ n 01 i=2 O11 °1 ”A 4
A + 21+» All .... . . * ,.
other components as in LB above
A b , * C **
. , ’ D‘- F‘T {—3. + \__l_. _. A) 21.11.}. 2(_1 _. __1_) 34%.]
A + A++:A, “ ” °1 °11 O1 3:, O1 011 acA
B inert
D2 RT
5 no;
A, B 3 fi * C **
b a
A + A+-A DA 5% [6; + (EL‘ ‘ El) “"li+ ““l%—'
2 1 11 1 8 C, BC
A A
A + A2++ A3
* ** **5
ac 3 c 3c
- .1. _ .1 111 111 111
B inert (0111 01) (ma—E—1F + 2 -.—;E_¥ + 3 a )1
A ' A CA
RT
DB

n02

 

no

first one or two polymerizations, all the equilibrium
constants might be assumed equal. It might also
reasonably be assumed that no polymer carries more than
a certain number of tags, say three or four at most.
This would make computation much easier. The physical
insight of equations of this type is considerable, as
will be seen in later discussion.

Binary Mutual Diffusion--
Nonassociated Systems

 

 

For a binary nonassociated system with components

A and B, equation (11) gives the flux of component A:

v'- CA BuA C CAVA auA + CBVB BuB
A 82 GB az

 

(81)

Substituting the definition of chemical potential gives

 

 

 

J v'= BE.[_ SA a Ln aA + C (CAVA 3 Ln aA +
A n 0A az A 0A 82
C V 8 Ln a

+ 8 B 82 B)] (82)

B
By the chain rule

a Ln aA = _£ 3 Ln aA 8 CA (83)
32 C 8 Ln CA 8z

“2:8 = .3: : :2 :3

B B

Substituting (85) and (86) into (84) gives

 

 

J V = -D 3C = 52 [_ _l 3 Ln aA + CAVA 3 Ln aB
A AB 02 n GA 8 Ln Cn CB 3 Ln bB
+ CAVA : in :A] ~ (85)
CA n A

From the definition of mole fraction and partial molar

volume the following hold:

 

8 Ln X X,
a'Ln CA = c; (86)
A B B
8 Ln XB = XA (87)
3 L C“ C V
n D A A

Substituting (86) and (87) into equation (85) and making

use of the fact that CAVA + CBVB = 1 one obtains the well

known Hartley-Crank Equation [20]:

RT‘ B A A
D = — E— —] --.——-— (88>
AB n CA GB 3 Ln XA

This equation has been used with some success in
predicting mutual diffusivities in non-associating
solutions. It was originally derived under the assumption
that the molar volumes were constant, but this is not a

necessary condition.

Taking the limits as

comparing to equation (#5)

A2

XA + O and as XB + O, and

we see that

Lim .
D _ = ——| = D (89)
XA+O AB noA A
Lim _ RT _ *
XB+O DAB ‘ noB ‘ DB (90)

Binary Mutual Diffusion-—
Associated Systems

 

 

Consider a system of two components A and B, in

which component A undergoes the simple dimerization

A + A ++ A2

and the second component B is inert. The flux of A is

 

 

V , Cl aul Cll Bull Clvl 3“1
JA - ' o n 82 - 20 n 32 + CA( o 32
1 11 l
Cllvll a“11 C2V2 3“2
+ o 82 + o 32) (91)
11 2

where the true species have been numbered as previously.
Proceeding as in the derivation of the Hartley-Crank

Equation, using the definition of the chemical potential,

the chain rule, the relation CAVA

the mutual diffusivity is found

+ CBVé = l, and the

2V

assumption V11 = A’

to be

43

 

x x O X 0
RT A l 1 ll 1 3 Ln a
D =—-—[——+X..(-—-——+ ——-)l————— (92)
AB n 02 D XA 01 XA all 8 Ln X

Detailed derivation of this result is given by Anderson,

<3r Wirth [34], and will not be reproduced here.

Consider a system of two components A and B, in

vfluich the dimerization reaction
A + B ++ AB

The flux of A is given here by

 

occurs.
v _ C1 3“1 C12 8“12 Clvl 3“1
JA - ' o n 32 ' o n 32 + CA< o 82
1 l2 1

 

 

+ C12V12 01112 + C2V2,3“2)
012 32 02 32
By’ the same process, with the assumption that
VBJB = VA + V5 this leads to the diffusivity:
o o
m X X
DAB=E%[FLJ—XB+E££~XA
l A 2 B
o 2
+ 1 X12 (XA ‘ XB) 3 Ln a (94)
X X 3 Ln X

 

°12 A B

Detailed derivation is again given by Anderson [1] or

Wirth [3n].

 

(93)

44

Note that equations (92) and (94) differ from the
Hartley-Crank equation only in the addition of an extra
term. In the case of self-association, this term is
positive, and predicts that the mutual diffusivity is
more than that predicted by the Hartley-Crank equation.
In the case of cross-association, this term predicts a
smaller diffusivity.

It is generally true that the activity term in the
Hartley-Crank Equation over-corrects. That is, when the
system shows positive deviations from Raoult's Law, the
thermodynamic correction predicts the diffusivity to be
less than it would be if the solution were ideal. In
many of these cases, the experimentally measured diffusi-
vities are greater than those predicted by the Hartley-
Crank Equation, though still less than for an ideal
solution. When the system shows negative deviations
from Raoult‘s Law, the thermodynamic correction predicts
diffusivities higher than for an ideal solution. In
these cases, the measured mutual diffusivities are found
to be less than those predicted by the Hartley-Crank
Equation, but still larger than for an ideal solution.

Equations (94) and (96) reduce the magnitude of the
deviation of the Hartley-Crank Equation from ideality.

In a system where association takes place, they should be
better predictors of diffusivity than the Hartley-Crank

Equation. This has been found to be so for several

 

45

systems. However, in some cases there is strong evidence
that the components do not associate significantly, and
the Hartley-Crank Equation still tends to over—correct.
Two such systems will be presented here. Some
theoretical explanation must still be found for this
discrepancy.

Ternary Mutual Diffusivities--
Nonassociated Systems

 

 

There have been two major approaches to multi-
component diffusion. Onsager [28] proposed a set of
equations for an N-component system relating the flux
of each component to the concentration gradients of all

the components, thereby defining N2 diffusion coefficients:

J = g D 30'
1 j 13 -—l i = 1, 2,....N (95)

Then, based upon the theories of irreversible thermo-
dynamics, he showed that only (N - l)2 of these diffusion
coefficients were independent. These diffusion
coefficients, however, are not easily measured.

Baldwin, Dunlop and Gosting [3] therefore proposed
a different description, involving only N - 1 independent
fluxes, and (N - l)2 diffusivities:

Jul

J

0
D15 ——l 1 = l, 2,....N - l (96)

32

_l a
i 1

46

These diffusivities are not the same as the Onsager

diffusivities, but are related by the expressions

{71'
.7 ' '=l,2,....N-l (97)

Since then, Costing and coworkers have presented
several methods for experimentally determining the Dij's

[13, 14]. It is preferred, therefore, to relate hydro—

chynamic theory to the Costing diffusivities. This will

bee done for a nonassociated ternary system to demonstrate

true method. It is also useful to develop equations

puredicting the phenomenological coefficients of the
Cuisager theory, to show that hydrodynamic theory predicts
tine validity of the Onsager Reciprocal Relations.

From equation (ll), the fluxes are

 

 

 

V - Cl Bul A CiVl Bul C2Vé 3u2 C3V3 3u3
Jl - - o n 32 + °l( o 32 + o 82 + o 32)
l l 2 3 .
(98)
J V = - C2 Bul + c (Clvl Bul + C2V2 8u2 + C3V3 3H3)
2 o2n 32 2 Ol 82 02 82 03 82
(99)

13 = - 8.1. La - :2. L...
32 C3 32 C3 82

 

47

Combining equations (98), (99) and (100) to eliminate the

gradient of chemical potential of component 3 gives

 

 

v c- l-V.C VQC, an , c c V V an
51 = " a: “—0; l) + 3‘3 5: ' "—1 2C3; ‘ 53'] ““322 (101)
l 3 ° 9 3 2
J v = _ clcztv; _ El] 8ul _ 32 [(l-C2V2) + V3C2] 8H2 (102)
2 n 0 ol 82 n 02 G3 82

Mathematically, the total derivative of the chemical
potential can be given in terms of the partials of all
the independent variables. For a ternary system at

constant temperature and pressure, there are only two

 

 

 

 

 

independent variables, which may be taken as Cl and C2.
Therefore,
8ul = 8ul 8Cl + 8ul 802 (103‘
82 8 l 2 8C2 82 )
8u2 = 8u2 8Cl + 8u2 8C2 (10“)
82 8Cl 82 8C2 82

Substituting these into equations (101) and (102) yields

7 V C C 8n

 

 

 

 

 

v _ Ci (i-Clvl) C1V31 25“1 3 2 l 2 2 Cl
J1"{T[T_+8Jac+(§_"a_) na—_}'§T
l 3 l 3 2 l
C — 1 "n
_ {_l [(1 clvl + clv3] all + clc2 v; _‘V2) 8u2} at,
n Cl 03 802 n 03 02 8C2 82

(105)

II O .

 

 

 

 

v _ 0102 v3 V1 8111 C2I_(l—C2V2) c273 3112 301
J2 - 4-? <5— “ 5—)T + “at o + 0 33c 82
1 1 2 3 1
_{clc2 (v3— _ [haul + oath-02V” + 057313112} 302
0 01 C2 n 02 03 8C2 82
(106)

From the definition of chemical potential, these can be

 

 

 

 

 

 

written
J V = - 52 {C r(l-Clvl) + ClV3] 8 Ln a1.
1 n 1 Cl 03 801
3 2 1
RT r(l-Clvl) ClV3 8 Ln al
' _fi {GIL o + o 3 8C
1 3 l
V V 8 Ln a 8C
3 2 2 2
+ C C (—— ——) } (107)
l 2 03 02 802 82
JV=-3?-{CC(Y-§-—Y33Lnal+
2 n l 2 03 02 8Cl
+ C [(l'c2v2) + C2V'2j 801 8 Ln a2 } 801
2 02 03 82 801 82

 

“9

V 'V 3 Ln a

 

 

 

 

 

RT 3 l l
- ——-{c c <—— - -—
n l 2 03 Cl 8Cl
+ 02[(l'C2V2) + C2V3] 8 Lralfla2 8:2 (108)
02 03 C2 2
Comparing these to the defining equations (96) gives the FT
diffusivities predicted by hydrodynamic theory:
ll AA n l Cl 03. 801
V ‘V 8 Ln a
3 2 2
+ 0.0 — - ) } (109)
l 2 O3 02 3Cl
12 AB n 1 Cl 03 802
V V 8 Ln a
3 2 2 -
+ C C —— — —— } (110)
l 2 03 02 801
D =D =BE{CC (E-EBLnal
21 BA n l 2 03 Cl 801
1 c V‘ C'V 3 Ln a
+ c2 [( 2 2) + 2 3] 2} (111)

(112)

 

Kett [23] derived these equations, and generalized them

to a system of N components, obtaining

 

D = c; [(1—vici) + civN] 3:;
13 n Oi ON 8C3
N-l cch Vk V? a k
k=l ” ON Ok 3
k#i

The theory of irreversible thermodynamics states
that the rate of entropy production in the ternary
diffusing system is
V 8“2

T ——-= — J ——— J ——— J- ———

dt 1 82 2 82 3 82

However, equation (100) can be used to eliminate u3, amd

the constant volume relationship lel + J2V2 + J3V3 = O
can be used to eliminate J3:
T 9§ = J V Y + J V Y (11%)

 

2 8p.
1 = 2
Y = - Z (6 + l 1) —4l
i 8:1 13 C353 32 5 ={0 J21
13 1 3:1

c...
ll
[1“
K
+
t“
K;

c_.
II
L'"
K:
+

L

Substituting the expressions for Y from (115) into

1
equations (116) and (ll?) and rearranging yields

V 8“1 3“2

J1 ’ -(aLll + YL12) 82 ‘ (BL11 + 5L12) 82
8p 8p

V.. 1. __l _ ._i§
J2 “ ”(8L21 ' YL22) 82 (BL21 + 5L22) 82

where a, B, y, and 5 are defined by

l l 2 l
a=l+———-—— B=_..___
C3V3 C3V3

C3 3 C3 3

Equating coefficients between (118), (119) and (101),

(102) yields four independent equations for the Li

J's:

(115)

(116)

(117)

(118)

(119)

 

 

 

 

 

 

l l) l
Llla + Ll2Y —E E C + O 1 (120)
l 3
c c V V
_ __l_2_ _3_ _ .2.-
L118 + L12‘S ' n [o o J (121)
3 2
c c ‘V ‘V .
_ l 2 ,-_3_ _ _2_ g...
L210L + L22Y ' n L03 0 1 (122) g
c 1-V c v c '
_ _2 ( 2 2) 3 2
L218 + L226 - n E O + ————O l (123)
2 3
The Onsager Reciprocal Relation states that L12 = L21.
To test this, solve equations (120) through (123) to
obtain
—- - - la - --—E + 18
n o o n o o
L = 3 2 l 3 (124)
12 a6 - BY
0102 [X1 V1:lcs C2 [(l-V2C2) + V3C2]
n o ' 3— ' n o G Y
L = 3 1 2 3 (125)
21 a3"; BY

lV1 + C2V2+ C3V3= l, and the defining

expressions for a, 8, y, and 6 we may obtain from

Using the relation C

equations (124) and (125) the desired result:

12 —

21

C

l

C2V1(1‘C1V1)

0'

1n

53

C C

_ 1 2

Vé(1-CZVE)
0'21] 3

Therefore, hydrodynamic theory states that Onsager's

Reciprocal Relation is valid for ternary isothermal dif—

fusion in a nonassociating system.

The only required

assumption is that of constant molar volume, used in

obtaining equations (101) and (102).

Miller [26] has developed equations which allow

Onsager's Reciprocal Relation to be tested experimentally:

where

 

 

= a D12 - c Dll
ad - bc
d D21 - b D22
ad - bc
Clvl 8ul
(l + 5‘5?) 35—
3 3 1
02V? 8u2
(1+é—V‘)'é'é"
3 3 l
C V 8n
1 1 l
(l + ) -—-
C3V3 8C2

Q)
“I:
[\J

c)
O

Q)
"C:
F"

a)
O

Q)
I:
N

co
0
m

(127)

(128)

(129)

(130)

(131)

d

 

=(l+C_2_;,§.)::a+clV2.a_u_];
C3 3 8C2 C3V3 8 2

Ternary Mutual Diffusion--
Associated Systems

 

(132)

Kett [23] has developed equations for the ternary

diffusivities in a ternary system of components A, B and

C subject to the dimerization equilibrium

where

AA

AB

A +_B ++ AB

component C is inert.

Kett's equations are given here without derivation:

 

 

 

 

c c- V'c an
F l (1 V c ) + ‘2 (1 v c ) + 3 A] l
1 _ _
cl 1 A 012 12 A o3n 8CA
V c c c- _ V’c c an
E 202 A + 0 i2 (1’V12CA) + 30A B] 302
2n 12n 3n A
— 2
C C V C 8n
1 — 12 — 3 A 1
[o n (l’VlCA) + o n (l V12CA) + o,n 3 8C
- 12 3 B
v c c c v c c an
E“ 8 i A + o 12 (1‘V12CA) + 3oAnBJ 302
2 12 3 B

(133)

(134)

 

‘37.
.13

 

 

BA "

02 _ c VCB
E 32—; ”'Vzca) + —— ————n—-J 5—1— (135)

+

V c c c 'V c c aul

D = [- -——l—§ + 12 (l-V12CB) + o

 

O'

— 2
C _ C _ V C 8n
+ [_—_2 (l-V ) + ——-——l2 (l-V C ) + 3 B J 2

c (136)
G2” 2 B 012” 12 B

 

By making the assumption that V12 =‘V + VB he obtained

A
c c V c C‘V
.. - _.:.L__B__ -- _. 42.45.... -—
LBA ‘ LAB ’ " oln <1 VlCA) 02 (l V208)
c C V c
A B 3 — — l2 - —
+ o3n (l'VlCA'V2CB) + 3:;fi(14vl2CA) (l-V12CB) (137)

thus verifying Onsager's Reciprocal Relation for this
simple associated system. The assumption of constant
molar volume leads to the assumption made above, so this
verification is exactly as reliable as the previous
case. Kett [23] also developed similar equations and
verified the Onsager Reciprocal Relation for the self—
dimerization system of A, B and C, where A is subject to

the equilibrium

56

A + A ++ A

and B and C are inert.

Equations of this type now will be derived for a

slightly more complex type of associated system.

sider the ternary system with competing equilibria pre-

viously discussed under Tracer Diffusion.

same nomenclature, the fluxes in ternary diffusion are

 

 

 

 

 

 

 

 

Using the

 

 

given by
J V = _ .31 iii _ 012 Bul2 _ C13 3“13
A oln 82 012” 82 01311 82
n 01 82 012 82 013 32
87 a c V
1 2 2 .12 1 .3.1 .13
02 82 o 82
3
J V = _ C2 3“2 _ C12 Bu12
B o2n 82 012” 82
CB C1V1 3“1 + C 2V12 a“12 C13V13 3“13
+ '71- E o 32 o 32 + o 32
1 12 13

 

 

(138)

(139)

For this system

 

 

 

8“12 = 8“1
82 82
8u13 = 8ul
82 82

Therefore, equations

C

V 1 _
UJA - -[EI(l-CAV1) +
c
12 —
-[-—(1-c V ) -
012 A 12
c
V 12
nJ - -[-——(l-c V )
B 012 B 12
c
-[5-1—2-(1-cBV2> +
12
c c V
_ B 3 3J 3“3
G3 82

57

 

 

 

 

a“2 .
8u3
1
(138) and (139) become
0 c an IT
———(l-C V 1) + ———(l-C V )]———
012 A 12 013 A 13 32
C C V Bu C C C V an
A 2 2 2 B A 3 3 3
J -[ (l-C V )- J
02 82 5'13 A 13 03 82
(142)
_ CBCI 1 _ CB013V1333‘J1
01 013 82
012 B 12 82 013
(143)

By combining the stoichiometric relationships

 

 

A l l2 13
CE = 02 + 012 (144)
CC = C3 + Cl3
with the Gibbs-Duhem equation, we obtain ?
211-313.111-323“? (1145) ‘1
82 CC 82 CC 82

Since the chemical potentials are functions of CA and CB

only, the expression for the total derivative gives

 

 

B (146)

 

 

 

 

(147)

Substituting (145), (146) and (147) into equations (142)
and (143) yields, after considerable algebraic manipula-

tion

 

 

 

an c c c
nJA = -[,Cl{O l(1- -cAVl ) + 012(1— —cAV12 ) + (1-55)(1-CAV #)
A 112 C13
c503 V3 23112 012 CB (:13 __
+ .C o } + ac {o (l-CAV12) ' 5" (l-CAV13)
c 3 A 12 c 13
_ CA02V2 + CACBC3V3 Joe, _ [3“1 {El(1-C'V )
o2 Cco3 oz BOB 01 A l
2C _.
c 0 CA 3V
12 A 3
+ (1-c V ) + (1———)(1— c )— }
012 A 12 cC AV 3 oi 13 Acc3 3
8“2 C12 1 "B C1 — CAC2V2
+ at {o (l'tAV12) "5‘ 8""(1‘CAVB> ' o
B 12 c 13 2
c c c V ac
+ ‘éfié—l'i} 1 .3 (1A8)
CO3 02

6O

 

 

 

 

 

 

 

 

an _
r‘JB "' ' [’a‘t‘l {FEM-CBV12) ' CB( <2: 1 + lg 13)
A 12 1 13
c c c. V. c V Bu C C
A B 3 13 3 3 2 12 V 2 '-
+ ( 1 ),+M{ (1CV)+-—(l—CV>
CC 013 O3 °VA 0‘12 A 12 °2 B 2
c 2 c V c V ac a c
+ 2(13 13 + 3 3)}3 A A _ {£1 {lg—(141v )
C 013 03 OZ A 012 D 12
VlCl V13C13 CAGE Cl3VlB C3V3
”an, + o >+c<o “'3’”
1 l3 l3 3
311,) c C
_ 12 — 2 "
+ {—(1 c v ) + ~<l—C V )
EC; 012 A 12 O2 3 2
CB2 C13V13 C3V3 3C3
+ C ( C + C ) j 82 (149)
C 13 3

Comparing this to the defining equations for the

diffusivities gives the desired results:

 

 

 

 

 

 

 

 

c c c
1 1 A 12 A 13
D — — [——(1-c V ) + (1-c V, ) + (1-:— (1—c v )
AA n 01 A l 012 A 12 C 013 A 13
CA2C3V3 auj 1 ch chl,
+ 3n‘+-£ (l—CV)-n “(lCV)
Cc°3 acA n 012 A 12 0013 A 13
CAC2V AoBC3V3 8u2 A
_ o + —-c G 3 BC (lSU)
2 c 3 A
c V.c. V c
1 12 1 1 13 13
D = — [———(1—c V, > - C-\ + ——————>
BB n 012 B 12 B Cl Gl3
cAcB 013V13 C3V3 Bul 1 012 _
+ C ( o + o )1 BC + F [a (l'CBVl2)
c 13 3 B 12
c _ c - c V V c an
+ 53(1—CBV2) + g ( 13 13 + g 3)] —53 (151)
2 C 13 3 B

Toe cross-coefficients are given by the same expressions,

except that the differentiation of the chemical potential

is with respect to the other component.

The validity of the Onsager Reciprocal Relation will
now be demonstrated fcr this system.

As before, the rate
of entropy production is given by

 

 

 

 

Bun 8n Bu” 8% 8n
g§ _ _ 1 _ 1 2 _ T 3 _ 12 _ 13 .
T dt J1 32 ”2 éz U3 32 J12 82 J13 32 (152)
Applying equations (1“1

) and (142) gives

d8 8“1 3“2
T E? ‘ ’(J1 + J12 + J13) az ‘ (J2 + J12) az
3u3
8p?
Equation (146) allows —§§ to be eliminated:
C C 8p
m ds- r V .1: ~ .1 -___1
1 5E — -LJl + 412 - d3 + (1-C ) J13J 82
C C
CD 3u2
- LJ2 + J12 - q (.43 + J13)] T-Z— (1514)
The assumption of constant volume
lel + J2V2 + U3V3 + J12V12 + J13Vl3 = O (193)
allows J3 to be eliminated from equation (154):
T d8 - P* “J R' NJ (1")
a? ’ ‘ d1 ‘ W 2 ‘ J12 ’ 13 9°

where the expressions P, Q, R, W are defined by

 

 

 

Applying equations

(141) and (142) gives

 

 

d5 341 8“2
* a? = ’(Ji + J12 + J13) az ‘ (J2 + J12) az
3u3
_ (03 + 013) BZ (193)
an
Equation (146) allows —53 to be eliminated:
C C 3p
m d8 _ F T i "’ ‘ i l
1 at ’ -LJ1 + “12 ‘ on 03 + (*'c ) J131 82
C C
C: an?
_ ' _ .z T 1 — \
The assumption of constant volume
’ - _ J _ — _ = R'—
alvl + J2V2 + 3V3 + J12V12 + J13V13 0 (1,3)

allows J3 to be eliminated from equation (154):

l - QJ2 - Rdl2 — WJl3

where the expressions P, Q, R, w are defined by

(156)

63

Bul C V aul CBVl 3u2
——-—-——— + ————'-r“.
CV3 82 ‘ CCVB oz

8L‘2 C3V2 aue CAV2 3“i

Q‘ W+W7§+QV§TE

 

 

 

 

 

 

 

82 32 CC 3 oZ CCVé 32
w = 3“1 + °“2 + CAV12 “1 + (l_ig aul + CBV13 °“2
Bz 32 CCV3 2 CC 32 CC‘V’3 Bz
_ .Cji 3.31.2
CC 2

JA = Jl + J12 + Jl3
JB = J2 + Jl2
yields
m 9§ = — J P - J Q - J (R-P-Q) - J (w-P> (157)
‘ dt A B l2 13
Now if we assume that Vl2 = Vl + V2 and Vl3 = Vl + V3,
leaving
T g_s_=_J (3141 + CAVl 3“1 + CBVl 3112
dt A 32 CCVB az CCVS az
-J(31‘2+E£.Y_2_3_‘11+C_BY32‘_2_) (158)
B 32 GOV: az COVE az

64

Irreversible thermodynamics also states that the fluxes

are related to the phenomenological coefficients:

J v= _ L (3“1 + CAVl 3“1 + CBVl 3“2
A AA 82 C V 32 C V 82
C 3 C 3
— L (ill-a + CAV2 :3; + CBV2 :33 (159)
AB 82 Cnv 82 C V 82
C 3 C 3
J v= _ L (3“1 + CAV1 aul + CBVl 3“2)
B BA 82 C W 82 C v 82
C 3 C 3
- L (8u2 + CAV2 aul + CBV2 8u2) (160
BB 32 C V 82 C V 32
C 3 C 3
Carrying out the multiplications in equations (159)

and (160), equating coefficients to equations (142) and

(143) and solving for LAB and LBA gives
c c V c C‘V
.. —_1_1.3__- 2A :
LAB ‘ LBA ‘ ' oln (l‘VlCA) ' 02n (1’ 2°B)
c c V
A B 3 — — _
O3” (VlcA + V2CB 1)
C
12 — —
+ g——fi(l-V12CA)(l-V12CB)
12
C13 — —
+ 8——fi(l'V13CA)(l'V13CB) (161)

13

[’Y‘a‘

1nus, hydrodynamic theory predicts the validity of the
Onsager Reciprocal Relation for this associated system,
under the assumptions of constant volume and that the

volume of the dimer is equal to the sum of the volumes

of the component monomers.

EXPERIMENTAL

Tracer Diffusivities

 

Tracer diffusivities for this work were measured by
means of the capillary technique, as modified by Wirth
[8]. In the basic capillary technique, a capillary of
known length, with one end closed, is filled with a
solution containing tagged molecules of one component.
This capillary is then immersed in a relatively large
volume of a solution with the same chemical composition,
but containing no tagged molecules. Diffusion is then
allowed to proceed for a period of time, after which the
capillary is removed from the bulk solution and emptied.
The relative amounts of tagged material before and after
the experiment are determined. The boundary value problem
for diffusive transfer from the capillary is then solved
to give the change in the concentration of tagged
molecules as a function of time, capillary dimensions and
tracer diffusivity. Since the time and capillary dimensions
are known, the tracer diffusivity can then be found from
the change in concentration of tagged material.

Ordinarily, the molecules are tagged with a radio-

active isotope. In this case it is easiest to measure

66'

67

the total amount of radioactivity present rather than the
concentration. This presents no difficulty, however.

There are four main sources of error in this basic
technique:

a. Inaccuracies in determining precisely the
amount of radioactivity before and after diffusion. This
is particularly important in determining the initial
count rate.

b. Proper maintoinance of the conditions of the
boundary value problem during the experiment. This means
that there must be no convective mixing within the
capillary during the experiment, and no material may be
transferred by any means other than diffusion.

c. Immersion effects. Material must not be washed
out the end of the capillary by the turbulence created in
the bulk solution when immersing the capillary or removing
it at the end of the experiment.

d. Convective transfer during the experiment.
Convection near the end of the capillary must be strong
enough to maintain the boundary condition of zero con-
centration at the end of the capillary. Yet it must not
bee so strong that it washes material from a segment of the
capillary, thus effectively shortening the length of the
capillary during the experiment.

The two latter problems are due to the open end of

the capillary, and are difficult to correct as long as the

68

end is open. The other two problems are generally less
serious.

To correct for (c) and (d), Wirth covered the open
end of the capillary with a very thin (0.007 in.) porous
glass disc. This changed the boundary value problem,
introducing a resistance term at the end instead of a
constant concentration. Since material could diffuse
through the disc, but could not flow through, this pro-
cedure effectively eliminated convection from the
capillary. However, since the resistance of the frit had
to be calibrated by measuring a known diffusivity, a new
possible source of error was introduced.

The error due to (b) can be largely eliminated by
making the bulk solution slightly less dense than that
in the capillary. Then, as diffusion proceeds, a density
gradient is established, which tends to eliminate convec-
tion within the capillary. This unfortunately introduces
the possibility of some mutual diffusion occurring along
with the tracer diffusion. It has been shown by Van Geet
and Adamson [31] that if the concentration difference
between bulk and capillary solutions is greater for tracer
diffusion than for ordinary diffusion, the tracer flux will
be much greater than the ordinary diffusive flux. Since
the difference in the concentration gradients was quite

large in these experiments, the author feels that any error

69

introduced from simultaneous mutual diffusion will be
covered by the calibration of the resistance of the glass
frit.

Inaccuracies introduced by (a) can be decreased
only by careful experimental technique. Wirth's [34]
original technique was modified in this study only
slightly to improve the accuracy of the initial count.

The procedure for an experimental run is given in

some detail in Appendix A.

Calculation of Tracer Diffusivities

 

Let the closed end of the capillary be designated
z = O, and the open end be designated 2 = L, the length
of the capillary. If transfer of material within the
capillary is only due to diffusion, the well-known
diffusion equation holds. The initial concentration is
constant throughout the capillary. At the closed end, a
material balance will show that the concentration
gradient must be zero. If, at the Open end there is a
constant resistance to flow, the following boundary

value problem holds:

1 (162)

 

 

*
0C,
5.C. 1; 2‘ - O for 2 = C t > 0
OZ 3 —
x
r— 2 D * as. N *
DOG. : - l _. .L r‘ 9" ') _ - qu
v n * A* v-
1.C. ~C, = C, for t - 0, o 5.2 5.2
1,0
x a ,
wnere C is Cu concentration 01 tagged component i,
n * .
t lS the time, 2 is the distance coordinate, C,O is the
*3
o o a - * a a n 4- .
initial value of C. and n is_the constant resistance to

n -‘ O A ‘r‘ ’- ‘ 'r I"; '3‘. «.‘f‘ A r‘ ‘2 q - .Y
transfer from the open end 01 Cue Capillary.

This boundary value problem can be easily solved

by separatioh of variables, to give the concentration as

 

* a v-
C, m sin 1,2 2 *
.L r" L; \ ‘1
———— = 22 ' - . ~- - ex —A D 2 “OS A 2
* _,‘A C + Sin A 2 cos 1 2 p( n i ’° n ‘
C. n-i n n n
1,0 '
(164)
where Ar is given by the solution of
A
'A " "“ * A
CO: K 4." - :4
n I L n

A detailed solution of the boundary-value problem is

given in Appendix III of reference [343

F.)

If equation (164) is integrated over the length 0

"3

the capi lary, a very use-ul ratio results:

71

 

0* m sin2 8 D *t
2‘ i
_£%EKE = 22 [ . n exp(-B -—§-J] (165)
Ci 0 n=1 Bn‘an + Sin 8n cos 8n) n L

where an is given by the solution of

*
RD.

_ l
cot 8n - Sn L

 

*
where C is the average concentration in the

i,ave

*
capillary, as defined by the expression I: Ci dV =

*
Ci V. This ratio is the ratio of the final count to
,ave
the initial count measured for the capillary in the
experiment.
Since transfer through the frit is diffusive, as

soon as the process reaches steady-state the resistance of
the frit becomes inversely prOportional to the dif-

fusivity:

R-—d—'g’
Di

The constant of proportionality a depends only upon*the

RD
i is

 

pore geometry of the frit. Therefore the group
dependent only upon geometry of the experimental
apparatus, and can be determined by some calibration

technique.

72 ‘

From equation (165) a plot can be made of

 

 

* * *
Ci av Di t RDi
——f——E vs 2 for several values of L . Then
i,o L

from an experiment with a chemical whose self—diffusivity

is known, the value of R may be determined. Wirth did *
RD.
this, using carbon tetrachloride. He determined that ~31—

’1':-

had a value of 0.012, with a variation from capillary to
capillary which would lead to a .7% maximum variation in

measured tracer diffusivity. Consequently, the value of

*
RD

L was taken as 0.012.

 

*

3|:

*

C, D t RD

Using a plot of ‘ avg vs 12 for L = .012,
L

 

 

the tracer diffusivity is determined from the count rates

resulting from each experiment. The ratio of final count

 

*
C
rate to initial count rate is equal to -£¢3KE. The value
C
it * 1,0
Ci av Di t
of -—:¥—5 fixes the value of L2 . Since t and L are
i o

3

*
known, the diffusivity Di is easily calculated.

Mutual Diffusivities
Mutual diffusivities were measured in this labora-
tory by means of a Mach-Zehnder [4,5] interferometer.

This instrument is shown schematically in Figure B-l of

73

Appendix B. A collimated monochromatic light beam is
split into two beams by a half—silvered mirror. One
beam is passed through a solution in which diffusion is
occurring. The other beam is passed through a reference
solution in which there is no concentration gradient.
When the two beams are recombined, if the optical path
lengths are only Slightly different, interference pro-

duces

h)

fringe pattern.
Since the optical path length is dependent upon the
refractive index of the medium, it can be shown that the

fringe pattern formed by the recombination of the beams

ve index vs position in the

*Jn

represents a plot of refraCt

diffusing system. The refractive index is in turn'related
to the composition of the system. By photographically

H,

be

Fl
’53
d
"f

recording the changes ringe pattern with time,

1

the changes in compositior (and hence the diffusivities)

..

can be determined.

a

he diffusion cell was constructed so that a step-
change initial condition could be approximated, and so
that diffusion would be one—dimensional along the vertical
axis. The cell was filled from the bottom with the

denser of two solutions varying slightly in composition.
The less dense solution was slowly introduced down the
wall of the cell, forming a layer above the denser
solution. When the cell was full, and had reached

e uilibrium temperature in the interferOJeter thermostat
b ,

71;

solution was removed slowly through slits on opposite
sides of the cell, and replaced in the cell from top and
bottom; After some time, a steady state was reached, in
which the solution above the slit was of one composition,
and that below the slit of another. The thickness of the
boundary determined how closely the step-change approxi-
mation was obeyed. In practice, the boundary could be
made small enough that it closely approximated the con-
centration distribution after a few seconds of diffusion
from a true step—change initial condition.

The flow into and out of the cell was stopped, and
free diffusion from this initial condition occurred,
which was followed photographically. Details of an
experimental run are given in Appendix B.

Analysis of_Results of Binary Mutual
Diffusion Experiment

 

 

The problem of one-dimensional free diffusion in
an infinite medium is an old one. It was solved by
Wiener [33] in 1893. If the initial position of the
boundary is designated as 2 = O, and the initial distribu-
tion of concentration gradients is Gaussian, the solution

for the gradient in terms of position.and time is given

by
AC
dC o ' 2
_ g —- exp(_z /u D t) (166)
dz 2’?D_—t' AB

AB

 

- 75

provided the diffusivity is constant. If the initial
concentration difference is small enough, both these
conditions will be met. Furthermore, the refractive
index difference may be considered a linear function of
the concentration difference for small AC. This solution

may then be written

3% = _.___A_r_1___ exp {—22/4 DABt) (167)
2/NDABt

This may be integrated to give the refractive

index difference between the points z = 0 and z = 2.
n2 - no 1
Ar = 5 erf (2/ V4DABt ) (168)

Solving this for 2 gives

n
z = VED, t erf.l (Z—E————9) (169)

The photographic image can be considered a plot of
refractive index vs position in the cell. The total
refractive index difference between any two points is
proportional to the number of fringes crossed by a
vertical line between the two points. The fringe number
can then be used as a measure of refractive index. Call
the fringe number of a reference point in the straight

line portion of the photograph fringe number zero. The

76

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.............. a J

77

total number of fringes crossed by the vertical line is
J. The fringe number of the point z = O is J/2, from the
choice of coordinates in the boundary value problem.

The difference between the refractive indexes of the

point 2 = 23 and 2 = 0 is given by

0 I ’O = .
.-%5_— “J (170)
The distance between any two fringes numbered 3 and k is

n1 — n

n.-n
z - 2 = /4D Lt [erf-1(2—JEE——9) - erf 1(2‘éfifi"'9)3

(171)

The actual distance in the cell is not the same as that
measured on the photograph, but differs by the magnifica-

tion factor of the camera:

2, = M2. ‘ (172)
d J

t
where aj is the measured distance, 23 the true distance

and M the magnification factor of the camera. Com-
bining equations (170), (171) and (172) leads to the

desired result:

 

4M2D t = E 23 - 2k 12 (173)
AB err‘l(—1——2 'J) - err'l(———2k‘J>

78

The true time of diffusion is not the measured time,

since the initial boundary was not a perfect step-change:

where tm is the measured time, t is the true time, and

t the true initial time. If the right side of equation

0
(173) is plotted vs tm, the slope of the line will give

the diffusivity, and the intercept the true initial time:

2
slope 4M DAB

(174)

intercept - 4M2 D t

AB 0

Calculation of Binary Mutual
Diffusivities

 

The photographic plate was measured by a microsc0pe
with a traveling eyepiece, capable of measuring down to
0.0001 cm. The total number of fringes was counted and
recorded. Then a set of ten fringe numbers was chosen,
five of which were higher than J/2 and five of which were
lower. These were chosen so as not to extend into the
region of curved fringes near the edge of the diffusion
boundary. These were paired, and the right side of
equation (173) calculated for each pair.

For each exposure, 23' and Zk’ were measured for
each pair of fringes. From the measurements and the

previous calculations, five values of the right side of

equation (173) were calculated and averaged. The average

79

values were then plotted against tm as described above.
A straight line was fit to these points by the method

of least-squares. If the correlation coefficient was
less than .995, the run was rejected (although generally

it was above .999). Otherwise, the value of D was

AB
calculated from the least-squares slope and intercept.

Analysis of Results of Ternary
Mutual Diffusion Experiment

 

 

Fujita and Costing [13] have shown that the ternary
diffusivities defined by equations (98) can be determined
experimentally from knowledge of the behavior of the
refractive index gradient curves as diffusion proceeds.
Their method involves measuring the second moment and
the height-area ratio of graphs of %% vs 2 at several
times during the experiment, and from these determining
the reduced second moment and the reduced height-area
ratio. This is done for several different initial
composition differences, and the graphs of reduced
second moment and reduced height-area ratio vs refractive
index fraction are then used to calculate the diffusivities.

A typical plot of refractive index gradient vs
position is shown on the following page. Here 20 is the
centroid of the curve, and 22 is the maximum value of

d2
max

dn

32’ which is at the centroid for Gaussian curves.

80

The centroid is defined by

if” 2(33) d2
(175)

° f°° (%)d2

-00

It can easily be seen th .at the denominator is equal to
the total refractive index change across the boundary,

which is proportional to the number of fringes 0.

Therefore
2 = .l___ foo 2(92) C12 (176)
1 d2

where A is the proportionality constant.

The second moment is defined by

M(Z-Z >2 (g 2) dz 1 m
a = x3 I
cog—h z)dz

 

(ca—20>2 <3 3) dz

(177)

81

and the total area under the curve is

co

An = f (331—21) d2 =)(J (178)

area

.CD

The refractive index gradient can be determined
from the photographic plate as follows. Near the center

of the boundary, where the fringes are almost straight,

%% can be approximated from the distance between two

fringes:

8n

(a ) A

02 = (179)
m 2 = ZJ+1 + J ZJ+l - J

 

 

In the curved portions of the pattern near the edges of
the boundary, this approximation does not hold, and the
value of g?“ must be determined by measuring the tangent

to the curve:

d2 dy dy

QB_=9£a%—Y.=9£tane (180)
m m

The value of gg-can be determined by measuring the

distance between two fringes in the y direction:

n - n A(j+1 - J) A
(.12. = 1+1 41 = = (181)

dy yj+l-yj yJ+l-yj yJ+l-yj

Equation (180) can now be written

82

 

tan 6 (182)

Q:
N
%
L1
+
H
I
V
C.»
€HV

where A and w are proportionality constants representing
the change in refractive index per fringe in the z
direction, and the distance between fringes in the y
direction respectively.

The distances measured on the photograph are not
true distances, so equations (176) and (177) must be

corrected for the magnification factor of the camera:

 

<—) dzm (183)

 

dzm (184)

Note that in substituting for %%— the constant A cancels.
m

The reduced second moment is defined by

E?
l\)

D2m = 2? . (185)

and the reduced height-area ratio is defined by

 

 

 

- 2 2
' le), (AJ)
D = = (186)
A 1m: {3-3 12 41rtM2 [z A __ z 32
max » 3+1 3

max

83

The measured time is not the true time, but t = tm + tO

so these can be written

 

 

2 _
—2 ‘ D2m m + D2m t0 (187)
J2
= D t + D t (188)
“WMZ E l - z 32 A m A 0
3+1 J max

m
D2m and DA can be calculated by plotting 5; and
the left side of equation (188) vs tm. The slopes will be
D2m and DA respectively, and the intercepts will give the

true initial time. The left side of equation (188) and
m
_3 can be calculated from measurements of the photographic

2
plate. This would ordinarily be rather difficult, but a
computer program has been written for this. This
program, the data deck structure, and the procedure for
measuring the plate are given in Appendix C. Sample
refractive index gradient curves and plots of IEE-Ivs tm
are given in Appendix D, with the experimental results.
In a ternary system, the refractive index can be
expressed as a function of the concentrations of any two
components, for example n = n(CA,CB). This in turn)
can be expressed as a Taylor series expansion in terms
of CA and CB. For small enough concentration differences,
the higher order terms of the expansion can be dropped,

and

84

An = RAACA + RBACB

Defining the refractive index fraction of component A by.

R AC

A A
a = (189)
A RAACA + RBACB

 

we see that

d + d = l (190)

The values of RA and RB can be determined from
measurements of An at several different AC's by a least-
squares technique. Since accurate direct measurements
of An require relatively large concentration differences,
the preferred method is to determine the values of RA'
and RB. defined by the equation

! l
J = RA ACA + RB ACB (191)

I I
where RA = ARA and RB = ARB.

This will allow smaller composition differences to be

used, since J can be measured more precisely than An.
. -AC
By equation (191), we see that if J A is plotted vs.

 

AC ' ‘
-7f3 , the result should be a straight line, with slope

85

R!
fiTE-and intercept §%-. As can be seen in figure D-10,

A A
this is true for the ternary system chloroform acetone-
benzene which will be investigated in this work. Note

R R
that .75.: fig and that only this ratio is needed in

R B B
defining the refractive index fraction.

Fujita and Gosting [13] have shown that plots of
reduced second moments and the reciprocal of the square-
root of the reduced height-area ratio vs. refractive index

fraction of one component should be straight lines:

Dzm = 82m GA + 12m (192)
l —
WA ‘ SA 0‘A * IA (193)

where 32m and SA are the slopes and 12m and IA are the

intercepts at dA = 0. Their proof is based on the same
assumptions which have been made here, and which hold
whenever the concentration differences are small.

For convenience in notation, define the intercepts

at dA = l by the expressions

L2m = I + 32m ~ (194)

LA = 1A + sA (195)

where

reference [23].

diffusivities is given in Appendix C.

86
Their expressions for the diffusivities are

L L2m S2m IA

 

 

 

 

 

 

 

 

2
D = _ JDljj + L2m J D131 + SA
AA 32
m
t I2m S2m LA
D = - DlJ + 12m DiJ + sA
BB 32
m
AB 31 i2m ”33’
R1
DBA 2; (”2m ' DAA)
[DH]15 is the root of the cubic equation

3/2
lDljl + (12m ‘ IA ‘EZQ [D

A detailed derivation of these equations is given by

Fujita and Gosting, or may be found in Appendix II of
1
(200) for [Dul’2 and then calculates the four ternary

the plots of equations (192) and (193) for the system

studied in this work may be seen in Appendix D.

(196)

(197)

(198)

(199)

(200)

A computer program which solves equation

The linearity of

RESULTS AND DISCUSSION

The following systems were studied experimentally

in this work:

8..

Tracer diffusivity of 2—butanone in the
system 2—butanone - carbon tetrachloride
Tracer diffusivity of p-benzoquinone in the
system p-benzoquinone - benzene

Mutual diffusivity in the system
p-benzoquinone — benzene

Tracer diffusivity of diethyl ehter in the
system ether - carbon tetrachloride

Mutual diffusivity in the system diethyl
ether — carbon tetrachloride

Mutual diffusivity in the system carbon
tetrachloride - chloroform

Mutual diffusivity in the system benzene -
chloroform

Ternary mutual diffusion at equimolar composition

in the system acetone - benzene - chloroform

Experimental results, and intermediate determinations for

the ternary system, are given in Appendixes D through F.

The discussion of results will be organized by type

of diffusivity studied, rather than by composition of the

systems studied.

87

88

Tracer Diffusivities

Equation (82) predicts that for a self—associating
component in a binary system, the product of the tracer
diffusivity and the viscosity should have its highest
value when that component is very dilute. As the con-
centration increases, the ratio of polymers to monomers
present in the system will increase, and the tracer
diffusivity will decrease.

In binary system with cross-association the situa-
tion is slightly different. As can be seen from the

equilibrium constant expression,

‘the ratio of dimers to monomers of component A is pro-
portional to the mole fraction of component B. Hence
‘the percentage of A molecules which are tied up in the
(dimers is highest when component A is very dilute, i.e.
XMhen xB + 1. Therefore, the tracer diffusivity-
‘Viscosity product of a component is lowest when that
Ccnnponent is extremely dilute, and increases as the
Concentrat ion increases .

*
Figure 1 shows the variation of the DAn product

for associating components in three systems. In the

Systems ethanol - carbon tetrachloride and acetic acid

carbon tetrachloride hydrogen bonding is quite strong

7, dynes

D n x 10

89

 

 

Mole Fraction Component A

A
Figure l.-—Din vs Mole Fraction for Associating
Components.

|
|
O
l
|
|
8
\\
2 o-P°\
\
\\
\
\\
\\‘
\0\
O\
\ \
C O
E . \\
\ \\
\\\ \\U\.
.~"" ‘::o——"O————.
1.0... .M A.
-ir—- Diethyl Ether (A)
——E+—- Chloroform (B)
-—O-- Ethanol (A) - CClu (B)
-.—o—- Acetic Acid (A) - CClu (B)
0 1r 1 1 1 4
o .2 .4 .73 .8 l.

90

between ethanol molecules and acetic acid molecules.
Carbon tetrachloride, on the other hand, is probably
quite inert. The curves for these two systems have

*
exactly the shape predicted by equation (80). The DAn

product is highest when x + 0, and decreases as

A

x + l.

A
In the system ether - chloroform, spectroscopic

evidence [15] suggests that hydrogen bonding occurs

between ether and chloroform to form dimers with the

form

Presumably, steric hindrance prevents the formation of
larger ploymers in this system. Ether—chloroform is
therefore a cross-associating system. The curves in

Figure 1 agree with the predictions of equation (55).
s

in
component is extremely dilute, and increases as the

The D product for each component is lowest when that
concentration increases.

I Equation (45) predicts that for a system where
neither component associates appreciably, the tracer
diffusivity - viscosity product will be a constant
independent of composition. Figure 2 shows the tracer

diffusivity - viscosity product for several systems in

91

 

 

 

 

 

 

 

Mole Fraction Component A

*
Figure 2.-—Din vs Mole Fraction for Nonassociating

Components.

:-—n———n-- D--n———U— —O--———g-9—-- —--§
(.1 V' 0 V A Q
a _ f ’ C
p L 4. .I I; I I-
.01
——€+—— Ethyl Iodide (A) [7]
__.__. Butyl Iodide (B) [7]
——4+—— Benzene (A) [34]
-—0—- Carbon Tetrachloride (B) [34]
——a——- Chloroform (A) [34]
——1F—— Carbon Tetrachloride (B) [34]
t i ‘T : a
0 0.2 0.4 0.6 0.8 1.0

91

 

 

 

 

 

 

 

*
Figure 2.--Din vs Mole Fraction for Nonassociating

Mole Fraction Component A

Components.

UJ _
T’ x: ngL. 11.7:21. ii: at IL’E 1p
m“ !
E3 1.0+
>4
C
*
c:
-—£+—- Ethyl Iodide (A) [7]
-—4F—— Butyl Iodide (B) [7]
——<+—— Benzene (A) [3“]
-—0—-— Carbon Tetrachloride (B) [34]
.——a—-— Chloroform (A) [34]
——1F—— Carbon Tetrachloride (B) [34]
O i f t : a
0 0.2 0.4 0.6 0.8 1.0

92

which there is good reason to believe that there is no
association. The normal iodides for instance are
essentially non—polar, saturated, and contain no groups
active enough to form hydrogen bonds. Carbon
tetrachloride is also non-polar, and the electron
clouds of the chlorines are quite inert to hydrogen
bonding, even in an electron-donor capacity.

As predicted by equation (45) the D:n products for
these systems are straight lines. Furthermore, the D:n
product for 0C1“ is the same for all the systems given
here. This can be taken as supporting evidence for the
assumption that diffusing species behave like particles
flowing through a continuous medium. The diffusion
process is influenced by the viscosity of the medium,
but not by the character of the molecules which comprise
the medium.

Spectroscopic studies have suggested that ketones,
'being polar molecules, undergo some dipole-dipole inter-
actions which lead to the formation of self-polymers in
ssolution. Anderson [1] successfully applied the self-
(dimerization model to explain the positive deviation
:from Raoult's Law in the system 2-butanone - carbon
‘tetrachloride. He then used equation (92) to fit
experimentally measured mutual diffusivity data with

excellent results. Wirth [34] later measured the tracer

93

diffusivities of carbon tetrachloride in this system,
confirming the fact that carbon tetrachloride did not
associate.

In this work, the tracer diffusivities of 2-butanone
were measured, hoping to verify the self-association
model. ‘Experimental results for this system are given
in Appendix E, and shown in Figure 3.

The tracer diffusivity - viscosity products for
s

in
products are constant throughout the entire concentra-

this system are shown in Figure 4. Since the D

tion range, it must be concluded that there is no associa-
tion in this system, at least with respect to diffusion.
The dipole-dipole interactions observed spectroscopically
apparently are not strong enough to hold the dimers
together against the shear forces they presumably undergo
while diffusing. This would indicate that the inability
of equation (88) to predict mutual diffusivities in this
system is not due to the formation of polymers. A
possible cause would be inaccuracies in the vapor-liquid
data in the literature. The system clearly warrants
further study.

Spectroscopic studies have shown that highly con-
jugated molecules with electron withdrawing groups

adjacent to the conjugation, such as

cm2/sec

5
or DAB x 10 ,

D

l\)
O
l

94

3.0-

 

 

 

1.0m
. D ex erimentall
AB, p
—--- Hartley—Crank Equation
*
D DA 2—butanone
*
0 DB carbon tetrachloride [34]
O : : : : 4
O 0.2 0.4 0.6 0.8 1.0

Mole Fraction 2—Butanone

Figure 3.—-Mutual and Tracer Diffusivities for the System
2—Butanone - Carbon Tetrachloride at 25°C.

 

 

95

 

 

 

 

.0"
t 0 a O-
O
.0“
*
-——€F——- DAn 2-Butanone
*
———O———- DB” Carbon Tetrachloride [34]
0 i i f, i as
0 0.2 0.4 0.6 0.8 1.0

Mole Fraction 2—Butanone

Figure 4.-—Tracer Diffusivity — Viscosity Products for
the System 2-Butanone — Carbon Tetrachloride at
25°C.

"flu.

O i\‘02

x/ \\\., ¥ / , ~.

g E p-benzoquinone * \_, 1,3,5-trinitro
4,// Nd;\// N02 benzene
'0

can undergo charge-transfer interactions with donor

molecules, usually aromatics, which stabilize the

structure ’::\C
0:," ,eeo

It has been established that p-benzoquinone will
associate to form dimers in solution with aromatics
[2, 11] and equilibrium constants have been measured for
several of these systems.-

In an effort to experimentally verify equations
(150) through (151) for a ternary system with competing
equilibria, it was decided to study the system
p-benzoquinone — benzene — p-xylene. It was expected
that the quinone would form dimers with benzene and xylene,
and that no other associations would occur.

Mutual and tracer diffusivities were studied for
the component binary system quinone-benzene. Since
quinone is only slightly soluble in benzene at 25°C, the
results cover only the solubility range. Experimental

results are given in Appendix E and shown in Figure 5.

2
r DAB x 105, cm /sec

l-'

97

 

.0+
0

 

 

.0"
._49__ D
AB
. *
DA
*
O DAn
0 L l l A l
l 1 l I I
0 .01 .02 .03 .04 .05

Mole Fraction p—benzoquinone

Figure 5.--Mutual and Tracer Diffusivities in the System
p-benzoquinone (A) — benzene (B) at 25°C.

98

The tracer diffusivity-viscosity product for
quinone is also given in Figure 5. The variation of
this DZn product should be that predicted by equation
(55), since this system is considered to have only cross—
association. That is, the DZn product should increase
as the concentration of quinone increases. As can be
seen in the figure, it does not increase, but decreases
instead. This was interpreted as some sort of interaction
leading to self-association of quinone which masked the
effect of the cross-association.

The change in the D:n product for a small change
in concentration is much greater for self-association
than for cross-association. It is possible that if the
concentration of quinone could be increased, the cross-
association effect would again become predominant. In
any event, the associations present are too complex to
be treated by the equations developed here, and work in
the ternary system was not carried further.

Tracer diffusivities were measured for ether in the
system diethyl ether — carbon tetrachloride across the
entire composition range, and for carbon tetrachloride
at the two endpoints. Results for this system are given
in Appendix E, and shown in Figure 6. There were con-
siderable experimental difficulties in working with this
system, due to the high volatility, the low viscosity,

and the surface-wetting characteristics of solutions with

4.; IA

99

 

 

Figure 6.-—Mutual and Tracer Diffusivities in the System

Mole Fraction Diethyl Ether

Diethyl Ether — Carbon Tetrachloride at 25°C.

8.0.-
7.0._
6.0__
O +
(D
U)
\.
(\l
S 5.0,,
Lhc)fl
H +
x
B
Q “.0‘- +
2..
0
2|:
Q
3.0-r +
2.0., g
7 DAB
1.0-L ,
¢DA
0 : : : : 4
0 .2 .4 .6 .8 1.0

 

100

high ether content. As a result, the uncertainty of
this data is higher than for the other systems studied,
as indicated by the bars on Figure 6.

It was assumed that the ether — carbon tetra-
chloride system would be a simple nonassociated system,
and tracer diffusivities would be as predicted by
equation (45). In fact, the shape of the DZn product
curve for ether is more like what one would expect for
a cross-associated system. Since the carbon tetra-
chloride is non-polar, and its chlorines do not form
hydrogen bonds, this phenomenon is rather difficult to
explain.

Being somewhat unfamiliar with the mechanics of
charge-transfer complexing, the author hesitates to
eliminate this possibility, but it does seem unlikely.
Furthermore, over a period of time the bulk solution
discolored, indicating a reaction of some sort pro-
ceeding. It is possible that the reaction (though not
extensive and rather slow) indicates that some inter—
molecular interactions were occuring beyond the usual
attractive and repulsive forces. Another alternative is
that the assumption of a continuous medium breaks down
here. This is supported by the fact that the Dgn
product of CClu changes only about 10% over the concen-
tration range, while that for ether changes about 30%.

Again, this system warrants further study. Investigation

101

of other properties besides diffusion might also provide

some insight.

Mutual Diffusivities

As part of a study of the ternary system ether -
chloroform - carbon tetrachloride, Wirth [34] measured
tracer diffusivities for both chloroform and carbon
tetrachloride in the component binary chloroform - carbon
tetrachloride. These measurements showed that, as expected,
both chloroform and carbon tetrachloride are nonassociated
in this system.

The Hartley-Crank Equation, equation (88), should
describe mutual diffusion in this system. Wirth [34]
measured mutual diffusivities in this system to
experimentally verify this equation. He encountered some
experimental difficulties, and scatter of data cast some
doubt on his results. The best data he could obtain from
his results, however, showed that the Hartley-Crank
Equation is inadequate to describe this system. The
shape of the Hartley-Crank curve is wrong when compared
to Wirth's data. It was of interest then to attempt to
duplicate Wirth's data, to determine whether thel
discrepency is truly in the equation, or whether it might
be in the experimental results.

The author encountered less experimental difficulty
in measuring this system. Experimental results are

given in Appendix E, and shown (along with Wirth's

102

results) in Figure 7. As can be seen, the author's
results agree quite well with the best results obtained
by Wirth. The Hartley-Crank Equation is definitely
inadequate for describing this system.

The discrepancy in this case is very hard to
explain. The assumption of a continuous medium is
probably a good one, since otherwise the effects would
have shown up in the tracer diffusivities as well. The
activity data reported in the literature used in calcu-
lating the thermodynamic correction factor appear to be
quite good. The system is only slightly non—ideal, so
the correction factor is not too large in any case.
There is definitely no association, as can be seen from
the Dzn products. This phenomenon is puzzling, and will
probably require further investigation to provide an
explanation.

The Hartley-Crank Equation also fails in another
nonassociated system, 2-butanone - carbon tetrachloride.
In this case, however, the predicted mutual diffusivity
curve has the correct shape, differing only in the
magnitude of the correction from ideality (see Figure 3).
The author suggests that this may be due to a slight-
error in the activity data, from the experimental vapor-
liquid equilibrium measurements of Fowler and Norris

[12].

103

 

 

 

2.0
O
(D
U)
\.
(\J
E
0
m0
O
H
x
‘13;
Q 1.0 "'
a
0
3|:
Q
--0 -— DAB’ this work
- 9-- DAB’ Wirth [3A]
"‘a“‘ DA
*
——e— DB
0 : : : t 4
0 .2 .4 .6 .8 1.0

Mole Fraction Chloroform

Figure 7.--Mutual and Tracer Diffusivities in the
System Chloroform (A) — Carbon Tetrachloride
(B) at 25°C.

104

The author measured mutual diffusivities in the
system benzene - chloroform, to further test the Hartley-
Crank Equation. Experimental results are given in
Appendix E, and shown in Figure 8.‘ It was expected that
this would be a simple nonassociated system. Although
tracer diffusivities were not measured in this system,
the self—diffusivities of both components are available.
When the D:n product of the pure component is compared

to the D Bn product when that component is extremely

A
dilute, the results indicate that both components are
nonassociated.

As can be seen in Figure 8, the Hartley-Crank
Equation again fails for this system. The shape of the
curve is qualitatively correct, but the correction is
again too much. The error in this case is probably too
large to attribute to inaccurate activity data.

Further investigation in this system is warranted,
particularly measurement of the tracer diffusivities
over the entire concentration range, to make certain
there is no association.

The author has also measured mutual diffusivities
in the system diethyl ether - carbon tetrachloride.
Activity data are not available for this system, there—
fore it cannot be used to test the Hartley-Crank

Equation. Furthermore, experimental difficulties

(previously described under Tracer Diffusivities) caused

105

 

 

 

 

 

3.0-r
/ / \ \
/ \
/ O o 7‘ \\
/ O \\
(5\\ O \
\
e 1
2.0“
0 .'
(D
(I)
\C
(\J
E
0
m0
0
r_|
x
m
<
c:
1.01
0 Experimental results
---- Hartley-Crank Equation
Ideal system
0 .2 .4 .6 .8 1.0

Mole Fraction Benzene

Figure 8.—-Mutual Diffusivities in the System Benzene
(A) - Chloroform (B) at 25°C.

106

considerable scatter in the data, especially near the
center of the concentration range. Experimental
results are given in Appendix E, and shown in Figure 6.
As a measure of the uncertainty, the standard deviation
of the data are listed in Table E-3, and are indicated
by bars on Figure 6.
Mutual diffusivities were also measured for the a
system p-benzoquinone - benzene, up to the solubility .
limit. Results are given in Appendix E, and shown in
Figure 5. Again, no activity data are available. The 'fi
complex associations present in this system precluded
testing the hydrodynamic equations in any case.

Error Analysis--Mutual and
Tracer Diffusion

 

 

Bidlack [27] and Kett [16] found that for the
instrument used in this study, the experimental precision
was i1% for volatile liquids such as used here. This
was based upon several runs on aquaeous solutions of
sucrose. These runs were compared to determinations
made by Gosting [17] on the sucrose - water system, with
agreement within i0.5%. They therefore conservatively
estimated the precision of the method using this inter-
ferometer as 21%.

This author accepts the figure of i1% for the
precision of the method and the instrument. Since the

experimental procedure was not changed from earlier

107

procedures, the only remaining source of error would be
that introduced by the experimenter. The author takes the
agreement between his data and that collected by Wirth for
the system chloroform - carbon tetrachloride as evidence
that no systematic error has been introduced which would
give consistently high or low experimental diffusivities.

At several compositions in the systems chloroform -
carbon tetrachloride and benzene - chloroform mutual
diffusivities were measured two or more times. At all
these compositions, the values obtained agreed within
12%, and in most cases within 21%. The author takes this
as evidence that random error introduced by the experi-
menter is within the precision specified for the method
by Bidlack and Kett. The experimental precision for the
studies in this work will therefore be taken as 21%.

This figure does not apply to the system ether -
carbon tetrachloride, because of experimental diffi-
culties felt to be inherent in this system, which have
been discussed previously. In this system four or more,
determinations were made at each composition, and
averaged. The averages are reported, along with the
standard deviation of the data, in Appendix E.

Wirth [34] has shown that the modified capillary
technique used in this study has an experimental pre-
cision of 22%. This was shown.by comparing tracer
diffusivities at extreme dilution with mutual diffusivi-

ties extrapolated to zero concentration (which must be

108

identical according to equations (91) and (92)), and by
repeated runs for the same composition and comparing the
reproducibility.

The author accepts this as the experimental precision
of the method using this experimental apparatus. To
determine the amount of error introduced by changing the
experimenter, the author reproduced the self diffusivity of
carbon tetrachloride (which Wirth used for calibrating the
cells), with a deviation of about 21%. Further evidence
is the comparison between mutual and tracer experimental
diffusivities at extreme dilution in the systems
2-butanone - carbon tetrachloride, and p-benzoquinone -
benzene. The author concludes that the experimental
error introduced into the method by changing the experi-
menter is within the experimental precision reported by
Wirth. Tracer diffusivities reported here are therefore
assumed to be accurate to within 22%.

Again, this does not apply to the system ether -
carbon tetrachloride. Experimental difficulties here make
the results somewhat more uncertain. Determination of
precision is rather difficult. The values at extreme
dilution are within 25% of mutual diffusivities (which
are themselves uncertain). The author estimates tracer
diffusivities in this system are accurate within 25%,

and are so reported in Appendix E.

‘4

109

TernaryADiffusivities

 

Ternary diffusivities were measured in the system
acetone — benzene - chloroform, at an average composition

of x = x = x = .333. Typical curves showing the

A B C
change of the refractive index gradient throughout the

run are given in Figure D-l. Values of D2m and DA and.
the time correction factors are given in Appendix D.
The linearity of equations (187) and (188), which are
used to evaluate the reduced quantities, can be seen in
Figures D-2 and D-3.

and V DA vs refractive index

fractions for all three independent choices of components

Plots of D2m

are given in Figures D—4 through D-9. (Ternary Dif-
fusivities can be expressed in three different ways,
depending on which components are considered, i.e.,

D D D D or D D D D or D D

BA’ BB AA’ AC’ CA’ CC BB’
As will be pointed out later one set of

AA’ AB’ BC’

D D

CB’ CC'
diffusivities may be more advantageous in testing the
hydrodynamic model and Onsager's Reciprocal Relation

than the other two sets.) The slopes and intercepts of
these lines were determined by a least-squares analysis,
and are given in Appendix D. These slopes and intercepts
were then used with the computer program given in
Appendix C to determine the diffusivities, which are

given in Table 2 for the optimal choice of components

for this system.

110

 

 

 

 

TABLE 2.-—Predicted and Experimental Ternary Diffusivities
in the System Chloroform (C) - Acetone (A) - Benzene (B)
at 25°C.

Predicted EXperimental 95% Confidence

O A ‘- n /
>30) DCC 5-674 .5070 11.02
P U)
fi\\ D "'4 ‘ + 1
>04. A — oh I. — o
H E CA 99 1 55 7 5
U) 0 A
a , D - .942 — .80 _ .815

AC

(Hm
e40 -,
3,3 DM 2.515 1.74 21.52

x I‘ll“.
H
m'
C) U)
”€98 L12
5L3 ET -3.561 2.337 29.88
oz)
8:: L
o .
E94 21 2 .
C: O l l
on)
n
D...

 

characteristics. As can be see. in Figure 9 and Figure

0
O

10, the scatter in the data is large enough to mask any
curvature due to association. It is probable that there
may be some cross—association in the binary acetone -

chloroform, but the other two bi.aries are felt to be

In any event, the curvature due to association
is not likely to be extreme, since the end points vary
by only 25% in the binary systems. It was therefore

assumed that the ternary system could be considered a

dynes

3

D n x 10

111

 

 

2.0»
o D '
U o
[3 O D O
E: ’ ' ' ' . l o o
E Q U
0 o
3
O o O
o
1.0.
*
DAn, Acetone, McCall and Douglass [25]
*
. DAn, Acetone, Anderson and Babb [l8]
*
D DBn, Chloroform, McCall and Douglass [25]
*
O DBn, Chloroform, Anderson and Babb [18]
O s i ; : 1
O 0.2 0.4 0.6 0.8 1.0

Mole Fraction Acetone

*
Figure 9.——Din vs Mole Fraction in the System Acetone -
Chloroform at 25°C.

112

 

 

2.0»
<9 0
o 8 o
o o o
4 :
o o o
o ‘ .
U)
(1)
5::
>5
'6
n 1.0l
[x
O
H
N
3k .C
Q
9%
0 DAn, acetone, McCall and Douglass [25]
4(-
‘ [th benzene, McCall and Douglass [25]
0 ; : s + :
o 0.2 0.14 0.6 0.8 1.0

Mole Fraction Acetone

*
Figure lO.——Din vs Mole Fraction in the System Acetone —
Benzene at 25°C.

113

non-associated system, at least as an approximation.
Equations (lll) through (114) can'then be used to predict
the ternary diffusivities. These equations depend on

the assumption that the molar volumes are constant.

Since the composition differences within the diffusion
cell were kept very small, the author feels that this
assumption has been met experimentally.

Activity data for the three binaries were fit to
Margules equations and then combined to give ternary
isothermal activity data (as discussed on pages 18-21).
The friction factors were taken to be the weighted
averages of the friction factors at the end-points in
the various binaries. Viscosity was measured with a
Canon-Fenske viscometer. These quantities were then
used with the computer program in Appendix C to predict
the ternary diffusivities, which are given in Table 2
along with the experimentally measured values.

The 95% confidence levels of the measured data,
which will be determined in the next section are also
listed with the experimental data. It can be seen
that the predicted values of the diffusivities fall well
within the 95% confidence limits. The author therefore
feels that the experimental determinations support
hydrodynamic theory. The predicted values of the

phenomenological coefficients L and L2]- are also

12
within the 95% confidence limits of the measured values.

A...— ILA... _ . __.

FJ
F"
.t‘:

Within experimenta' precision, this can be taken as
empirical verification of the Onsager Reciprocal
Relations.

The 95% confidence levels are quite large for this
set of experimental data. This will be discussed in the
next section in detail. It will be shown that the
values of the cross—coefficients are extremely sensitive
to the experimentally measured intercepts, and that a
very slight error in determining the intercepts can
lead to an extreme error in the cross-coefficients,
as well as a significant error in the main coefficient.

From this sensitivity analysis, and a consideration
of theexperimental data, suggestions will be made for
modifying this procedure. The author believes that
through a thorough investigation of certain factors
leading to experimental uncertainties in the present
method, techniques can be developed which will allow this
method to give 95% confidence levels within 20% or so
for the cross-coefficients. This would then give a
rigorous test of the hydrodynamic model and the Onsager

Reciprocal Relation.

Error Analysis—~Ternary Diffusion

 

Since ternary diffusion has been studied in so few
systems, and since the time involved in making a complete
determination at any one composition is so long,

0

determination of experimental preCision by comparison to

115

published data is virtually impossible. The number of
reference systems is also quite small. In the past, the
usual procedure was simply to determine the experimental
uncertainties in the measurement of the various slopes
and intercepts used in calculating the experimental
ternary diffusivities.

Kett [23], for instance, reported 95% confidence
levels of a percent or so, and concluded (implicitly)
that his experimental diffusivities were of the same
order of precision, a percent or so. In fact, this
confidence level would lead to a much larger confidence
level for the cross-diffusivities than he implied. Later
evaluation of his data showed that the confidence levels
were actually somewhat larger than he reported, which
would lead to even more error in the diffusivities.

The errors in the cross-coefficient resulting from
a 1% error in the value of the intercepts can be as
large as 200%. This is because the calculation of the
cross-diffusivities involves subtraction of two rather
large numbers to obtain a small one, so that uncertainties
in the larger numbers are greatly magnified. Furthermore,
errors in the main coefficients as large as 20% can result
from a 1% error in the intercepts. It would seem quite
worthwhile then to look at the sensitivity coefficients
of the ternary diffusivities, which relate the change in
a calculated diffusivity to a change in a measured

parameter.

116

If a dependent variable is a function of several
independent variables, the functidnal form usually
involves several arbitrary parameters. The values of
the parameters for a given physical system are usually
determined by measuring the dependent variable at
several values of the independent variables. The
experimental "best" values of the parameters are then
assumed to be those which give the least-square error
when fitted to the data of the experimental measurements.

If the equation is of the form

y = f(al a2 ... a x2 ...x ) (201)

where the ai's are the parameters and the xi's are the
independent variables, then the sensitivity coefficients

are defined by

 

a? iaj = l: 2: 00- n
ya = (a; )a x K = l, 2, ... m (202
i i J’ K j # i

The sensitivity coefficients measure the change in the

dependent variable with a change in the parameters, and

are themselves functions of the independent variables.
Ternary diffusivities can be treated as dependent

variables whose values depend upon independent variables

(component mole fractions, temperature and pressure)

I

and the parameter I SA’ and S2m‘ The functional

A’ 2m’
form of this dependence is given by equations (196)

....J
...!
N

through (200). Since the functional form is rather
complex, analytical evaluation of the sensitivity
coefficients is rather difficult. They can easily be
determined numerically with the aid of a computer,
however. A small change is made in the value of one
of the slopes or intercepts, keeping the others constant,
and the change in the diffusivities is noted. This has
been done for the system measured in this study.

If the sensitivity coefficient is multiplied by

bitrary parameter, a reduced sensi-

p)
*5

the value of the

tivity coefficient may be defined

 

s: s a. y = a. < df) (203)
i i i a

2

This gives the chan~e in the diflusivities for a one—
percent change in the parameter. This is useful, since
if the percentage uncertainty in experimentally
measured parameters is known, the uncertainty in the
diffusivities can be determined. These same arguments
also hold for the phenomenological coefficients used to
test the Onsager Reciprocal Relations.

If the experimental data are to be used to test a
proposed model, it would be best if the sensitivity
coefficients with respect to the measured parameters
were as small as possible. In the case of ternary dif-

fusivities, three independent choices of components may

be made. The sensitivity of the main and cross-
diffusivities will not necessarily be the same for each
choice. It would be best then to choose those components

for which the sensiti

<;
} 1
Cf
'. I
(D
U)
m
*3
(D

lowest. It can be seen

from Table 3 that in the s stem acetone (A) - benzene‘

%

(B) - chloroform (C), the sensitivity coefficients are
generally lower when the set of diffusivities DCC’ DCA’
DAC and DAA is Chosen to describe diffusion. This is the
basis for the choice made in ,reparing Table 2.

If the 95% confidence levels for the parameters

are known, then the 95% confidence levels for the dif-

Flo

fus vities can be approximated from the sensitivity

coefficients. By assuming that the sensitivity
coefficients are constant for different values of the

parameters, upper a.d lower limits for the diffusivities
t

may be calcula

 

95% limits of D =

(20A)
a, >

(.1.
H
Cf
Hwn
%
H

Confidence limits on the parameters I , I , S and S
A 2m A 2 51

may be determined from the variances of the parameter

(which are determined during the least—squares analyses)

by means of a statistical t—test:

.. . yF'—————-
95%, limits on a, 2 hi (205)

... e

ll
l+
CT
(I)
S1

V

119

TABLE 3.——Reduced Sensitivity Coefficients, sgi in the

System Acetone (A) - Benzene (B) - Chloroform (C) for
all Choiced of Coordinates.

 

 

a
i I2m IA S2m SA
y
DAA + 12 + 18 + 03 - Ol
DAB — 09 - 10 +.O3 +.03 -5
x10 “
DBA + 12 +.24 0 0 0 0
DBB - 09 - l3 +.O3 + 04 ;
DAA - 19 + 34 0 0 + 01
D - 08 + 12 - 02 + 02
AC x10-5
DCA + 38 -.78 0 0 - 02
DCC + 16 - 27 + 03 -.05
DBB - 21 — 36 + 03 - 04
D - ll - 23 0 0 0 0
BC xlo 5
DCB + A2 + 65 - 08 + 07
DCC + 2H +.42 O O 0.0

 

where t stands for the statistical parameter from the
t-test, and s:(a1) is the statistical estimate of
variance of the parameter a1, as determined from the

least-squares analysis. The 95% confidence limits on

120

the diffusivities were determined by these formulae,
and are given with the values of the diffusivities in
Table 2.

To provide a rigorous test of hydrodynamic theory
and of the Onsager Reciprocal Relations, it would be
necessary to reduce the 95% confidence levels on the
main coefficients to about 10% or so of the values of
those coefficients, and the 95% confidence levels on
the cross-coefficients to at least 50% or so of the
values of the cross-coefficients. Since the sensitivity
coefficients for the intercepts are so high, it would
be necessary to reduce the variances of those intercepts
to within a few tenths of a percent. It would also be
necessary to reduce the value of t from the t-test.
Since t decreases with the number of degrees of freedom
(i.e. experimental measurements made) at a given con-
fidence level, a statistically large number of measure-
ments should be made for every set of diffusivities
desired.

Since t approaches a constant value as the number
of measurements increases, it becomes apparent that the
sample variance must also be reduced. Concisely, this
means that more precise measurements must be made, as
well as more of them.

This means that the spread in the data seen in

l

the plots of D and MDA vs a must be reduced. The

2m

121

reasons for the spread in the data are difficult to

determine. One probably important cause of this spread

is the quality of the initial boundary. It can be seen

from Table A th t the true initial time determined from

equation (187) and the 321 vs t curves is not the same
e

as that determined from quation (188) and the

1 - w-fi ~ 9 I 0 1 ~
vs t durves. ii the initial boundary nad been
m
VD

A
a true step change, and the timer started the instant

 

that flow from the cell was stopped, the true initial

W

l—~l
c1

time would have been t” = 0.

1.;

s assumed that the

m

initial boundary is such as would have been formed by

‘

diffusion from a step—change for a short period of time.

This would have resulted in a true initial boundary in
which the refractive index gradient curve was Gaussian,

and the true initial time would have been the same whether

determined from equation (18?) or (188).
It can be seen that the measured refractive index

gradient curves in some runs are obviously not Gaussian.
The curves are slightly skewed to one side or the other.
Ternary iffusion from a step change boundary always
gives skew curves, except at two times during the run

when they are true Gaussian curves. However, in order

M

f-’

for equations (187) and (188) to apply, the true init a
distributions must be Gaussian. (Note that this is the

true initial di

(1)

tribution, not the experimental boundary.)

The author believes that the error introduced by a

122

TABLE 4.--True Initial Times for Ternary Runs, as
Predicted by Equations (187) and (188).

 

 

to’ sec. to, sec.

Run Number eq. (187) eq. (188)
60 -20 —43.8
62 -23 -37.1
63 -90 -55.2
65 -56 "”9°2
66 -13 ' 7'8
68 _52 -65.7
69 _74 -72.8
70 —58 -“7.9

 

non—step-change boundary can be related mathematically
to the difference in the initial times determined from
the two curves, and possibly one measurement of the
refractive index gradient during the run. He was not
able to derive such a relation, however.

It certainly seems reasonable, however, to use the
difference between the time corrections for the two
curves as a criterion for rejecting a run. If the two
initial times varied by more than a certain amount, the
run would be rejected. Determination of what difference
should lead to rejection will probably take considerable

study and experimentation.

It might also
possible methods of

this boundary could

be worthwhile to investigate other

forming the original boundary. If
be improved, the better approximation

to a step-change would undoubtedly lead to better

.1

.-

 

results, and less spread in the D2fi and /3_ vs a
H .‘
A
curves.
In summary, t'e experimental results for the

system acetone —

benzene - chloroform support

the

predictions of hydrodynamic theory, within the experi—

H)

sion o

F—‘.

mental prec
extreme sensitivity

method presently is

’5

C)
' 5
u.
m

test of the ge

is as valid as many

The author believes

enough, through fur
rigorous test of th

C
:3
(D
'5
Fl
ti
<
(D
U)
Cf
' 1
0'4

the present method. Due to the
of the cross-coefficients, the
rigorous
Reciprocal Relation, although it
previously published veriiications.
that the method can be improved
ation, to give a

.- .

‘eCiprocal Relation.

C)
:3
m
n)
0‘}
(I)
*3

SUMMARY

Hydrodynamic theory has been used to derive equa-
tions describing the effects of composition on mutual
and tracer diffusion in certain associated and non-
associated liquid systems. These equations have been
tested by experimental measurements of binary mutual,
ternary mutual and tracer diffusivities.

Tracer diffusivities have generally verified the
predictions of hydrodynamic theory quite well (ether -
carbon tetrachloride being an exception). As the per-
centage of a component which is associated into
complexes increases, the tracer diffusivity - viscosity
product decreases, and vice—versa.

Although many systems have been found for which
hydrodynamic theory does apply quite well, three non-
associated systems are presented here which seem to be
exceptions. In non-associated systems, the Hartley-
Crank equation should describe mutual diffusion. In
benzene - chloroform and 2-butanone - carbon tetrachloride,
the Hartley-Crank equation qualtitatively predicts the
shape of the mutual diffusivity curve, but fails
quantitatively. In chloroform - carbon tetrachloride,

it fails qualitatively as well. These failures may well
124

125

be due to inaccurate vapor — liquid equilibrium data,
however. In the ternary system acetone - benzene -
chloroform (which is here considered to be non-

associated) hydrodynamic theory predicts the ternary
diffusivities, within experimental precision. Unfor-
tunately, experimental uncertainty is rather large, and
this may not be considered a rigorous test of the

theory. Within experimental precision, hydrodynamic

theory also predicts the validity of the Onsager Reciprocal
Relation.

The author feels that this experimental uncertainty
is due primarily to the difficulty of forming a good
boundary within the diffusion cell, which is critical in
measuring ternary diffusivities. Suggestions are given
for future investigations to reduce the experimental error

and provide a more rigorous test of hydrodynamic theory.

The author suggests that future work in liquid

non—electrolyte di“lusion is needed in four particular

areas: (a) improvement of experimental methods and
techniques, so that difquivities may be measured more

precisely, reliably, and hopefully more easily than is
now possible; (b) more systems need to be studied to
support conclusions which have been previously arrived
at on the basis of a small number of studies; (c) those
systems which seem to offer contradictions to hydro-

dynamic theory need to be studied more carefully;

(d) further theoretical work needs to be done,

*0
O
U)
U)
...-I.
0'
}__1

<<:

+4.
{1)
I

extending the principles used here to continuous assoc

tion, or simultaneous self-a sociation and.cross—

U)

association for example.

Experimental problems which lead to low precision
in measurements were discussed considerably under the
Ternary Diffusion error analysis. The author feels that
a thorough study of the effects of different boundary
conditions, different methods of forming the initial
boundary, and possibly a new mathematical treatment for
obtaining diffusivities from refractive index gradient
curves would be a self—contained and quite worthwhile

research program.

127

Besides this work on ternary diffusivities,
however, the author feels that much improvement could
still be made in the techniques used for measuring
binary mutual and tracer diffusivities. For example, a
technique which would totally eliminate convection in the
capillary, perhaps by using a porous capillary instead
of an open one, would be well worth developing. Or an
interferometer which used a laser light source and a
better set of lenses, or for which a better boundary
could be established, would be worth investigating.

Although the systems so far studied have generally
supported hydrodynamic theory, there is not enough
evidence to conclusively say that it is correct. This
is a general problem in liquid diffusion--there simply
has not been enough raw data generated in the past to
thoroughly test any new theory prOposed except for a
comparatively small number of cases. For instance, to
the author's knowledge, there has never been published a
complete study of any ternary system (mutual and tracer
isotherms of all three component binaries, ternary mutual
diffusion isotherms, and ternary tracer diffusion, to say
nothing of the effect of temperature on all of these).

Systems such as 2-butanone — carbon tetrachloride,
and chloroform - carbon tetrachloride, which appear to
contradict hydrodynamic theory (or at least present

ambiguities to be resolved) should be studied more

carefully. It is quite possible that studies of
phenomena other than difiusion would be very useful
here. It has been recently proposed that the chlorine
atoms of carbon tetrachloride engage in a limited form
of hydrogen bonding with alcohols, and therefore they
might cause some very weak bonding effects in these

to the breakdown of the Hartley-

Q.

systems which lea
Crank equation. This seems unlikely, but perhaps

spectroscopic studies directed at this particular

’0

henomenon might provi‘e some useful information.

1

Theoretical work based upon lydrodynamic theory

J

could be directed at finding a simplification bf equation
(80) and its generalization which could be applied to
tracer diffusivity in a ~ystem with extensive association,
such as ethanol - hexane, r aniline - benzene. Or
equations could be developed for application to systems
like aniline — toluol where there is both self-
association and cross-association. Equations for con-

tinuous self-association as pplied to tracer diffusion

pa

could be related to imilar equations for mutual

U)

. -

e

H,
Fl

0..

diffusion. This is again open for much investigation.

3
3
my

D
E

Di
D l

129

APPENDIX A

130

...-1 7...
x
0

F1”, Fl. :fi _

Figure n-l lS a schematic diagra
capillaly. SpeCific details regarding

.‘1
C)
p)
|. J
F.J
§<
Cf
0

prepared gravimetif

by means of a CLristian and Becker
balance.

i al

or

mole fraction on a large analy c
solutions were degassed just beiore

out #0 C to remove

CY

minutes at a

U)
F.)-
X
O
Q)
'U
[.1-
|._J
*4
(l)
}_J
(D
U)
2'
(D
’5
(D
* b
t I
H
l._J
(D
Q.
0'
%
cr
f.)

.L.

in the screw cap.

 

Torbal

balance.

an experiment

from the

of

The capill

fiI-r'fi'fiv n}
1 cirruSIoh

~

m of the diffusion

dimensions,

C).
(I)
U J
}_J
FS
(D
a)
O
O
B
’0
O
(D
I J.
C i’
H.
O
:3
(I)
2
(D
*5
(D

fraction,

torsion

Bulk solutions were prepared to within 1.005

The tracer
for 15

solution.*

‘e following procedure:

malleable nickel
to insure
and placed

ary was inserted

*
Air bubbles coming out of solution during the run

and drifting up to the glass disc were
experimental difficulties. They would
the capillary as they drifted upwards,
changed the resistance constant in the
condition at the capillary end.

F“
W
..J

one of the principle
mix the contents of
and probably
boundary

132

EV FRIT HOLDER

gar-z...— GLASS FRIT

l

I

I

I .

. M CAPILLARY
l
l
I

 

 

 

I
I
I
W
i‘
i
I
l
'_

FOlL DISC

   

SCREW CAP

 

 

Figure A-l.--Schematic Diagram of the Modified
Capillary Cell.

F.)
LA)
LA)

into the screw cap, and tightened with a pair

of pliers to insure a good seal to the foil

The capillary was immersed to within one half

inch in the bulk solution, and the bulk con-

equilibrate at 25°C for one hour.

The capillaries were then filled with degassed

(h

tracer <olution by a .50 cc tuberculin syringe,
with a 19 ga needle cut to the same length as
the capillary. Great care was taken to avoid
trapping air bubbles in the capillary. The
syringe was emptied of air bubbles by inverting
and ejeCting some of the solution. The syringe
was then inserted in the capillary, and with—
drawn siowiy, discharging solution as it was
removed. A puddle of tracer solution was left
covering the top of the capillary.

The glass frit was then filled by dipping it

.‘

n the bulk solution, and allowing the excess to

H-

drain away. The frit was then carefully placed
on top of the capillary, in such a way that no
air bubbles were trapped, and with the puddle of
tracer solution covering the frit.

The frit holder was then placed over the end of
the capillary. By this time the excess tracer

m

mentioned above had generally evaporated. lo

13A

prevent the frit from drying out and trapping
air, the depression in the frit holder was
filled with tracer solution. The entire
assembly was then placed in the bulk solution,
washing away the excess tracer solution in

the process.

After all six capillaries had been filled, the top
was placed on the bulk container, leaving the vent open
until the vapor pressure could drive out the excess
air. The bulk container was then placed in the thermostat,
and the time recorded. The filling process took about

half an hour for all six capillaries;

1

After tnree to five days, depending upon the
diffusivity, the bulk solution was removed from the
thermostat. The six capillaries were emptied into nylon
counting vials as follows:

I. 5 cc of scin illation fluid was placed in the
bottom of the counting vial, and a syringe was
filled with l0 cc of scintillation fluid for
step number (4) below.

2. The capillary was removed from the bulk
solution, and the frit holder carefully removed
without disturbing the frit. The excess
solution on the frit was allowed to evaporate.

3. The capillary was inverted into the counting

vial, which washed the frit from the end. The

end of the capillary was kept beneath the

surface to avoid flash evaporation of tracer

A. The conical depression in the screw cap was
filled with fluid from the syringe mentioned

in step I). The foil disc was then punctured

U‘
C).

y the syringe nee le, and tracer solution
I

f lary with scintillation

FA

ushed from the capi
fluid. When 2 or 3 cc remained in the syringe,
the capillary was removed from the vial, and
washed off with the remaining scintillation
fluid, so that all the tracer solution was in
the vial.

5. The coun ing vial was capped, and identified
by the capillary number. It was shaken gently
to thoroughly mix the contents, and the glass
frit was removed. The glass frit was washed
in acetone to remove scintillation material in
preparation for the next experiment.

for all six capillaries,

{)1

This procedure w s repeate

O

Q)

and the time recorded. The emptying proceSs took about
20 minutes.

The capillaries and screw caps were then cleaned
with acetone, dried, and prepared as before. Initial

counts were then prepared as follows:

F!
U)
0\

A counting vial was prepared as in step (1)

that the puddle was kept as sma

P

rocedure.

filled with tracer solution

J

the filiin

}

procedure, except

09

F»!

l as possible,

ideally covering only the opening of the

capillary.*

The excess tracer

solution was allowed to

evaporate until level with the surface of the

O
-

. l
‘

(D

F]
m
'5

CE.

'0

inverted into th

in the previous

C

(D

apillary was then quickly
counting vial, and flushed as

mptying procedure. The vial was

then identified by the capillary number as an

initial count.

 

*This procedure had to be modified slightly for
containing a high concentration of ether, since
these wetted the surface of the capillary extensively.
The solutions would not evaporate level with the surface,
but would form a depression into the capillary before all
the solution had evaporated from the surface.

The capillaries were filled with a solution for
which wetting was not a problem, and emptied in the usual
manner. They were then refilled with the same solution,
and allowed to evaporate to the same estimated depth as

solutions

the problem solutions.

Emptying these and comparing the

count rates from the two sets gives an estimate of the

ratio of the true
for the problem solutions.

initial count to the measured counts

Since the depth of the depression could only be
estimated by eye, this procedure introduces some error,
but it is at least a random error, due to the estimation,
rather than a systematic error due to incorrect initial

counts. The diffusivities

for these solutions are

reported with an uncertainty of :5%, instead of the

usual i2%.

137

The equipment was th n cleaned and prepared for

The initial and final counts were then counted
using a Packard Tri-Carb scintillation counter. Since
some of the chemicals used, notably carbon tetrachloride
and p-benzoqui.one, acted as quenching agents, the gain
of the scintillation counter was reset before every
counting session to give optimum count rates.

When the data from the count rates had been

of the five closest values of the

C);

analyzed, if the sprea

tracer diffusivity (as calculated for the six capillaries)

‘l

was reater than 5%, i.e. 12.5% from the mean Value, the

09

experiment w discarded. This was to eliminate those

9)
U)

runs in which there may have been convective mixing
within the capillaries. Since the magnitude of such
effects depends upon when during the experiment they
occurred, convection would cause a spread in the values-
of the diffusivity as well as an increase in the apparent

1 I

value. It was felt that this screening procedure would

1

eliminate those runs in which convection occurred.

3)

"U

m
7.
U
H

138

J
‘1

L1)

Figure B-l is a schematic diagram of the Mach-
Zehnder interferometer used for measuring the binary
and ternary mutual diffusivities. Figure B-2 is a
diagram of the diffusion cell. Specific details of
construction are given by Bidlack [A], as are instruc-
tions for alignment and adjustment of mirrors to produce
the proper fringe pattern.

Two solutions with slightly different compositions
were prepared gravimetrically using a Christian and Becker
Torbal torsion balance. The difference in composition
of the two solutions was generally .01 mole fraction for
binary systems, but could be varied to give the proper
number of fringes, depending upon the refractive indexes
of the two components. The cell was filled according to
the following procedure:

1. The plunger was placed in the filling syringe,

and all valves were closed, except valve 2.

2. Approximately 40 cc of the denser solution were

placed in reservoir B, and the top of the

139

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141

glass solution reservoirs
made from 50 cc syringes

[‘__j filling E] A [[‘% B[£
_ syringe

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

valve 2 valve 1
\ cell
cell body
window “*
$-
valve A
/;;/ valve 3
boundary ’//4
sharpening
slits
siphon valve 5

Figure B-2.——Diagram of Diffusion Cell.

i'i2

reservoir was covered with aluminum foil to
retard evaporation.
Valve 5 was Opened, and solution allowed to

fill the cell to just below the level of the
slit. The filling syringe was then used to
draw fluid back and forth through valve 5 to
remove air bubbles trapped near the valve

stem. Solution was then allowed to fill the

cell to about one hal inch above the slit.

 

Valve 5 was then closed.

 

Valve A was then opened. Solution was forced
through valve A by the filling syringe, until
the solution level in the cell was just above
the slit. Care was taken not to force any

air from the cell into the siphon line. Valve A
was then closed, and valve 5 opened. Solution
was allowed to flow into the cell until the
level was again on half inch above the slit.
Step (A) was repeated until fluid flowed from

the siphon line, to insure that the line was

0
O
'3
FC)
|._J
(D
C I‘
(D

ly filled up to the tee in the line.
The process of steps (A) and (5) was then

repeated for valve 3, to fill the other side

of the siphon line.

10.

1A3

Valve 1 was opened, and the plunger removed
from the filling syringe. At this point, the
siphon was checked by slightly opening valves

3 and A consecutively to make sure fluid would
flow freely from the cell. Valves 3 and A

were left closed after checking the siphon.
Valve 2 was closed, and the filling syringe
filled with the less dense solution. The
plunger was then replaced.

Valve 2 was then opened very slightly, and
solution was allowed to flow very slowly down
the wall of the cell. The flow rate was kept
very slow until the level was an inch or so
above the slit, to avoid turbulence and mixing
at the boundary. After this time, valve 2 was
opened a little to allow solution to flow in
more freely. To stabilize the boundary, valves
3 and A were opened so that solution flowed
through the siphon at a rate of one drop every
two or three seconds.

When solution began to appear in reservoir A,
valves 3 and A were again closed. Solution was
forced back and forth through valve 1 to remove
any air bubbles from the valve stem. With
liquid above the bottom of reservoir A, valve

2 was then closed.

 

 

The less dense solution was then added to

FJ
Fl

reservoir A until the level was even with that
in reservoir B. Reservoir A was then covered
with aluminum foil to retard evaporation.
The diffusion cell was now ready to be placed in the
water bath for the experiment.

Before the cell was placed in the water bath, the
fringe pattern was checked to make sure the fringes were
straight, vertical and in focus. It was usually found
that they had drifted slightly away from the vertical
since the last experiment. This could almost always be
corrected by making a fine adjustment of mirror 3.

The cell was then placed in the water bath. Valves
1 and 5 were opened several turns each. Valve three was
then opened until the flow rate from the siphon was
approximately one drop every six seconds. Valve A was
then opened until the flow rate was one drop every three
seconds. It was important that the flow rate be the same

_from each side of the cell maintain flat boundary. It

()1

was also important that the flow rate into the top and

1

bottom of the cell be the same so that the initial

(‘f

distribution of concentra ion gradients would be

symmetric about the boundary.
It usually took about 20 to 30 minutes for the
cell to reach the equilibrium temperature, and for a good

77'

boundary to form. wne

i

. the boundary had formed, valves

p

15‘

 

' rted. A series

w
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of seven pictures, at intervals of 2 minutes were taken.
(In the faster diffusing systems, the intervals were
somewhat shorter.) In some cases, valves 3, 4 and 5
were again opened, anOther boundary formed, and a second
set of pictures taken.

The photographic plate, a Kodak Type M plate, was
developed by the following procedure:

1. The plate was developed for 5 minutes in

2. The devel op me.t was then stopped by a one-

(1')

minute oak, with continual a in tap

0‘!

water.
3. The image was then fixed by a 5-minute soak,
with ir termittent agitation, in Ko da k Rapidfix
4. The plate was then removed from the fixer, and

washed for about one minute under running

F4)

water, and then allowed to dry or at least

The photographic plates were extremely sensitive to light,

1

solute darkness (no safe

0‘

and had to be handled in a
light) throu gho out the entire procedure, until the fixing
step had been co.pl eted.

A new plate was then inserted into the film holder,
making sure that the er*u lsion side of the plate faced

outward. This was irport ant because the thickness of the

..J
:i
0\

plate was enough to throw the image out of focus, and
perhaps change the magnification factor of the camera.
The developer and fixer were replaced after every ten
runs, in order to maintain a consistently high imag

quality in develonment.

 

 

 

1147

COFRUTLR PROGRAMS FOR ATA ANALYSIS

U

The following computer program uses measured
values of the slopes and intercepts to calculate experi-
mental values of the ternary diffusivities. The program

language is FORTRAN IV, with specific deck structure

.1:
H
0:)
C)
O
O
O
’ I
U
{:1
(‘1‘
(D
*5
O
* b
c 1‘
1’3
(D
O
0
FJ

forl the IB lege of Engineering,

 

. .. n . . 2 L
hiChigan State niversity. One constant, R /R1, mUSb

be specified within the program at the designated point.

-‘

Other data (the lepes and intercepts) is read by the

computer.

// J$B

// RflR TDIRF
*IECS (CARD, 14A
*EXT END RD RRECIS
*NCNRRZC ESS RRflG

*ZNE NERD l\iLoL
AM

-REAL 12M, IA, L2x, LA
CEMMZN x
READ (2,10
FERMAT (A E
LA = IA + SA
L2M = 12M + 3

IF (12M) 3, u,

WRITE (3, 20)

o FERMAT lHl, 6

2% IA, 82M, SA

HFJ
o

3

:2N, IA, s2M, SA, L2M, LA
3 15.5)
2

R /R1 MUST BE SPECIFIED

LL U3 Q?

3

2

C

C TI R AL
C AT THIS REINTR
C

e.g. R = .A2999
1A8

CALL PRTS (I2M,
II = -(X**2 + L
322 (X**2 + I
31 = (I2M — 32
D2l = (L21 — o;
N.:T3 (3,AC) 31;
LRIT: (3, AI) 31
WHITE (3, 42) D2
WRIIE (3, 43) 32
AO F RMAT (83 3AA
AI FZRMAT (83 3A3 =
A2 T RMAT (83 23A =
A3 FCRMAT (8H DEB =
GE TC 1
A A CENT '33
CALL EXIT
END
// FTR PRTS
*EXTENDED PRECZSZEN
"CNL WORD INTEGERS
*NCNPRCCTSS TRLCRAM
*LIST SEURCE PRLTRAM
UBRZUTZNE PRTS
IN ms TN A<A>,
EM N X

pgawqp><z<vtjm

l = N + l
(l) l

(2) CT2L — C
(3) = C

(A) = - s2M/SA)
C N IS DEGREE ZR
C P IS PfiLYNEMI.L
C C? IS DERIVATIVE

X = .CCCCCI

10 CALL SYND (A, B,
XPRE = X
X = XP.E - P/DP
I l T -

CQNT INUE
RETURN
END

11

IA, S2M, SA)
2N * X + (L2H * IA *
2M * X + (IZM * LA *
2) * R
l) / T
2
I
2
T‘Tj“ :\
3 L¥D¢Jj
, Lij.j)
n-I— :\
3 L-j-J)
, LID.D)
(CT2N, CTA S2M, SA)
3(4), C(A)

**2

RZLYNZVIAL
IVHLUA- L?

AT

S2M/SA))/S2M
SZM/SA))/S2M

 

 

//

//
*CC

mol
the

dat

FER
‘RT“ TEN;
*;\v: 11

~¢\;

*N NPROC

YIN
DE
wZF

*L:ST sz

SUBR UTLNE SYND

AE

F“
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L.)

g
Ni.)

3 PRECISIZY
D INTEGERS
ESS PRZGEAM
URCE FRAGRAM

[IF ...4 A
\“3 ‘4': La
“*K:n::f 4(4), 3(4), C(4)
1) = ski)
1 I 2, N;
z) = 4 I) + X * : <:-;>
1\ _ - ~\.
J./ -' 23 J-/
2 2
\ \
) I

[11.0 /'\ &/‘\/‘\ 8
H

L) r]

m w L) *U O U C) LU U [u L

5‘

XEQ TD

The
a non a
ar volum

cinary

m1
8.. ine

ll
[1)

(
I
\
72
\J-
\
J
)

(1 H
LU
1/. ()A ll

vities

H

following program predicts ternary diffu

U}

"sociated system from friction coefficients,

, viscosities and the Nargules constants of

U)
:1
C
U)
c I
U‘
(D
*3
(D
Q)
Q.
[1)
U)
} J
:3

p

xfi “V“; fl r—T‘ ~ ‘V
systems. inese constart

program language is FORTRAN IV, with specific

deck structure for th IBM 1800 computer of the College

of"

203

*2 4 _- .- L.“ V. fi..,\ , h .- '
angineering, Micaigau state univerSity.

// J23
// F23
:Izcs

:NQN;D

*EXTE:T3

*UNE

FRED

JED 1J1“ ..UJ. ¢\‘

WEED ImeGERS

REAL oni, LNG2, LNA, LNB
READ (2, 200) 412, A21, 413, A31, A23 A32, 0
FzRALT (7? 10.5)

READ (2, 201) ETA

 

 

f \)
C)

{\J

OOCDO

NH

Mk4

LLLM4T (F10 5)

IL (ETA) 1, 2, 1

LL40 (2, 202)V1, V2, V3

LL43 (2, 202) 3:01, :02, s:c3

LL40 (2, 202) X4, X3, 10

32344? (3L10.5)

31c1 RLPLLLLNTS LT/(LFICT :ZN CfiEFF)

 

VZLJ = 1m 4 01 + XL * V2 + X 4 V3
C4 = X4/VZLL
03 = 1:3/VZLL
00 = XC/VZLD
LXG1 = 2*X4*XL*(421-X4‘421-XL*412) + XL*XL*412
+ 2*X4*X0*(431—X4*431—Xc*413) + XC*XC*A13
+ (X3*Xc—24X4*XL*X0) * (421 + 413 + 432 - 0)
X4? = X4
LP = XL
X0? = c
LNG2 = 2 *XL*X X4* (412— 1:L*412— X4*421) + X4*X4*421
+ 2.*XB*XKC* (432-1:L*432-Xc*423) + XC*X0*423
+ (X4*Xc-2. *X4*XL*XC)*(421 + 413 + 432 - C)
LNA = LNGl + 4L00 (X4)
LNB = LNG2 + 4L00 (XB)
VCXLX ; VoLL - (X4 + .003) * V1 - XL * V2
cmst = ONE N/V3
TMZLs = X4 + XB + .003 + CMOLs
X4 = (X4 + .OO3)/TM6LS
X8 = XB/TMZLS
X0 = c1quS/T40L s
LNGl = 2. *X4*XL*(421-X4*421—XB*412) + XL*X3*412
+ 2.*XA*XC*(ABl—XA*A31—XC*Al3) + XC*XC*413
+ (XL*Xc-2.*X4*XL*XC) * (421 + 413 + 432 - C)
LNG2 = 2.*XL*X4*(412—XB*412—X4*421) + X4*X4*421
+ 2.*XL*XC*(432-XB*432-Xc*423) + XC*XC*A23
+ (X4*Xc - 2.*X4*XL*XC) * (421 + 413 + 432 - 0)

DADCA = (LNGl + ALOGCXA) - LNA) * VOLD/.003
DBDCA = LNG2 + ALOG<XB) - LNB) * VOLD/.003

VCNEW=VOLD - XAP * V1 - (XBP + .003) * V2
CMOLS = VCNEW/VS '
TMQLS = XAP + NB? + CMELS + .003

NA = XAP/TMfiLS
SE = (X3? + .003)/TMOLS
X0 = CMflLS/TMflLS

OOOOO
\DCDNO\U‘I

+4 r4 F4 44 14

Nt—J

[1) II)
U U

DU

:2: 2‘
34343
in t1) [1]

nwar+

()1Hu

>< '11
:1:-

XB+

+||+Il

+

1) L13 :17 f‘J

2.:
j.—
9
O

qjqunjm

00000

152
1 = 2. 4X44XL4(421— X44421-XL4412) + XL4XL4412
+ 2.4144X04(431-X44431— X04413) + X04X04413
+ (XL4X0-2.4X44 L4X0) 4 (421+413+432-0)
2 = 2.4tL4t44(412—XL4412- X4+421) + 44X44421
+ 2.4XL4XC4’432-XL40 32— XC*'A 23) + X04X04423
+ (X44X0—2. X44XL4X0) 4 (421*413+432—0)
CL = (LNGl + 4L00<X4> - LNA) 4 V0L0/.003
CL = (LNGQ + 4L00(XL) — LNB) 4 VOLD/.003
(3,100)
(3,101) 04004, 0L304, L400L, DLDCL
(1' L4204 33004 343031 DLDCL T)
4P
3?
CP
04004 ET4*CA*(SIGl*(l.—Vl*SIGl) + SIG3*V3*CA)
LLLc4/LT442 40L4(V340103-V243102)
L4LCL LTA40L4(Q*14(1.-V143101) + S1034V340.
LLLLL/LL'4 H40 4(V343103-V243102)
DJDCA/ZTA*"“*CS* V3” G3—Vl*SIGl)
LLLC4/Lt442:4(s*22 (1,-V240L) + SIG3*V3*CB‘
LLL B/F”‘*“A*C *(V3*SIG3—Vl*SIGl)
DBD L/L1.40L4(s 024(1.—V240L) + SIG3*V3*CB)
(3,105)
(3,106) 044
(3,107) 34L
(3,108) :34
(3,109) DEB
(13 , // 'TLL LLL010TLD DIFFVSIVITIES ARE
(1L , 1L44 = 1 ,.L 15 5)
(1L , 104L - 1 , L 15.5)
(1H , 'DBA ' , F 15.5)
(1L , 'DBB = 1 , L 15.5)
1
‘XIT

 

.2“

n
U

APPEI‘QDIX

153

APPENDIX D

oncentrations in the system
* (B) - Benzene(C)

 

C')
:D

m

C)
C)

<l

:13)

<I <3
C ) [1'1

' {I

Q l :13

:D

t I]
@

 

O
[D

' l
a

O
O

.3

(I)

4.097 moles/Lit-
b.097 moles/Lit.
H.097 moles/Lit.
.0807 Lit/mole
.0740 Lit/mole

.0894 Lit/mole

-7

1.290 x 10 dynes
n A “A-7
$.Hy3 X LU dynes

.487 CD

.431 RT
.0701 RT
.0978 RT

.3260 RT

 

154

 

155

 

 

 

.00.
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no
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0
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0 . 0
o
O
O
. O
o
o o
o
o o 0 o
o . o
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X
m

Figure D—l.——Typical Refractive Index Gradient Curves
(taken from run #65).

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156

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157

 

 

 

1. u j—
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D 200 400 600 800

tm, seconds

Figure D-3.——Linearity of Equation (188).

158

 

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159

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BETfiii.

 

.Amv mflmNflmm I A<v

 

onumo¢ I ADV anomOhoano Empmmm msu CH on m> ENQII.wIQ mhsmfim
oa
H+ o.o HI mI mI :I
w n L 0 w 4 o
Io.H
Io.m
O
.-o.m
O
IrOo:
. o.m

 

mag

SOT X

161

 

ii“
3>

 

 

 

o
200‘~
g:
1001f
O + ., + . ‘ I . + I 1,
_u -3 -2 -1 0 0 +1
0‘0
Figure D-7.-- —£— vs do in the System Chloroform (C) - Acetone
“DA

(A) — Benzene (B).

162

 

.A<v mCOpmo< I on anm0hoago I Amy mcmNCmm Empmmm mnu CH 6 m> Em

mun.muo wasmfim
m6

m+ m+ 0.0 as m: m: a:

1-H
+

 

fi—

lb
#b
O

 

 

163

 

 

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.23 onumo< I ADV ELOMOQOHQU I Amv mcmwcmm Emumzm map 5” a w> IHI II.mIQ mhswfim
ma
m+ m+ H+ o.o HI NI mI :I
.r i i c i u w u o
..00H
0
o
O ..oom

 

IL“
r—i Q

l6u

 

 

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IhI
o<
:oo.+ moo.+ moo.+ Hoo.+ 0.0 Hoo.I moo.I moo.I
:Hoo.
:moo.
Imoo.
IrJOO .

 

 

L.moo.

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x0
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system

8
5

 

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Q.

 

 

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r
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orm(C)

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051101701

 

E. J

5
,4

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71

x0
«1.

Ann
Qu

 

APPENDIX E

 

166

Téikalle E—1.--Experimenta1 B'
£33ystem.Chloroform(A) — C

b

{oiokIlIbM
wtwvmwm
flattOtt

‘TAEBILIS E-2.--Experimental Binary Mutual Diffusivities in the

X

APPENDIX

E

EXPERIMENTAL RESULTS

nary

1 Mutual Diffusivities in the
arbon Tetrachloride(B) at 25°C.

D

AB

x 105, cm2/sec'

1.557
1.572
1.680
1.757
1.779
1.976
2.007

System Benzene(A) - Chloroform(B) at 25°C.

D

AB

x 105, cm2/sec

 

IX
\
O.C)C)ES
0.C)C)ES
0.2?(3'7
0.207
0-31453
0-3‘H-53
0.4-$314
0.7'8355
0.785
0.9‘5953
_\

2.345
2.359
.1414
.uua
.u95
.u19
.396
.3ua
.344
.265

NNNNNNNN

 

167

 

...J
O\
00

TABLE E-3.-—Experimenta1 Binary Mutual Diffusivities in the
System Ether(A) - Carbon Tetrachloride(B) at 25°C.

 

XA DAB x 105, cm2/sec
0.005 1.50 i .05
0.200 1.95 i .06
0.1400 2.38 i .114
0.600 2.99 i .20
0-800 3.76 2 .3‘4
0. 995 u 2:

 

.59 .07 ”“3

TABLE E—A.-—Experimenta1 Binary Mutual Diffusivities in the
System p-benzoquinone(A) - Benzene(B) at 25°C.

 

 

 

 

 

5 2 i
steink DAB x 10 , cm /sec
0.0041 2.20
0.0057 2.16
0.0132 2.09
0.0239 1.98
0.0266 1.95
0.0315 1.96
0.01430 2L9u
TAE31613 E-S.-—Experimental Tracer Diffusivities in the

System 2-Butanone(A) — Carbon Tetrachloride(B) at 25°C.

 

 

J<1X D: x 105, cm2/sec D; x 105, cm2/sec
\
“-000 1.320
8-837 1.611
0.518 2.260
0' '78 2.720
0.325 3.183
' 82 3.300
0995 2.973

/

FJ
0\
KC)

P3

‘AES ‘E-6.--Experimenta1 Tracer Diffusivities in the

T17
fSysytenIp-benzoquinone(A) - Benzene(B) at 25°C.

 

 

XA DA x 10 , cm /sec 08x 10 , cm /sec
0.0000 2037
<0.0010 2.%2
0.0103 201
0.0464 201

 

'TAEHJS E—7.--Experimenta1 Tracer Diffusivities in the
Ehystem.Ether (A) - Carbon Tetrachloride(B) at 25°C.

 

 

* *
xA DA x 105, cm2/sec Df3x 105, cm2/sec
0.000 1.32
0.024 1.629 t .082
0.024 1.674 t .084
0.332 2.636 t .132
0.530 3.796 t .190
0.850 5.383 t .269
0.999 4.395 i .220
1.000 7.91 r .396

 

TABLE E-8.--Experimenta1 Density and Viscosity in the
System p-benzoquinone(A) - Benzene(B) at 25°C.

 

 

XA density, g/ml viscosity, cp
0.0011 0.8738 0.600
0.0103 0.8767 0.608
0.0283 0.8826 0.617
0.0387 0.8861 0.629
0.0464 0.8896 0.631

 

 

170

TABLEIL9.-—Experimenta1 Density and Viscosity in the
EWstem Ether(A) - Carbon Tetrachloride(B) at 25°C.

 

 

 

 

 

 

XA density, g/ml viscosity, cp
0.000 1.585 0.913
0.1851 1.4177 0.677
0.3119 1.3047 0.445
0.4808 1.1536 0.583 I
1.000 0.7074 0.225
TABLE E-10.-—Experimenta1 Density and Viscosity in the _ II
System Acetone(A) — Benzene(B) - Chloroform(C) at 25°C.
xA xB XC density, g/ml viscosity, cp
0.328 0.339 0.333 1.0457 0.4869

0.339 0.332 0.329 1.0427 0.4870

APPENDIX E

 

171

APPENDIX F

THERMODYNAMIC DATA

TABLE F-1.--Activity Data for Acetone (1) — Benzene (2)
at 25°C (from Timmermans [30]).

 

 

 

 

X lny2, lnyz, lnyl, lnyl,
' 2 exp. eq. (24) exp. eq. (25)
0.1251 .5002 .4827 .0011 .0060
0.2500 .3937 .3896 .0265 .0278
0.3652 .3122 .3022 .0662 .0669
0.5550 .1829 .1672 .1802 .1835
0.7150 .0898 .0750 .3400 .3452
0.8249 .0659 .0300 .4846 .4966
0.8862 .0387 .0131 .5793 .5971
0.9500 .0301 .0026 .7169 .7148
Margules constants: A12 = .8169
A21 = .5685

TABLE F-2.--Activity Data for Acetone (1) — Chloroform
(3) at 25°C (from Hildebrand and Scott [21]).

 

 

 

X3 1ny3, lny3, 1nyl, lnyl,
exp. eq. (24) exp. eq. (25)
.0600 -.6733 -.6771 -.0101 -.0019
.1840 -.5276 -.5502 -.0202 -.0198
.2630 -.4308 -.4700 -.0513 -.0431
.3610 -.3711 ~.3723 -.0943 -.0873
.4240 -.3285 —.3126 -.1278 -.1259
.5080 -.2614 -.2379 —.l985 -.1913
.5810 —.l985 -.1788 -.2877 -.2621
.6620 -.1287 -.1208 -.3857 -.3574
.8020 -.0513 -.0441 -.5798 -.5682
.9180 —.0101 -.0079 -.7765 -.7917
Margules constants: A13 = -.9791
A31 - -.7372

172

173

TABLE F-3.-—Activity Data for Benzene (2) - Chloroform (3)
at 25°C (from Timmermans [30]).

 

 

 

 

--15

x2 1nY2 1nY2 1nY3 1ny3
exp. eq. (24) exp. eq. (25)
0.1340 -.3439 -.2248 +.0119 —.0066
0.2600 —.2837 -.1548 +.0109 -.0237
0.3180 -.1767 -.1278 -.Ol31 -.0347
0.6400 -.0598 —.0299 -.0845 -.l225
0.7160 -.0284 -.0178 -.1301 -.1480
0.8660 -.0202 -.0036 -.1532 -.2012
Margu1es Constants: A23 - -.3180
A32 = -.2500

TABLE F-4.-—Constants for use in equations (20) through
(24) for the system Acetone (1) - Benzene (2) - Chloroform
(3) at 25°C.

 

Al2 = .8169
A21 = .5685
A13 = -.9791
A31 = -.7372
A23 = -.3180
A32 = -.2500

C = 0.0

 

APPENDIX G

 

Capitals

 

A

C

APPENDIX G

NOMENCLATURE

interaction perameter a]

concentration, interaction parameter

 

diffusivity

Onsager diffusivity

reduced second moment

reduced height-area ratio

energy of vaporization

force

Gibbs free energy

enthalpy of mixing

intercept of second—moment curve at a = 0.
intercept of height-area ratio curve at a = 0.
flux, fringe number

equilibrium constant

phenomenological coefficient; length of capillary
intercept of second-moment curve at a = 1.0
intercept of height-area ratio curve at a = 1.0
magnification factor of camera

number of species in solution

simplifying constants

175

+<EZ<1P3

Small

176

gas law constant; refractive index constant;
simplifying constant

entropy

slope of second-moment curve
slope of height—area ratio curve
statistical variance

reduced sensitivity coefficient
temperature

volume

simplifying constant

thermodynamic parameter

activity; arbitrary parameter; Miller coefficient
Miller coefficients

arbitrary function

fringe number

moment

number of moles; refractive index

time; statistical t-test parameter

velocity

mole fraction

distance

Greek Capital

 

A

2

difference

summation

 

Greek small

 

177

prOportionality constant; refractive-index

simplifying constant

activity coefficient; simplifying constant

solubility parameter; Kroneker delta;

constant in friction coefficient

 

activity-defined equilibrium constant
laboratory-fixed coordinates
medium-fixed coordinates

activity-coefficient—defined equilibrium

chemical components A, B, C
species i, 3
species 1, 2, 3

driving force

a fraction;

8 simplifying constant

8n eigen value

Y

0

simplifying constant

n viscosity

6 time

A proportionality constant
An eigen value

p chemical potential

0 constant 3.14159 ...
o

0 volume fraction

Subscripts

a refers to

0 refers to

m refers to

y refers to

constant

A,B,C refers to
1,3 refers to
jl,2,3 refers to
d refers to
r refers to

resisting force

 

 

5 refers
V refers
3 refers
2 refers
0 refers
ave refers
max refers
Superscripts
m refers
V refers
0 refers
* refers
' refers

to

to

to

to

to

to

t0

120

to

to

to

178

sphere

volume-fixed coordinates
fringe number j

distance 2

initial value

average value

maximum value

medium-fixed reference plane
volume—fixed reference plane
standard-state
tagged species

related quantity

molar property

 

BIBLIOGRAPHY

 

179

BIBLIOGRAPHY

Ph.D. Thesis, University of

1. Anderson, D. K.,
Washington (1958).
nd Keefer, R. M., J. Am. Chem. Soc.,

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Baldwin, R. L., Dunlop, J., and Gosting, L. J.,

3' P.
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Bidlack, D. L., Ph.D. Thesis, Michigan State University,

4.
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5. Caldwell, c. s. ‘
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6. Carman, P. 0., J. Phys. Chem., 12. 1707 (1968).
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11. Foster, R. T., a
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Fowler, R. T., and Norris, G. S., J. App. Chem., 5,

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Gosting, L. J., and Morris, M. 5., J. Am. Chem. Soc.,

Hardt, A. P., Anderson, D. K., Rathbun, R., Mar, B. W.,
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Hartley, G.

S. and Crank, K., Trans. Faraday Soc.,
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Hildebrand, J. H. and Scott, R. L., The Solubility
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Johnson, P. A. and Babb, A. L., Chem. Rev., 55, 387
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Kett, T. K., Ph.D. Thesis, Michigan State University
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Kohnstamm, P. and Van Dalfson, B. M., Proc. Akad.
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McCall, D. W. and Douglass, D. C., J. Phys. Chem.,
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Nikol'skii, S. S., Teoretisheskaya i Eksperimantal'naya
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Onsager, L., Phy. Rev., 51, 405 (1931).
Sutherland, W., Phil. Mar., 5, 784 (1905).
Timmermans, J., The Physico-Chemical Constants of
Binary Systems in Concentrated Solutions, Interscience

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Van Geet, A. L., and Adamson, A. W., J. Chem. Phys.,
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Van Laar, «3., Z. Physik. Chem., 15, 723 (1910).

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182

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Wohl, K., Trans. A. I. Ch. B., 5;, 215 (1946).

 

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