TRACER AND MUTUAL DIFFUSION
IN ISOTHERMAL LIQUID SYSTEMS
WITH SPECIFIC ASSOCIATION

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
GUY B. WIRTH
1968

 

   
    

  

LIBRARY

Michigan 3 if: (F:
University

  

“—fi‘ .—.

all... _..?

This is to certify that the

thesis entitled

TRACER AND MUTUAL DIFFUSION IN ISOTHERMAL
LIQUID SYSTEMS WITH SPECIFIC ASSOCIATION

presented bg

Guy B. Wirth

has been accepted towards fulfillment
of the requirements for

KIM—D;— degree in an inee ring

WM

Major professor

 

Date June 14, 1968

0-169

 

n‘ .4.- "

‘ ‘ fine.“_

I Y IIN‘BING BY I
HUM: & SUNS' I
800K nmnrnv mp

nnnnnnnn olnu‘nu

"IIIGM". IICUIGII
t . a

.n’

     

 

ABSTRACT

TRACER AND MUTUAL DIFFUSION IN ISOTHERMAL
LIQUID SYSTEMS WITH SPECIFIC ASSOCIATION

by Guy B. Wirth

A theoretical and eXperimental study was made to
investigate the effects of specific association on mutual
and tracer diffusion. The Hartley and Crank (1) intrinsic
diffusion approach was used to describe the diffusion
mechanism from which equations were derived which relate
the mutual and tracer diffusivities to the thermodynamic
and molecular prOperties of the systems of interest.

For associated systems, it is necessary to con-
sider diffusion of clusters as well as the individual
molecules. A general chemical model, suggested by
Nikolskii (2), was used to calculate the concentration
of these clusters and the solution activities.

Three types of associated systems were studied:

(a) binary mixtures of A and B where A associates
to form a dimer cluster and B remains inert

(b) binary mixtures of A and B where A associates
with B to form a bimolecular cluster

(C) ternary mixtures of A, B and C where A aSSO"
ciates with B as in (b) and C remains inert.

Guy B. Wirth

A Mach-Zehnder diffusiometer was used to measure
the mutual diffusion coefficients. Tracer diffusion co-
efficients were determined by a modified capillary method,
develOped to minimize errors inherent in previous
capillary studies using Carbon—l4 tracers. The accuracy
of the capillary method was found to be 12%.

Data were collected for six systems at 25°C:

(I) benzene-carbon tetrachloride--an example of
a nonassociating system

(2) chloroform - carbon tetrachloride-~an example
of a nonassociating system

(3) methyl ethyl ketone-carbon tetrachloride-—a
type (a) associated system

(4) acetic acid - carbon tetrachloride-~a type
(a) associated system

(5) ether—chloroform--a type (b) associated
system

(6) ether-chloroform - carbon tetrachloride--a
type (c) associated system.

These data agree well with values predicted by
the equations derived in this work. Deviations can be
attributed to the inability of the models to take into
account nonideal behavior as well as inaccuracy in the

eXperimental values of diffusion coefficients and act1v1ties.

 

1G. S. Hartley and J. Crank, Trans. Faraday Soc.,
32. 801 (1949).

2S. S. Nikol'skii, Theoreticheskaya i
EkSperimental'naya Khimiya, g (3), 343 (1966).

TRACER AND MUTUAL DIFFUSION IN ISOTHERMAL

LIQUID SYSTEMS WITH SPECIFIC ASSOCIATION

By

1‘1
\ -2

Guy Bi Wirth

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Chemical Engineering

1968

ACKNOWLEDGMENT

The author wishes to eXpress his appreciation to
Dr. Donald K. Anderson for his guidance and encouragement
during the course of this work.

The author is indebted to the Division of
Engineering Research of the College of Engineering at
Michigan State University and to the Dow Chemical Company
for providing financial support.

Appreciation is extended to the author's sister,
Carol, for her help in the preparation of this manuscript.

The understanding and patience of the author's
family and expecially his wife, Darlene, is sincerely

appreciated.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENT . . . . . . . . . . . . . . .
LIST OF FIGURES . . . . . . . . . . . . . .
LIST OF TABLES . . . . . . . . . . . . . . .
INTRODUCTION . . . . . . . . . . . . . . . .
BACKGROUND . . . . . . . . . . . . . . . . .

Intrinsic Diffusion . . . . . . . . . . .
Planes of Reference . . . . . . . . . . .
Mutual Diffusion in Binary Systems . . . .
Tracer Diffusion . . . . . . . . . . . .

Diffusion in Multicomponent Systems . . .
Thermodynamics of Solutions . . . . . . .

THEORY I O O O O O O O O O O I O O O O O O 0

Concentration Dependence of Diffusivities
--Associated Systems . . . . . . . . . .
Selection of Systems . . . . . . . . . . .

EXPERIMENTAL METHOD 0 O O O O O O O O O O 0

Modified Capillary Cell . . . . . . . . .
Procedure . . . . . . . . . . . . . . . .
Calculation of Diffusion Coefficients . .
Calibration of the Cells . . . . . . . . .
Analysis of Radioactive Samples . . . . .
Materials . . . . . . . . . . . . . . . .
Discussion of Error . . . . . . . . . . .

RESULTS AND DISCUSSION . . . . . . . . . . .

Theoretical . . . . . . . . . . . . . . .
Experimental 0 O . O C O O C O O O O O O 0

CONCLUSIONS . g o o o o o o o o o o 0 0 0 0

iii

ii

vii

10
21
24
3O
32

46

46
63

74

75
78
8O
82
85
86
89

91

91
96

114

FUTURE WORK .

Theoretical
Experimental

APPENDIX I

APPENDIX II
APPENDIX III

APPENDIX IV

APPENDIX V

APPENDIX VI

APPENDIX VII

APPENDIX VIII

NOMENCLATURE

BIBLIOGRAPHY

Fortran Compute

determination o

Constants .

Experimental Procedure
Solution of Equation (226)

Fortran Computer Program for
Diffusion Coefficient

Calculations

Determination of Friction
Coefficients From Tracer

Diffusivities

Experimental Tr
at 25°C . . .

Experimental Mutual Diffusion Data

at 25°C . .

Experimental De
Data at 25°C

iv

1:

acer Diffusion Data

nsity and Viscosity

Program for the
f Equilibrium

Page
116

116
117

120

121
123

128

129

132

135

136
137

141

 

LIST OF FIGURES
Figure Page
1. Activity as a Function of Mole Fraction for
the Methyl Ethyl Ketone - Carbon Tetra—

Chloride system C O O O I O O O O O O O O 0 66

2. Total Pressure as a Function of Mole Fraction
Ether for the Ether - Chloroform System . . 69

3. Logarithm K as a Function of l/T for the
Ether - Chloroform System . . . . . . . . . 71

4. Heat of Mixing as a Function of Mole
Fraction for the Ether-Chloroform System . 72

5. Schematic Diagram of the Modified Capillary

cell 0 O O O O I O I O O O O O O O O O O O 77
6. Diffusion Cell Coordinates . . . . . . . . . 81
2
* * ' * L and
7. C iavg/Ci0 as a Function of Dle/
RD*/L O O O O O O O I O O O O 0 O O O O O O 83
1
8. Count Rate as a Function of Mole Fraction
for the Methyl Ethyl Ketone - Carbon
Tetrachloride System . . . . . . . . . . . 87
9. Count Rate as a Function of Mole Fraction
for the Chloroform - Carbon Tetrachloride 88
SYStem O O O O O O O O 0 O O O I O O 0 O G
10. Density and Viscosity of the Benzene - 97
Carbon Tetrachloride System at 25°C. . . .
11. Density and Viscosity of the Chloroform - 98
Carbon Tetrachloride System at 25 C. . . .
12. Tracer and Mutual Diffusion Coefficients for
the Benzene-Carbon Tetrachloride System 99

at 25°C. 0 O O O O O C C O O C O O O c O O

Figure Page

13. Tracer and Mutual Diffusion Coefficients
for the Chloroform - Carbon Tetra-
chloride System at 25°C. . . . . . . . . 100

14. Tracer and Mutual Diffusion Coefficients
for the Methyl Ethyl Ketone - Carbon
Tetrachloride System at 25°C. . . . . . . 103

15. Tracer and Mutual Diffusion Coefficients
for the Acetic Acid - Carbon Tetra-
chloride System at 25°C. . . . . . . . . 106

16. Tracer and Mutual Diffusion Coefficients
for the Ether - Chloroform System at
25°C. 0 C O O O O O O O O O O O O O O O O 108

17. Density and Viscosity of the Ether —
Chloroform — Carbon Tetrachloride System
at 25°C. I O I O I O O O O 0 O O O O O O 111

18. Tracer Diffusion Coefficients for the
Ether - Chloroform - Carbon Tetrachloride

System at 25°C. . . . . . . . . . . . . . 112

vi

LIST OF TABLES

Table page
I Capillary Cell Calibration Results . . . . 84
II Summary of Diffusion Equations . . . . . . 92
III Tracer Diffusion Data for the Benzene-Carbon
Tetrachloride System . . . . . . . . . . . 132
IV Tracer Diffusion Data for the Chloroform -
Carbon Tetrachloride System . . . . . . . 132
V Tracer Diffusion Data for the Methyl Ethyl
Ketone - Carbon Tetrachloride System . . . 133
VI Tracer Diffusion Data for the Acetic Acid -
Carbon Tetrachloride System . . . . . . . 133
VII Tracer Diffusion Data for the Ether - 133
Chloroform System . . . . . . . . . . . .
VIII Tracer Diffusion Data for the Ether -
Chloroform - Carbon Tetrachloride 134
System . . . . . . . . . . . . . . . . . .
IX Mutual Diffusion Data for the Benzene - 135
Carbon Tetrachloride System . . . . . . .
X Mutual Diffusion Data for the Chloroform - 135
Carbon Tetrachloride System . . . . . . .
XI Density and Viscosity Data for the Benzene - 136
Carbon Tetrachloride System . . . . . . .
XII Density and Viscosity Data for the Chloro- 136
form - Carbon Tetrachloride System . . . .
XIII Density and Viscosity Data for the Ether -
Chloroform - Carbon Tetrachloride 136

SYStem o o o o o o o o o o c 0

vii

INTRODUCTION

Interest in liquid diffusion had been maintained
for over a century. Since the diffusion process is one
of the fundamental aspects of liquid behavior which must
be described by a liquid state theory, diffusion rates
have been useful in testing such theories. The importance
of understanding diffusion has also increased as more
basic consideration is being given to various mass trans-
fer processes where liquid diffusion is one of the
important factors.

In the usual sense, diffusion of mass refers to
the dissipation of a chemical concentration gradient by
molecular transfer with no overall mass flow caused by
forces external to the diffusion system. Thus, if two
miscible liquids, containing the same components but
having different concentrations, are placed together, the
molecules of the components will diffuse from the higher
to lower concentration until the solution is uniform. ThlS
diffusion process results from the thermal energy of the
molecules which gives rise to their Brownian movement.

Fick (28) was one of the first to mathematically
describe diffusion. By analogy to conductive heat trans-

fer, he deduced that the rate of transfer of a material

.,_Jn__ _ _ _

‘-P'“*"“fiwaur
J.

 

was proportional to its concentration gradient at constant

temperature and pressure. That is:

'3 = -Dvc (1)
where 3 is the flux (i.e., the rate of transfer) of
material across a reference plane of unit area and VC is
the three dimensional concentration gradient. The constant
of proportionality, D, is called the diffusion coefficient
and was first thought to be constant for each system at
constant temperature and pressure. But, it is now known
to be a function of concentration. Equation (1) is usually
simplified to the one-dimensional form given by equation
(2) because of the difficulty in obtaining measurements

for diffusion in more than one direction.

— .. 39.

Either equation (1) or (2) is used to define the diffusion
coefficient.

Tracer diffusion is different than ordinary
diffusion in that it is observed when there are no chem-
ical concentration gradients present. To better visualize
this, consider a solution of uniform chemical composition
in which part of the molecules of one component is labelled.
If there is a higher concentration of labelled molecules in
one region of this solution than in another, measurable
diffusion of these labelled molecules will take place even

though the solution is chemically uniform. The flux of

“1““ “‘" “-‘=¥£2‘.F.J

labelled molecules is measured in the laboratory.
Frequently this labelling is accomplished with a radio—
active tracer, from which the name tracer diffusion is
derived. As in ordinary diffusion, the tracer diffusion
coefficient is defined by equation (2). For example,
denoting the labelled component by *, the defining rela-

tion is:

J* = -D* —- (3)

where D* is the tracer diffusion coefficient. Like the
ordinary diffusion coefficient, it also is a function of
concentration.

Studies have been published recently concerning
the concentration dependence of the diffusion coefficients
in nonideal solutions in which nonideality is caused by
association of the molecules. In an associated mixture,
two or more molecules may cluster together and diffuse
through solution as a unit instead of as single molecules.

In this study, relationships between tracer and
ordinary diffusion coefficients are derived for systems
where the molecules associate to form simple clusters of
two molecules. Experimental measurements of the diffu—

sion coefficients were used to verify these relationships.

4-. ‘uua up. ‘-‘.-

BACKGROUND

Intrinsic Diffusion

For a system to be at equilibrium at constant
temperature and pressure, the chemical potential must be
uniform throughout the system. For example, a system
having two phases will be in equilibrium only if the
chemical potential of each species is the same in both
phases. If the chemical potentials of a component are
not equal in both phases, there will be a spontaneous
migration or, prOperly, diffusion of that component from
the phase of high potential to the phase of lower poetn-
tial. This diffusion will continue until the chemical
potentials, but not necessarily the concentrations, are
equal in both phases.

The diffusion process will also occur within a
single phase. The molecules will diffuse from a region
of high concentration and high potential to a region of
low concentration and low potential until the concentra-
tion and the chemical potential are uniform. The rate
0f approach to equilibrium will depend upon the differ-
ence in chemical potential gradient. Therefore, it is

reasonable to state that the chemical potential gradient

4

 

is the quantity which drives the system to equilibrium,

that is, it is the driving force for diffusion.

[driving force for diffusion] = - 3%; (4)

A nonuniform chemical potential can be thought to

my

‘
:-

exert a force on a molecule which causes the molecule of

the i—th component to move with a velocity, Vim’ with
respect to the surrounding medium. A molecule moving in I
this fashion is said to be undergoing intrinsic diffusion.
As this molecule diffuses, momentum transfer to
the surrounding medium results in a resisting drag force
which is exerted on the molecule in a direction, Opposite
to that of the velocity Vim' If the molecule is Spherical
and if the surrounding medium is assumed to be a continuum,
Stokes' law applies and

[resisting force for dissusion] = ~6nrirnfim (S)

where ri is the molecular radius and n is the viscosity
of the surrounding medium. When the molecule diffuses

with a constant velocity (i.e., it experiences no accel—
eration), the sum of the resisting force and the driving

force for diffusion must be zero.1 From equations (4)

1For a more exact description of nonsteady-state
diffusion, a term associated with the acceleration of the
molecule should be included. Although such terms may be
important in some cases, they become negligible for the
small chemical potential gradients encountered in experi-
ments designed to measure diffusion coefficients (42).

and (5) this sum is
Sui
- —§; - N67rrinvim = 0 (6)
where N is Avrogadro's number. Solving this equation for
vim and then multiplying by Ci’ the concentration of the
i-th component in moles/volume, yields:
Ci aui

ivim = m 73' ‘7’

Note the Civim is the molar flux of 1. And, Since vim IS
the velocity relative to the medium, Civim will be the

molar flux relative to the medium, Ji . Hence,

m
J = “Ci LL11 (8)
im 6nrinN 32
From thermodynamics,
_ = , 9
pi “io NlenAl ( )

where Ai is the activity of i and “i0 is the chemical
pOtential of i in the standard state-—a function of temp—

erature and pressure only. Using equation (9), one can

rewrite equation (8) as

 

__ kT alnAi 33$ . (10)
Jim I 6nrlfi alnCi 82

Comparing this equation to equation (2), one can see that
the bracketed term corresponds to a diffusion coefficient.
Since the flux, Jim’ is with respect to the medium, the

diffusion coefficient is defined with respect to the

 

medium.

alnA.
kT 1 (ll)

Dim = 6nr.n alnC.
1 1

In dilute solution, the activity of the solute is

proportional to its mole fraction. Further, its mole

fraction is

 

Ci
Xi = Esolvent (12)
so that
alnAi = 1. (13)
alnCi

Thus, equation (11) can be written as

kT (14)

Dim = 6wr n °
' i

 

This equation, known as the Stokes - Einstein equation,

was derived independently by Einstein (24) and Sutherland
(64). It has found application in the diffusion of
colloidal particles and macromolecules in low molecular

weight solvents.
However, if the radius of the diffusing molecule

is smaller than the molecules of the solvent, the molecule
can no longer be considered to diffuse through a continuum

Rather, the medium must be viewed as containing particles

Also, most molecules are not spherical

of finite size.
Thus, the

and their shape will effect their mobility.

resisting force for diffusion may not be given by Stokes'

law.

 

The idea that the diffusion coefficient is
inversely proportional to viscosity is an old one. In

1858 Wiedemann (68) observed that for a given solute in
a series of solvents, the product Drivaried less than D

alone. In view of this observation, it may be assumed

that, even where Stokes' law is not obeyed, the resistance

to diffusion can be separated into a product of viscosity

and a parameter, f, which depends very little on viscos-

the resisting force for diffusion is:

ity (33). Now,
(15)

[resisting force for diffusion]
This friction co-

where fi is a friction coefficient.

efficient will, in general, be concentration dependent

and also dependent upon the size and shape of the molecule

By calculating fi for several solutes in one solvent,

Anderson (1) found that it is approximately proportional

to the square of molecular radius. Bidlack (7) confirmed

Anderson's findings and also concluded that fi depends

upon molecular shape, but only for very unsymmetrical

molecules. In any case, equation (15) will be accepted

as the resisting force.

Again, one can sum the driving and resisting

forces for diffusion to obtain:

aui
... ...—32 — finv1m = 0. (16)
From equation (16) it follows, as before, that:
alnA
l - (17)

kT

D . ::
1m fir131nCi

 

 

This diffusion coefficient, which Hartley and

Crank (33) called the intrinsic diffusion coefficient,

will be used later in this work.
Intrinsic diffusion in liquids and solids may be

clarified by identifying it with the widely known kinetic

theory of diffusion developed by Eyring and co-workers

According to this theory, the

(25, 26, 27, 39, 54, 62).

diffusing molecule spends most of its time oscillating

within a "cage" of other molecules. At relatively infre—

quent intervals, such a molecule receives sufficient energy
to break through the potential barrier which limits its

motion and moves in a random direction into another "cage."

This diffusion process thus consists of a series of
activated, random jumps relative to the loose quasi—

crystalline lattice of the liquid or the better-defined

lattice of the solid. The theoretical treatments of the

above authors give the flux of the diffusing molecules

with respect to the surrounding molecules. Diffusion

coefficients calculated from this theory should be identi-

fied with intrinsic diffusion coefficients.
From the foregoing discussion, it can be said that

the diffusion of molecules relative to their surroundings,

i.e., relative to the surrounding medium, is one of the
basic mechanisms for liquid and solid diffusion. Further,

intrinsic diffusion is seen to occur because of the ran-

dom motion of the molecules.

 

10

Planes of Reference

While Fick (28) extended the laws of heat

conduction to the problem of diffusion in a simple manner,

a complication arises in connection with diffusion. In

a diffusing mixture, the velocity of the individual

2 are different and as a result, there is a

This bulk velocity itself
To

speCies, Vic,
bulk velocity of the fluid.
contributes to the overall flux of each Species.
separate pure diffusive flux from the overall flux, it

follows that the flux of a species should be measured
relative to a plane which moves with the bulk velocity.

There are several ways to define the bulk velocity so it

is necessary to carefully specify the reference plane
across which the flux is measured. Since the concept of
the reference plane is basic to the definition of the

diffusion coefficient, it is appropriate to discuss some

In this discussion, the following relations will

C
p= Z oi (18)

i=1

of them.

be required:

2The velocity, viC does not refer to the velocity
of a particular molecule of the i~th component; rather it
refers to the average velocity of the molecules in a small
volume of the liquid. This velocity is a function of

position and time.

 

and
c
1 = Z CV. (19.)
i 1
i=1
where p is the density of the mixture (g/cc), p is the
is

mass concentration of the i-th component (g/cc), Vl
the partial molal volume of the i-th component (cc/mole)

and c is the total number components in the mixture.

Mass Fixed Reference Plane The mass fixed refer—
ence plane is defined aS a plane through which there is
no net transfer of mass. From this word definition, it

follows that
(20)

C
Z MiJiM = 0
i=1

where JiM is the molar flux of i relative to the mass

fixed plane and Mi is the molecular weight of i.

The meaning of this reference plane may be

clarified from the standpoint of experiment by determin—

ing its velocity and the flux JiM in terms of fluxes

relative to laboratory coordinates. The mass flux of the

i-th component through the mass fixed plane is determined

by the velocity of the component, ViM’ relative to that

plane. That is,
MiJiM = iniM° (21’

But, ViM is the difference between the velocity of the

i-th component relative to fixed coordinates and the

 

12

velocity of the mass fixed plane relative to fixed

coordinates
Vim = Vic ’ VMC (22’
the velocity of the mass fixed plane, is called

where VMC,
Combining the last two equa—

the mass average velocity.
tions yields:

V

...)- (23)

MiJiM = pi(vic -
Summing equation (23) over all Species and combining the

result with equation (20) gives:

C
i=1

Or, after solving for GMC’

 

c
E: p.v.
_ ._ 1 1C .
YMC — 1—1 p (25)

Identifying pivic as the mass flux of 1 relative to fixed
coordinates, one can see that the mass average velocity

is the net flux of mass across a coordinate fixed plane

divided by the solution density.

C
2: OLCL
_ _ 0— 1 1C

 

<

Since both p and Jic are functions of position and time,

will also be so dependent.

VMc

 

13

To demonstrate the contribution of bulk flow to

the overall flux of a component, eXpand equation (23) to

(27)

or

M.J

1 1c = i 1M (28)

From this last result, one can see that the mass flux

the bulk motion.

Number Fixed Reference Plane The number fixed
reference plane is defined as a plane through which there
is no net flux of molecules; or, equivalently, a plane

through which there is no net flux of moles. Therefore,

reference plane. From the definition of this plane, it

follows that

C
E: JEN = o (29)
i=1

Where JiN is the molar flux of the i-th component rela-

tive to the number fixed plane.
The relationship of this reference plane to

experiment can be seen by noting that the flux

_ = . - ‘ (30)
JiN ‘ CiviN Ci(VlC VNC)

14

where VNC, the velocity of the number fixed reference
plane, is the number average velocity. Summing equation
(30) over all components and combining the resulting

equation with equation (29) yields:

C
VNC = Z Crivic
i=1
-c—-—— (31)
:c.
1
i=1

Since Jic = Civic’

c
VNC = :Jic
i=1 . (32)
c
:c.
1
i=1
Thus, the number average velocity, at a position in the
fluid, is the net molar flux through a coordinate fixed

plane at that position divided by the total molar concen-

tration of the solution.

The effect of bulk motion can be seen by eXpanding
equation (30) to

J. =J. +c.§-’ . (33)

Again, the overall flux relative to a coordinate fixed

Plane is the flux relative to the number fixed plane plus

15

a term resulting from the bulk motion of the solution.
(However, this bulk motion of the fluid is defined dif-
ferently than in equation [28].)

Component Fixed Reference Plane The component

 

fixed reference plane is defined as a plane through which
there is no flux of one particular component. Usually,
the reference component is taken as the solvent so this
reference plane is often called the solvent fixed refer-

ence plane. From the definition,

J = O (34)
00

where the first subscript, o, designates the reference
component and the second the component fixed reference
plane. The flux of any other component, i, with reSpect

to this reference plane is given by:

J. = C.v. = C-

V. - v
10 i 10 1 1C oC

‘ ) (35)

 

where VOC is the component average velocity. Writing

this equation for the reference component, one obtains:

J = C

v - v
00 0 0C oC

 

‘ ). (36)

Substituting equation (34) into equation (36) and solving
f — ' :
or v0C yields

= 0 0C . (37)
CC C

 

<I

16

Or, since JOC = COVOC’ this equation can be written as

rewritten as

J. = J. + c.3

where the contribution to the overall flux of i by bulk
motion is evident.

.Vglume Fixed Reference Plaee The volume fixed
reference plane is defined as a plane through which there
is no net flux of volume. Noting that the volume flux of
the i-th component relative to the volume fixed plane is

JivVi, it follows from the definition that:

C
ZJini = o (40)
i=1

Where Jiv is the molar flux relative to the volume fixed

reference plane.
To Show the relationship between this reference
Plane and the coordinate fixed plane, write the flux as:

= = . - 37 (41)
Jiv Civiv Ci(VlC vc)

17

where ch is the volume average velocity. This equation

can be rewritten as:

tain an eXpression for V C' multiply equation (42) by V:

V
and sum over all components. The result is:

C C C
ZJicVi = ZJiVVi + ch Z CiVi (43)
i=1 i=1 i=1

C
ch = Z JiCVi. (44)

Thus, one sees that _VC is the total volume flux across
a coordinate fixed plane and as such, it will be zero
When the total volume of the system is constant. In that

case, equation (42) becomes:

= (45)
Jic Jiv

and the coordinate fixed and volume fixed plane become
identical. Experimentally, constant volume conditions
can be maintained by having very small concentration

differences in the system. Therefore, the volume fixed

18

plane has had wide application in the literature and will
be used in this work to define diffusion coefficients.
For a binary system, two diffusion coefficients

can be defined by the following equations:

BCA
JAv = -DAV 32 (46)
BCB
JBv = -DBV 82 ° (47)
However, by using the fact that:
VABCA + VBBCB = O (48)

along with equation (40), it can be shown that:

D = D E D (49)

AV BV AB'

The coefficient, D is called the mutual diffusion co-

AB’
efficient and from this point on, it will imply that the
volume fixed reference plane is used.

Medium Fixed Reference Plane In the discussion
of the previous reference planes, the velocity of the
various components were measured relative to the mass,
number, component or volume average velocity. However,
it was shown in the previous discussion of intrinsic

diffusion that the velocity of a component relative to its

surrounding medium is more significant. This is the

velocity referred to in Stokes' law (equation [5]).

Therefore, it is desirable to define a new reference plane

19

fixed with respect to the medium. (i.e., it has the same
velocity as the medium.) Material would be transferred
through such a plane by intrinsic diffusion only. Thus,
the medium fixed reference plane is defined as a plane
through which material is transfered by random molecular
motion only.

The medium average velocity, V can be written

mC’
as:

va = v. - v. (50)

and the flux of the i—th component relative to the medium

fixed plane is then:

(51)

 

 

or, as before:

= .— 52
JiC Jim + CiVmC ( )

where Civ is a flux resulting from the bulk motion of

mC
the liquid. The kinetic theory discussed previously
suggests a molecular mechanism for the bulk motion (33).
Molecules do not always jump into "cages" which are large
enough to accommodate them. They pass through an acti-
vated state of high potential energy to a new position
which, although it has lower potential energy than the
activated state, has higher energy than its original

position. AS a result, the new set of neighboring mole—

cules is pushed slightly further apart to reduce potential

20

energy. The total effect over a large number of such
processes is a small increase in hydrostatic pressure
which is relieved by a bulk flow of the liquid.

Although this plane of reference is significant
in terms of the molecular mechanism of diffusion, diffu-
sion coefficients which are defined relative to it cannot
be directly measured from eXperiment as can the mutual
diffusion coefficient. However, the medium and volume
fixed planes can be related in a useful way by multiplying

equation (52) by Vi and summing over all components. Thus:
c c c
Z JiCVi = X Jimvi 4‘ Vmc Z CiVi' (53)
i=1 i=1 i=1

Combination of this equation with equation (19) and (44)

gives the result:

C

I — =_ _ _ o 54
E: Jimvi ch Vmc ( )
i=1

The utility of the last equation will be seen in the next
section. Hartley and Crank (33) were one of the first to
formally discuss this plane of reference. However, they

limited their work to constant volume systems in which

ch = 0'

21

Mutual Diffusion in Binary Systems

measured with respect to the volume fixed reference plane.
The mutual diffusion coefficient, DAB’ is the coefficient
usually calculated from diffusion experiments. From the
point of view of interpreting diffusion coefficients in
terms of molecular motions, DAB appears to be complicated
by the bulk motion of the liquid. On the other hand,
intrinsic diffusion, i.e., diffusion with respect to the
medium fixed reference plane, is not complicated by the
bulk motion. Therefore, it is desirable to explore the
relation between mutual and intrinsic diffusion. The
basis for the following discussion is an important work
by Hartley and Crank (33).

The flux of component A relative to the volume

fixed reference plane is given by:

= 5
JAv CAVAV ( 5)

or:

= - ‘ . 56)
JAv CA VAC ch (

 

 

To this equation, add and subtract CAEmC° Thus,

J = C

V" ). (57)
AV A

VAC ‘ Vmc) ’ CA(ch ' mC

 

But,

5 (58)

22

So,

JAv = JAm ’ CA

 

VVC - VmC)‘ (59)

At this point, use equation (54) with equation (59) to get:

“EFL-I,

JAv = JAm “ CAIJAmVA + JBmVB)° (60)

From the definitions of the diffusion coefficients, 4

_ _ ..45
JAv " DAB 32 (61)
J - —D 335 (62)
Am “ Am 82
and,
acB
JBm = -DBm 82 ° (63)

These last three equations are now combined with equation
(60) to yield:

acA acA _ acA

DAB 52 = DAm 52 - CAVADBm 52 _

C

— B . 64
CAVBDBm “E ( )

Further, from equation (48),

B - IA . (65)
v

23

So, equation (64) becomes:

 

DAB = DAmIl - CAVA + DBmCAVA (66)
which, when combined with equation (19), becomes: N
DAB '-'—' CBVBDAm + CAVADBm. (67) ~:"‘

Equation (67) can be modified by substitution for ‘

DAm and DBm from equation (17) to give:

D -121 13.13:sz .133??sz .(68,
AB " 1) f ainc B B fB alncB A A

A A

From the definition of mole fraction and equation (19),

it can be shown that:

a1nx X

 

 

 

 

1 A = E (69)
8 nCA CBVB
and:
Blnx X
B
31 = it (70)
nCB CAVA

Therefore, equation (68) can be rearranged to the well

known Hartley~Crank equation (33):

E2 XB XA BlnAA

D -= + —— - (71)
AB 1) I; fB SInxA

24

81nA
The activity term, FIHX—’ that can be factored out because
A a1nAA ainAB
the Gibbs-Duhem relation requires that FIHXXI= FIHX; .

While Hartley and Crank (33) restricted their equation
to systems having constant volume, no such restriction

was necessary in the derivation presented here.

' . V I;: - {E'EJJ

Tracer Diffusion ;

Tracer diffusion refers to diffusion in a l
chemically uniform mixture. It frequently is observed
by labelling some of the molecules of one component with
an isotopic form of one of their constituent atoms. Then
a concentration gradient of labelled molecules is
established while maintaining the total concentration at
a constant value. After diffusion starts, the concentra-
tion gradient of labelled molecules will be dissipated by
random molecular motion. The intrinsic flux of labelled
molecules can be found by summing the driving and re-
sisting forces to diffusion as done above. The result is:

BlnAt 8C?
J* = _ kT i i (72)
im nff BIan 82

 

where superscript, *, indicates labelled molecules. Again,
the term in the brackets is identified as the intrinsic
diffusion coefficient, in this case, of the labelled

molecules of the i-th component:

25

*
D* kT SlnAi

im = f? 31nC¥ (73)
i i

 

Generally, the fact that a molecule is labelled
does not change its physical properties appreciably.3

Therefore, it is reasonable to assume that the physical

1.“ ~
-... ‘fi crux,
. - -' ..‘aa-

prOperties of the labelled and unlabelled molecules of
the same component are identical so that f: = £1. The

activity of the labelled Species is:
*= 'k *
Ai ‘y. Xi (74)

where'y: is the activity coefficient and is constant Since
the chemical concentration is uniform. Therefore, differ-

entiation of equation (74) gives:

BlnA:
SInXE

= l. (75)

The concentration of labelled Species in moles/cc, C3, is
related to its mole fraction, X3, by:

X?
l
* = o

C

: X.V.
l 1

i=1

3This is not a good assumption for molecules of
low molecular weight like H2 or even H20 where the label

can significantly change their molecular weight and size.

26

Because the chemical concentration is uniform, the
denominator of equation (76) is constant and independent

of X3. Hence, equation (76) can be differentiated to

yield:

_ 1;]!

Blnxt
i
alnC

41?.

 

* = 1. (77)
i

Multiplication of equation (75) by equation (77) gives

the result:

 

alnA:
HIKE? = l. (78)

Thus, the intrinsic diffusion coefficient for the labelled
molecules becomes:

D‘!‘ = FIE-T- - (79)
i

im

Another result of the uniform chemical concentra-
tion is that all the reference planes discussed previously
are the same. That is, they all have the same velocity.
This may be shown for the mass, number, component and

volume fixed planes by substituting the relation,

J. = C

1C into equations (26), (32), (38) and (44).

ivoC'
Because Vic must be zero for every component4 when all

the chemical concentration gradients are zero, the re-

sulting equations are readily simplified by using equations

A

4 0
There are no external forces on the system which
may cause convective flow, etc.

27

(18) and (19) and lead to:

It requires a Slightly different argument to show that
the medium average velocity, EEC, is the same as the other

reference planes. Recall that

c
‘ - ‘ = V.
ch Vmc Z Jim 1 ‘54)
i-l
and that the intrinsic flux
BCi
Jim = - Dim _§E . (81)

Consider, for a moment, the components which are not
labelled. Since their concentrations are uniform, it is
apparent from equation (81) that their intrinsic fluxes
are all zero. Therefore, the right side of equation (54)
is the sum over the labelled and unlabelled.i-th compo-
nent. Thus,

— - — = * _. + u V (82)
ch Vmc Jimvl Jim 1

Where superscript, u, designates unlabelled i molecules.
The overall concentration of the i-th component is the

sum:

_ u (83)
C — C3 + Ci

‘M“__' .
‘a-A- - . -.:
‘d ‘ .
..

 

28

and is constant. Hence,

L1
8C; 3Ci
"7572‘=“Tz' (84)

Also, the physical prOperties of the labelled and un—

labelled molecules are the same so that Dim = DIm' Then,

it follows from equation (81), (82) and (84) that:

v = v
mC

“.13- — ‘M-‘_‘ "' , ‘1
‘

VC' (85)

And finally, equations (80) and (85), taken together,

Show that

with reSpect to any one of them. In particular, it can be
defined with respect to the medium fixed reference plane.
Then the tracer diffusion coefficient equals the intrinsic
diffusion coefficient of the labelled molecules and from

equation (79).

D. = (TEE . (87)
1 .

For a binary mixture, the two tracer diffusion coefficients

are:

kT (88)

 

29

and

* = .... .
DB (89)

These last two eXpressionS can be combined with
equation (71) (the Hartley-Crank equation) to obtain a
relationship between the mutual and tracer diffusion co-

efficients in binary mixtures.

 

gold-silver alloys. Since that time, investigators have
attempted to experimentally verify this equation but with—
out great success. Their failures generally resulted
from large eXperimental errors and, as will be pointed
out later, a poor choice of systems.

The three diffusion coefficients in equation (90)
are concentration dependent and are, in general, not

equal. However, the limiting values of the mutual diffu-

limit D = Dg (91)

...-.. - ----:—. 3.

 

30

and

limit D = D* (92)
x y 0 AB A

A

as can readily be seen from equation (90).

Diffusion in Multicomponent Systems

The usual method of expressing diffusive fluxes
in binary systems by means of a single mutual diffusion
coefficient is Simple and effective. In many practical
cases, however, it is necessary to deal with diffusion
processes occurring in multicomponent systems. A general
description of these is, as would be expected, far from
Simple.

Gosting and co-workers suggested (5) and experi-
mentally verified (23) that the fluxes JAV and JBV of

solutes A and B in a solvent C may be described for the

volume fixed plane of reference by the two equations:5

 

 

 

 

 

 

aCA acB
JAv = ' DAM 82 c c DABV 82 c c (93)
B’ c A’ c
and
ac (ac
A B
JBv ’ ‘ DBAv 32 c c ' DBBv azIC c (94)
B’ c A’ c

. 5For simplicity, the ensuing discussion will con-
31der ternary systems. However, it can be extended to a
C component system in a straight forward fashion (5).

31

where DAAV and DBBv are the main difquion coeffiCients

and DABV and DBAV

The important thing to note here is that to describe mutual

are the cross-term diffusion coefficients.

diffusion, more than one mutual diffusion coefficient must
be defined. Thus, the relatively simple analysis which
led to equation (71) for binary mixtures must be modified
for multicomponent mixtures (41). However, mutual diffu—
Sion in multicomponent systems will not be considered in
this work.

Tracer diffusion in multicomponent mixtures will
be important here. The tracer diffusive flux of a compo-
nent in a multicomponent mixture can still be expressed
in terms of one tracer diffusion coefficient. This can
be shown by applying equation (93) to the labelled
molecules of, say, component A in a ternary mixture. Such
a system can be thought of as a quaternary system of Au,
A*, B and C. Recall that in the case of tracer diffusion,
the concentration gradients of all the unlabelled compo-

nents are zero and for the labelled component.

 

 

 

 

 

u
8C3 _ BCA
32 — - _§2 . (95)
Hence, equation (93) reduces to
ac;
- _ ** _ 5‘:
J21. " DAA DAA 32 (96)

 

 

32

where 0AA is a main diffusion coefficient and DAA is a
cross term diffusion coefficient. In mutual diffusion,
the cross term coefficients are an order of magnitude

less than the main coefficients (23). While they have

not been measured for tracer diffusion, one can see that
the difference, DAA - DAA , can be identified as the tracer
diffusion coefficient by comparing equation (96) to equa—
tion (3). Therefore, in a multicomponent system, there

is only one tracer diffusion coefficient for each compo-
nent. This is fortunate for it permits the analysis used

to derive equation (87) to be used for a multicomponent

system.

Thermodynamics of Solutions

The previous discussions have shown that the use
of the chemical potential gradient as the driving force
for diffusion intimately ties thermodynamics and solution
ideality to the diffusion process. Therefore, a brief
review of solution thermodynamics follows.

Ideal Solutions An ideal solution may be defined

 

as a solution in which the various pure components in-
volved do not experience any modification of properties
other than that of dilution. Further, ideal solutions
must obey Raoult's law throughout the range of concentra-
tion. Raoult's law states that the partial vapor pressure
of any component is prOportional to the mole fraction of

that component.

pk = Pka (97)

In terms of the interaction between molecules of
a solution, an ideal solution is one in which the inter—
action between molecules of different substances in the
mixture is the same as between the molecules in the pure
component. The solvent in any binary solution, therefore,
obeys Raoult's law in the limit as the concentration of
the solute goes to zero.

Nonideal Solutions While the concept of the ideal

 

solution is extremely useful, very few systems obey
Raoult's law over the whole concentration range. The
method usually adOpted in dealing with nonideal solutions
is to find a number, Yk’ which, when multiplied by the
mole fraction of the particular Species, makes applicable

a relation having the form of Raoult's law. Thus,
pk = Pkykxk. (98)

The number, Yk’ is the activity coefficient and is a
function of composition. The quantity, Yka, is the
activity of the k-th component.

The total pressure of a system is the sum of the

partial pressures of all components, as shown in equation

(99).

C
7‘ = Z Pkkak (99)
k=l

34

Unlike the ideal solution, the total pressure of a
nonideal solution is not a linear function of mole frac-
tion and negative as well as positive deviations from this
relationship are observed.

In nonideal solutions, the intermolecular attrac-
tive energy between unlike molecules is different from
that between like molecules. If the interaction between,
say, A and B molecules is larger than either A — B or
B - B interactions, the tendency of each of the molecules
to escape from the liquid to the vapor phase is reduced
and there are negative deviations from Raoult's law.
Similarly, if the A - B interaction energy is smaller than
either the A - A or B - B interaction energy, positive
deviations result.

Molecular interactions have been accounted for in
two ways; firstly, due to the effect of molecular sizes
and nonspecific, nonbonding intermolecular forces and
secondly, due to association.

In the first, molecules are thought to interact
through van der Waal or London forces for example.
Hildebrand, SE 31. (34, 35, 37), called attention to a
particular class of these nonideal mixtures which
Hildebrand (34) first called "regular solutions." As con-
ceived by Hildebrand, these solutions differ from other
nonideal solutions in that thermal energy still suffices

to give nearly random molecular mixing. So, such solutions

35

are characterized by the absence of any specific
interactions such as association and by equal sized
molecules. Hildebrand and Scott (36) found, for regular

binary solutions:

I]

BlnA '1

ainx: — 1 RIF XAXB (100) I

and j
AHmiX = BXAXB (101) l

where B is a constant for a system. It is related to the
difference in the energy of attraction of unlike molecules
as compared with the mean of the energies for pairs of
like molecules.‘ Lewis and Randall (45) point out that
equations (100) and (101) are in reasonable agreement with
experimental data even if the component molecules are
somewhat different in size.

The second model of solution nonideality considers
the molecules to interact with relatively strong specific
forces to form weakly bonded clusters of molecules. The
individual molecules are thought to be in equilibrium with
the clusters in the same way that reactants are in equili-
brium with products in a conventional chemical reaction.
Therefore, this approach is frequently called the chemical

model of solution nonideality.

36

Chemical Model of Solution Nonidealityfi Dolezalek
(19) was the first to prOpose the chemical model. He was
able to successfully apply the model to several nonideal
systems. For example, he eXplained the nonideality of

the ether-chloroform system by assuming that one ether

—..—_:-:- 1.9,

~-_ 4‘

molecule associates with one chloroform molecule to form ~(

...-‘13:“ . ’1'.

a bimolecular cluster. Then, assuming ideal solution be-
havior for this mixture of the ether molecules, chloroform
molecules and clusters, he predicted the total pressure

of the system as a function of composition to within the
eXperimental accuracy of the data--using only one arbi—
trary parameter the equilibrium constant, K, for
dimerization.

Dolezalek's success was, in part, fortuitous
because early proponents of the chemical model were
hindered by limited knowledge of molecular interactions
and, in particular, of hydrogen bonding. This ignorance
precluded a priori decisions as to the nature of the
clusters to be expected. Hildebrand and Scott (36) Showed
that application of the chemical model to any general
system can lead to results which are obviously inconsis—
tent with presently accepted theories. For example,
deviations of the argon-nitrogen system were eXplained
by Dolezalek (20) using the chemical model with the un-

likely assumption that the argon molecules associate.

37

Hildebrand and Scott (36) further point out that
in spite of such inconsistencies, the basic premises of
the chemical model are still valid when independent
verification of association can be found. Such informa-
tion can come from other physical properties of solutions
such as the absorption of light at frequencies attribut-
able to the formation of hydrogen bonds, etc.

Dolezalek (19) derived the equations of the
chemical model separately for each type of specific in-
teraction. However, it is possible to arrive at general-
ized equations with the more basic approach suggested by
Nikol'skii (52).

Consider a system in which there are different
chemical species, some of which may be inert and do not
associate at all while the remainder are at reaction
equilibrium with respect to one another. Let P be the
number of phases and r the number of independent reactions

of the type

2 vijjk = 0 k = 1,2,-——-,r (102)

where ij is the stoichiometric coefficient of the j-th
Species, in the k-th reactor. The coefficients ij will
be taken as positive if the j-th species is produced in

the k-th reaction and negative if it is destroyed by this

.1
.
h'u’v‘"

, ‘___._+',.._ :4;

w: u a. fa"? '_

38

reaction. If the j—th Species does not take part in the
k-th reaction, then ij = O. The r reactions will be
independent if the rank of the r x n matrix composed of

the coefficients ij is equal to the number of equations,

r. That is, there are stoichiometric coefficients such

...z. "51.,
.
. §
i—‘ .I‘a'

 

that ‘
11 V13 _____ lr '
I
V12- ~'-1 ----- 1 "
G = i : I ¢ 0. (103)
I I
I ...... .
Vr1 Vr2 vfr

 

 

Since the state of the system is specified by P
values of temperature, P values of pressure and P x (n—l)
mole fractions, P x (n+1) variables must be specified to
describe the system. But, there are (P-l) equalities of
temperature, (P-l) equalities of pressure, n x (P-l)
equalities of chemical potential of the n species between
the P phases and r conditions of chemical reaction equili-

brium shown in equation (104).

T)

v. u. = o k = 1 2,---- r 104
E 3k] I I ( )
i=1

Thus, the total number of equations between the variables
is (n+2) x (P-l) +r. And, in terms of Gibb's phase rule,

the degrees of freedom are n - r + 2 - P.

39

Making the same analysis of a system in which no
reaction occurs, the degrees of freedom are found to be
c + 2 - P where c is the number of components. By compar-
ison of the two expressions, it is seen that c = n - r.
In addition, Denbigh (18) has shown that in a reacting
system, n — r is the minimum number of substances which
must be available in the laboratory to prepare the system.
So, the number of components is:

c E n - r. (105)

For this study, then, the components will refer to the
various substances weighed out in the laboratory to pre-
pare the system and the Species will refer to the entities
present in solution after equilibrium is attained. The
monomer of a component will refer to that species con-
taining one molecule of the component.

Call the r Species which initially are absent the
basic species. They are formed upon mixing components
whose monomers may be the residual n - r Species or only
some of these as is the case when some components are
inert. Designate the number of moles of components ini-

tially present by Ml,---,Mn_r. The moles of speCies

existing at any time will be taken as ml---,mr for the

basic Species and ml,---,mn_r for the remaining speCies.

As a result of the k-th reaction, the masses of

Species are changed according to the equation:

40

£ = $3. = —————— = Amh‘ = (S (106)
Vk1 sz ”kn k
k = l,---,r

where Amkj is the change in numbers of moles of the j-th
species due to the k-th reaction. The total change in

the number of moles of the j-th species is then

r
Amj = 22' ij6k j = 1,2,---,n. (107)
k=l

The number of moles of the j-th basic Species is
r

~ = ' = ———, . 08
n3 2: ij5k 3 1,2, n (1 )
k=l

r
_ _ - = ,2,---,n-r. (109)
M. — mj Z ij 6k 3 1

Solution of equation (108) for 6k yields:

(5 k = 1,2,---,r (110)

k

 

r

X Akjmj
_ 3:1
— G

 

41

where Akj is the algebraic complement of the element vkj
in the G determinant. Substitution of equation (110) into

equation (107) gives:

 

r
Mj = Inj - Z djkmk j = l,2,---,n-r (111)
k=l
where
I: Viink
- i=1 - (112)
O‘jk “ G
For the n-r nonbasic Species, mj = 0 (There was non

originally present.) and
r
— _ ~ ' = 1,2 --— n-r (113)
mj - z ajkmk J I I
k=l

The mole fraction of the j-th component is de-

fined as

M
x. ———l—— j = l,2,-—-,n—r (114)

L.)
:3

-r
M.
3
3:1

and the mole fraction of the k-th species as

42

m .

X.E~L j

J n
5:...
J

i=1

1.2.—--,n. (115)

Equations (111), (114) and (115) can be solved Simultan-

eously to yield:

r
Xi ‘ Z 0‘ijk
k

 

=1 .
Xj = Hit r j = l,2,---,n-r (116)
Z Xj ‘ Z “jkxk
j=l k=l

Again, the ~ indicates a basic Species.
Nikol'skii (52) has shown that the activity of a
component is equal to the activity of its monomer.
Further, if the phase of interest is liquid and if the
vapor above it can be regarded as an ideal gas mixture

with no association occurring in it, then

A? = A9 = ——l. j = l,2,---,n-r (117)

Where the superscripts c and n indicate component and
Species respectively. The quantities pj and pjo are,
reSpectively, the partial pressure of the j-th Species
above the mixture and above the pure component in its

standard state. Since the gas mixture in equilibrium

 

with the liquid is assumed ideal, the activity of the

j-th component relative to the pure component standard

state is:
x.PB
A; = ‘fial j = l,2,---,n-r (118)
j

where P; is the vapor pressure of the pure j-th component
and P? that of the pure j-th monomer. The ratio, Pg/Pg,

can be calculated from the fact that A; = 1 when Xj = 1.

Thus,
P9
—% = gl— (119)
Pj jo

where xjo is the mole fraction of the j-th monomer at the

standard state. The activity now can be written as:

X .
A; = §_l j = l,2,---,n-r (120)
jo

In addition to the above composition relations,

there are the conditions of equilibrium:

kj k = l,2’__—'r (121)

Where Kk is the equilibrium constant of the k-th reaction.
This equilibrium constant varies with temperature accord-

ing to the well-known van't Hoff equation:

 

44

Bank AHrk
g—TET = - _“E_ k = l,2,---,r. (122)
T

The quantity AHrk is the standard enthalpy change due to

the k-th reaction. It will, in general, be a function of

temperature, but usually this dependence is not great.
Another quantity which will prove useful is

defined by the following equation:

C.
x3? = fifi‘1_ j = l,2,---,n (123)
Ck
k=l

where c. is the concentration (moles/liter) of the j-th
Species and Ci that of the i-th component. This expres-

sion can be rewritten in terms of known quantities as:

j (124)

j = 1,2,---,n.

The n-r conditions of equation (116) and the r

conditions of equation (121) permit the mole fractions

of all n Species to be determined as a function of the

equilibrium constants and mole fractions of the components.

45

Then, via equation (120) the activities of the components

can be related to the same quantities. The resulting

expression for A? can be used to calculate vapor pressures
of associated solutions by assigning appropriate values to
Alternately, Kk can be determined by finding the

Kk.
value such that equation (120) best fits the available

activity data.

 

THEORY

Concentration Dependence of Diffusivities-—
Associated Systems

There is evidence which indicates that equation
(90) is valid in the case of metallic solid solutions

(16, 59, 60) where movement of simple atoms is involved.

However, data for associated liquid solutions have con-

flicted with this equation. The works of Babb and co—

Carman and Miller (12) and Hammond and

workers (32),

Stokes (31) all indicate the conflict. These investigators

gave the probable explanation when they pointed out that
associated liquids can form hydrogen—bonded clusters and

diffusion may involve movement of these clusters.

Further, they noted that the thermodynamic factor

in equation (90) always over—corrects when applied to

associated systems. In other words, positive deviations

alnA

from Raoult's law corresponding to ————-
BlnxA

< 1 lead to

calculated values of DAB which are too small and negative

alnA
deviations corresponding to 5137—-> 1 lead to values which
A

46

 

47

are too large. Positive deviations can be correlated

with a tendency to form clusters of one type of molecule

and negative deviations suggest clusters consisting of

different types of molecules. According to the chemical

model, these clusters are in equilibrium and the concen-

tration of each Species can be calculated. Therefore, it

is possible, in principle, to modify the derivation of
equation (90) to take the association into account (1).
In reality, the equations of the chemical model are not

analytically soluble when clusters of five or more

molecules exist in solution. The only readily soluble

cases are for Simple clusters of two molecules.

Three of these relatively Simple types of associa-

tion will be discussed below. The tracer diffusivity

expressions for these three cases have been independently
developed over the last three years by the author.

However, Stokes (63) derived similar equations for type

(a) and more recently, Carman (10, 11) published diffusion
expressions for both types (a) and (b).
Type (a) A binary mixture of A and B where compo-

nent A associates to form a dimer cluster and component

B is inert.
Thermodynamic Relations——For this case, the

following quantities can be specified by inspection:

 

c = 2, (125)
r = 1, (126)
n = 3 (127)
v11 = -2 (128)
Oil = l (129)
and
v12 = O. (130)

Specification of these quantities permits equations (116),

(120), (121) and (124) to be solved for the mole fractions

of the Species and the activities. The following equa—

tions result:

 

x
K = A; (131)
25.1
X . 33:4
1 1+Y’ (132)
2
2XA 4KXA
X231-m-W, (133)
(1+Y)
2
- 4KxA _
xll = —————3, (134)
‘ (1+Y)
ZXA (1+Y)
x0 = , (135)

l (1+Y)2 + 4KX§

 

49

2xA (1+Y)
(1+Y)2 + 4KXA

4KXA
i1? = , (137)
(1+Y)2 + 4KX§

 

Ac = XA (1 + [1+4K]l/2) (138)
A 1+)!
and
2
AS = 1 - ;:% - —:Ef§§ (139)
(1+Y)
where

(140)

K:
II

1/2
I1 ’ 4KXA IXA ZII
It is now possible to differentiate equation (138)

to obtain:

 

(141)
4KX
alnAC l + A
____A = ____1:1_
alnxA Y
and

Differentiation of equation (139) with reSpect to XB

comparison of the result with equation (141) demonstrates

SlnAg BlnAg
that alnX - as required by the Gibbs—Duhem relation.

 

A aian

Diffusion Relations--Recall that the total flux

of a component is calculated by adding the effects of

intrinsic diffusion and bulk motion. For component A,

 

50

the total flux is the sum of the contributions resulting
from diffusion of monomers, dimers and bulk motion. That
is,

JAv = J1m + 2J11m + CA IVVC ' Vme) (142)

where in this case J and Jllm are the intrinsic fluxes

lm

of monomers and dimers, respectively, and

 

ch ‘ va = ‘ (Jlmvl I J2mV2 + J11mV11 ° (143’
As before,
aci
Jim = ‘ Dim W (144)
and
ac
A
= - ——— - 145
JAV DAB 82 ( )

Now, by substituting equation (143), (144) and

(145) into equation (142) and solving for the mutual

diffusion coefficient, one obtains:

' ~ BC
3C1 + 2 D Scll _ C V l
DAB ‘ Dlm TEX" 11m acA A 1 1m acA

 

 

c: acll “ 3C2 °
vllbllm ‘36; + v2D2m To;

 

51

The molar volume of the dimer, V11, is assumed to be:

<22

11 = 27A. (147)

Substitution of equation (147) into equation (146) followed

by rearrangement of terms yields:

 

 

 

8cl ~ 8511
D = _ —
AB Dlm ScA + 2D11m acA l CAVA I
(148)
8c2 _
D2m SCA CAVB'
N ' — _ = =
ow, the two relations, CAVA + CBVB l and dCA/dCB
-Vé/V , are used to reduce equation (148) to:
acl ~ 8511 _
DAB = Dlm Fe; + 2D11m "FEX’ CBVB +
(149)
3c2 _
Dzm ac CAVA
A
Equation (17) is now combined with (149) to give:
D _ kT cl Blnxl + 2cll alnxll X +
AB 7 77 f c alnx ~ BlnX B
1 A A fllCA A
(150)
Blnx2

”I
NH

EInXB xA

 

52

By differentiating equation (131) to get:

81nx alnx
ll _ 2 l
BTXA * mg (151)

and by using equation (120) in conjunction with the Gibbs-

Duhem relation to give

Blnxl BlnA

 

_._.___ = = = ——-— , (152)
BlnXA BInXA 31an 81nXB
equation (150) may be written as:
4~ I 31 AC
D =£I___Cl+a—_Cllx+—lxI-———nA (153)
AB n fch {.1ch B f2 AJ ainxA

For ease of computation, it is useful to define psuedo

mole fractions by equation (123). Thus, the final result
is:

kT
AB I“:

(154)

 

U
II
I
I'hl X
+

as first derived by Anderson (1).

Tracer diffusion is concerned with the flux of
labelled molecules of one particular component. There—
fore, there will be a mixture of dimer clusters containing
one and two labels in addition to the labelled monomers.

The total concentration of labelled A molecules is given by:

53

f- (155)

Since there is no bulk flow in the case of tracer diffu-
sion, an eXpression for the flux of labelled A molecules

is obtained by adding the fluxes of the various labelled

species:
* = * ~ * ~ *
JA Jlm + Jlm + 2J1:m (156)
and as before,
3c* ~ 8611 ~ 851i
= * * ** .
DA Dlm Fe: + Dllm "To; + 2Dllm ‘66; (157)

Assuming that a dimer containing two labelled molecules

has the same friction coefficient as one containing one

 

 

 

labelled molecule, D11; = D11; and equation (157) becomes:
3c* 85 * 85**
1 ~ 11 11
D* = D * + D ** + 2 * - (158)
A 1m Sc; 11m BCA BCA
From equation (155),
” * ~** 8c*
3311.23311=1-3Ci (159)
*
so that:
” sci ” (160)
. it
DA = 91* DlIm it; + Dum

 

 

 

Ill

5 l

(I)

'14.

54

It is now assumed that the labelled A molecules
are distributed such that the ratio of members of labelled
monomers to the number of labelled A molecules will equal
the ratio of the total number of monomers to the total
number of A molecules. Of course, this is not an assump-
tion if the labelled molecules are identical to the un-
labelled molecules. In terms of concentrations, this

equality is:

(161)

1.9.8
I

(3 o

a. Is

Since the concentration is uniform throughout the solution,

it is apparent that the latter ratio is constant. So,

 

8c* c
CA A
Equation (160) can now be written as:
x‘l’ ~
— - ~ * — * 163
DA" 01;: D11m XA+Dllm ( I
When xi is defined by equation (123). Recalling that
~kT (79)
* =
Dim 7E1 ’

the final result is:

 

kT l 1 X? 1
DA _ 7T (f; ‘ §_—) XA + :—- ° (164)
11 f11

The inert component, B, diffuses only as monomers

so one can write simply:

* = *
JB J2m (165)
from which
kT 1
* = __ __ .
DB n f2 (166)

A relatively simple relation between the tracer

and mutual diffusion coefficients can be obtained by

combining equations (154), (164), and (166). This rela-
tion is:
kT X11 XB alnAA
._ 'k . _ — _ 0
DAB - XADE 'I' XBDA 'I' 2 n ~ X W (167)
fll A A

Stokes (63) and later Carman (10, 11) derived equations

very similar to this equation.
Type (b) A binary mixture of A and B where A
associates with B to form bimolecular clusters.

Thermodynamic Relations--The principles involved

in TYPE (b) are the same as in Type (a). To avoid DGGd‘

less repetition, many equations will be presented without

 

56

detailed eXplanation. The following quantities are the

same as Type (a):

c = 2, (168)

r = l, (169)
and

n = 3. (170)

But, the stoichiometric coefficients for Type (b) are:

v11 = —1, (171)

V11 = 1, (172)
and

012 = -l. (173)

The equations of the chemical model can now be

solved to get:

 

 

 

 

K = 12 , (174)
X1X2
x - x + z
X = A B , (175)
l 1 + z
_ + z
X _ XB XA , (176)
2 _ l + Z
s 1 — z (177)
X "' I

12 ‘ 1 z

 

0;

firm:

57

 

 

 

 

 

X0 = X _ 1 - z
1 A 2 ’ (178)
o 1 - z
X = _
2 XB 2 ' (179)
X o = l - z
12 2 ’ (180)
AC = xA - xB + z
A l + z (181)
and
AC = xB - xA + z
B l + z (182)
where
K + l (183)

Also, the following equation can be derived from equation

(134):

C
alnAA _ l
SInXA Z '

(184)

Equations (181) and (182) can be shown, by direct differ-

entiation, to satisfy the Gibbs-Duhem relation.
Diffusion Relations—-The total flux of component
A is given by:

vVC - vmC (185)

 

 

JAv = J1m I J12m + CA

58

where Jlm and J12m are the intrinsic fluxes of the monomer

and cluster, respectively, and

ch ' Vmc = VlJ

V2 J12m + V2J2m. (186)

The partial molar volume of the cluster has been approxi-

mated by:

(187)

Proceeding by the same steps presented in Type (2), the

(188)

 

Again, Anderson (1) was the first to derive this equation.

Tracer diffusion in this case is less complicated
than in Type (a) because the dimer can carry only one
label. Thus the total concentration of labelled A

molecules is:
~ * (189)
+ €12

 

59

and the flux of these molecules is simply:

J: = Jlfi + J13. (190)

Again, the final result follows immediately by

the process discussed for the Type (a) associated system:

0
kT X .
D*=__l_:l_._l.+.:l_ - (191)
A 7‘ f1 f XA f
12 12

The same process applied to component B yields:

P

I X
D§=§1[?1_~AX_2.~1. (192,
2 f12 B f12

Combination of equations (188), (191) and (192)

gives:
C
x O BlnA
kT 12 A
DAB XADB + XBDA 2 r) E BlnXA (193)

as also derived by Carman (10, 11).

Type (c) A ternary mixture of A, B and C where
A associates with B and C is inert.

Thermodynamic Relations--Here there are three

components, four Species and one reaction. So,

C = 3, (194)
n = 4’ (195)
(196)

 

60

V11 = '1' (197)
611 = l (198)
v12 = -l (199)
and 013 = 0. (200)

These conditions combined with equations (116),

(120), (121) and (124) of the chemical model yield:

K = I (201)

 

X = , (202)

 

x - x - -———I + U
X = B A K + , (203)

- - U
l 2xc I K + I (204)
12 KX

c (205)

 

 

 

 

 

 

 

 

 

 

 

1 _ KXC _ U
o _ K + 1
X1 ‘ XA ‘ 2 . (206)
1 _ KxC _ U
X0 = x _ K + 1
2 B 2 ' (207)
1 KxC _ U
3" = K+1 '
12 2 (208)
C _
x3 _ xC (209)
x
x - x - C + U
AC _ A B K + 1
A — KxC ' (210)
l I K + 1 I U
Xc
c _ xB ' xA ' R‘I‘I + U
AB ‘ Kx (211)
c
1 + K + 1 + U
a
nd C 2XC
AC = KxC (212)
1 + K + 1 + U
where
2KxC szg 4KXAXB 1/2
U = l ‘ K‘I‘I + IE-I_IF§ ‘ 'R‘T‘I (213)

Diffusion Relations--Recall that Simple flow

equations, such as equation (2), are not applicable to

mutual diffusion in ternary systems. But, since there

 

62

are no concentration gradients of components in tracer
diffusion, a single tracer diffusion coefficient still
suffices to describe the process. Then, the intrinsic
diffusion model can be applied to describe tracer diffu-
sion as before.

In fact, the addition of the third inert compo-
nent to a Type (b) associating system does not change the
diffusion equations from the form of those in the previous

case (Type [b]). The flux of labelled A molecules is

still:
.. ~ *
J3 - J1; + J12m (214)
and the flux of labelled B and C molecules is:
- ~ *
Jg — J2; + J12m (215)
and
, = * 216
J5 J3m ( )

The counterpart of equation (76) for this ternary

system is:

(217)

 

Ci":
1 -—
XAVA+XBB cc

. . . *
where the denominator 18 again independent of X1' Hence,

BlnA?
i

-———— - ' 79 is valid as before.
BlnCI l and equation ( )

63

Equations (2l4), (215) and (216) can now be

simplified to:

1" X0
._ M 1 1 1 1
DIE—7 f—--~—-—)-,-+:-—- (218)
1 £12 A £12
.0 1
kT l l 2 l 1
05:7]— f—-~———-3g—+-J (219)
~ 2 £12 B fl2
and
kT
0* = --— ° (220)
C nf3

Although the form of these equations is the same as their
counterparts in Type (b) [equations (191) and (192)], the
composition variables are different because of the pres—

ence of the third component.

Selection of Systems

Usually, there are several systems which one
might gues to associate in a given way, but few whose
activity and spectroscopic data have been reported in the
literature. Since these data are required to make firm
decisions as to the exact nature of the association, the
number of systems suitable for study is greatly reduced.
Another factor to be considered is the availability of

labelled material. Suprisingly, many common chemicals

 

64

are not available from commercial tracer suppliers. Thus,
it is frequently a problem to find a system which can be
studied. In this section the six systems that were in-
vestigated are presented along with evidence of their
association behavior.

Chloroform (A) - Carbon Tetrachloride (B) The
isothermal vapor-liquid equilibrium data by McGlashan,
Prue and Sainsbury (49) coupled with the absence of a
temperature effect on the heat of mixing of the system
(13) indicates that chloroform — carbon tetrachloride
mixtures are nonideal and nonassociative. Krishnaiah,
g3 31. (44) concluded further that the system is regular.
By fitting heat of mixing data (13) to equation (101) a
value of B was determined with which the thermodynamic

factor can be calculated from equation (100) as:

alnAg (221)
————= - . x. .
alnX 1. O 74XA B
A
Benzene (A) - Carbon Tetrachloride (B) Staveley

 

and co-workers (61) report that this system behaves as a
regular solution and as such is nonassociative. Lewis
and Randall (45) fit the thermodynamic data by Staveley

(61) to obtain the B factor in equation (101). Using

their value:

65

BlnA

glfii— = 1. ~ 0.23XAX

3’0

B (222)

3’

Methyl Ethyl Ketone (A) - Carbon Tetrachloride (B)

Affsprung and co-workers (46) investigated self-association

3":
A.
of ketones in carbon tetrachloride. Their results indicate ‘I
that methyl ethyl ketone dimerizes in carbon tetrachloride. 1
I
They proposed that the dimers are formed by dipole—dipole L
interactions of the type shown below. E
x? = (.3
R': :/R
O = C
\R.

Such antiparalled arrangements are feasible as long as
the groups R and R' are small enough to permit the close
approach of the two carbonyl groups. Such is the case
for methyl ethyl ketone. This system is an example of a
Type (a) associated system.

Equations (138) and (139) were fitted to the
experimental activity data of Fowler and Norris (29) by
finding the value of K which gave the best fit. A non-
linear least square computer program was used to accom—
plish this task. The best value of K was found to be
5.62 which agrees closely with the value of 5.60 found by
Anderson (1). Figure l is a plot which compares the
results of equations (138) and (139) to the data. It

can be seen that the equations predict the data very well.

66

 

L0

0.8"

0.6 r-

0.4 (-

ACTIVITY

 

00.0

Figure l.

 

l 1

‘1 l
0.2 0.4 _ 0.6 0.8 1.0
MOLE FRACTION KETONE

Activity as a Function of Mole Fraction for the
Methyl Ethyl Ketone - Carbon Tetrachloride Sys-
tem. 0 Data, Reference (29); Calculated
from Equations (138) and (139).

 

67

Acetic Acid (A) - Carbon Tetrachloride (B) There
is a great deal of evidence which indicates that acetic
acid strongly associates in solution. In fact, the asso—
ciation is so pronounced that in solutions concentrated

in acetic acid, it “polymerizes” to very large clusters

, .

(53). Spectrosc0pic data of Barrow and Yerger (6) suggest

__-__.—-—=— .31.,

WW... ‘ . I

that acetic acid associates to form only dimers when the

 

concentration is below 0.01 mole fraction acetic acid.
For these low concentrations, this binary mixture is an
example of a Type (a) associated system. A schematic of
an acid dimer is shown below.

¢O---HO\

CH3C CCH

\OH-—-oo 3

Barrow (6), Davies (17) and Wenograd (67) deter-
mined the dimerization equilibrium constant for acetic
acid in dilute solutions. Their values varied from 930
(mole fraction)-1 by Barrow to 55,000 (mole fraction)”1
by Davies. Hence, it is not possible to arrive at a
meaningful value for K from the literature. Therefore,
the equilibrium constant will be determined later from
consideration of the diffusion data.

Ether (A) - Chloroform (B) Lord and co-workers
(47) and Gerbier and Gerbier (30) studied this system

using I. R. and Raman spectrosc0pic techniques,

68

respectively. Both investigations showed that the two

components associate to form a bimolecular cluster as

shown below.

C H
/ 2 5

C2H5

Cl3

This system is an example of a Type (b) associated system.
Given this specific association, other physical

phenomena can be predicted. Since only total pressure

versus composition data are available for this system,

it is not possible to determine the equilibrium constant,

K, from the activities of each component. But, it is

possible to calculate the total pressure of the system

by combining equations (99), (181) and (182) to obtain

an eXpression for the total pressure in terms of mole

fractions of the components and the equilibrium constant.

= C C C C
W PAAA + PBAA (223)

The best value of K was determined by fitting this equa-
tion to the total pressure data with the computor program
shown in Appendix I. It can be seen from Figure 2 that

equation (223) predicts the total pressure very well.

 

III‘EI Icclr‘c E<E“‘ .(‘I'C'Il

69

700

600

500

400

300

 

200-

TOTAL VAPOR PRESSURE, 1r (mm Hg)

I00“

 

 

 

 

 

 

() I 1 J, l
0.0 0.2 0.4 0.6 0.8 )0

MOLE FRACTION ETHER

 

Figure 2. Total Pressure as a Function of Mole Fraction
for the Ether - Chloroform System. 0 Data,
Reference (22); Calculated from Equation
(223).

 

70

that
aan _ AHR (122)
3(l/T) R ’

straight line with a lepe of - -§B . Figure 3 is such
a plot. From a least squares fit to the data, it is found
that AHR = 2.72 (k cal/mole); a very reasonable hydrogen
bond energy. Also from this fit, K = 2.86 (mole fraction)-1
at 25°C. This is in fair agreement with the value of 2.73
(mole fraction)-1 suggested by Anderson (1).

Anderson (1) showed that if the three species,
ether monomer, chloroform monomer and cluster, mix ideally,

the heat of mixing for the mixture is given by:

= “’ 0 224

AHmix AHRX12. ( )

Equation (180) and the value of AHR found above were used
With equation (224) to calculate AHmix as a function of
mole fraction. The results are compared to the data of

Hirobe (38) on Figure 4. The agreement between the two

is very good.

h .IZO— Iroquu—Hu wldozv V.- oLIZ/VLIWZOMV ED—IE—I:DOMU

 

EQUILIBRIUM CONSTANT, K (MOLE FRACTION")

9" 95399509
0 00009

.4>
9

SN
0

I“
o

 

“02.6

Figure 3.

71

    
 
  

I L

2.8 3.0 3.2 3.4 3.6 3.8

RECIPROCAL OF ABSOLUTE TEMPERATURE,
I / T. CK")

Logarithm K as a Function of l/T for the Ether -
Chloroform System. 0 Calculated from Data,
References (21), (22), (43), (50), (58);

Least Square fit.

T

700

r

600

500)

400 t

T

300

cc) / mole

- 200

AH“,

I

I00

 

Figure 4.

72

 

 

‘fio‘. .31 ‘

l

J _l g
0.2 0.4 0.6 0.8 )0
MOLE FRACTION ETHER

Heat of Mixing as a Function of Mole Fraction
for the Ether - Chloroform System. 0 Data,
Reference (38): Calculated from Equation
(224).

73

Ether (A) — Chloroform (B) - Carbon Tetrachloride
(E) To the author's knowledge, none of the physico-
chemical prOperties of this system have been studied.
However, one can Speculate as to the association that
should be observed. Notice that this ternary system can
be formed by adding carbon tetrachloride to the binary
ether—chloroform system. Assuming that the carbon tetra-
chloride is completely inert, its addition to a binary
system should have no effect other than dilution. Then
the nature of the ether-chloroform association should be
the same as a Type (b) associating system, and the value
of the equilibrium constant should not change from the
value observed for the ether—chloroform mixtures, 2.86

(mole fraction)_l.

rm

 

2:: f: ‘-

EXPERIMENTAL METHOD

The Open-end capillary method for measurement of
tracer diffusion coefficients is one of the earliest and
most widely used methOds (4, 56, 66). The principle of
this method is as follows. A capillary of known length,
L, closed at one end, is filled with a solution of known
concentration. Some of the molecules of one component
are labelled with an isotopic form of one of the constitu-
ent atoms. The capillary is immersed in a solution of the
same chemical concentration but which contains no labelled
molecules. The volume of the unlabelled bulk solution is
much larger than the labelled capillary solution. There-
fore, the concentration of labelled molecules remains
essentially zero in the bulk solution as they diffuse from
the capillary. Natural convection, caused by slight
temperature variations or mechanical Vibrations, sweeps
the labelled molecules from the neighborhood of the
capillary mouth and maintains zero concentration at the
mouth.

This process is described mathematically by the

following differential equation and boundary conditions:

74

'. 8‘10:
..l

,‘myw’p

j,

 

75

 

 

ac; 820;
...—...: 'k
()0 Di a2 (a)
2
ac:
= O at z = O, t>0 (b)
02 Tn
(225) 2.
C; = at z = L, t>0 (c) E
I
C? = C? at 0 = O, 0<z<L (d) L

If measurements are made by the Open—end method,
there are two sources of error which may affect the diffu-

sion coefficients (56):

(l) The immersion effect--When a cell is immersed
in the unlabelled bulk solution, there occurs
a mixing of the capillary solution and the
bulk solution, and initial condition (225)
(d) is not fulfilled.

(2) The A2 effect--If the velocity of the convec-
tion currents across the capillary mouth is
too large, there will be a length, A2, at
the open end of the capillary in which the
concentration of labelled molecules is zero.
Then boundary condition (225) (c) is not
fulfilled.

Cursory experiments with open—end capillaries indicated

that the second source of error was very large.

Modified Capillary Cell

To minimize the immersion and 02 effects and to
facilitate the radioactive analysis of the cell contents

after a diffusion run, a modified capillary cell, shown

76

in Figure 5, was devised. The cell consisted of a
capillary tube, covered at one end with a thin porous
glass frit. Diffusion occured through the pores of the
frit but convective tranSport, and hence, the immersion
and A2 effects, were eliminated by the small size of the

pores. The other end of the capillary was sealed with a

777:3

metal foil disk, held in place with a funnel-shaped screw I

cap. After a diffusion run, the cell contents were

In... 3" ‘_-I\

analyzed with a standard liquid scintillation technique L;
(55). This method requires the cell contents to be
dissolved in a liquid scintillation medium. Therefore,
the metal foil seal was punctured with a medical syringe
needle and the cell contents flushed, with scintillation
liquid, into a counting vial. The funnel—shaped design
of the screw cap permitted a liquid seal to be maintained
around the syringe needle during the flushing process,
thereby preventing loss of capillary contents.

The screw caps were made of Type 304 Stainless
Steel, chrome plated for corrosion resistance. A teflon
washer was inserted into each cap to insure complete
closure of the capillary end.

Disks of 0.001 inch thick metal foil were used to
form the seal at the closed end of the capillary. The
disks were thin enough to be easily punctured with a

medical syringe needle.

 

 

 

Figure 5. Schematic Diagram of the Modified Capillary

Cell.

 

  

 

77

um,- FRIT HOLDER

gm GLASS F RIT

M— CAPILLARY

 

FOIL DISC

SCREW CAP

78

The capillary tubes were constructed of teflon.
The six tubes were machined to 6.78 cm. in length, each
having the same length to within 0.1 mm. The bore of
each capillary was 0.2 cm. Threads were machined on the
capillaries so that they could be screwed into the caps.
Each tube was individually numbered.

The glass frits were manufactured by the Kontes
Glass Company and were of their "M" porosity. The frits
were hand-lapped to a thickness of 0.047 t 0.004 cm. and
numbered to correspond to a capillary tube. In all ex—
periments, cells were assembled using frits and capillaries
having corresponding numbers.

The frit holders were made of teflon; machined to
fit tightly when pushed over the capillary end. The
tapered lip held the frit in place while obstructing the
capillary end as little as possible. Six cells of this

design were used in this project.

Procedure

Mixtures were made up gravimetrically from the
pure components. Bulk solutions were prepared by weigh—
ing on a large analytical balance to the nearest 50
milligrams. Approximately 800 cc. of mixture were re—
quired to fill the bulk solution containers. Labelled
capillary solutions were prepared in 20 cc. screw cap

glass vials with metal foil lined lids. The amount of

79

each component was determined to the nearest tenth of a
milligram. An additional foil cap was added to the vials
to prevent evaporation during weighing. Solutions were
transferred with glass medical syringes by inserting the

syringe needle through a pin hole in the foil cap.

Three capillary cells were run simultaneously in
the same bulk solution. The tracer diffusion coefficients
determined from each cell were averaged and one number
reported for the run. A detailed procedure for a run is

presented in Appendix II.

The mutual diffusion coefficients were determined
with a Mach-Zehnder diffusiometer. The diffusiometer
measures the refractive index of the solution in a glass-
windowed diffusion cell with an Optical interferometer.
The instrument was patterned after a similar diffusiometer
described by Caldwell, Hall and Babb (8, 9) and was con-
structed by Bidlack (7). Bidlack presents very complete
Operation instructions and data analysis procedures, so
they will not be discussed here.

A Cannon-Fenske capillary tube viscosimeter was
used to determine the viscosity of solutions. It was
calibrated with Spectre-quality carbon tetrachloride and
benzene, following the procedure outlined by Daniels and

Alberty (15).

-..-cal” war- 1.-..

.. '4.

80

Calculation of Diffusion Coefficients

The large AZ effect noted for the open—end
capillaries showed that there were relatively strong con—
vection currents in the bulk solution. Therefore, it is
assumed that these currents maintain zero concentration
of labelled molecules at the exterior surface of the frit.
If the frit is very thin, a quasi steady state concentra-
tion distribution is rapidly attained in the frit after
diffusion starts. Then the frit can be considered a
resistance to diffusion and the following equation de-

scribes the process.

 

ac; 8203
‘50 = Di —‘2 (a)
02
ac;
TZ=Oatz=O, 8>0 (b)
(226)
‘k
-D* 8C1 l C* at z = L 0>0 (c)
i 02 R i '

where R is the resistance of the frit. The length variable,

2, is defined in Figure 6.

81

 

 

 

 

 

Figure 6. Diffusion Cell Coordinates

The solution of equation (226) follows by separa—
tion of variables as shown in Appendix III. The result
is:

00

2
C? SIN 8 D?
lavg _ 2 n ex _ 2 10
- . p B (a)
Cio n=1 8n (8n + SINBnCOTBn) n L2

 

 

where 8n is defined by: (227)
RD:
COTBn = 8n L . (b)
The term Ciavg is the average concentration of labelled

molecules in the capillary after a time 0.

 

82

At first inspection, it seems very difficult to
obtain a value of D: from experimentally determined 0, L
and Ciavg/Cio' However, the problem can be greatly
simplified by examination of diffusion in the frit. Trans-
port of material across the frit occurs by ordinary
molecular diffusion through the liquid-filled pores.
Hence, the resistance of the frit, R, is inversely prOpor-
tional to the diffusivity, DE. The prOportionality con-
stant will be a function of the frit porosity and thickness.
As a result, the group RDz/L is independent of D: and is a
function of cell geometry only. Now, only the exponential
portion of equation (227) (a) contains DE.

A computer program, shown in Appendix IV, was
written to calculate C? /Cio for various values of

iavg

2 * * * 1 tted versus
Die/L and RDi/L. Then, Ciavg/Cio was p o

DEB/L2 for particular RDg/L as shown in Figure 7. Using
a value of RDg/L, determined by calibration of the cells,

and eXperimentally determined values of Ciavg/Cio' 0 and

L, D: was obtained from this plot.

Calibration of the Cells

Rathbun and Babb (57) analyzed the results of

several investigations and concluded that the tracer

diffusion coefficient of pure carbon tetrachloride was

-5 The six cells employed in this

1.32 x 10 (cm.2/sec.).

study were calibrated using this as a standard.

CTAve /C-

0.8)

0.7-

 

0.6

0.5

0.3

0.!-

 

 

 

 

 

0.0O

' * *
Figure 7. Ciavg/Cio a

83

R0?/ L =O.I

I
805; L=0.0I2
RDi / L-0.o

I I I I I

0.2 0.4 0.6 0.8 )0

0:" G/L‘

s a Function of D*i'/L2 and RDE/L.

84

The cells were calibrated as follows. Values of
2 .
1k t *
Ciavg/Cio and Die/L were calculated for three experimental
observations of each cell. Then three values of the cell
constant, RDi/L were determined from Figure 7 and

averaged for each cell. Table I is a listing of these

average cell constants for the six cells.

Table I. Capillary Cell Calibration Results

(a)

 

Cell RDE/L
1 0.012
2 0.014
3 0.011
4 0.014
5 0.013
6 0.010

0.012 average

(a) average of three determinations

. ' i * ’

It is apparent from Figure 7 that Ciavg/Cio is
nearly independent of RDg/L when RDE/L is as small as
0.012. In fact, the t 9% deviation of the cell constants
listed in Table I corresponds to only a i 0.7% deviation
in D? when C? /C# = 0.5. Since this deviation in the

i iavg io
diffusion coefficient is not significant, the cell con-

stants of the six cells were averaged and one value, 0.012,

85

used for all cells. Later, to determine diffusion
coefficients, three cells were run simultaneously and the
three resulting values of D: were averaged. This process
greatly reduces any error caused by using one call con-

stant for all cells.

Analysis of Radioactive Samples

At the end of a diffusion run, the contents of
each capillary were analyzed by a Packard Tri-Carb liquid
scintillation spectro—photometer. The scintillation
solution consisted of 4 grams of PPO (disphenyl oxazole)
and 50 milligrams of POPOP (1, 4 of di 2, 5 phenyl-oxasole
benzene) dissolved in 1 liter of toluene as suggested by
Price (55).

It is generally assumed that the count rate
measured by a counter is prOportional to the amount of
labelled material present in the sample. However, this
assumption is not always a good one. Changes in the
chemical composition of the sample can cause varied de-
grees of absorption (quenching) of the light pulses
emitted by the scintillation liquid. Also, some of the
older counters give results which are dependent upon the
count rate.

To study the effects of quenching, experiments
were made on mixtures containing various mole fractions.

The count rate from a given quantity of solution (0.15 cc.)

86

was determined in each case. The results are shown in
Figures 8 and 9 for two systems: methyl ethyl ketone -
carbon tetrachloride and chloroform - carbon tetrachloride.
Carbon tetrachloride was the labelled component in each
case. The linear graphs indicate that there is no change
in quenching with composition of the capillary solution.
The eXperiments were then repeated for solutions
with 1/2 the activity in the first experiments. At each
mole fraction, the ratio of counts obtained from samples
with two different activities remained constant. This
demonstrated that the counts from mixtures having differ—
ent amounts of radioactivity but the same composition
were quenched in the same proportion. Hence, the observed
count rates were proportional to the concentration of
labelled molecules and could be substituted directly into
equation (227) to obtain the diffusing coefficients.
Based on the results for the above two systems, this con-
clusion is assumed valid for all the systems of interest.
Except for samples very dilute in tracer, a total
of one million counts was recorded for each sample. This
reduced the statistical error involved in the count rates

to a negligible amount,

Materials
Spectroquality chemicals were used when calibrat—
ing the diffusion cells and the viscometer. Analytical

reagent grade chemicals were used elsewhere. The

87

 

8.0

 

7.0

S”
o

9‘
O

:b
9

3.0 t

2.0

COUNTS PER MINUTE x I0“;

)0

 

 

 

I I I I

0.2 0.4 0.6 0.8 )0
MOLE FRACTION KETONE

 

Figure 8. Count Rate as a Function of Mole Fraction for

the Methyl Ethyl Ketone - Carbon Tetrachloride
Svstem.

88

 

 

COUNTS PER MINUTE x I0"

 

 

 

 

 

0.00.0 (fa 0.4 0L6 0:8 10

MOLE FRACTION CHLOROFORM

Figure 9. Count Rate as a Function of Mole Fraction for
the Chloroform - Carbon Tetrachloride System.

89

refractive indexes of the materials were measured and
compared to literature values (65) as a purity check.
Carbon - l4 labelled materials were obtained from
the Nuclear Equipment Chemical Corporation and the Nuclear-
Chicago Corporation. The manufacturers' statement of

purity was accepted with no further purity check.

Discussion of Error

The accuracy of a given run of three cells is
governed by the calibration of the cells and the experi-
mental precision of the method. The cell calibration is
based on the tracer diffusivity of pure carbon tetrachlo-
ride suggested by Rathbun and Babb (57). Since they
obtained this value from consideration of several investi-
gations, it is probably accurate and the resulting cell
calibration is, therefore, reliable. However, further
evidence of the accuracy of the cell calibration can be
obtained from the tracer diffusion data collected in this
work. In dilute binary solutions, theory predicts that
tracer and mutual diffusivities are equal. For five
binary mixtures studied here, intercepts of the mutual
and tracer diffusivity curves were determined by graph-

ically extrapolating to zero concentration. The inter-

cepts agree to within 0.5% for the five systems. Based

on these results, errors in the tracer diffusion coeffi—

cients resulting from uncertainties in calibration are

estimated to be 0.5%.

90

The accuracy of one given run of three cells is,
therefore, affected more by the precision of the method.
A measure of the precision was obtained by making several
duplicate runs for six different solutions. The repro-
ducibility of these runs ranged from i 0.7% for pure
carbon tetrachloride to t 2% for pure benzene with an
average precision of t 1.25%. Therefore, the accuracy

of the method was assumed to be within i 2.0%.

RESULTS AND DISCUSSION

Theoretical

From the application of the intrinsic diffusion
concept, the medium fixed reference plane and the chemical
model of solution nonideality, equations were derived which
express the tracer and mutual diffusion coefficients in
terms of concentration dependent quantities and friction
coefficients-—one for each species in solution. These
equations (see Table II) can be empirically fit to
experimental diffusion data by adjusting the values of
the friction coefficients to give the best prediction.
However, one can justly argue that, given enough arbi-
trary constants, any curve can be closely approximated.
Therefore, the fact that these equations can be made to
fit the data does not necessarily constitute support of
the concepts from which they were derived. If, on the
other hand, the friction coefficients were determined
from independent measurements and the equations still

predicted the diffusion coefficients accurately, the bases

of the equations are probably valid. It is shown in

Appendix V that, for the systems studied in this work,

the frictional coefficients can be determined strictly

91

hue ®m>rb
WCOWUUSHUHQ CCHmwnmemwflA~ M0 Mlhqw:::~am.u IHIH. OHQH...~.

 

92

 

 

 

 

 

 

 

 

 

 

 

mcoflumsvm COHUSMMHQ mo mumfifism

. HH mHQMB

mam mx mam me if m
aNmHV III? I... ..III '2 I l. l U *n—
H mx H H 1 ex
0
NHM fin A NHW HM I C AN
35.: inl+fl IMIIIH' H'MH«Q m...¢ mvcm¢
o L
HH
< fl m
C C
Ahmav «Mle mm MMh_Mm m + mamx + madx u mdo
m
3.8: H“): n mo
0
Ham <X Ham MN m F» d
Avmav IMI + Ml IMI I MI ; IN M «Q BHUCH m .d...< m cam d
x
fixCHm ¢ m m fl md
Acme m ..o x + «o x_ u o
083
m
3.8 aiwlc. n mm
«a. a
Ammv HMI M «Q CORROHUOMM< 02 m was <_
msowuwsqm sown:MMHQ soHumHoommfi musmsomfiou
mo mmha

 

 

 

 

 

 

 

 

 

 

uv..J~J-u IV~.LPU..V \ I ~ u. ...~.,-..~I

93

7 . I‘NEINN‘
H. - .

w I ..l
./

. .I;
Ii’ii4 "lava: ll...“ 1.11.. 1.1! v.)- ICD “nap

 

 

 

 

 

 

 

IIIIIIIIIIIIJ
m
o Imkuu
A NNV Bx «Q
a... LWRLWUL ...-..
H Nx H H ex .
o
NH NH
m m w H L
a x z m l» 4 . U can m m
III II III I II II n bumcfi U m...¢
AmHNv fl H + AX A H M “Q Bx «D
o
a: mHm c «a ma
AmmHv ex Hm NH) mm m I momx + mo x_ n o
msofiumsqm soflwsmwflo GOHUOH00mm¢ musmsomfioo
mo meme

OmncHUcoo ..HH meme

 

94

from tracer diffusion intercepts alone; a source
independent of mutual diffusion measurements. Thus, it
follows that the relations which are most significant are
those between mutual and tracer diffusivities, equations
(90), (167), and (193). The tracer diffusion relations
[equations (87), (164), (166), (191), (192), (218), (219)
and (220)] are significant in that the degree to which
they predict the concentration dependence of the tracer
diffusivities across the concentration range reflects on
the concentration dependence of the friction coefficients
and the accuracy of the chemical model in predicting the
association of the systems.

It was pointed out earlier that the activity term
in equation (90) over corrects when it is applied to an
associated system. Similar equations derived here for

associated systems predict this over correction.

 

ainAg
= * *
DAB [XADB + xEDA} 8171’)? (90)
7 0 X BlnAC
kT 1 B A
= * * —— ——— ——
f A A
11
i 0 BlnAC
kT 12 A
_ * ‘k _ _
, DAB XADB + xBDA 2 T1 E ainx (193)

95

Equation (167) describes diffusion in a binary
mixture where component A associates with itself to form
a dimer cluster. In such solutions, the thermodynamic
factor is less than unity and equation (90) would predict

a diffusion coefficient which is too small. Equation (167),

X2
I...
pfihg<

on the other hand, has the additional term, 2 g; l
11

H1?

which increases the value predicted by equation (90) and
the over correction is reduced.

A similar circumstance arises when equation (193)
is applied to binary systems where the components asso-
ciate with each other. The activity term is greater than
unity for such mixtures and equation (90) would predict a

mutual diffusivity which would be too large. The term,

xe
NO

-2fl1
n

 

, lowers this value in equation (193) and better

H11
N

l
agreement with experiment is eXpected.

Both equation (167) and (193) reduce to the form
of equation (90) when there is no association, since the
psuedo mole fraction of the dimer cluster is zero in each
case. Also, both of these equations have the correct

limiting behavior as either XA or XB approaches zero.

equals D* as X or X

That is, the limit of D A B A'

AB

respectively, approaches zero.

96

Experimental

Density and viscosity data for the benzene-carbon
tetrachloride and chloroform — carbon tetrachloride sys-
tems are shown on Figures 10 and 11. Diffusion coefficient
data for these systems are shown on Figures 12 and 13.
Friction coefficients were determined from the tracer
diffusivities and viscosities of the pure component as
described in Appendix V. The tracer diffusion coefficients
of each component were then calculated from equations (88)
and (89) assuming the friction coefficients to be indepen-

dent of concentration and plotted on Figures 12 and 13.

kT
D* =-——— (88)
A an
kT
* = ___
DB nfB (89)

It can be seen that the agreement between theoretical and
eXperimental values is very good across the entire concen-
tration range for both systems. Tracer diffusivities of
benzene in carbon tetrachloride mixtures by Johnson and
Babb (40) are also plotted on Figure 12. While less pre-
cise than the data collected in this study, they are in
general agreement with the curve calculated from equation

(88). From these results, one can conclude that equations

 

 

 

 

97

 

 

 

 

0.9
C
2.0.
C
- ~08
l.5*
" C
'8
\ (07
CF
.
).
1::
3 I0)
DJ
CD
~06
0.500 0.2 0.4 0.6 0.8 100'5

MOLE FRACTION BENZENE

Figure 10. Density and Viscosity of the Benzene - Carbon~

Tetrachloride System at 25°C.

n Vignngitv Data-

0 Density Data:

(centipoise)

VISCOSITY

 

 

 

 

 

“*I‘I‘l

98

 

 

 

 

 

0.9
I)
|.60B
C
“0.8
0
I55-
3‘ 7.:
a, 07 g
V .8
O
)-
':. >-
m I—
L211 I.50L g
C)
(0.6 8
S
O
>
L450.0 0.2 0.4 0:6 087 100'5

MOLE FRACTION CHLOROFORM

Figure 11. Density and Viscosity of the Chloroform -
Carbon Tetrachloride System at 25°C. 0 Density
Data: 0 Viscosity nzu-a-

99

 

2.5

x (0° (cmzl sec)

DIFFUSION COEFFICIENT

 

 

 

 

 

I I

 

 

Figure-12.

I l
0.2 0.4 0.6 0.8 )0
MOLE FRACTION BENZENE

Tracer and Mutual Diffusion Coefficients for-
the Benzene - Carbon Tetrachloride System at
25°C. 0 Tracer Diffusion Data for Benzene;

O Tracer Diffusion Data for Benzene, Reference
(40); =1Tracer Diffusion Data for Carbon Tetra-
chloride; A Mutual Diffusion Data; Equa-
tion (88); _ ___ Equation (89); -----

Rnnafinn (90).

 

100

 

x 10" (cm’lsec)

DIFFUSION COEFFICIENT

 

I.0

 

 

 

 

0.0

Figure 13.

1 _l I I
0.2 0.4 0.6 0.8 )0
MOLE FRACTION CHLOROFORM

Tracer and Mutual Diffusidn Coefficients for

the Chloroform - Carbon Tetrachloride System

at 25°C. 0 Tracer Diffusion Data for Chloro-
form; 0 Tracer Diffusion Data for Carbon
Tetrachloride; A Mutual Diffusion Data;

Equation (88); ----- Equation (89); ___
Equation (90).

101

(88) and (89) describe the tracer diffusivities of the
benzene - carbon tetrachloride and chloroform—carbon
tetrachloride systems very well with constant friction
coefficients.

The behavior of the mutual diffusion coefficients
are predicted by equation (90) is also shown on Figures

12 and 13.

D =

* *
AB XADB + XBDA 5162‘ (90)

C
[ BlnAA
A

The activity terms were calculated from equations (221)
and (222). Equation (90) predicts the mutual diffusiv-
ities of the benzene - carbon tetrachloride system very
well. As can be seen from Figure 13, this is not the
case for chloroform-carbon tetrachloride mixtures where
mutual diffusivities calculated by equation (90) differ
by as much as 10% from the eXperimental data. This dis-
crepancy can be attributed to an error in the measurement
of the mutual diffusivities for this system. Although the
data form a smooth curve with intercepts which are con-
sistent with the tracer diffusion intercepts, attempts to
duplicate these data were not successful. Cordes and
Steinmeien (14) also measured mutual diffusivities in
this system. Their data differ from the data collected

here and are not consistent with the tracer diffusivities

102

at infinite dilution. Therefore, it is not reasonable to
attribute the poor agreement to fault of equation (90).

EXperimental results for the methyl ethyl keton —
carbon tetrachloride system are shown on Figure 14.
Anderson's (3) viscosity and mutual diffusion data were
used for this system.

Equation (166) was applied to the tracer diffusion
data of carbon tetrachloride. As can be seen from Figure
14, there is excellent agreement between predicted and

experimental results.

_ kT
03 _ fif— (166)
2
i0
D. 2 .122 _l. _ ___} .1. + ___} (164)
A 9 f1 f XA f
11 11
~ 0 X alnAC
. kT 1 B A
= * .. __ ___. __
DAB XADB + XBDA + 2 T) E XA EIHEX (167)

Labelled methyl ethyl ketone was not available from com-
mercial suppliers so the tracer diffusivities of this
component could not be measured. However, it is possible
to predict these data from the tracer diffusivities for
carbon tetrachloride and the mutual diffusivities. The
friction coefficient of the carbon tetrachloride molecules,

f2, was determined from the tracer diffusivities of carbon

103

 

x (0" (cm‘/ sec)

 

DIFFUSION COEFFICIENT

 

 

 

0'00.0 01.2 0.4 0.6 0.8 I.O

 

MOLE FRACTION METHYL ETHYL KETONE

Figure 14. Tracer and Mutual Diffusion Coefficients fer
the Methyl Ethyl Ketone - Carbon Tetrachloride
System at 25°C. 0 Tracer Diffusion Data for
Carbon Tetrachloride; 0 Mutual Diffusion Data,
Reference (3); Equation (166); -----
Equation (167); _____ ___ Equation (164).

 

104

tetrachloride. Then, equations (164) and (167), together,
were fit to the mutual diffusion data to get the best
values of f1 and E11' The activity term was calculated
from equation (141). The curve for the tracer diffusion
of the ketone, in Figure 14, was then calculated from
equation (164).

Although this curve has not been quantitatively
verified, it is qualitatively correct. At low ketone
concentrations, the diffusion coefficient is relatively
high because the ketone is diffusing as individual mole—
cules. As the ketone concentration increases, the diffu-
sion coefficient decreases as the ketone molecules form
the larger and more slowly diffusing dimer clusters. At
the same time, the viscosity of the solution decreases,
causing the diffusion coefficient to increase. In this
case, both effects cancel and the diffusion coefficient
is relatively constant after an initial decrease.

Diffusion data for the acetic acid-carbon tetra—
chloride system are shown on Figure 14. The viscosity and
mutual diffusion data are by Anderson and Babb (3).

A reliable value of the equilibrium constant is
not available in the literature. So, it was necessary to

~

fll and K from the diffusion data. To

and K, equation (164) and (167) were empiri-

find f f

~

f

1’ 2’

find f1' 11

cally fit to the mutual diffusivities and the tracer

diffusivities of acetic acid. f2 was calculated from the

105

tracer diffusion coefficient of pure carbon tetrachloride
and equation (166). The curves shown on Figure 15 were
calculated from equations (164) and (167) using K = 4000
(mole fraction)_l, a value within the range of the litera-
ture values.

It can be seen that, while the equations predict
the qualitative behavior of the data, quantitatively there
is disagreement. Also, the disagreement is generally
greatest at higher acid concentrations.

The disagreement can be attributed to two effects.
First the acid molecules may associate to form higher
order clusters. Since the concentration of these higher
clusters increases with acid concentration, it follows
that the disagreement between the dimer model and eXperi-
ment should also be greatest at the higher concentrations.
Second, the chemical model may not accurately describe the
activity of the system. The thermodynamic term, as calcu-
lated by equation (141), is 0.45 at XA = 0.00642. While
this seems abnormally small, the activity data available
in the literature cannot be used as a check because they
were not taken at low concentrations (73).

Thus, it would appear that little can be concluded
as to the validity of equations (164) and (167) from the
results of this system. However, three points should be
restated. First, the equilibrium constant determined from

the data is within the range of literature values. Second,

106

MOLE FRACTION ACID x IO'

 

 

 

 
     
 

 

  

0.0 0.2 0.4_ 06 0.8 IO
T I F I
p—Intercept
3.0"
|
I
‘6
\
\
\
2.0)1 \
. \
3 ‘ \\*’;i
g N ‘
\. ~-—--—-———
. \- O
'6 \-_9__-__
2 IO
Q3

 

 

I. I I

. J
0.0 0.2 0.4 0.6 0.8 (.0

00 l

MOLE FRACTION ACID

Figure 15. Tracer and Mutual Diffusion Coefficients for the Acetic Acid
Carbon Tetrachloride System at 25°C. 0 Tracer Diffusion Data
for Acetic Acid; 0, Mutual Diffusion Data, Reference
(3); -------- Equation (161+); ...... ... ...... Equation (167).

 

107

the diffusion equations can be fit to the qualitative
behavior of the data. And third, this system is a rigid

test of any model because the diffusion coefficient more than
doubles in a concentration span of less than 1%. Thus,

it is felt that this system does indicate the validity of

the diffusion equations even though they do not predict

the data well.

Diffusion data for the ether-chloroform system
are shown on Figure 16. Again, the mutual diffusion and
viscosity data are those of Anderson and Babb (2). fl
and E12 were determined from the intercepts of the ether
tracer diffusion data and f2 and E12 were determined from
the intercepts of the chloroform tracer diffusion data as
described in Appendix V. The two values of flz, determined
in this manner, were different by approximately 3%.
Therefore, an average E12 was used with fl and f2 in
equations (191), (192) and (193) to predict the mutual

and tracer diffusivities over the whole concentration

range.
1 X0

D.=I$I -11___l-:.1_ (191)

A ” E-1_ f XA f

12 12

k l XO 1
Dg=l?l_~___%_:_ (192)

n 2 f B f

I0.0

9°
0

9’
o

DIFFUSION COEFFICIENT x I05 (cmz/sec)
.45
O

0.0

Figure 16.

108

 

 

 

 

 

I

 

I I l
0.0 0.2 0.4 0.6 0.8 LO

MOLE FRACTION ETHER

Tracer and Mutual Diffusion Coefficients for

the Ether - Chloroform System at 25°C. 0

Tracer Diffusion Data for Ether; O Tracer

Diffusion Data for Chloroform; A Mutual Diffu—

sion Data, Reference (2): ----- Equation (191);
Equation (192); ___ _ ___ Equation (193).‘

109

 

~ 0 C
81 A
D = x D* + x 0* - 2 53 X12 n A (193)
AB A B B A I) ~ alnX

The plots of these equations are shown in Figure 16. The
activity term in equation (193) was calculated from equa—
tion (184). The agreement between data and the predicted
values is very good for the mutual diffusivities and the

tracer diffusivities of ether. However, a small but con—

sistant error is noted for the tracer diffusion of chloro

form. There are two factors which could cause this error.
First, the chemical model may not adequately describe
association in the system. This is not likely for it can
be used to predict the tracer diffusivities of ether very
well. The second eXplanation for this error is suggested
by comparing the effective molecular sizes as indicated

by the molar volumes of the pure components. The molar
volumes of ether and chloroform are 104 and 81 cc/mole,
respectively. Thus, chloroform is a relatively small
molecule surrounded by larger molecules of ether and
ether-chloroform clusters. As discussed earlier, the
friction coefficients should be constant when the medium
surrounding a diffusing molecule can be considered a con—
tinuum. Since the chloroform molecules are smaller than
the molecules in the surrounding medium, the medium cannot

be considered a continuum and, therefore, the friction

110

coefficient of chloroform is a function of concentration.
This system provides very strong support for equations
(191), (192) and (193).

Viscosity and density data for the ether-chloro-
form-carbon tetrachloride system are presented in Figure
17 and the tracer diffusivity data in Figure 18. For
simplicity, the mole fraction of the carbon tetrachloride
was 0.5 for all measurements. Since this is a ternary
system, the procedure discussed in Appendix V cannot be
used to evaluate the friction coefficients from the data
taken for this system. Therefore, the friction coeffi-
cients found for the binary ether—chloroform system were
used. Tracer diffusivities for ether and chloroform were
calculated from equation (218) and (219), using these

friction coefficients, and plotted in Figure 18.

X0
kT 1 1 __1_

1
* = __ __ _ ___. _
DA n £1 E XA + E (218)
12 12
O
kT 1 1 x2 1
D* = — - :— — + :— (219)
B 0 f2— f xB f
12 12

It can be seen from this figure that the agreement between
theory and eXperiment is excellent for ether but, as in
the ether-chloroform system, it is not as good for
chloroform. Again, a comparison of molar volumes (104,

97 and 81 (cc/mole), respectively, for ether, carbon

111

 

 

 

 

0.7
MOLE FRACTION CARBON TETRACHLORIDE
‘7 .. =05
C
I6—
0
I.5— -0.6
g" ,4. ' 7)?
g ' .8.
E» 8
T 8.
)- I.3T 0
I; >-
3 ~05 t:
LIJ U)
0 8
I.2- g
>
U A
to ' ‘ ‘ 1 0.4
0.0 01 0.2 0.3 0.4 0.5
MOLE FRACTION ETHER
Figure 17.

Density and Viscosity of the Ether - Chloro-

form - Carbon Tetrachloride System at 25°C.
0 Density Data; 0 Viscosity Data.

112

 

x I05 (cm‘/sec)

TRACER DIFFUSION COEFFICIENT

 

4.0 t

 

MOLE FRACTION CARBON TETRACHLORID L
=0.5

 

 

 

|.C) I I I I
0.0 DJ 0.2 0.3 0.4 0.5
MOLE FRACTION ETHER
Figure 18. Tracer Diffusion Coefficients for the Ether-

Chloroform - Carbon Tetrachloride System at
25°C. 0 Tracer Diffusion Data for Ether; 0
Tracer Diffusion Data for Chloroform;
Equation (218); ----- Equation (219).

113

tetrachloride and chloroform) indicates that chloroform
is a smaller molecule diffusing through relatively large
molecules. The resulting concentration dependence of the
chloroform friction coefficient accounts for the disagree—
ment between theory and eXperiment. The tracer diffu-
sivities of ether and chloroform in this ternary system
can be predicted from the tracer diffusivities of the same
two components in binary mixtures.

The eXperimental tracer diffusion, mutual diffu-
sion, density and viscosity data collected in this study
are tabulated in Appendixes VI, VII, and VIII, respectively,

for detailed reference.

CONCLUSIONS

It can be concluded from this study that the
intrinsic mechanism of liquid diffusion and the concept
of the medium fixed reference can be combined with the
chemical model of solution nonideality to derive equations
which accurately predict tracer and mutual diffusion
coefficients in systems whose components associate in
simple, specific ways. To make these predictions, it is
necessary to specify friction coefficients-—one for each
diffusing species. This study shows that in a binary
mixture the friction coefficients can be determined
directly from the intercepts of tracer diffusion coeffi-
cients alone. Further, it is concluded that the friction
coefficient of a component, whose molecules are smaller
than those of the other components, is a function of
composition.

This study provides evidence which substantiates
the model that in binary mixtures, ether and chloroform
cross associate to form bimolecular clusters. Tracer
diffusion data indicate that the same cross association
of ether and chloroform occurs when nonassociating carbon

tetrachloride is added to binary ether-chloroform mixtures.

114

115

The friction factors for ether and chloroform are
concluded to be the same in the binary ether—chloroform
system and the ternary ether-chloroform—carbon tetra—
chloride system. It is also shown that the friction
factor of chloroform is concentration dependent for both
the binary and ternary systems.

The Hartley-Crank equation (equation (71)) and
the Darken equation (equation (90)) are shown to be valid
for nonassociated binary systems.

It is the opinion of this author that the intrinsic
model of diffusion is very likely the true mechanism by
which diffusion occurs. This opinion is based on the fact
that the intrinsic model leads to equations from which
mutual diffusivities can be predicted from measurements of
tracer diffusivities and viscosity and, in the case of
associated systems, from an equilibrium constant which can
be determined from thermodynamic data. It would seem more
than fortuitous that this would be the case if the intrin-

sic model were not correct.

FUTURE WORK

Theory

More work should be done in the deveIOpment of the
chemical model to remove one of its primary faults; the
assumption that the species form an ideal solution. The
assumption was made to derive an expression for the compo—
nent activities in terms of measurable quantities.

One method of attack can be seen by recalling
that the activity of a component is equal to the activity
of its monomeric species. Since the component activities
can be calculated from vapor pressure data, for example,
and since the mole fractions of the species can be deter-
mined from spectrosc0pic techniques, the activity coeffi-
cients of the monomeric species can be determined as a
function of composition.

Most likely, the species form a nearly ideal
solution, so any simple, thermodynamically consistent set
of equations (multi-component van Laar, Porter, etc.) can
be fit to the activity coefficient data. The resulting
expressions could then be used to modify equation (120)
to give a more accurate thermodynamic description of the

system of interest.

116

117

The most logical extention of the intrinsic model
of diffusion is to multicomponent systems. Kett and
Anderson (76) have made a study of the intrinsic model as
applied to ordinary diffusion in ternary systems. It
would also be of interest to investigate relationships

between tracer and ordinary diffusion for these systems.

Experimental

The accuracy of the capillary technique could be
increased if the average concentration (in equation (227))
could be measured several times during the run. A tech—
nique of this type was used by Mills (51) for materials
labelled with an isotope whose nuclear emission is capable
of penetrating the walls of the capillary. Counters were
placed outside the capillary. When carbon - 14 or tritium
labelled materials are used, absorption of the relatively
soft beta particles in even the thinnest capillary walls
precludes the use of Mills' method.

Instead, the capillary itself could be constructed
of a scintillation crystal that is inert to the solvent
action of the system of interest. Calcium floride is one
material that is resistant to many mixtures of this type

studied here. The crystal could be viewed from the side

by a photomultiplier detection system and the count rate

determined at prescribed time intervals.

118

Anticipated problems are in the areas of
contamination of the crystal with radioactive materials
and poor conversion of radioactive disintegrations to
detectable light pulses.

Another area for future experiment work is in the

area of activity determination. To check the validit of

BlnAA

___ I
81nXA

must be known quite accurately. Much of the activity data

the diffusion equations, the thermodynamic term,

in the literature is very bad. Therefore, it would be
desirable to have good activity data on carefully selected

systems.

APPENDIXES

APPENDIX I

Fortran Computer Program for the Determination
of Equilibrium Constants

C THIS PROGRAM FITS TOTAL PRESSURE DATA TO THE TYPE (B)

C ASSOCIATION MODEL. NOMENCLATURE: W = WEIGHT FRACTION,
C X=MOLE FRACTION, A=VAP. PRES. OF A, B=VAP. PRES. OF B,
C PI=TOTAL PRES., AK=EQUIL. CONSTANT

SZZXX = 0.
SXX = 0.
8Z2 = 0.
SXXZ = 0.

READ 100, T
READ 102, A, B
DO 1 J=1, 9

READ 102, WA, PI

WB = 1.-WA

XA = WA

SB - 1.-XA

XX=XA*XB

A = ((XA-XB)*(A-B)-PI)/(PI-A-B)
22 = z*z

szzxx = szzxx +z2*xx

sxx = sxx +xx

522 = 822 + 22

sxx2 = sxx2 + xx*xx
1 PRINT 103, XA,XX,Z2
SLOPE = (9.*SZZXX-SXX*SZZ)/(9.*SXX2-SXX*SXX)
AK = -SLOPE/(SLOPE+4)
PRINT 100, T
2 PRINT 102, SLOPE, AK
100 FORMAT (F7.4)
102 FORMAT (2F7.4)
103 FORMAT (3F7.4)
END

120

(l)

(2)

(3)

(4)

(5)

(6)

APPENDI X I I

Experimental Procedure

Bulk solution of the desired concentration was
prepared and placed in a bulk solution container.

The container was in turn placed in a constant
temperature bath for at least one hour before the

run was started.

Capillary solution of the same chemical concentration
as the bulk solution was weighed out. A small portion
of one component consisted of labelled material, added
with the component.

The capillary solution was degassed by allowing the
solution to equilibrate at approximately 30°C.

The capillaries and frits were washed three times
with acetone and dried. Three cells were assembled.
The bulk solution was removed from the water bath.
Each cell was filled in turn and placed into the bulk
solution. The container was then returned to the
water bath and diffusion allowed to proceed for three
to six days.

After the run, 5 cc. of scintillation liquid were

measured into each of three counting vials.

121

(7)

(8)

(9)

122

Each capillary was then emptied by the following

procedure.

(a)
(b)

(C)

(d)

(e)

The frit holder was removed.
The cell was inverted into the counting vial.

The funnel-shaped portion of the screw cap
was filled with scintillation liquid.

The metal foil disk was punctured and the
capillary flushed with 10 cc. of scintillation
liquid.

The porous frit was removed from the counting
vial.

The cells were cleaned, reassembled, filled with

capillary solution and emptied as per the above

procedure.

The vials were placed in the scintillation counter

and the activities were determined. If the count

rate of any one of the three samples deviated more

than 5% from the mean activity of the three, the run

was discarded.

APPENDIX III

Solution of Equation (226)

2

 

 

 

 

ac; 3 CE
39 = Di 2 (a)

82

ac;
82 = O at z = O, 6>0 (b)
(226)
ac; 1

-D3 82 = R C: at z = L, 6>O (c)
C1 = Cio at 6 = 0, L>z>0 (d)

Following the general procedure of separation of

variables, a product solution is assumed of the form

CI = F(z)- W(6). Then, the differential equation becomes:
1 d2F 1 aw 2
F ___—2 = W d—e— = - X , (III-l)
dz 1

a constant. Or,

d F + AZF - o (III-2)

 

123

124

and
d? 2
IE 4' D:A q} = 0. (III-3)
The product of the solutions of these two equations
is:
C3 = (klcos(Az) + k2 sin (Az)eXp(-A2D: ). (III-4)
Also,

so:
a: = (kzxcos(Az) - k, Asin (Az)eXp(—A2D; ). (III-5)

 

To satisfy boundary condition (226) (b), k2 from equation
(III—5) must be zero. Then with k2 = 0, combine equation

(III—4) and (III-5) with boundary condition (226) (c) with

the following result:

2 . _ _ l _ 2 * _
-eXp(-A D:6)k,ASIn(AL) — fifif eXp( A Die)klcos(AL) (III 6)

Or, after simplifying,

= * -
COT(AnL) RDiAn (III 7)

where the subscript n on A indicates the infinite number

of roots.
Since the differential equation is linear, this

infinite number of solutions may be added. Then, from

125

equations (III-4),

* = _ _
Ci 2 kneXp( AnDie)cos(Anz) (III 8)

n=1

Now, substitute this eXpression into boundary

condition (226) (b), multiply both sides of the resulting

equation by cosAmzdz and integrate between 2 = O and L.
Then,
L L
CEOXCOS(Amz)dz =Jr2: knCOS(Anz)COS(Amz)dz (III-9)
n:
o 0

When m ¢ n, the integral on the right side is zero and
when m = n, it is:

E + —l— SIN(AnL)COS(AnL)) (III-10)

k (
n 2 2An

The integral on the left is:

—lSIN(AnL). (III—ll)

A
n

So,

* L
2Ci0 SIN(An )

k = . o (III-12)
n AnL + SIN(AnL)COS(AnL)

 

126

The final solution is then:

 

 

l “ SINA L
* eXp
0 n=1 AnL + SIN(AnL)COS(AnL)

(III-13)

_2*
( AnDie)COS(Anz).

The solution in this form is not useful, for the
concentration ratio is not conveniently measured as a
function of position and time. It is more convenient to
measure the ratio of average concentration to initial
concentration as a function of time. An expression for

the average concentration can be obtained by combining

(III-l3) and (III-l4).

*dz
* = 1 (111-14)

j?
o

Ciavg j? dz
0

C

The result is:

 

 

Ciav w SINZBn
——E?§ = 2 1:1 &n(8n + siancoan) exP
(227)(a)
D?6
2 1
-8n 2

127
where
n n (III-l6)

Equation (III-7) can now be rewritten as:

RD;
COTBn = Sn L (227)(b)

 

By defining the new variable, Bn’ the concentration ratio

can be written as a convenient function of two dimension—

RD? D?8
1 1
less groups, __f and ——§ .

APPENDIX IV

Fortran Computer Program for Diffusion
Coefficient Calculations

c R=NUMBER OF TERMS IN THE SUM
c w = RD*/L
c AN=BETA N
PRINT 100
D0 8 J=1, 100
R = J
R = R/SO
DT = R
w = 50.0
R = 1.0
CAVG = 0.0

DO 11 = 1,10
7 AN = (2. *R-l.)*3.l4l6/2.
3 G=AN-(AN/W-COTF (AN))/(l./W+1./SINF(AN))**2)

IF (ABSF(AN/W-COTF(AN))-.00001)2,2,4
4 AN = G
GO TO 3
2 AN = G
SUM = (SINF(AN))**2/(AN*(AN+(SINF(AN))*(COSF(AN))))*

lEXPF(-AN*AN*DT)
CAVG=CAVG+SUM
IF(SUM/CAVG-.OOOOl)5,5,6
6 R=R+l.0
GO TO 7
5 CAVG = 2.*CAVG
PRINT lOl,CAVG,W,DT,R
R=1.
CAVG=0.0
1 w=w+5.0
8 CONTINUE
100 FORMAT(*0*10X*CAVG*1OX*W*12X*DTL2*lOX*R*)
101 FORMAT (*0*,4(7x,F7,4))
END

128

APPENDIX V

Determination of Friction Coefficients
From Tracer Diffusivities

If the diffusion relations derived in this work
are to be used for quantitative predictions, the friction
coefficients, fi, must be specified. For components of
a nonassociated system or for a nonassociated component
of an otherwise associated system, the tracer diffusiv—

ities are given by equation (79)

kT
I 79
D1 PIT ( )
1
Given viscosity and tracer diffusion data, one can calcu-
late fi at any concentration.
The Situation is more complex for an associated

component of a mixture. When component A associates it—

self to form dimer clusters,

0
-kT —l-—l-——-l—+—l——. (164)

Here it is not possible to calculate a value for both fl

and E11 from viscosity and tracer diffusion data at any

one concentration. However, the frictional coeff1c1ents

129

130

can be calculated directly from the eXperimental tracer
diffusion intercepts. Using equation (135) and

l' Hospital's rule, it can be shown that

O
I . X
limit 1 _
XA+O x ‘ 1' (V'l)

P

It follows directly from equations (V—l) and (164) that

limit D* = -——. (V—2)
XA+O A flr‘

Therefore, fl can easily be calculated from the intercept

 

 

of DA at XA = 0. Also from equation (135),
0

limit f1 _ 1 + (1+4K)1/2 (V_3)

XA+1 xA 1 + (l+4K)l/2 + 4K
and hence,

1/2 I

a. = —- 751-1 12‘1302 + 4K + ——: )1
Xz+l 1 fl]. 1 + ( + 11

~

The friction coefficient, fll' can now be calculated from

this equation after fl has been determined from equation

(v-2).

131

The following limiting eXpressions are applicable
when the two components associate with each other to form

a dimer cluster:

 

limit D* = ——— (V-S)
xA+1 A nf1 '
. . kT
llITllt D5 = FF- , (V-6)
XB+1 ' 2
limit D1: = 1%: K i 1 f—l— + 715— (v-7)
X +0 1 f
A 12
and
. . kT 1 1 K
11m1t D* = — + :— (V-8)
XB+O B r] K + I I; £12

After fl and f2 have been calculated from equations (V-5)

and (V-6), f can be calculated from either equation

12
(V-7) or equation (V-8). If different values are obtained,

an average E12 can be used.

 

0.0
0.216
0.26
0.34
0.523
0.677
0.82
1.00

 

0.0

0.08
0.18
0.29
0.42
0.50
0.75
0.76
0.98
1.00

APPENDIX VI

EXperimental Tracer Diffusion
Data at 25°C.

 

 

 

 

Table III. Tracer Diffusion Data for the
Benzene (A) - Carbon Tetrachloride (B) System
D3 x 105 (cmZ/Sec) DE x 105 (cm2/sec)
-- 1.32
1.57 _-
1.58 -_
-- 1.48
1.76 __
-- 1.71
2.00 _-
2.14 __
Table IV. Tracer Diffusion Data for the
Chloroform (A) — Carbon Tetrachloride (B) System
2
DA x 105 (dmz/sec) DE x 105 (cm /sec)
-_ 1.32
1.57 --
_- 1.49
1.82 --
__ 1.73
2.02 --
2.23 --
-_ 2.04
__ 2.16
2.44 --

132

 

133

 

 

 

0.0

0.027
0.027
0.101
0.201
0.642
0.642

 

0.0

0.04
0.42
0.44
0.65
0.82
0.85
0.95
1.00

Table V. Tracer Diffusion Data for the Methyl
Ethyl Ketone (A) - Carbon Tetrachloride (B) System
X D* x 105 (cmz/sec)
A B
0.0 1.32
0.25 1.55
0.40 1.88
0.50 2.00
0.73 2.35
Table VI. Tracer Diffusion Data for the Acetic
Acid (A) - Carbon Tetrachloride (B) System
DA x 105 (cmz/Sec) DE x 105 (cmZ/sec)
__ 1.32
2.31
2.33
1.80
1.65
1.27
1.28
Table VII. Tracer Diffusion Data for the.

Ether (A) - Chloroform (B) System

 

 

D; x 105 (cmZ/sec) DE x 105 (cm2/sec)
-- 2.44
2.15 2.44
3.05 --
-- 2.65
-- 2.80
6.23 --
-- 3.55
—- 4.13
8.75 --

134

Table VIII. Tracer Diffusion Data for the Ether
(A) - Chloroform (B) - Carbon Tetrachloride (C) System

 

 

 

[xC — 0.5]

XA Dg x 105 (cmZ/sec) DE x 105 (cmZ/sec)
0.0 -- 2.03
0.1 2.32 --
0.1253 -- 1.99
0.2505 2.81 --
0.2510 -- 2.11
0.3411 -- 2.13
0.3751 -- 2.17
0.3742 3.38 --
0.4511 - 2-30

0.5014 4.62 --

APPENDI X VI I
EXperimental Mutual Diffusion Data at 25°C.

(X is the average concentration of the solutions
above and below the boundary in the diffusion cell.)

Table IX.
Benzene (A)

Mutual Diffusion Data for the
- Carbon Tetrachloride (B) System

 

Chloroform (A)

5 2
XA DAB x 10 (cm /sec)
0.0150 1.450
0.2548 1.439
0.4936 1.451
0.7656 1.755
0.9964 1.945
Table X. Mutual Diffusion Data for the

— Carbon Tetrachloride

 

5 2
XA DAB x 10 (cm /Sec)
0.0265 1-505
0.4311 1.644
0.7389 1.752
0.8255 1.796
0.0966 1.881
0.9418 1.972

135

(B) System

APPENDIX VIII

Experimental Density and Viscosity Data at 25°C.

XA
0.0
0.0301
0.1711
0.2953
0.5606
0.7864
1.0

 

C

xA
0.0
0.1118
0.2318
0.4915
0.8450
1.0

 

Table XI. Density and Viscosity Data for the
Benzene (A) - Carbon Tetrachloride (B) System

n (Centipoise) p (g/cc)

0.8888 1.5945

0.8794 1.5639

0.8393 1.4745

0.7992 1.3902

0.7205 1.2036

0.6563 1.0386

0.5970 0.8736

Table XII.
hloroform

 

 

Density and Viscosity Data for the

(A)

- Carbon Tetrachloride (B)

T1(Centipoise)

System

0 (g/cc)

 

 

0.8888
0.8263
0.7611
0.6608
0.5680
0.5401

1.5945
1.5744
1.5600
1.5322
1.4955
1.4798

Table XIII. Density and Viscosity Data for the Ether
(A) - Chloroform (B) - Carbon Tetrachloride (B) System

 

 

 

XA r1(Centipoise) O (g/cc)
0.0 0.6475 1.5307
0.1031 0.6298 1.4428
0.2501 0.5702 1.3288
0.3495 0.5189 1.2509
0.5010 0.4343 1.1370

136

AH
m

AH

im
iM
iN

io

NOMENCLATURE

algebraic complement of the element, 0 ., in
the G determinant k3

activity

constant in equations (100) and (101)
concentration of a Species, (moles/cc)
number of components

concentration of a component (moles/cc)
mutual diffusion coefficient (cmz/sec)
tracer diffusion coefficient (cmZ/sec)
intrinsic diffusion coefficient (cm2/sec)

a diffusion coefficient de ined relative to
the volume fixed plant (cm /Sec)

friction coefficient (cm)

determinant of stoichiometric coefficients
heat of mixing (calories/mole)

heat of reaction (calories/mole)

molar flux relative to the medium fixed
reference plant

molar flux relative to the mass fixed reference
plane

molar flux relative to the number fixed
reference plane

molar flux relative to the component fixed
reference plane

137

’3 W W 1"!

<l

iC

im

iM

iN

138

Boltzmann's constant

equilibrium constant

capillary length (cm)

molecular weight of the i-th component
change in the number of moles of a component
change in the number of moles of a Species
number of species

Avogadro's number

vapor pressure of a component

partial pressure

the j-th species taking part in the k-th
association reaction

Stokes' radius of a molecule (equation (5))
number of association reactions

resistance of a glass frit to diffusion
universal gas constant

absolute temperature

function defined by equation (213)

partial molar volume (cc/mole)

velocity of the i-th Species relative to the
coordinate fixed plane

velocity of the i-th Species relative to the
medium fixed plane

velocity of the i-th Species relative to the
mass fixed plane

velocity of the i-th Species relative to the
number fixed plane

139
velocity of the i-th species relative to the
component fixed plane

velocity of the i-th species relative to the
volume fixed plane

medium average velocity

mass average velocity

number average velocity

component average velocity

volume average velocity

mole fraction of a component

mole fraction of a species

function defined by equation (140)
distance variable defined by Figure 6
function defined by equation (183)
function defined by equation (112)
parameter defined by equation (III-16)
a function of distance only

activity coefficient

the extent of reaction

viscosity

time

n-th root of equation (III-7)

chemical potential

stoichiometric coefficient of the j-th species
in the k-th reaction

total vapor pressure

140

O density
W a function of time only
Subscripts
A,B,C, refer to components
i,j,k running indexes
AVG refers to the average value of a parameter
0 refers to a reference condition like standard

state, zero time, etc.

Superscripts

* refers to an isotOpically labelled molecule

~

refers to a basic species (cluster)

0 refers to a psuedo quantity

c refers to a component

n refers to a species

u refers to an unlabelled molecule of the labelled

component in tracer diffusion

10.
ll.

12.

13.

14.

BIBLIOGRAPHY

Anderson, D. K., Ph.D. Thesis, University of
Washington (1960).

Anderson, D. K. and Babb, A. L., J. Phys. Chem.,
66, 1281 (1961).

Anderson, D. K. and Babb, A. L., J. Phys. Chem.,
66, 899 (1962).

Anderson, J. S. and Saddington, K., J. Chem. Soc.
Suppl., 381 (1949).

Baldwin, R. L., Dunlop, P. J. and Gosting, L. J.,
J. Am. Chem. Soc., 11, 5235 (1955).

Barrow, G. M. and Yerger, E. A., J. Am. Chem. Soc.,
16, 5248 (1954).

Bidlack, D. L., Ph.D. Thesis, Michigan State
University (1966).

Caldwell, C. S., Ph.D. Thesis, University of
Washington (1955).

Caldwell, C. S., Hall, J. R. and Babb, A. L., Rev.
Sci. Instr., 36, 816 (1957).

Carman, P. C., J. Phys. Chem., 16 (10), 3355 (1966).
Carman, P. C., J. Phys. Chem., 11 (8), 2565 (1967).

Carman, P. C. and Miller, L., Trans. Faraday Soc.,
66, 1838 (1959).

Cheeseman, G. H. and Whitaker, A. M. B., Proc. Roy.
Soc. (London), A226, 406 (1952).

Cordes, H. and Steinmeien, M., Z. Phys. Chem., 62,
335 (1966).

141

142

15. Daniels, P., Mathews, J. H., Williams, J. W.,
Bender, P. and Alberty, R. A., Experimental
Physical Chemistry, McGraw Hill, N. Y., 1956,
p. 62.

 

 

160 Darken, Lo So, Trans. A. I. M. E0, 175’ 184 (1948)::

17. Davies, M., Jones, P., Patnaik, D. and Moelwyn -
Hughes, B. A. I J. Chem. SOC. I a, 1249 (1951) o

18. Denbigh, K., The Principles 66 Chemical Equilibrium,
University Press, Cambridge (1955).

 

 

 

l9. Dolezalek, F., Z. Physik. Chem., 66, 727 (1908).
20. Dolezalek, F., Z. Physik. Chem., 26, 585 (1918).

21. Dolezalek, F. and Schulze, A., Z. Phys. Chem., A83,
45 (1913).

22. Dolezalek, F. and Speidel, F., Z. Phys. Chem., A94,
72 (1920).

23. DunlOp, P. J. and Gosting, L. J., J. Am. Chem. Soc.
21, 5738 (1955).

24. Einstein, A., Ann. Physik. 11(4), 549 (1905).

25. Ewell, R. H. and Eyring, H., J. Chem. Phys., 6L 726
(1937).

26. Eyring, H., J. Chem. Phys., 6, 283 (1936)°

27. Eyring, H. and Hirschfelder, J., J. Phys. Chem.,
41.249 (1937).

28. Pick, A., Pogg. Ann., 26, 59 (1855).

29. Fowler, R. T. and Norris, G. S., J. App. Chem., 5,
266 (1955). ‘

30. Gerbier, M. M. and Gerbier, J., Compt. rend., 248,
669 (1959).

31. Hammond, B. R. and Stokes, R. H., Trans. Faraday Soc.,
‘62, 781 (1956).

32. Hardt, A. P., Anderson, D. K., Rathbun, R., Mar, B. w.
and Babb, A. L., J. Phys. Chem., 63, 2059 (1959).

33.

34.

35.

360

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

143

Hartley, G. S. and Crank, K., Trans. Faraday Soc.,
66, 801 (1949).

Hildebrand, J. H., J. Am. Chem. Soc., 61, 66 (1929).
Hildebrand, J. H., J. Am. Chem. Soc., 61, 866 (1935).
Hildebrand, J. H. and Scott, R. L., The Solubility

.9: Non-electrolytes, 3rd ed., Reifihold Pub. Corp.,
New York (1950)?

 

 

Hildebrand, J. H. and WOod, S. E., J. Chem. Phys.,
1.817 (1933).

Hirobe, H., J. Fac. Sci., Tokyo, 1, 155 (1925).

Hirschfelder, J., Stevenson, D., and Eyring, H.,
J. Chem. Phys., 6, 896 (1937).

Johnson, P. A. and Babb, A. L., J. Phys. Chem., 66,
14 (1956).

Kett, T. K. and Anderson, D. K., Ph.D. Thesis,
Michigan State University (1968).

Kirkwood, J. G., Baldwin, R. L., DunlOp, P. J.,
Gosting, L. J. and Kegeles, G., J. Chem. Phys.
66 (5), 1505 (1960).

Kohstamm, P. and van Dalfsen, B. M., Proc. Akad.
Wetenschappen, Amsterdam, 6, 156 (1901).

Krishnaiah, M., Pai, M. U. and Sastri, S. R. S.,
J. Chem. Eng. Data, 16, 117 (1965).

Lewis, G. N. and Randall, M., Thermodynamics, 2nd ed.,
McGraw-Hill Publishing Co., Inc., N. Y. (1961).

Lin, T. F., Christian, 8. D. and Affsprung, H. E.,
J. Phys. Chem., 11, 968 (1967).

Lord, R. C., Nolin, B. and Stidham, H. D., J. Am.
Cham. Soc., 11, 1365 (1955).

Markuzin, N. P. Zhurnal Prinkladnoi Khimii, 39(8),
1769 (1966). _-

McGlashan, M. L., Prue, J. E. and Sainsbury, I. E. 8.,
Trans. Faraday Soc., 66, 1284, (1954).

50.

51.

52.

53.

54.

55.

56.

57.

58°
59.

60.

61.

62.

63.
64C

65.

66.
67.

68.

144

Michand, F., Ann. Phys., IX-6, 223 (1916).

Mills, R. and Godbole, E., Aust. J. Chem., 11(1)
(1958).

Nikolskii, S. S., Teoreticheskaya i Eksperimental'
naya Khimiya, 1(3), 343 (1966).

Pimentel, G. C. and McClellan, A. L., The Hydrogen
Bond, W. H. Freemand and Co., San Francisco

T1860).

Powell, R. E., Roseveare, W. E., and Eyring, H.,
Ind. Eng. Chem., 66, 430 (1941).

 

Price, W. J., Nuclear Radiation Detection, McGraw
Hill, N. Y. (1958).

 

Rastas, J. and Kivalo, P., Acta, Polytech. Scand.
66’(l964).

Rathbun, R. E. and Babb, A. L., J. Phys. Chem. 66,
1072 (1961).

Schmidt, G. C. Z. Phys. Chem. A121, 221 (1926).
Seith and Kottman, Naturwiss, 66, 41 (1952).

Smigelskas, and Kirkendall, Trans. Amer. Inst. Min.
Met. Eng., 171, 130 (1947).

Staveley, L. A. K., Tupman, W. I. and Hart, K. R.,
Trans. Faraday Soc., 61, 323 (1955).

Stearn, A. E., Irish, E. M., and Eyringe, H.,
J. Phys. Chem., 66, 981 (1940).

Stokes, R. H., J. Phys. Chem., 66(11), 4012 (1965).
Sutherland, W., Phil. Mar, 6, 784 (1905).
Timmermans, J., Physico--Chemical Constants of Pure

Compounds, Elsevier Publishing Company, Ific.,
N. Y. (1950).

 

 

 

Wang, J. H., J. Am. Chem. Soc. 3, 510 (1951).

Wenograd, J., J. Am. Chem. Soc., 16, 5844 (1957).

Wiedemann, G., POgg. Ann., 104, 170 (1858).