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Fluctuation thermodynamic properties of nonelectrolyte liquid mixtures

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Fluctuation thermodynamic properties of nonelectrolyte liquid mixtures
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Non-electrolyte liquid mixtures
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Wooley, Robert Joseph, 1954-
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xiii, 319 leaves : ill., photographs ; 28 cm.

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Carbon ( jstor )
Chemicals ( jstor )
Compressibility ( jstor )
Correlation coefficients ( jstor )
Data lines ( jstor )
Database models ( jstor )
Fringe ( jstor )
Light refraction ( jstor )
Mathematical independent variables ( jstor )
Parametric models ( jstor )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1987.
Bibliography:
Includes bibliographical references (leaves 311-318).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Robert Joseph Wooley.

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FLUCTUATION THERMODYNAMIC PROPERTIES
OF NONELECTROLYTE LIQUID MIXTURES









BY

ROBERT JOSEPH WOOLEY


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1987





























to Sue, for her patience and

encouragement.















ACKNOWLEDGEMENTS


I would like to thank Dr. J. P. O'Connell for his help and guidance throughout

this study. Further, I wish to thank Dr. Robley C. Williams, Jr., of Vanderbilt

University for his enthusiastic help and many thoughtful suggestions on how to install

a laser on an old ultracentrifuge. His sincere interest in a project not actually in his

field was inspiring.

I also wish to thank Dr. Fricke who had his differential refractometer fixed so

that I could use it and Dr. Mike Young and his lab workers for tolerating my presence

and allowing me unlimited free use of their microcomparator for many long hours of

measurements.

I am grateful to Stewart Goldfarb for making his Dortmund VLE database

available to me and to the Valdosta State College Library who allowed me to check

out VLE data books that were not available anywhere in the State of Florida.

I would also like to thank Sue Wooley for her assistance in typing, proofreading

and formatting this final document.















TABLE OF CONTENTS



Page
ACKNOWLEDGEMENTS ............................. ........ iii

KEY TO SYM BOLS ........................................... vii

A BSTR A CT ................................................... xii

CHAPTERS

1 INTRODUCTION .................................. 1

2 DETERMINATION OF SOLUTION
THERMODYNAMIC PROPERTIES BY RAYLEIGH
LIGHT SCATTERING ............................. 6

Introduction ....................................... 6
Background ....................................... 7
Rigorous Thermodynamic Fluctuation Development ...... 11
Light Scattering in Relaxing Solutions .................. 15
Calculation of Thermodynamic Properties .............. 18
Fluctuation Properties ............................ 18
Activity Coefficients ............................ 18
Activity Coefficients from Equation (2-46) .......... 19
Comparisons with Phase Equilibrium Measurements .. 23
Sensitivity of Activity Coefficient Calculations .......... 25
Sum m ary .......................................... 30

3 CONCENTRATION DERIVATIVES OF DILUTE
SOLUTION ACTIVITY COEFFICIENTS USING AN
ANALYTICAL ULTRACENTRIFUGE .............. 32

Introduction ....................................... 32
Equipm ent ........................................ 40
A pparatus ..................................... 40
Composition Detection ........................... 41
System Selection .................................... 45
Error A analysis ..................................... 49
Sum m ary .......................................... 52










4 EXPERIMENTAL MEASUREMENTS
OF DILUTE SOLUTION ACTIVITY
COEFFIEIENT DERIVATIVES ..................... 53

Introduction ....................................... 53
Carbon Disulfide/Acetone System ..................... 54
Chloroform/Acetone System .......................... 60
Carbon Tetrachloride/Acetone System .................. 64
Benzene/Acetonitrile System ........................ 68
Carbon Tetrachloride/Methanol System ................. 72
Summary of Experimental Results ..................... 76
Pressure Effects on the Results ...................... 77
The Effect of Imputities on Ultracentrifugation ......... 81
Sum m ary .......................................... 82

5 DIRECT CORRELATION FUNCTION INTEGRAL
DATABASE AND MODELING .................... 83

Introduction ....................................... 83
DCFI Database .................................... 87
Isothermal Compressibility Data ................... 91
Thermodynamic Property Modeling .................... 99
DCFI M odeling ................................. 99
Activity Coefficient Modeling from DCFIs .......... 100
Sum m ary .......................................... 115

6 SUM M ARY ....................................... 119

APPENDICIES

A ULTRACENTRIFUGE OPERATION ................ 124

B LASER LIGHT SOURCE FOR MODEL E
ULTRACENTRIFUGE INSTALLATION
AND ALIGNMENT ............................... 138

C FORTRAN PROGRAM FOR CONVERSION OF
FRINGE MEASUREMENTS TO COMPOSITION
PROFILES IN THE ULTRACENTRIFUGE
FRNGCNV AND FRNGCNVP ..................... 149

D EXCESS VOLUME AND PARTIAL MOLAR
VOLUME CALCULATIONS ....................... 165

E CALCULATION OF INITIAL
ULTRACENTRIFUGE SPEED ..................... 167










F RESULTS OF EXPERIMENTAL
ULTRACENTRIFUGE MEASUREMENTS
OF FIVE BINARY SYSTEMS ...................... 169

G FORTRAN PROGRAMS FOR THE CALCULATION
FOR PVT CALCULATIONS AND DATA
REGRESSION ..................................... 228

H DIRECT CORRELATION FUNCTION
INTEGRAL DATABASE ........................... 247

I FORTRAN PROGRAMS FOR REGRESSION OF
ACTIVITY COEFFICIENT DATA TO
EQUATIONS 5-39a AND 5-39b ........................ 306

BIBLIOGRAPHY .............................................. 311

BIOGRAPHICAL SKETCH ..................................... 319















KEY TO SYMBOLS


A Composition profile in ultracentrifuge uncertainty analysis.

A(k) Relaxation correction to J(k).

A12 Margules Binary Parameter.

A21 Margules Binary Parameter.

B(k) Relaxation correction to J(k).

Bij Matrix Element, Eq. 2-18.

Bk Characteristic parameter, Eq. 2-26.

Cp Heat capacity at constant P.

Cv Heat capacity at constant V.

Cn Direct Correlation Function Integral between two type 1 compounds.

C12 Direct Correlation Function Integral between a type 1 and a type 2

compound.

C22 Direct Correlation Function Integral between two type 2 compounds.

C' Characteristic parameter, Eq. 5-19.

CM Characteristic parameter of a mixture, Eq. 5-19.

Ck Characteristic parameter, Eq. 2-26.

AC Grouping of direct correlation function integrals.

D Activity coefficient derivative in ultracentrifuge uncertainty analysis.

D Diffusion coefficient.

F Factor in Eq. 2-48.









G12 NRTL binary parameter.

G21 NRTL binary parameter.

J Ratio of Rayleigh peak to the Brillouin peak.

J(0) J corrected to 0 wave vector.

J(k) J as measured.

JID J ideal.

K Grouping of variables, Eq. 2-44b.

MI Molecular weight, component 1.

NR Random number.

P Pressure.

R Gas constant.

R90 Rayleigh ratio measured at 90.

Rc Composition contribution to Rayleigh Ratio.

Rc,D Composition and density cross contribution to Rayleigh Ratio.

RD Density contribution to Rayleigh Ratio.

RIs Isotropic portion of Rayleigh ratio.

AST Total entropy change.

S Entropy.

T Temperature.

T* Characteristic parameter, Eq. 5-19.

TL Characteristic parameter of a mixture, Eq. 5-19.

Ui Uncertainty of variable i.

V Volume.

Vi Partial molar volume of component i.

a Binary parameter, Eq. 5-39.









ai Typical variables in uncertainty analysis.

b Binary parameter, Eq. 5-39.

c Speed of light.

cij Direct correlation function.

d Approach to equilibrium.

f Parameter in Eq. 2-33.

gij Radial distribution function.

hij Total correlation function.

h7- Excess enthalpy.

ic Intensity of Rayleigh peak.

iB Intensity of Brillouin peak.

j Fringe number.

Ajp Change in fringe number due to pressure.

k Boltzman constant.

kv Wave vector.

k Compression.

n Refractive Index.

Anp Refractive Index change due to pressure.

r Parameter in Eq. 2-43, Chapter 2.

r Radius from center of Rotor, Chapter 3.

t 8 Time to Equilibrium.

v Infinite frequency velocity of sound.

Vo Zero frequency velocity of sound.

vE Excess Volume.

vs Velocity of sound.









w Width of centrifuge cell.

xi Mole fraction, component i.

xO() Parameter in Eq. 2-6.

y() Parameter in Eq. 2-5.

y(2) Parameter in Eq. 2-9.

ap Coefficient of thermal expansion.

Ai Portion of scattered light transferred from side peak to the

central peak.

8 Approach to equilibrium.

e Dielectric constant.

^Yi Activity coefficient of component i.

'Yc Cp / Cv.

^y Infinite dilution activity coefficient.

Ks Adiabatic compressibility.

KT Isothermal compressibility

K( Excess isothermal compressibility.

X or Xo Wave length of incident light.

Ili Chemical potential of component i.

Av Frequency of peak separation.

v Frequency of incident light.

p Density.

Pi Concentration of component i.

pU Depolarization ratio.

p. Characteristic parameter, Eq. 5-19.

pL Characteristic parameter of a mixture, Eq. 5-19.














T

0

(0

v

x

((Ae)2}

((Ae)C)

<(A)DIAB)

((Ap)2)

((AX2)2)

Superscripts

0


NRTL parameter.

NRTL parameter.

Relaxation time.

Scattering angle.

Rotational speed.

Volume of centrifuge cell.

Reduced bulk modulus.

Total mean square fluctuation in dielectric constant.

Central peak contribution to the dielectric fluctuation.

Adiabatic contribution to the dielectric fluctuation.

Mean square density fluctuation.

Mean square composition fluctuation.



Pure component property, or initial concentration.

Ideal property.

Excess property.















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

FLUCTUATION THERMODYNAMIC PROPERTIES
OF NONELECTROLYTE LIQUID MIXTURES

By

Robert Joseph Wooley

December 1987

Chairman: John P. O'Connell
Major Department: Chemical Engineering

The behavior of the concentration derivatives of the pressure and component

activity coefficients has been studied via their connection to integrals of the statistical

mechanical direct correlation function (DCFI). Consideration of their direct

experimental accessibility with light scattering and equilibrium sedimentation in an

ultracentrifuge was made and dilute solution data have been taken with the latter

method. A new binary solution model, based on simplified forms of the DCFI and

reduced bulk modulus, has achieved good agreement with a newly compiled database

of DCFI and fluctuation properties for nonelectrolytes.

The data needed to establish a new theory based on fluctuation solution properties

do not include reliable values of activity coefficient derivatives at low concentrations.

These are usually obtained by differentiation of a specific model with parameters

fitted to vapor-liquid equilibrium data with results greatly affected by the model

chosen. As alternatives, two experimental methods for direct determination of the

activity coefficient derivative have been explored.









It has been concluded that measurement of concentration fluctuations of binary

systems with light scattering techniques is not useful for activity coefficients because

of sensitivity to experimental error and excessive requirements of ancillary data.

Equilibrium sedimentation in an analytical ultracentrifuge has been judged more

promising and measurements have been made with a Beckman Model E apparatus

with a newly installed laser optical system for Rayleigh fringe detection.

Measurements have been made of five systems that meet well-defined criteria for

refractive index and density differences as well as solution nonideality.

The measured chemical potential derivatives have been found to be lower than

those calculated from VLE models. However, in each system a single multiplicative

factor generally brings the results to within experimental error of the VLE models.

These values support the low concentration extrapolations of the Wilson and NRTL

Equations. It was not possible to distinguish the better of these models for activity

coefficient derivatives.

A database of DCFI values for 28 strongly nonideal binary systems has been

assembled using liquid compression and excess volume data along with Wilson and

NRTL parameters fitted to VLE data. Modeling of the DCFIs and activity

coefficients was attempted. While the density dependence is extremely complex, a

successful approach for activity coefficients is to separate the activity coefficient

derivative into a function of DCFIs, 1/AC, and the reduced bulk modulus, x. The

results are generally insensitive to the mathematical form chosen for 1/AC so a linear

expression has been used along with an idealized variation of x to obtain a new binary

activity coefficient model. Least squares fits to all systems in the database yielded

excellent results.















CHAPTER 1

INTRODUCTION

With continually increasing energy, raw material, and capital costs, chemical

process design engineers are constantly called upon to lower the costs associated with

the design and operation of chemical plants. Recent advances in computer simulation

technology allow the process engineer to rapidly assess the feasibility and economics

of new processes and process modifications, thus helping reduce capital and operating

costs with more optimum designs. However, while these computer simulators can

easily describe unit operations and converge flowsheets with complex recycles, they

are limited by their ability to model and predict the physical properties needed for a

reliable design. Further, physical property calculations, especially phase equilibria

properties, usually consume a considerable amount of the computer time in process

simulation (O'Connell, 1983). Finally, most empirical models for phase equilibrium

calculations can require exhaustive amounts of binary and sometimes multicomponent

data, especially if the compounds are highly non-ideal or unusual. As a result, new

and different processes, such as those found in synthetic fuels and bioprocessing,

continue to tax the currently available models. The consequence of all of this is that

computer simulation requires simple and reliable models for process design of current

and future systems.

To meet this demand, new thermodynamic property models, free of common

simplifying assumptions, such as pairwise additivity of intermolecular forces, rigid

molecules, etc. are needed. The fluctuation solution approach (Kirkwood and Buff,

1951; O'Connell, 1971; 1981) offers these advantages. Kirkwood and Buff (1951)

formulated the density derivatives of the chemical potential and total system pressure









in terms of integrals of the radial distribution function. O'Connell (1971) used these

formulations and the Ornstein and Zernike (1914) equations to give the

thermodynamic properties in terms of the direct correlation function integrals (DCFI).

These direct correlation functions appear relatively insensitive to the details of

intermolecular interactions and may be modeled with simple functions. Additionally,

because the thermodynamic properties are obtained by integration, the results can be

much less sensitive to their parameterization. For example, Mathias and O'Connell

(1979, 1981) found that this approach worked very well for gases dissolved in liquids.

With models for the direct correlation function integrals, the solution density and

the chemical potential, essential in phase equilibrium calculations, can be calculated or

predicted from the pressure, temperature and composition. In particular, such a

model may allow for accurate modeling of the activity coefficient of highly non-ideal

systems.

The DCFIs modeled by Mathias and O'Connell (1979, 1981) work well for liquids

containing supercritical components (dissolved gases) but apparently not as well for

condensed-phase systems (Campanella, 1984). Campanella made an attempt to model

the DCFI for vapor-liquid equilibria (VLE) and liquid-liquid equilibria (LLE) systems

in a fashion similar to that of Mathias (1978), using a hard sphere term and a

perturbation term. He found in VLE systems that the hard sphere term did not show

adequate compositional variation. Further, at low concentrations the data used to

calculate DCFIs were not accurate enough to properly determine the compositional

dependence in that region. His work was hindered by having to calculate activity

coefficient derivatives indirectly by differentiating either tabulated phase equilibria

data or data from a model, such as Wilson (1964). This has lead to uncertainty in the

actual compositional behavior of DCFIs.

The objective of the present work has been to explore the compositional behavior

of the DCFI in more detail. The relationships between DCFIs and mole fractions and









measurable binary quantities of isothermal compressibility, KT, partial molar volume,

Vi, volume, V, and activity coefficient derivatives, lead to DCFI values

from experiment. Binary data have been collected (aided by the recent compilation of

isothermal compressibilities by Huang (1986)) on non-ideal systems and a database of

DCFIs for a wide variety of systems has been created.

A problem in developing an accurate DCFI database is the collection of accurate

activity coefficient derivative data. Typically, this quantity would be calculated by

differentiating a specific activity coefficient model whose parameters are fitted to

vapor-liquid equilibrium data. However, as Campanella (1984) has pointed out, there

can be a significant difference in the activity coefficient derivative depending on the

model chosen to fit the VLE data. This effect is especially apparent in the low

concentration regions where the various activity coefficient models can give

significantly different values. Also, extrapolations of finite concentration VLE data

do not necessarily agree with directly measured infinite dilution activity coefficients

(Loblen and Prausnitz, 1982; Schreiber and Eckert, 1971), suggesting further

uncertainty in reliability of the activity coefficient models fitted to VLE data in this

region.

To eliminate the need for an intervening activity coefficient model between

experimental VLE data and the activity coefficient derivative, methods have been

explored for experimental measurements that directly yield the activity coefficient

derivative, especially in dilute regions. In addition to its use in DCFI model

development, such data could be used by others in establishing new activity

coefficient models.

Two methods for determining the activity coefficient derivatives directly have

been considered. The potential use of Rayleigh light scattering is explored in Chapter

2. Several authors (Coumou and Makor, 1964; Brown et al., 1978; Miller, 1967; Miller

and Lee, 1973; Maguire et al., 1981) have described procedures for finding









thermodynamic properties from light scattering. These involve measuring the

contribution of concentration fluctuations to the light scattering either with the ratio

of the total light scattered at an angle of 900 to the incident light or the ratio of the

scattered central Rayleigh peak to Brillouin side peaks. This leads to several different

methods for connecting light scattering measurements to thermodynamic properties,

many involving approximations that are subject to question. Therefore, a thorough

analysis of the procedure and errors involved in obtaining activity coefficients using

light scattering data has been performed. The method requires a considerable amount

of auxiliary data and appears to be limited in its accuracy while requiring a very high

refractive index difference for the chemicals being measured.

The second method considered for measuring activity coefficient derivatives is by

ultracentrifugation as described in Chapters 3 and 4. This procedure has been used

previously to measure activity coefficients (Cullinan and Lenczyk, 1969; Rau, 1975;

Johnson et al., 1959). The composition profile established during centrifugation

readily leads to the activity coefficient derivative. Chapter 3 describes the theoretical

basis and expected errors involved in determining activity coefficients from

ultracentrifugation. The method has the advantage over light scattering techniques of

only needing mixture refractive index and density data to support the centrifuge

measurements. It does require that the binary pair have a suitable difference in

density, but the refractive index difference need not be extremely high.

Chapter 4 describes the experimental measurements that have been made on five

systems: carbon disulfide in acetone, chloroform in acetone, carbon tetrachloride in

acetone, carbon tetrachloride in methanol and benzene in acetonitrile.

Ultracentrifuge measurements have been made in the 1.5% to 10% (mole) range at

temperatures from 100 C to 350 C. In support of those experiments, mixture refractive

index measurements have been taken in a differential refractometer on all systems.

Chemical potential derivatives and activity coefficient derivatives from these









measurements have been compared with vapor-liquid equilibrium data fit to the

Wilson (1964) and NRTL (Renon and Prausnitz, 1968) models.

Once activity coefficient derivatives have been obtained, a database of DCFIs can

be developed. As described in Chapter 5 sixteen non-ideal systems selected by

Campanella (1984) for study plus twelve other non-ideal systems have been examined.

For each of these systems, excess volume data, pure component compressibility data

and vapor-liquid equilibrium data were known. For nineteen of the systems, mixture

compressibility data are also available.

The DCFI can be conveniently divided into volumetric and activity coefficient

derivative terms. To calculate an "ideal" DCFI, an ideal mixing rule is assumed for the

volume and isothermal compressibility while the activity coefficient derivative term is

set to zero. By subtracting this ideal DCFI from the real DCFI, determined from

experimental quantities, an excess quantity, resembling the activity coefficient

derivative term, remains. Models for this excess DCFI can lead to the compositional

variation of these theoretically based quantities as well as aid in future modeling

efforts. Campanella (1984) pointed out that most of the activity coefficient models

have a common form when written in terms of the composition derivative. Chapter 5

explores the possibilities of modeling the excess DCFI with empirical forms.

The constant pressure activity coefficient derivative of a binary solution can be

written in terms of the three pairwise DCFIs. This grouping of DCFIs has suggested

a new empirical relationship for activity coefficients and excess Gibbs energies.















CHAPTER 2

DETERMINATION OF SOLUTION THERMODYNAMIC PROPERTIES
BY RAYLEIGH LIGHT SCATTERING

Introduction

The scattering of light by a fluid medium is caused by the electric field associated

with the light inducing periodic oscillations of the electrons in the sample. This leads

to energy being reradiated as light, scattered at different frequencies and angles

relative to the incident (Oster, 1948). By observing this scattered light as a function of

the scattering angle, information about the sample can be deduced. In particular, a

contribution to the light scattering by solutions is due to concentration fluctuations

which can be related to the concentration derivative of the chemical potential or,

equivalently, the activity coefficient (Miller, 1967). The major advantage of such an

analysis is that the material is not affected by the experiment and only very small

amounts of the sample are required.

Such measurements are attractive as an aid for the development of models for

solution behavior. In fluctuation solution theory (Kirkwood and Buff, 1951; O'Connell,

1971; 1981) the concentration derivative of the chemical potential, along with the

isothermal compressibility, and the partial molar volume are the properties needed to

determine the three pair-correlation function integrals of a binary solution. While

there are often sufficient data from vapor pressure measurements to accurately

calculate the composition derivative of the chemical potential at midrange

concentrations of the components, the accuracy deteriorates at low concentrations.

To develop a solution model based on correlation function integrals, non-traditional









means of getting this derivative must be considered. An attractive candidate appears

to be light scattering, since it leads directly to the derivative.

Previous work in this area has focused on using the technique to obtain excess

Gibbs energies by double integration of the concentration derivative obtained from

the scattering (Brown et al., 1978; Maguire et al., 1981). The conclusion was that

phase-equilibrium measurements were easier and more accurate for giving this

quantity. In particular Miller and Lee (1973) pointed out that a refractive index

difference of at least 0.2 between the pure components is required in a system to

produce reliable results by light scattering. If that is truly the case, most common

organic systems of interest would be eliminated.

Several different analyses connecting the light-scattering measurements to the

desired quantities have appeared (Brown et al., 1978; Maguire et al., 1981; Coumou and

Mackor, 1964; Miller, 1967; Miller and Lee, 1973), many involving approximations that

are subject to question. To quantitatively test whether the method can be used, the

system cyclohexane/benzene, a typical system of lower refractive index difference (An

= 0.07) for which all of the auxiliary properties are available, was examined here.

This chapter describes the basis and procedures necessary to obtain information

on excess Gibbs energies, activity coefficients and their derivatives. Included is a

sensitivity analysis to decide on the method's utility as well as delineation of all the

thermodynamic property information required for the technique.

Background

When light is scattered from a solution, there exists a central, unshifted peak of

highest intensity, known as the Rayleigh peak, and two shifted (Brillouin) peaks of

lesser intensity on either side. The concentration fluctuations contribute only to the

central Rayleigh peak (Miller, 1967). Einstein (1908) originally developed the Rayleigh

Ratio, the ratio of the intensity of the Rayleigh Peak to that of the incident light.

From Einstein's theory, the Rayleigh Ratio measured at a scattering angle of 90 can









be written as a function of the mean squared fluctuation of the dielectric constant, e.


r2
Rs 24 V((AE)2) (2-1


where R1s is the measurable isotropic portion of the Rayleigh Ratio at 900. If the

scattered light is not all depolarized, the isotropic portion must be extracted from the

total scattering with the appropriate Cabannes (1929) factor.



RIS= Ro90 "7p (2-2
( 6 + 6 pu


where p, is the depolarization ratio. For pure components ((Ae)2) was originally

approximated as a function of density only.



<(AE)') = ((Ap)2) (2-3


Coumou et al. (1964) pointed out that another thermodynamic variable (temperature or

pressure) would be required to describe this fluctuation properly. To determine which

of these variables would be most appropriate, they wrote ((Ae)2) as a function of

density and temperature as well as of density and pressure. The fluctuation in E as a

function of temperature and density is



V((AE)2) =kT (] (2-4



where



y() [KT 1 ][x(1)]2 (2-5
KS









x)= 11 +1 + r + !S (2-6
ap aT aP ap paT aP

with KT and Ks are the isothermal and adiabatic compressibilities, respectively, and n is

the refractive index. This treatment assumes that


S(de 1 aE 2n ( an (2-7
Idp K aP KT I aP T

It is also possible to write ((Ae)2) as a function of density and pressure



V((Ae)2 kTKT [1+ y(2)] (2-8



where



y(2) = + [x)]2 [1 x()]2 (2-9
X KS


and the approximation


pdE 1 E 2n an (2-10
SIp J aT p ap IT p

has been used. With data for the partial derivatives of refractive index and

compressibilities, Coumou et al. (1964) evaluated values of y0) and y(2) and compared

them to V((Ae)2) for representative substances. They found that y(2) was 2% to 10% of

V((Ae)2), whereas y(1) was 0.01% to 0.1%. Therefore, the fluctuations of pure

component dielectric constants are best described as a function of density and

temperature because the corrections can be ignored. Combining equations (2-4) and

(2-1), neglecting y(1) and changing variables (de = 2n dn) gives an expression for

scattering by a pure component:








SRD = 2-2kTn2 ( an (2-
Rs R- X4KT (211


For solutions, Coumou and Mackor (1964) described the isotropic Rayleigh scattering
as a function of density and concentration.

Ris = RD + Rc (2-12

Kirkwood and Goldberg (1950) gave the concentration contribution as


2w 2kT ( On )n/ (/0p12
Rc = 4 V x, n J- (2-13


However, Brown et al. (1978) concluded that a cross term, RCD, is required to represent

the interaction of the density and the concentration effects. Such a term had

previously been written by Dezelic (1973),

R IT 2kTKTXlX2 )T (OE )(
RcD = X4 p (2-14
2\ Iap aX2P

Using the same logic as Coumou et al. (1964), the density derivative of the dielectric

constant can be replaced with the pressure derivative of the refractive index, yielding


S2-rr2kTXlX2n2 On n (2-15
CD 4 aP X2J


Thus the full connection of experiment to chemical potential derivative can be made

from equations (2-11), (2-13), (2-15) and (2-16).


RIs = RD + Rc + RC,D (2-16


The value for Rc yields the desired concentration derivative.









While the added cross term of Brown et al. (1978) improved results over the earlier

work of Coumou and Mackor (1964) for some systems, their general contention was

that the method was not as accurate as standard phase equilibrium methods for

activity coefficient determination. This is true because of the many ancillary

measurements that must be made along with the light scattering intensity to determine

the activity coefficient. These other properties include

Rgo, the Rayleigh Ratio at scattering angle of 900,

pU, the depolarization ratio, calculated from polarizabilities,

p, the solution density,

KT, the isothermal compressibility,

n, the refractive index,

the composition derivative of the refractive index,
ax2 Ap
8an2
the pressure derivative of the refractive index.

Equation (2-16) gives a relationship between light scattering measurements and

thermodynamic properties which uses approximations in the form of the cross-term

and of the density derivative of the refractive index. A parallel, but more general

development has also been done.

Rigorous Thermodynamic Fluctuation Development

The rigorous treatment purposely by-passes the density and concentration

crossterms (Miller, 1967). Instead, fluctuations in total entropy from variations in the

temperature, volume, entropy, pressure, chemical potential of the solute and the

number of moles of the solute are considered (Landau and Liftshitz, 1980).


-1
AST = 1 (ATAS APAV + Aj.2Ap2) (2-17
2T


This fluctuation in total entropy can be written as a general function of any three of

the state variables listed above,








13 3
AS 1 i1 Bi xi xj
2 i=1 i=1


where


(xi xj) = k B-


(2-18


(2-19


Choosing the three independent variables as T, P, and P2 = x2 p, gives the others as



AS AT + AP+ Ap (2-20
SaT 2 aP Tp aP2 )PT


AV= AT+ 8 AP + Ap2 (2-21
laT p,P2 aP Jp2 ap2 PIT


AiL2 L) AT+ 1v AP + fL2 Ap2 (2-22
I T pp a, 9P p aP2 P T

By inserting equations (2-20)-(2-22) into equation (2-17) and using elementary

thermodynamics, we find


l[Cp
2 T 2


(AT)2- 2VATAP + T (AP)2 + (Ap2)2
ST T T lP2


Using equations (2-18) and (2-23), the following fluctuations can be identified,


kT2
((AT)2) -


(ATAP) -
KTCV


((AP)2) -=
VKS


(2-23





(2-24a



(2-24b


(2-24c









((Ap,)2) = k T (2-24d



(ATAp2) = (APAp2) = 0 (2-24e


The same connection between this fluctuation theory and light scattering can be made

through equations (2-1) and (2-2).

It is known that adiabatic pressure fluctuations do not mix with entropy and

composition fluctuations; they contribute only to the Brillouin peaks. These are given

by



((AE)1DIAB)= f ((AP)2) (2-25


Substituting


=a( aJ TVtp + a( (2-26
SE 2 2 2


results in



C ^/YC a P'P2


+2kT2 ( ) + kV (2-27
KTCV 'T Pp2 I KsV aP 2T

Using an equation of the form of (2-1) yields the ratio of the intensity of the Brillouin

peaks to the incident light from equation (2-27). However, a more useful form exists.

The entire dielectric fluctuation is, in terms of fluctuations of the state variables, T, P,

and p2,








((AE)2) )2 ((AT)2)) + 2 (ATAP)
aT 2 T (P tp


+ ( ((AP)2) + ((A)2) (2-28
dP JT,p P2 T,P

Here, only one of the three cross terms is non-zero. From equations (2-24a-e), this can
be converted to


kT2 a + 2kTT P9(e aE
Cv T p,p2 KTCV IT Jpp 2 P2
k-T & E P2 )TP
+ -)2 + kT (2-29
VKs aP T, p2 aP2T P

The contribution to the central or Rayleigh peak is given by


((Ae)2) = ((Ae)2) ((Ae)2DIAB) (2-30


Subtracting (2-27) from (2-29) leads to


C) kT2 + k nT P2 (2-31
( ) V aJ k 1 JP,p 2/ aP 2 )TP ,

Because the ratio of the Rayleigh to the Brillouin peaks allows the use of two
intensities that are more similar than those of scattered to incident light, their ratio, J,
is used.


J = ic / 2 iB = ((Ae)2) / ((AE)2DIAB) (2-32


Defining the variable f in terms of x() from equation (2-6) (Miller and Lee, 1968) gives


2x_() yc [x(112
1 x(+) (1 x0())2









Equation (2-32) can be rewritten to include both the temperature and pressure
derivatives of the refractive index


/c 1 Cp (an 2 / n 2(
J + f (c -1) x1 x(1 +fTc) (2-34
1l+ fyc T Tx2 aT 8x2TP


where


Cp
T/c (2-35
Cv

Often (Miller, 1967; Coumou and Mackor, 1964), it is assumed that x'(1 = 0, giving


C=Ycl +n 2c/ [( xn i2( a 2 ](
JT= c x-/1 + [[c 1) xTP] (2-36
T I dxa )T[,]

The equation connecting light scattering data to thermodynamic properties is
(2-34) or (2-36). It assumes that there are no internal molecular relaxation processes
that contribute to the scattering. Unfortunately, this is not normally the case.

Light Scattering in Relaxing Solutions
Miller and Lee (1968) noted that internal molecular relaxations in a liquid can
transfer energy from the central to the outer peaks. Fishman and Mountain (1970)

showed how to correct the dynamic changes in pressure, temperature and

concentration for this effect using linearized hydrodynamic equations for momentum,

energy and diffusion transport. Miller and Lee (1973) implemented the Fishman and
Mountain expressions to find the portion of the scattered light transferred from the
side peak to the central peak, (Ai). Then, the measured ratio of the Rayleigh to the

Brillouin peaks, J(k), becomes


J(0) + A,
J(k) = (2-37
1 Ai









where J(0) is the ratio to be used in equations (2-34) and (2-36), and Ai is

A, = A(k) J(0) + B(k) (2-38

with


A(k) = [(v2 v2) k T2 + ()4- ( )2] / [ (vskVT)2 + (! 4j (2-39a



B(k) = [(v v) k) 72 + ( 2 [(vskvT)2 + ()] (2-39b


Here, the physical quantities are

vo: infinite frequency velocity of sound

vo: zero frequency velocity of sound

vs: velocity of sound in solution

T: single relaxation time for the internal degrees of freedom

kv: change in wave vector by the relaxation

The value of vs can be determined from the frequency shift, or the separation, Av,

between the Rayleigh and Brillouin peaks.


cAu
VS-
2v sin(0/2) (2-40


where the additional physical properties are
v: frequency of incident light

c: speed of light
0: scattering angle

The value of kv can be obtained from the incident light wave length Xo and 0.


kv = ) sin(o/2) (2-41










The ratio of scattered peaks, J(0), for use in equations (2-34) or (2-36) can be written in

terms of the measured ratio, J(k) and the relaxation functions, A(k) and B(k):


J) = J(k) [1 B(k)] B(k)
J(0) = (2-42
1 + A(k) + J(k) A(k)

Combination of equations (2-42) and (2-34) gives a completely rigorous treatment.

Fishman and Mountain (1970) point out that A(k) is small and can sometimes be

neglected, allowing measurement of J(k) at several angles to be used to establish B(k)

and J(0) simultaneously. This eliminates the need for relaxation data but requires a

light scattering apparatus that can measure scattering at different angles. An

alternative simplification was attempted by Miller and Lee (1973). They ignore A(k)

and force T, v0 and vo to vary linearly with concentration. They then obtain



J(k) = c r-1 + (yc- 1) rKxi x2 1 + x2 a ] (2-43



where


2
r 2
v2 (2-44a


CK an an



This more general treatment arrives at equations (2-42) and (2-34) without

approximations. There are four different treatments available to relate light

scattering to thermodynamic properties. Three of the four methods involve

approximations which can lead to different results, but require less input data. Our

objective was to do the most complete analysis possible and determine what








quantitative information can be obtained while listing the total amount of input

information required for such a treatment.

Calculation of Thermodynamic Properties
Fluctuation Properties

Recently, several authors (Iwasaka et al., 1976; Kato et al., 1982; Kato and
Fujiyama, 1976; Kato, 1984) have used the early theory of Miller (1967) to describe the

concentration fluctuations in solutions from light scattering measurements. Iwasaka,

et al. (1976) calculated the concentration fluctuations, V((Ax2)2) by solving for ((Ap2)2)

in equation (2-28) for the system, CCI4 / CS2. Miller and Lee (1968; 1973) indicated that

both these substances are relaxing fluids, an effect ignored by Iwasaka and Kato. Kato

(1984) calculated the Kirkwood-Buff (1951) integrals over the total correlation function

from chemical potential derivatives and other fluctuation properties. As we establish

in more detail later, the stated accuracy of about 10% in the light scattering

measurement is not sufficient for determination of reliable chemical potential

derivatives, even when relaxation is included. In fact, without consideration of

relaxation effects, the results are expected to be seriously in error.

Activity Coefficients

Most investigators ultimately use a formulation to calculate solution activity

coefficients. The relationship of most generality and utility for the activity

coefficient derivative is


( alny2 c 1 Cpx an /ran r [,r -[nc- 1 l (245
lax,] =lif1 RT2 [[x// "Tjj/ l['J(0)--1 (2-45
ax2 1+ f11 c RT' ax2 aT 1+ ffycJ X2

where J(0) is defined by equation (2-42). The most recent use of this complete theory

of Miller and Lee and of Fishman and Mountain is that of Maguire et al. (1981), who

used J(0) values from light scattering to obtain activity coefficients and excess Gibbs

energies to compare with those from experimental vapor-liquid equilibrium data.








The activity coefficient can be calculated from


Ix ( +fJID/ J(0 ]) + f'c /x2 dx2 (2-46



The quantity, JID, is a convenient grouping of variables that resembles J(0) for an

ideal solution:


TYc -1 + CpX1X2 an an ]2 (247
1 + fyc RT2 aX2 aT

where f is defined by equation (2-33). This is not the actual J(0) for an ideal solution

because there are still many data in JID that are dependent on the composition of the

real solution.

For comparison to vapor-liquid equilibria data, equation (2-34) can be used with

chemical potential derivatives calculated from VLE data to compare with J(0),

(JTHERM)*

Before discussing the results of the calculations by Maguire et al. (1981) and their

consequences, we recapitulate, in detail the elaborate procedure and extensive set of

property values required to obtain activity coefficients from light scattering data. We

also duplicate their calculations for the cyclohexane/benzene system, which was

chosen because most of the data have been measured. In other systems,

approximations would be required, leading to greater errors, we believe.

Activity Coefficients from Equation (2-46)

Listed below are the calculational steps used by Maguire et al. (1981) to obtain the

activity coefficients and excess Gibbs energies for the cyclohexane/benzene system.

In cases of ambiguity in the publication, we state our assumptions about what was

done.









1. J(k) from measured light scattering, at T, P, x (Table 2-1, Column 2)

From Table II of Maguire et al. (1981).

Assumed P = 1 bar, T = 296 K as in Brown et al. (1978).

2. kv from equation (2-41) (Table 2-1, Column 3)

a. Refractive index, n, at T, x (Table 2-1, Column 4).

Interpolated from Table 4 of Brown et al.(1978)

b. Wavelength of incident light, \X.

Assumed to be that of Brown et al. (1978), 5.46x107 m.

c. Scattering angle, 0.

Assumed to be 90 degrees.

3. Vs, from equation (240) (Table 2-1, Column 5)

a. Frequency of Brillouin Shift, Au not reported.

Used Figure 4 for vs(k) vs. composition of

Maguire et al.(1981). Table 2-2 lists values read off graph.

Values linearly interpolated to Table 2-1 compositions.

4. A(k) and B(k) from equations (2-39a,b) (Table 2-1, Columns 6 and 7)

a. vo at T, x. (Table 2-1, Column 8)

Pure benzene from measurements of Eastman et al. (1969).

Pure cyclohexane calculated using A(k) from Figure 3,

J(0) and J(k) from Table I, all from Maguire et al. (1981),

and B(k) calculated using equation (2-42).

Final value calculated from equation (2-39b).

Linear composition dependence assumed.

b. Vo at T, x (Table 2-1, column 9)

Pure benzene from measurements of Eastman et al. (1969).

Pure cyclohexane from measurements of Dorfmuller et al. (1976).









Table 2-1
Physical Properties Needed for Light Scattering Analysis
of Cyclohexane(1)/Benzene(2) Solutions


A(k) B(k)


x2 J(k) kv
m-1
x 10-7


V0 Vo T
m/s m/s sec
x 101


1 2 3 4 5 6 7 8 9 10

0.000 0.55 2.32 1.4285 1345.0 0.0166 0.0795 1395.7 1280.0 4.10
0.102 0.68 2.33 1.4321 1345.0 0.0391 0.0995 1411.5 1284.5 4.30
0.202 0.92 2.34 1.4366 1347.1 0.0588 0.1184 1426.9 1288.9 4.49
0.296 1.07 2.35 1.4412 1351.8 0.0736 0.1352 1441.4 1293.0 4.68
0.304 1.09 2.35 1.4417 1352.5 0.0744 0.1365 1442.6 1293.4 4.69
0.350 1.23 2.35 1.4442 1358.5 0.0764 0.1434 1449.7 1295.4 4.78
0.488 1.38 2.36 1.4532 1382.6 0.0738 0.1623 1471.0 1301.5 5.05
0.651 1.46 2.39 1.4656 1420.2 0.0584 0.1824 1496.2 1308.6 5.37
0.747 1.41 2.40 1.4739 1444.1 0.0474 0.1939 1511.0 1312.9 5.56
0.850 1.29 2.41 1.4836 1480.0 0.0222 0.2031 1526.9 1317.4 5.76
0.900 1.16 2.42 1.4887 1500.0 0.0071 0.2068 1534.6 1319.6 5.86
1.000 0.90 2.44 1.4990 1540.0 -0.0219 0.2140 1550.0 1324.0 6.05


Table 2-1 Continued

X2 J(0) J(0)t JID f CPr (a- (al} c-1 (an) KT PP

J/molK Pa' K-1 Pa-1 K-1
x 102 x l10 x 104 x 109 x 103

1 11 12 13 14 15 16 17 18 19 20 21

0.000 0.416 0.378 0.378 -1.47 155.0 0.0314 5.08 0.3700 -5.38 1.15 1.21
0.102 0.481 0.436 0.410 0.48 153.1 0.0394 5.12 0.3751 -5.46 1.13 1.21
0.202 0.623 0.571 0.458 2.24 151.2 0.0472 5.16 0.3801 -5.54 1.12 1.21
0.296 0.686 0.628 0.510 4.26 149.4 0.0546 5.20 0.3848 -5.61 1.10 1.21
0.304 0.696 0.638 0.513 4.64 149.2 0.0552 5.21 0.3852 -5.62 1.10 1.21
0.350 0.778 0.717 0.530 7.01 148.4 0.0588 5.24 0.3875 -5.66 1.08 1.21
0.488 0.845 0.776 0.616 7.55 145.7 0.0696 5.27 0.3944 -5.77 1.07 1.21
0.651 0.884 0.804 0.673 7.99 142.6 0.0823 5.29 0.4026 -5.93 1.04 1.21
0.747 0.846 0.760 0.669 7.38 140.8 0.0898 5.27 0.4074 -6.02 1.02 1.21
0.850 0.785 0.692 0.622 5.22 138.9 0.0979 5.24 0.4125 -6.14 1.01 1.21
0.900 0.703 0.606 0.571 4.24 137.9 0.1018 5.24 0.4150 -6.21 1.00 1.21
1.000 0.515 0.412 0.412 1.35 136.0 0.1096 5.23 0.4200 -6.38 0.99 1.21


tJ(0) Adjusted to Match JID at Pure Components









Linear composition dependence assumed.

c. T at T, x (Table 2-1, column 10).

Pure benzene calculated using A(k) from Figure 3 and

J(0) and J(k) from Table I, all from Maguire et al. (1981),

and B(k) was calculated using equation (2-42).

Final value calculated from equation (2-39b).

Pure cyclohexane from measurements of Dorfmuller et al. (1976).

Linear composition dependence assumed.

5. f from equation (2-33) (Table 2-1, Column 14)

x) from equation (2-5).

6. J(0) from equation (2-42) (Table 2-1, Columns 11 & 12)

Calculated values, Column 11.

Linearly adjusted to match pure component JD, Column 12

7. JID from equation (2-47) (Table 2-1, Column 13)

a. Cp at T, x (Table 2-1, Column 15)

Pure component values from Table III, Maguire et al. (1981)

Linear composition dependence assumed.

b. a- (Table 2-1, column 16)
8.X2 JT
Analytical equation of Brown et al. (1978)

c. a- (Table 2-1, Column 17).

Interpolated from Table 4, Brown et al. (1978)

d. (yc 1) or (Cp Cv)/Cv at T, x (Table 2-1, column 18)

Pure component values from Table I, Maguire et al. (1981).

Linear composition dependence assumed.
(a n
e. -) (Table 2-1, Column 19)

Interpolated from Table 2, Coumou and Mackor (1964).









f. KT (Table 2-1, Column 20)

Pure component data from Table III, Maguire et al. (1981).

Linear composition dependence assumed.

g. ap at T, x (Table 2-1, Column 21)

Pure component data from Table III, Maguire et al. (1981)

Linear composition dependence assumed.

8. Activity coefficients from equation (2-46) (Table 2-3, Column 3)

The integral was calculated using a trapezoidal rule.

Experimental data of Nagata (1962) (Table 2-3, Column 2)

Table 2-2
Speed of Sound in Cyclohexane/Benzene Mixtures
at 296 K and 1 atm

X2 VS X2 Vs
m/s m/s

0.0 1345 0.6 1410
0.1 1345 0.7 1430
0.2 1347 0.8 1460
0.3 1352 0.9 1500
0.4 1365 1.0 1540
0.5 1385


Comparisons with Phase Equilibrium Measurements

After duplicating the calculations of Maguire et al. (1981) in this way, a

quantitative comparison can be made of this method with vapor-liquid equilibrium

measurements. The values of In-y, are shown in Figure 2-1 and listed in Table 2-3. The

absolute average error is 0.25 or 25% in the activity coefficient. However, much

greater errors occur at low concentrations because the integration in equation (2-46)

accumulates errors. We expect that the results are not within the experimental

accuracy of the vapor-liquid equilibrium measurements, which should be correct to

within 2%. Figure 2-2 shows the scattering J(0) compared with JTHERM from analysis












1.4-


.3- ---- From Light

1.2Scattering
1.2 -
..... ....... From VLE Data

1.1 -

1-


0.9-


0.8-


In 2 0.7-

0.6-

0.5-


0.4-


0.3 -

0.2 "

0.1 "-'" .



0 0.2 0.4 0.6 0.8
Mole Fraction, x2
Figure 2-1 Activity Coefficients Calculated From Vapor-Liquid
Equilibrium Data of Nagata (1962) and From Light Scattering
Data For the Cyclohexane(1) / Benzene(2) System at 23 C.









of the vapor-liquid equilibrium measurements (Table 2-3, Column 6). These have an

average absolute difference of 0.035 or 5% which leads to the 0.25 average difference

in In-y2.


Table 2-3
Comparison of Activity Coefficients From
Light Scattering and Vapor-Liquid Equilibrium Data
For the System Cyclohexane(1)/Benzene(2) at 296 K


X2 ln/y2
from
VLEt


ln Y2
from
light
scatter


ln-y2
abs.
diff


L OlnY2)
from
VLEt


JTHERM
from
VLEt


data

1 2 3 4 5 6


0.000
0.102
0.202
0.296
0.304
0.350
0.488
0.651
0.747
0.850
0.900
1.000


0.335
0.275
0.222
0.176
0.172
0.151
0.096
0.046
0.025
0.009
0.004
0.000


1.378
0.965
0.624
0.424
0.412
0.344
0.188
0.087
0.050
0.021
0.009
0.000


1.043
0.689
0.403
0.248
0.240
0.192
0.092
0.041
0.026
0.012
0.005
0.000


-0.607
-0.561
-0.513
-0.465
-0.461
-0.436
-0.358
-0.256
-0.191
-0.117
-0.079
0.000


0.378
0.412
0.469
0.533
0.538
0.562
0.670
0.735
0.719
0.648
0.584
0.412


t Nagata (1962)


Sensitivity of Activity Coefficient Calculations


To determine the effect of uncertainties in the light scattering measurements, an

additive random error was introduced into the J(k) values and these were used in a

second set of calculations


J(k)ERROR = J(k)EXPT + F x NR


(2-48


where J(k)EXPT is the measured J(k) and NR is a random number between -0.5 and 0.5.

The factor, F, was adjusted until the standard deviation of the error between the















1.4 -. U
/ \
/ \
/ \
1.3 /-
/ \

1.2- / \





/ \
1. / 1
/ \

J /

0.9 ... ...



S...+" '...
0.8 -









0.4 .0



A JTMERM


0.37


Mole Fraction, x2

Figure 2-2 Various Rayleigh to Brillouin Scattering Ratios (J) For the
Cyclohexane(l)0.6 / / Benzene(2) System at 23 C.
Cyclohexane(1) I Benzene(2) System at 230 C.









J(k)ERROR and J(k)EXPT was either 0.05 or 0.10. The 12 random number values used had

an average of -0.113. This means that we introduced both random and systematic

errors.

The results of this numerical experiment are given in Figures 2-3 and 2-4 for the

calculated J(0) and In-y2 while Table 2-4 summarizes the results for the compositions of

Table 2-1. Part of the errors in J(k) are passed on to J(0) and, subsequently to ln(y2).

Since other input parameters, such as the speed of sound in the solution or the

relaxation parameters, may be difficult to find, we vary these individually by 5-20% so

their impact on J(0) and In(y2) can be detected. Table 2-5 shows that rough

approximations to these parameters, particularly for v T and B(k), will not be good

enough for obtaining activity coefficients as accurately as from vapor-liquid

equilibrium data.







Table 2-4
Errors in J(0) and Iny2 Caused by Random Errors in J(k)

Standard Deviation Standard Deviation Average Absolute
of in Error Error
Random Error in J(k) J(0) Calculated Iny2

0.00 0.041 0.249
0.05 0.046 0.243
0.10 0.092 0.598















0.9-


0.8-


0.7- 0


0.6-


0.5-
J
0.4- .. --


0
0.3-


0.2-
......... JTHERM
J(0)*
0.1 + 5% Error
0 10% Error

o I I I I I I
0 0.2 0.4 0.6 0.8 1
Mole Fraction, X2

Figure 2-3 Values of J(0)* Calculated From Measured Values of J(k) For the
System Cyclohexane(1) / Benzene(2) With Various Random
Errors Added at 230 C.

















.......... .......

-e----

- -4 -
- 4. --


Light Scatter
VLE Data
10 % Error
5% Error


/

I ....
.7.


-
I
I
I
I
I
I
I


I I
0.2


I I
0.4


I I
0.6


I I
0.8


Mole Fraction, x2
Activity Coefficient in the Cyclohexane(1) / Benzene(2) System
Calculated from Light Scattering, With Various Random Errors
Added to J(0)*.


In 12


Figure 2-4


I









Table 2-5
Error in In-y2 Caused by Bad Input Values
Standard Average
Deviation Absolute
Errored Table 1 In Error of Error
Inputs (%t) Values J(0) Iny2

All Original Values 0.033 0.249
Cyclohexane v,, 1100 (-15) 1280 0.063 0.330
Cyclohexane v
oCyclohexane v 1600 (+15) 1395 0.047 0.366

Cyclohexane T 1x10-1 4.1x10-" 0.160 2.02
Cyclohexane T 1x10-1 4.1x10-" 0.036 0.215
Benzene B(k) 0.205 (-4) 0.214 0.032 0.260
Benzene B(k) 0.225 (+4) 0.214 0.035 0.229
Cyclohexane B(k) 0.0625(-20) 0.0795 0.037 0.275
Cyclohexane B(k) 0.095 (+20) 0.0795 0.030 0.224

t Deviations from Table 1 Values

Summary

The measurement of concentration fluctuations from light scattering is a possible

route for obtaining activity coefficients. An investigation was made of

cyclohexane/benzene, a typical organic system, for which all necessary data for the

rigorous analysis, including relaxation information, were available. The results are

seriously in error from reliable vapor-liquid equilibrium data.

Miller and Lee (1973) state that large excess scattering (solution scattering over

pure component values) is needed to obtain activity coefficients. They point out that

systems with a pure component refractive index greater than 0.2 are required for

reliable results. For pure components here, J(0) is approximately TYc 1 (the

Landau-Placzek formula). The difference between the maximum J(0) observed for the

cyclohexane/benzene system and -ic 1 is approximately 0.40, evidently not great

enough to reliably determine activity coefficients. Miller and Lee (1973) indicate that

in systems with large pure component refractive index differences (An > 0.2) the


excess scattering can be adequate.









The results of this analysis, where An = 0.07, are consistent with the prediction of

Miller and Lee. The activity coefficients of the cyclohexane/benzene system cannot

be determined by light scattering even if all information is available. If the ancillary

information is erroneous, it is clear that the sensitivity of the results makes light

scattering unlikely to be viable for determination of activity coefficient derivatives in

any system. Finally, it should also be noted that instrument expense and dust-free

sample preparation are not insignificant factors to be considered in the

light-scattering experiment.

It might be appropriate to test the method with a system having An > 0.2 for

which all the auxiliary properties are available in order to determine the real

possibilities of the technique. However, such a high refractive index difference

requirement would limit the number of real systems so severely that another

measurement must be sought. Equilibrium sedimentation in an ultracentrifuge is such

a technique. As discussed in the next chapter, it can be used for systems where An >

0.06 and the extra required information is much less.















CHAPTER 3

CONCENTRATION DERIVATIVES OF DILUTE SOLUTION ACTIVITY
COEFFICIENTS USING AN ANALYTICAL ULTRACENTRIFUGE

Introduction

While light scattering is apparently not accurate enough to provide useful

information about solution activity coefficients, other possibilities besides traditional

vapor-liquid equilibrium (VLE) measurements (Gmehling et al., 1977, Wichterle et al.,

1973) need to be explored. In particular, accurate measurements need to be made in

the 2.5% to 10% (mole) range to determine activity coefficients and their composition

derivatives more accurately. Data in this region are of interest because, as Loblen and

Prausnitz (1982) and Schreiber and Eckert (1971) point out, extrapolations of finite

concentration VLE data to infinite dilution do not usually agree with the direct

measurements of -y by differential ebulliometry and chromatography. For example,

Figure 3-1 shows VLE data for the carbon disulfide/acetone system (Litvinov, 1952) fit

to the Wilson (1964), NRTL (Renon and Prausnitz, 1968) and Margules (Van Ness and

Abbott, 1982) activity coefficient models. Clearly, all of the models fit the mid-range

concentration data equally well. However, at low concentrations, the three equations

deviate from each another, the departure being characteristic of the equation.

The differences in these models are even more pronounced when examined at the

derivative level. As seen in Figure 3-2, when the same VLE data are fit to different

activity coefficient models, the compositional derivatives can be quite different. Here

the Margules model is not even monotonic. Besides the sensitivity of activity

coefficient derivatives to discriminate between models, such data lead directly to the

theoretical quantities of fluctuation solution theory (Kirkwood and Buff, 1951;









1.5

1.4
\
1.3 -\

1.2 ----- Wilson

1.1 Margules

NRTL
1 -
x A Litvinov, 1952
0.9-- VLE Data

0.8-

0.7 -

0.6-

0.5 -

0.4-

0.53

0.2-

0.1 -

0-

-0.1 1
-0.1 I I I 1 I I I
0 0.2 0.4 0.6 0.8

Mole Fraction, x,



Figure 3-1 Various Activity Coefficient Models Fit to the VLE Data of
Litvinov (1952) For the Carbon Disulfide(l) / Acetone(2) System
at 250 C.










0






/I




/


-1.2-
dI-1.4- /"
-0.6 -









-1.2- ,






-1.8 6

/
SWilson
-2 -
Morgule!

-2.2-
/ __ NRTL

-2.4- /
/
/

-2.6 -----

0 0.2 0.4 0.6 0.8

Mole Fraction, x,


Figure 3-2 Activity Coefficient Derivatives From Various Models Using
Parameters Fit to the VLE Data of Litvinov (1952). For the
Carbon Disulfide (1) / Acetone (2) System at 25 C.









O'Connell, 1971; 1981; Campanella, 1984). In particular, integrals of the statistical

mechanical direct correlation function (DCFI) can help future efforts toward solution

models. For example, Campanella (1984) has noted that the derivatives of common

activity coefficient models have a common mathematical form (ratios of polynomials

in mole fraction). Such observations could lead to an accurate and general empirical

activity coefficient model.

Thus, an experimental method which avoids use of a model and directly produces

values of activity coefficient derivatives in dilute solution would be highly desirable.

This would allow the determination of which, if any, of the activity coefficient

models are most accurate and provide a basis for future models. Equilibrium

sedimentation in an analytical ultracentrifuge provides such an experiment.

Equilibrium sedimentation distributes a heavier solute in a less dense solvent.

Centrifugal force acts to push the heavier solute to the bottom of the cell. Diffusive

forces tend to redistribute the solute. After a time these two forces balance each

other and establish a compositional profile which is lower in the solute nearer the top

of the cell. The experimental results are obtained from Rayleigh interference optics

which gives composition profile while the sample is still in motion. A specially

designed sample cell has two chambers or sectors, one is a reference which is filled

with solvent while the other has the sample mixture. Laser light is split to shine

through both chambers simultaneously. When recombined this light produces an

interference pattern corresponding to the radial refractive index differences of the

two cell chambers. The refractive index difference is directly related to the

composition. An illustration of the sample cell and light path is given in Figure 3-3.

Initially the solution in the cell is homogeneous and the ultracentrifuge is

operated in a steady fashion until equilibrium is established. After rotation starts, the

time to reach a degree of equilibrium distribution (t8) is found from










Ref Cell


Sample
Cell -


Center of
Rotation


Section


Recombined Laser
Light, Interference
Pattern

Window---

Cell --

Window-'--


Laser


x7~





I-I I-i


Ligh


Figure 3-3 Ultracentrifuge Double Sectored Cell Showing the- Location of
the Sample Solution and Reference Solvent Relative to the Laser
Light Path.


AA


A







(Ar)2 2
t8 -4T2-D r(1 8) (3-1

where D is the diffusion coefficient and 8 is the degree of equilibrium. For this
experiment, a 99.9% degree of equilibrium (8 = 0.999) can be considered adequate.
The solute (1) activity coefficient derivative is obtained from the isothermal
derivative of the chemical potential being zero at equilibrium. In the ultracentrifuge
the chemical potential is a function of distance of the solution from the axis of
rotation, r, the pressure, P, and the mole fraction of solute, xj.

dVL dr+ f-, dP+ "' dx = 0 (3-2
d Or TP,xl ( Trx OX TP.r

By substituting in known quantities for the partial derivatives, writing the pressure
derivative in a centrifugal field in terms of radius and rearranging, the activity
coefficient derivative can be found.
( R1 1 (alnh1
] =RT + J (3-3
ax1 T,,r Xl Ox T,P
LV-1 V1 (3-4
T -rr,x1

where V1 is the solute partial molar volume. In a centrifugal field,

dP = p wo2 r dr (3-5

where o is the angular velocity in radians per second.

(Ox'1) =-M, r (3-6
IOr JTPX1
where M, is the solute molecular weight. Substituting equations (3-3), (3-4), (3-5) and
(3-6) into equation (3-2) and rearranging gives

fOln71 _T (M -pV1) w2r (dr 1 (3-7
ax, RT Idxj J x1
In equation (3-7) the term (MI pV1) is called the sedimentation parameter. It is a









function only of the physical properties of the compounds used and is determined

from other measurements. The angular velocity, w, and radius are measured directly.

The concentration profile dr/dx1 is determined from the interference fringes.

High speed centrifugation has been used in the past by Cullinan and co-workers

(Cullinan, 1968; Cullinan and Lenczyk, 1969; Sethy and Cullinan, 1972 and Rau, 1975)

for obtaining activity coefficient data. However, they used a preparatory

ultracentrifuge, requiring that the machine be stopped and samples withdrawn by

syringe. Besides the possibility of causing disturbances in the distribution, this

technique limited their work to mid-range concentrations where vapor-liquid

equilibrium (VLE) measurements are of comparable accuracy and sensitivity. In their

initial work, Cullinan and Lenczyk (1969) used the method for the system

hexane-carbon tetrachloride. They assumed the sedimentation factor (M1 pV-)

remained constant over the range of hexane compositions (35% to 65%) and obtained

results within 5% of those calculated from VLE (Christian et al., 1960). Later, Sethy

and Cullinan (1972) and Rau (1975) used the same method on the carbon

tetrachloride-acetone system. By substituting the composition derivative of the

Wilson (1964) activity coefficient model into equation (3-7), centrifuge data over nearly

the entire composition range were regressed to obtain Wilson (1964) parameters.

Figure 3-4 shows activity coefficient derivatives of Rau's (1975) experiments on this

system. They are very similar to the results obtained using the Wilson (1964)

parameters reported by Prausnitz et al. (1967) indicating that the method is reliable for

midrange concentrations.

The ultracentrifuge, equipped with a conventional light source such as a mercury

arc lamp, has been used to measure thermodynamic properties and molecular weights

of biochemicals, polymers and salt solutions (Johnson et al., 1954; Richards and

Schachman, 1959; Johnson et al., 1959; Nichol and Winzor, 1976; Hwan et al., 1979; Hsu,

1981). Earlier investigators used Schlieren optics for the refractive index profile in the









-0.4-


-0.5-





-0.6-





-0.7 -

dirlyi
dlnX1

dx1

-0.8 -





-0.9 -


0.38


0 Rai, 1975


Prausnitz et al., 1967


--


0 0

fl


D
a


0.42


I I
0.46


0.5


0.54


I I
0.58


I I
0.62


Mole Fraction, x1

Figure 3-4 Activity Coefficient Derivative Data Calculated From the
Ultracentrifuge Data of Rau (1975) For the System Carbon
Tetrachloride(1) / Acetone(2). Also Shown is the Wilson Model
Fit to the Ultracentrifuge Data of Rau and to VLE Data, as
Given by Prausnitz (1967).


I









sedimented solution. More recent investigators (Richards and Schachman, 1959;

Johnson et al., 1959) point out that Rayleigh interference optics is more accurate for

equilibrium sedimentation experiments. Conventional light is not coherent enough to

give an interference pattern with the non-electrolyte, low molecular weight organic

systems of interest here. Williams (1972; 1978) describes a laser light source which

yields Rayleigh interference patterns from these systems.

Equipment

Apparatus

A Beckman Model E analytical ultracentrifuge (serial #685), modified with a laser

light source, was used in the present studies. This ultracentrifuge can spin a sample at

speeds from 10,000 rpm to greater than 50,000 rpm. The rotor used in these

experiments was an AN-D type (serial #3581). This is an aluminum rotor with a

maximum safe speed of 52,000 rpm when new. The high stress of continued operation

at high speed weakens the metal in the rotor, requiring that the maximum safe

operating speed be lowered with age. The sample cells were double sector (chamber)

type, 12 mm deep. The aluminum cell was manufactured by Beckman and the

titanium cell was custom manufactured by Central Machine Products, Gainesville, FL.

The sapphire cell windows were from Adolf Meller Company, Providence, RI. The

window liners were originally thin (15 mils) strips of PVC as suggested by Yphantis

(Ansevin et al., 1970) to cut down on the stress distortion of the windows at high

speed. However, the organic solvents studied here attacked the PVC. Therefore,

teflon of approximately the same thickness was substituted. All other cell parts were

standard Beckman elements. Appendix A gives more details of the cell assembly and

operation of the ultracentrifuge.

The ultracentrifuge has a vacuum chamber which can be evacuated to about 1

Kpa to allow the rotor to spin with very little air resistance which would raise the

rotor and sample temperature. The chamber is controlled at temperatures from 100 C









to 350 C 0.50 C using a heating element in the bottom of the chamber and a

refrigeration system. The temperature of the rotor is sensed by a thermistor in the

base of the rotor. For details of the thermistor calibration, see Appendix A.

The optical system used to detect the concentration profile is shown in Figure 3-5.

Its design was based on that of Williams (1972; 1978; 1985). Alignment was

accomplished by combining the procedures of several authors (Rees et al., 1974;

Richards et al., 1971a; 1971b; Gropper, 1964; Dyson, 1970) with helpful suggestions of

Williams (1985). The resulting alignment procedure is given in Appendix B. In short,

the optics consist of a Spectra-Physics 5 milliwatt HeNe laser (Model 105-1) mounted

on the ultracentrifuge frame. The laser light is expanded with a spatial filter and

passes through the rotating sample, just as conventional light would. The interference

pattern between the solution side of the cell and the pure solvent (reference) side is

established using a Rayleigh interference mask.

The fringe pattern was exposed on a strip of Technical Pan 2415, Estar-AH based

Kodak film held in place with a custom made film holder and developed using

standard darkroom techniques with Kodak D-76 developer and Kodak fixer. The

radial location of each shifted fringe is then determined with a Nikon Shadowgraph,

Model 6C (#7244) microcomparator. Two reference marks on the image, whose radial

locations are known exactly, yield the actual radial distance to each shifted fringe.

Composition Detection

From the photographic negative (illustrated in Figures 3-6 and 3-7) the

composition profile in the cell at equilibrium is determined. Each fringe, j, along the

radius, r, that is shifted from the horizontal is equivalent to a constant refractive

index change, An.

j(r) = An dl/ (3-8


where d is the cell depth and X is the light wavelength, 632.8 nm.











HeNe
Laser-7


Cylindrical
Lens




Camera
Lens Photo Plate

I- Condensing Lens


S-- Interference


Mask


Sample Cell
(in Rotor)




Collimating Lens


Mirro


Mirror


Figure 3-5 Schematic of the Laser Optics Used in the Ultracentrifuge for
Composition Detection.






























Figure 3-6 Laser Light Fringe Photo From the Ultracentrifuge, Showing a
Sample of Carbon Disulfide / Acetone. The Initial
Concentration of Carbon Disulfide was 3.56% (Mole) and the
Speed of Rotation was 21739 RPM.


Figure 3-7 Laser Light Fringe Photo From the Ultracentrifuge, Showing a
Sample of Carbon Disulfide / Acetone. The Initial
Concentration of Carbon Disulfide was 3.56% (Mole) and the
Speed of Rotation was 37000 RPM.









By measuring the horizontal distance between each of the fringes, a relative

refractive index can be determined as a function of position, r. Because the

concentration at the surface is finite, the actual refractive index profile and the

concentration profile must be determined by material balance. By separately

measuring the refractive index as a function of solute mole fraction, n(x1), and

knowing the initial mole fraction, xy', and volume of the cell, v, the actual composition

profile, xl(r), can be determined. The following equations give the conversion from

fringes to composition profile:

rt
rfw(r) d x,[n(r)] p[xl(r)] dr = v po x' (3-9
rb

where p is the solution density at x', w(r) is the cell width normal to the radius which

varies with radius in the cells used, and p[xl(r)] is the solution density at the position, r,

with solute mole fraction xl(r):

n(r) = j(r) X / d + n' (3-10

where n' is the refractive index at the top of the sample, at j=0. Equations (3-9) and

(3-10) are used to iteratively solve for the unknown n'. Then, equation (3-10) gives the

full composition profile in the cell. The analysis starting with the microcomparator

measurements at each fringe and leading to xl(r) are carried out with the FORTRAN

computer program FRNGCNV. This program is listed in Appendix C. It is fully

documented with comments and a sample input and output file.

While the program has the capability of approximating the mixture refractive

index from pure component values with several mixing rules, all final calculations

used new refractometer data fitted to a linear equation. These were carried out in a

Cromatix differential refractometer (Milton Roy Corporation, Model KMX-16). The

refractometer measures the difference between a solution of known composition and

the pure solvent. The light source is a HeNe laser of the same wavelength as that on









the ultracentrifuge eliminating any effect of wavelength on the refractive index. The

refractometer was first calibrated with known solutions of dried, reagent-grade sodium

chloride and deionized water using the data at 250 C of Kuis (1936).

Refractive index measurements were made on all systems used in the

ultracentrifuge in the range 0-10 mole percent solute. For carbon disulfide-acetone,

literature data from Campbell and Kartzmark (1973) and Loiseleur et al. (1967) exists.

The experimental data measured here compare favorably with the literature data. All

refractive index data were fitted to a first or second order polynomial in x1. The data

and expressions are given in the next chapter.

Once the composition profile is found, equation (3-11), which is a finite difference

approximation to equation (3-7), can be used directly, with the necessary physical

properties, to calculate the desired activity coefficient derivative.

SOln-1n (M p V-r) ( r1) 1 (3-11
a x, .r 2RT ((x0i (x,)j-,) (xOj
This calculation was carried out in a Lotus 123 spreadsheet. Details of the density

calculations required are given in Appendix D.

The initial speed for operation of the ultracentrifuge was estimated as described

in Appendix E so that about 15 total fringes would appear. In all cases each

concentration was run at two speeds to check for consistency. The first speed was

based on this estimate while the second speed was either 1.5 times greater or less than

the initial speed, depending on the actual number of fringes found.

System Selection

There are three major criteria to determine if a binary pair of chemicals could be

suitable for study by equilibrium sedimentation. First, because of the laser light

composition profile determination, an appropriate difference in refractive index must

exist between the two chemicals. Second, the sedimentation parameter (Mi pV1) must

be large enough for the solute to sediment at a detectable level in the solvent under an









accessible centrifugal force. Mixture density data must be available to determine this

accurately. Finally, the system should be at least moderately non-ideal. That is, the

dilute activity coefficient derivative should be sufficiently large that the result from

equation (3-11) yields a significant difference of the two large numbers. Additional

considerations are to have components of low volatility and toxicity. Here, no

compounds were chosen that have a normal boiling point below 490C.

Measuring fringe separation with the microcomparator indicated that 10 to 20

shifted fringes can be accurately found in a reasonable amount of time. This,

together with the range of rotational speed, gives the limits of pure component

refractive index difference and sedimentation parameter. A binary pair must have a

pure component refractive index difference of at least 0.06, and a sedimentation

parameter of at least 35 (measured at a solute concentration of 2%). Finally, an

infinite dilution activity coefficient greater than 2, or less than 1/2 insures that the

system is non-ideal enough.

The system carbon disulfide in acetone was studied because it is highly non-ideal

and had been thoroughly examined by Campanella (1984). The second system, carbon

tetrachloride in acetone, was studied by Cullinan and co-workers (Sethy and Cullinan,

1972; Rau, 1975) in an ultracentrifuge, which allowed a direct comparison with their

method.

Since both of the above systems showed a positive deviation from Raoult's law,

chloroform in acetone was selected to demonstrate the effects of negative deviations

from Raoult's law. While no sedimentation data exist for this system, there are many

vapor-liquid equilibrium literature data references.

An extensive search was made to identify two additional systems for which the

ultracentrifuge could supply useful dilute solution information. The following classes

of systems were identified as being interesting:









1. water/nitrile

2. water/amine

3. chloroalkene, alkene or alkane/cyclic ether

4. alcohol/alkane

5. amine/chlorinated alkane

6. nitrile/alkane or cyclic alkane

7. alcohol/chlorinated alkane

These were first evaluated by refractive index differences, An. In the first group,

water and benzonitrile were identified as the only common system with An > 0.06

(An = 0.19). However this system is only partially miscible. The candidate system

water/acetonitrile had a refractive index difference of only 0.03.

For the second group, triethyl amine was selected among the lower molecular

weight nonaromatic amines since refractive index generally increases with carbon

number. For water/triethyl amine, An = 0.06. Aromatic amines give larger An values

(e.g., for water/aniline An = 0.2) but these show immiscibility.

For the third group, several cyclic ethers (furan, tetrahydrofuran, pyran,

tetrahydropyran and 1,4-dioxane) were considered for solution with

tetrachloroethylene or trichloroethylene. However, the only pairs with a suitable

refractive index difference are tetrachloroethylene with either 1,4-dioxane (An = 0.09)

or tetrahydrofuran (An = 0.08). With pyran the only alkane and alkene with a large

enough An are heptane and hexene.

The best pair from the fourth group would involve a low carbon number alcohol

such as methanol, and a high carbon number alkane such as n-octane (An = 0.07).

For the fifth group the aliphatic amines and linear aliphatic monochlorides have

very similar refractive indexes. Thus, an aromatic amine, N-methyl aniline, was

chosen to be paired with chloroform (An = 0.12) or monochlorobutane (An = 0.09).









In the sixth group aliphatic nitriles and alkanes have nearly the same refractive

index. However, benzonitrile can be paired with either a linear or cyclic alkane, such

as with heptane An = 0.13 or cyclohexane An = 0.11. Also, the acetonitrile/benzene

system has An = 0.16.

Finally, chloroform and carbon tetrachloride have the highest refractive index of

chlorinated alkanes. Therefore, with any alcohol, particularly methanol, either of

these compounds has an adequate An.

The systems above were then evaluated by their ability to sediment as measured

by their sedimentation parameter. For screening purposes, the factor (M, pV1) was

approximated by (M1 po0mV) where the density (Po.02) was for a solute concentration

of 2% (mole).
Table 3-1
Sedimentation Factors For Various Binary Pairs
Binary Pair Sedimentation Factor
(M Po.02V?)
water/benzonitrile 0
water/triethyl amine 5
water/aniline 0
tetrachloroethylene/1,4-dioxane 59
tetrachloroethylene/tetrahydrofuran 38
methanol/n-octane 4
chloroform/methyl aniline 39
N-methyl aniline/chlorobutane 0.5
benzonitrile/heptane 30
benzonitrile/cyclohexane 20
benzene/acetonitrile 36
chloroform/methanol 55
carbon tetrachloride/methanol 75
chloroform/acetone 54
carbondisulfide/acetone 27
carbon tetrachloride/acetone 75

From the table above it is easily seen that the sedimentation parameter is less

than the desired 35 for many of the candidate systems. The only systems, not

previously selected, with a high enough sedimentation factor were tetrachloroethylene








with either 1,4-dioxane or tetrahydrofuran, N-methyl aniline with chloroform (or
carbon tetrachloride), chloroform (or carbon tetrachloride) with methanol and
benzene with acetonitrile. Final selection of benzene with acetonitrile and methanol
with carbon tetrachloride was based on low toxicity, and availability of solution

densities and compressibilities.

Error Analysis
Equation (3-7) describes the fundamental relationship between the derivative of
the activity coefficient and measurements in the ultracentrifuge. All quantities in
that equation except the derivative dr/dx1 are well defined as to their expected

uncertainty. This derivative must be calculated by an iterative integral method. For
that reason the analysis of error for this process is done in two steps. First, the effect

of variations in dr/dxl on the activity coefficient derivative is found. Then the error

to be expected in the dr/dxl term is determined.

An uncertainty analysis (Holman, 1971) can determine the uncertainty in the
activity coefficient derivative. This method assumes that a quantity in question, A,
can be written in terms of several independent variables, aj.

A = A(al, a2, a3 ..... a,) (3-12

If each variable ai has an uncertainty Ui, the uncertainty in A, UA can be written as


UA U1 + U 2 + ... f U (3-13
A 8a1 J a2 9 ) ) IS.a ))

In equation (3-7) the variables of interest are p, V1, w, r, T, x'. If the activity

coefficient derivative is defined as A and (dr/dxj) as D, equation (3-7) becomes

Saln-1 A =(M -pV-) 2r (3-14
ax, J RT x,








and equation (3-13) can be written for this specific case as

UA= A )IA )U ) + I UA 2A
ap PJ aV Vi J a) r J r
(A ( 1 (A 2 U'A 12 (3-15
aT UT ax+ I aD1D ] ())5

The measurement of refractive index as a function of composition was done in an
instrument of much greater accuracy than required for the ultracentrifuge experiment.
Thus, the uncertainty in this quantity is ignored. Each of the derivatives in equation
(3-15) can be evaluated from equation (3-14) and substituted into equation (3-15) giving

U V[( o2 r D p 2 r 2 + (2(Mr-pV) r U

+ (M2-pV) 2 (M-pV) r W U )2 +U 2
+ D(Ur +DUT)+ ) DU
RT Ur+ RT2 T (x)2 x

+ (Mj-pVj) r W2 2D] 2(3-16
RT UD(3-16

Uncertainties were assigned to each of the quantities in question as follows:
U = 0.1% UV = 0.1%

UT = 0.5 Co Ux, = 0.0002

U =0.1% U = 0.001 cm
W r
Typical values (from run 38-2 x' = 0.035 and run 29-1, xO = 0.109) were substituted

into equation (3-16) to illustrate the relationship of the uncertainty in D (dr/dx,) to

uncertainties in A -n The results are shown in Table 3-2.
O, xt1)








Table 3-2

Uncertainty in ) as a function of
( dr }
the Uncertainty in -
I dxj
x' = 0.035 x = 0.109
Ut % Ui U % Ui Ui%
0.1 0.2 3.5 .02 0.3
1.0 0.3 5.5 .05 1.0
5.0 1.0 20.8 .23 4.5

t % Uncertainty in -d
(. dx1
t Absolute and % Uncertainty in |-
1 ax, )
The error to be expected in (dr/dxl) will now be examined by introducing a

random error into its primary variables, xj, j and r. This error will be of an average

magnitude to be expected in the experiment. As an example, the fringe measurements

from run 38-2 (carbon disulfide/acetone) were used. The average error in r, as

indicated above, should be no more than 0.001 cm, but a worst case of 0.01 cm is used

here. Then the measurements on run 38-2, which has only 8.9 fringes, would have an

error of about 0.1 fringes. A summary of the contribution of various errors in the

experimental quantity (dr/dx1) and subsequently ( is given in Table 3-3 for the
-xi )
most severe case (x' = 0.035).

Table 3-3

Errors in and -
ax, dxj
x' = 0.035

Cause of Error U (lnyU (f
radius 5.5% 1%
j (fringes) 6.5% 1.3%
x4 initial solute mole frac. 3.5% 0.1%









The error due to radius measurement and fringe number are redundant, since they

are absorbed in the same effect, the number of fringes. The error introduced in x'

could be added to the other, yielding for a total maximum error in of less than

10% at low concentrations.

Summary

Direct measurements of the dilute solution activity coefficient derivative would

be valuable to settle discrepancies between extrapolated vapor-liquid equilibrium

measurements and direct measurements of infinite dilution activity coefficients, for

differences between activity coefficient models and for theoretical and modeling

purposes. The analytical ultracentrifuge has the capability of providing these.

The experimental equipment, including the laser light source for sensitive

Rayleigh Interference composition measurement, have been described. A complete

thermodynamic analysis of the ultracentrifuge has been given for the solute activity

coefficient derivative in terms of the measurable quantities in the ultracentrifuge. A

complete uncertainty analysis was conducted for the experiment. It pointed out that

with reasonable care the activity coefficient derivative can be determined to within

10% at concentrations near 2%.

Not all binary systems of chemicals are amenable to this technique. There must

be a moderate difference in refractive index of the pure components (An > 0.06), a

sufficient tendency of the solute to sediment in the solvent, [(M1 pV1) > 35] and

moderate non-ideality (-y > 2) so that the activity coefficient derivative is large

enough to be well determined. A thorough screening was conducted to select

interesting systems to meet these criteria. Five systems were chosen and both

sedimentation and refractive index measurements were made on them.














CHAPTER 4
EXPERIMENTAL MEASUREMENTS OF DILUTE
SOLUTION ACTIVITY COEFFICIENT DERIVATIVES

Introduction
Experimental measurements were made to obtain the composition derivative of

the solute chemical potentials and activity coefficients of five systems: carbon

disulfide in acetone, carbon tetrachloride in acetone, chloroform in acetone, benzene

in acetonitrile, and carbon tetrachloride in methanol. Experiments were conducted on

all systems over a range of composition, usually from 1.5% (mole) to 10% (mole) in the

solute (first component listed). Tests were usually conducted at three temperatures,

100, 250 and 350 C.

Equation (4-1) shows the basic relationship between the activity coefficient

derivative and experimentally accessible quantities

a 1/RT I (1n-, 1 (MI pV) 02 r ddn dr)
ax, ax, x1 RT dx dn(4-1

The density, p, and partial molar volume, V1, are obtained from literature data, the

refractive index/composition derivative (dn/dx1) was measured with a laser differential

refractometer as described in Chapter 3, and the refractive index profile, (dr/dn), was

obtained from the ultracentrifuge.

This chapter describes the experiments performed and presents the specific results

on each system. Comparisons with analyses from vapor-liquid equilibrium (VLE) data

are also made. Discussions of the influence of pressure due to cell loading and

composition and of impurities are given. Tables giving detailed results for each

experimental run are in Appendix F. These tables give the composition profile at









each fringe, all physical properties needed and the calculated activity coefficient

derivatives and chemical potential derivatives.

Carbon Disulfide/Acetone System

The carbon disulfide/acetone system was the initial system studied. As experience

was gained, inconsistent results were found for some of the measurements; these were

omitted. The final 21 runs adopted covered a range of carbon disulfide

concentrations from 1.05% (mole) to 10.9% (mole). All concentrations were run at

more than one speed to verify results. Two concentrations (10.9% and 2.07%) were run

at 100, 25 and 300 or 330 C. All other runs were at 250 C or 26.50 C.

A summary of the results for this system is in Table 4-1. Included are the overall

solute concentration, temperature, calculated activity coefficient derivative and

calculated chemical potential derivative. Also included are the activity coefficient

derivatives and chemical potential derivatives calculated from parameters fitted by

Gmehling et al. (1977) to the VLE data of Litvinov (1952) at 25 C for the Wilson (1964)

and NRTL (Renon and Prausnitz, 1968) excess free energy models. Excess enthalpy

data of Campbell et al. (1970) were used in equation (4-2) to calculate the activity

coefficient at 350 C from the 25 C VLE data of Litvinov (1952).


(In-)T=, = h 1 1 -1 + (lnY)T=T, (4-2
SR T2 T(2

Numerical composition derivatives of lny, were then calculated to verify the

consistency of the activity coefficient derivatives obtain from VLE data at different

temperatures contained in Table 4-1. The excess enthalpy data were compiled and

regressed by Gmehling et al. (1977). The chemical potential derivatives from the

centrifuge experiment and from the VLE are given in Figure 4-1.









Table 4-1
Summary of Carbon Disulfide(1)/Acetone(2)
Ultracentrifuge Experimental Results
and Comparison Values Calculated from Literature VLE Data
__ fa1ny_ ', 1n-,, ( Iny, a' alny1
xx T )1nl J/RT 1.l (/. RT [1/JIRT al
S axE I ax, E xxi xs WxsJ ax T) ( axT) l axl
C Expt Expt Wlsnt Wlsnt NRTLt NRTLt Wisnt


-19.03.0*
-16.81.2
-17.42.1
-12.21.4
-13.51.9'
-8.93.8'
- 15.221. 8
-7.11.2
-7.0-1. 1
-9.90.7
-9.50.5
-9.50.7
-9.40.9
-9.9-0.8
-7.72.4'
-6.60.3
-6.70.3
-3.60.2
-4.70.1
-4.40.1
-3.50.2


74.6
36.0
35.5
35.5
34.2
38.9
32.5
27.7
27.9
18.1
18.5
18.4
18.5
16.4
18.7
10.6
10.4
5.5
4.4
4.7
5.6


-2.466
-2.440

-2.433




-2.409

-2.387


92.73
50.87


-1.828
-1.825


93.36
51.48


-2.599
-2.567


45.78 -1.824 46.39 -2.560




32.72 -1.821 33.31 -2.532

25.71 -1.818 26.28 -2.506


0.01051
0.01876
0.01876
0.02074
0.02074
0.02074
0.02074
0.02846
0.02846
0.03558
0.03558
0.03558
0.03558
0.03767
0.03767
0.05806
0.05806
0.10949
0.10949
0.10949
0.10949


25
26.5
26.5
10
25
25
30
18.8
26.5
26.5
26.5
26.5
26.5
25
25
26.5
26.5
10
25
25
33


-2.319 14.90 -1.810 15.41 -2.427


-2.174


6.96 -1.792 7.34 -2.261


+ Standard deviations from various fringes.
t Calculated from VLE data of Litvinov (1952) at 250 C.
t Calculated from VLE data of Zawidzki (1900) at 350 C.
Appear inconsistent.

The results of the ultracentrifuge experiment are numerically different from the

VLE curve, though they have the same general trend with composition. Except for

five runs, a factor of 1.4 times the ultracentrifuge derivatives would make them


essentially coincide with the VLE values, as shown in Figure 4-1.


-2.380 24.16 -1.817


24.73 -2.499















100


90 -


80-


70-


60-
d _/RT

dx 5o -



40-


30-


20-


10-


0--
0.01


0.03 0.05 0.07 0.09


Mole Fraction, x1


Figure 4-1 Chemical Potential Derivatives From Ultracentrifuge Data (as
Measured and Adjusted) and From VLE Data (Litvinov, 1952, at
25 C) Fitted to the Wilson and NRTL Models, for the System
Carbon Disulfide(l) / Acetone(2).


0.11









Figure 4-2 shows derivatives of the activity coefficient. The filled rectangles are

calculated from the ultracentrifuge experiment, the lines are from various models

fitted to VLE data and the open diamonds are corrected centrifuge values. Again, it

is clear that, except for five points, the corrected data are within experimental error

of the calculations.

The five points that show the greatest deviation from the others and from the

VLE curve are probably in error. In one the concentration of solute was only 1.05%

(mole) and there was a blur in the interference pattern at the bottom of the cell. This

interrupted the fringe measurements and required extrapolation to determine the total

number of fringes present. Another was an early experiment for which all systems

were not necessarily operating properly. The others had less than the desired number

of fringes (7 to 20).

The two sets of measurements that were made at different temperatures do not

show a consistent change with temperature. The values at 2.1% (mole) marginally

indicate more negative activity coefficient derivatives as temperature increases, for

the 10.9% (mole) system, the results at temperatures above and below do not

encompass the 250 C value. Excess enthalpy and VLE data agree with the 2.1% (mole)

data, indicating a slightly more negative activity coefficient derivative at higher

temperatures.

It should be pointed out that refractive index/composition data have previously

been measured (Campbell and Kartzmark, 1973 and Loiseleur et al. 1967). As shown in

Figure 4-3 their results agree very well with the present measurements even though

their light source was the sodium line (589.3 nm) rather than the 632.8 nm laser light

used here. Thus, it appears that the refractive index measurements are reliable.

Measurements of mixture refractive index were made at 300 and 350 C and

extrapolated to other temperatures, as shown in Table 4-2.













13-

12- 0 Centrifuge Data
0 Adiustecr Data
10- VLE Wilson
S........................... VLE NRTL
8 -- ---- VLE Margules
6-
0 o
4-

2-
dlni o-_ -----------
dxi -2 ..... .................

-4-

-6 -
-8 -
-10-

-12-

-14-
-16-

-18-
-19-

-20-
-21 -
-22-
-23 -i I I I-,
0 0.02 0.04 0.06 0.08 0.1
Mole Fraction, xl



Figure 4-2 Activity Coefficient Derivatives From Ultracentrifuge Data (as
Measured and Adjusted) at 250 C and From VLE Data (Litvinov,
1952, at 250 C) Fitted to Various Models, for the System Carbon
Disulfide(1)/Acetone(2). Error Bars are Typical of All
Ultracentrifuge Runs.











1.384


1.382 -


1.38-


1.378-


1.376 -


1.374 -


1.372 -


1.37 -


1.368 -


1.366 -


1.364 -


1.362 -


1.36 -
6 a Expt. Data
1.358 / + Loiseleur et al. (1967)
Linear Fit
1.356 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Mole Fraction, x,

Figure 4-3 Experimental and Literature Refractive Index Data (at 25) Fit
to a Linear Equation in Mole Fraction for the System Carbon
Disulfide(l) / Acetone (24









Table 4-2
Refractive Index (n) Experimental Results
for the System Carbon Disulfide(1)/Acetone(2)

x, n n n
250 C 300 C 350 C
Extrapt Expt Expt

0.00000 1.35600t: 1.35395: 1.35183t:
0.00947 1.35786 1.35570 1.35354
0.02671 1.36045 1.35810 1.35574
0.02864 1.36106 1.35887 1.35667
0.05624 1.36629 1.36383 1.36137

t Values at 250 C were extrapolated linearly from the
experimental measurements at 300 and 350 C.
t Pure component refractive index data was taken
from Loiseleur et al. (1967) and Campbell and
Kartzmark (1977).



Chloroform/Acetone System

The chloroform/acetone system is the only system examined which had negative

deviations from Raoult's law. This means that the activity coefficient derivative will

be positive rather than negative as in the carbon disulfide/acetone case.

A summary of the results for this system is given in Table 4-3. A range of

concentrations from 1.9% (mole) to 8.7% (mole) (chloroform in acetone) was studied.

Each concentration was run at two or more different speeds to verify results. At one

concentration (5.3%) runs were made at both 26.50 C and 3730 C. All other runs were

made at 26.50 C. Table 4-3 also shows the activity coefficient derivatives and chemical

potential derivatives calculated from the VLE measurements of Rabinovich and

Nikolaev (1960) at 250 C using the Wilson, and NRTL model parameters from

Gmehling et al. (1977). The higher temperature VLE data of Zawidzki (1900) given in









Table 4-3, was verified as correct by scaling the VLE data of Rabinovich and

Nikolaev (1960) at 250 C to 350 C using the excess enthalpy data of Chevalier and Bares

(1969).

Table 4-3
Summary of Chloroform(1)/Acetone(2)
Ultracentrifuge Experimental Results
and Comparison Values Calculated from Literature VLE Data

ST f /RT f f RT f01 P^RT fn
ax0 J I( ax JRT a9x1) J at Y RT an aln a a RT ( xny,
1, (xx ) G, !,I x- ) Ox I ax, I (, xt) ( Ox, )

C Expt' Expt Wlsnt Wlsnt NRTLt NRTLt Wlsnt

0.01888 26.5 -14.83.0' 37.2 1.720 54.69 1.569 54.54 1.221
0.01888 26.5 -15.22.2* 36.5
0.02515 26.5 -7.81.2" 31.3 1.711 41.48 1.565 41.33 1.218
0.02515 26.5 -8.10.7' 30.9
0.03713 26.5 -6.31.6 20.4 1.694 28.63 1.559 28.49 1.210
0.03713 26.5 -6.60.7 20
0.03713 26.5 -6.92.6 19.8
0.05269 26.5 -3.80.6 15 1.672 20.65 1.552 20.53 1.201
0.05269 26.5 -3.70.5 15.1
0.05269 37.3 -4.40.6 14.4
0.06357 26.5 -2.70.5 12.9 1.657 17.39 1.547 17.28 1.194
0.06357 26.5 -2.20.5 13.5
0.06357 26.5 -2.70.6 13
0.08724 26.5 -1.70.4 9.7 1.625 13.09 1.540 13.00 1.179
0.08724 26.5 -1.70.4 9.6
+ Standard deviations from various fringes.
t Calculated from VLE data of Rabinovich and Nikolaev
(1960) at 25 C.
t Calculated from VLE data of Zawidzki (1900) at 350 C.
Appear inconsistent.

Figure 4-4 shows the chemical potential derivatives from the centrifuge and the

VLE data. Again if all of the ultracentrifuge values are multiplied by 1.4, the results

are within experimental error of those calculated from VLE data, as demonstrated in

Figure 4-4. Figure 4-5 shows the derivative of the activity coefficient calculated from

the ultracentrifuge experiments, from VLE data, and from the "corrected" values of































dc, /RT

dx1


25-



20-



15-



10-



5--
0.01
0.01


0.03


0.05 0.07
Mole Fraction, x,


0.09


Figure 4-4 Chemical Potential Derivatives From Ultracentrifuge Data (as
Measured and Adjusted) and From VLE Data (Rabinovich and
Nikolaev, 1960, at 25 C) Fitted to the Wilson and NRTL Models,
for the System Chloroform(l) / Acetone(2)












































Centrifuge Data
Adjusted Data
VLE Wilson
VLE NRTL
VLE Margules


1 I 1 1
0.04 0.06
Mole Fraction, x1


0.08
0.08


Figure 4-5 Activity Coefficient Derivatives From Ultracentrifuge Data (as
Measured and Adjusted) and From VLE Data (Rabinovich and
Nikolaev, 1960, at 25' C) Fitted to Various Models, for the System
Chloroform(l) / Acetone(2).


-2


-3-

-4-

-5-

-6-


dinI

dxl


-7-

-8 -



-10 -

-11 -

-12 -

-13 -



-15 -


-16


I I
0.02


0
0 0

--ooo..o.......... ......,.......................

~___ ___









the chemical potential derivatives. It is clear that the "as measured" results are

erroneous since they are negative.

One experiment (mole fraction chloroform was 0.053) was run at both 26.50 C and

37.30 C. Ultracentrifuge results give a lower activity coefficient derivative at 37.30 C

than at 26.50 C. The VLE data at 350 C (Zawidzki, 1900) and 250 C (Rabinovich and

Nikolaev, 1960) indicate the same trend. Negative excess enthalpy data of Chevalier

and Bares (1969) further support this behavior. The difference in activity coefficient

derivatives measured in the ultracentrifuge at 26.50 and 37.3 C is about 0.8. The VLE

data indicate a difference in derivatives at 250 and 350 C of 0.6.

There were no mixture refractive index data for this system available in the

literature. Pure acetone data were taken from Loiseleur et al. (1967) and Campbell and

Kartzmark (1973), while mixture data were measured at 300, the lowest temperature for

reliable results, and 35 C. Values at 250 C were extrapolated and all data are

summarized in Table 4-4.

Table 4-4
Refractive Index (n) Experimental Results
for the System Chloroform(1)/Acetone(2)

X1 n n n
250 C 300 C 350 C
Extrapt Expt Expt

0.00000 1.35600t 1.35395t 1.35180t
0.04076 1.35985 1.35779 1.35562
0.10279 1.36561 1.36350 1.36130
t Values at 250 C were extrapolated linearly from the
experimental measurements at 300 and 350 C.
t Pure component refractive index data were taken
from Loiseleur et al. (1967) and Campbell and
Kartzmark (1977).

Carbon Tetrachloride/Acetone System

For the carbon tetrachloride/acetone system measurements were made at three

compositions, and multiple speeds, from a low of 0.4% (mole) to a high of 7.3% (mole)









carbon tetrachloride, at 23 C for the low concentration point and 26.5 C for the

others. These are summarized in Table 4-5 and show the same trend as earlier

experiments. Figure 4-6 shows that the chemical potential derivatives from the

ultracentrifuge follow, but are consistently lower than, those calculated from the VLE

data of Brown and Smith (1957), which were adjusted from 45 C using heat of mixing

data from Brown and Fock (1957).

Table 4-5
Summary of Carbon Tetrachloride(1)/Acetone(2)
Ultracentrifuge Experimental Results
and Comparison Values Calculated from Literature VLE and hE Data
xaT axn__ (axJ/RT fan__ (ax1J/RT (alnxl ( 41)/RT (aln~yl
x0 T RT /RT /RT
1 Ixi ) ax) Fax, ) x, 8xx) lax-, Ox)
0C Expt' Expt Wlsnt Wlsnt NRTLt NRTLt Wlsnt

0.00429 23 -58.67.0' 161.2 -1.063 231.98 -0.972 232.07 -1.271
0.00429 23 -61.38.9' 159.7
0.02208 10.3 -6.72.0 38.2 -1.055 44.23 -0.971 44.32 -1.199
0.02208 26.5 -8.71.6 36.1
0.02208 26.5 -8.11.4 36.5
0.07246 26.5 -2.71.1 11 -1.033 12.77 -0.967 12.83 -1.086
0.07246 26.5 -2.80.3 10.9
0.07246 37.3 -3.20.8 10.5
+ Standard deviations from various fringes.
t Calculated from VLE data of Brown and Smith (1957) at
450 C.
t The VLE values at 450 from Brown and Smith (1957)
were adjust with the excess enthalpy data of Brown and
Fock (1957) to obtain a result at 250 C.
Appear inconsistent.

As Figure 4-6 also shows, if all of the ultracentrifuge values are multiplied by the

common factor of 1.2, the results correspond to the VLE data, particularly at the

higher concentration points. Figure 4-7 shows the resulting activity coefficient

derivatives for these experiments.

Two experimental ultracentrifuge runs on this system were conducted at

temperatures other than those shown in Figure 4-6. At a mole fraction of 0.0221











240


220


200


180


160


140


d, /RT

dx1


0 0.02 0.04 0.06
Mole Fraction, x


0.08


Figure 4-6 Chemical Potential Derivatives From Ultracentrifuge Data (as
Measured and Adjusted) and From VLE Data (Brown and
Smith, 1957, at 450) Fitted to the Wilson Model, for the System
Carbon Tetrachloride(1) / Acetone(2).














































- -


I I
0.02


0.04


I I
0.06


I I
0.08


Mole Fraction, x1


Activity Coefficient Derivatives From Ultracentrifuge Data
(Adjusted) and From VLE Data (Brown and Smith, 1957, at 45*)
Fitted to the Wilson Model, for the System Carbon
Tetrachloride(1) / Acetone(2).


Adj. Centrifuge Data 0 0
VLE Wilson
VLE- NRTL ...........
VLE Margules -


0.5-

0.4-

0.3 -

0.2-

0.1 -

0


dln

d -xi


-0.1 -

-0.2 -

-0.3 -

-0.4 -

-0.5 -

-0.6 -

-0.7 -


-0.8 -.


-0.9 -

-1 -

-1.1


Figure 4-7









carbon tetrachloride, runs at 100 C and 26.5 C were made, while at x, = 0.0725, runs

were made at 26.5 and 37.30 C. The results showed activity coefficient derivatives that

were statistically unchanged with temperature. Here the VLE and positive excess

enthalpy data (Brown and Smith, 1957; Brown and Fock, 1957) give more negative

derivatives with decreasing temperature.

There were no mixture refractive index data for this system available in the

literature. As before, experimental mixture data were measured at 300 and 35* C and

linearly extrapolated to other temperatures. This data is summarized in Table 4-6.

Table 4-6
Refractive Index (n) Experimental Results
for the System Carbon Tetrachloride(1)/Acetone(2)

x1 n n n
250 C 300 C 350 C
Extrapt Expt Expt

0.00000 1.35600t 1.35395t 1.35180t
0.04000 1.36144 1.35930 1.35706
0.07990 1.36679 1.36459 1.36230
t Values at 250 C were extrapolated linearly from the
experimental measurements at 300 and 350 C.
t Pure component refractive index data was taken
from Loiseleur et al. (1967) and Campbell and
Kartzmark (1977).

Benzene/Acetonitrile System

For the benzene/acetonitrile system, measurements were made from a low of

227% (mole) to a high of 8.6% (mole) benzene at 26.50 C and are summarized in Table

4-7. Again, as is shown in Figure 4-8, the chemical potential derivatives from the

ultracentrifuge follow, but are consistently lower than, those calculated from the VLE

data of Werner and Schuberth (1966) at 200 C (using the Wilson and NRTL equations

with parameters fit by Gmehling et al. 1977).











50



45



40



35



30
1dJ/RT

dx1
25



20



15



10


0.02


0.04 0.06


Mole Fraction, x,


Figure 4-8 Chemical Potential Derivatives From Ultracentrifuge Data (as
Measured and Adjusted) and From VLE Data (Werner and
Schuberth, 1966, at 200 C) Fitted to the Wilson and NRTL
Models, for the System Benzene(l) / Acetonitrile(2).


0.08









Table 4-7
Summary of Benzene(1)/Acetonitrile(2)
Ultracentrifuge Experimental Results
and Comparison Values Calculated from Literature VLE Data

S(aIny, (a8Q ny___ a l( ny __ iny
x( T ax1 a)/RT (l )/RT ln a_/RT ln
x J x axJ ax J ax ax, J x t ax
0C Expt Expt Wlsnt Wlsnt NRTLt NRTLt Wlsnt

0.02271 26.5 -8.6-1.1 35.1 -2.274 41.78 -2.166 41.89 -1.9973
0.02271 26.5 -8.02.0 35.8
0.03747 26.5 -3.41.7* 23.2 -2.221 24.45 -2.097 24.57 -1.9574
0.03747 26.5 -8.80.5' 17.8
0.04255 26.5 -5.41.3 18 -2.204 21.33 -2.074 21.46 -1.9441
0.04255 26.5 -8.31.1 15.1
0.04255 37.3 -8.61.2 14.8
0.05622 26.5 -6.20.5 11.5 -2.156 15.64 -2.015 15.78 -1.9082
0.05622 26.5 -5.30.5 12.4
0.05622 37.3 -6.60.8 11.1
0.05821 26.5 -3.72.5 13.4 -2.149 15.03 -2.006 15.18 -1.9030
0.06427 26.5 -3.92.1 11.6 -2.129 13.42 -1.981 13.57 -1.8873
0.06427 26.5 -3.20.7 12.3
0.08610 26.5 -3.40.3 8.2 -2.057 9.56 -1.895 9.72 -1.8322
+ Standard deviations from various fringes.
t Calculated from VLE data of Werner and Schuberth (1966) at 200 C.
t Calculated from VLE data of Palmer and Smith (1972) at 450 C.
Appear inconsistent.

Multiplying all of the ultracentrifuge values by 1.2 gives values which are within

experimental error of the VLE data (Figure 4-8).

The data for this system show much more scatter than the others, because lower

speeds and smaller samples yielded fewer fringes in most of the runs. These less

satisfactory conditions were necessary to minimize the extra spinning force on the

rotor due to the heavier mass of the titanium cell used for these chemicals. Activity

coefficient derivatives calculated from the "corrected" chemical potential derivatives

are given in Figure 4-9.

Two compositions of this system were studied at 37.3 C. While the uncertainty is

quite high, both runs appear to give more negative activity coefficient derivatives at




























- --- -. -.- -' -. r;1 4
....................


I0.02
) 0.02


I I
0.04
Mole


0.06
Fraction,


I I
0.08


0.1


Activity Coefficient Derivatives From Ultracentrifuge Data (as
Measured and Adjusted) and From VLE Data (Werner and
Schuberth, 1966, at 200 C) Fitted to Various Models, for the
System Benzene(l) / Acetonitrile(2).


0 Centrifuge Data
Adjusted Data
VLE Wilson
............. VLE NRTL
- -VLE Morgules


-2-


dinY1

dxl


-4-



-5-



-6-



-7-



-8-



-9 -


Figure 4-9


r w w









this higher temperature. The VLE data of Werner and Schuberth (1966) at 200 C and

of Palmer and Smith (1972) at 450 C indicate a trend in the opposite direction, i.e. less

negative derivative values at higher temperatures, consistent with the positive excess

enthalpy of Absood et al. (1976).

There were no mixture refractive index data for this system available in the

literature. The experimental data were measured at 300 and 350 C and extrapolated to

other temperatures. Pure component refractive index data for acetonitrile were not

available at different temperatures. The centrifuge data analysis can be done equally

well with absolute or differential mixture refractive index data. Therefore, for this

system, only the differential refractive index is reported in Table 4-8.

Table 4-8
Differential Refractive Index (An) Experimental
Results for the System Benzene(1)/Acetonitrile(2)

x, An An An
250 C 300 C 350 C
Extrapt Expt Expt

0.00000 0.00000 0.00000 0.00000
0.02032 0.00503 0.00499 0.00495
0.04022 0.00970 0.00962 0.00954
0.06668 0.01595 0.01581 0.01566
0.09176 0.02161 0.02142 0.02123
t Values at 250 C were extrapolated linearly from the
experimental measurements at 300 and 350 C.

Carbon Tetrachloride/Methanol System

For the carbon tetrachloride/methanol system, three compositions at 26.50 C were

studied from a low of 1.92% (mole) to a high of 4.9% (mole) carbon tetrachloride, as

summarized in Table 4-9. Again the results show the trend of earlier experiments. As

shown in Figure 4-10, the chemical potential derivatives from the centrifuge follow,

but are consistently below the curve calculated from VLE data using the Wilson and










\
\
\ .o.o.....


H\
I



\
\-


\\
l*.
*N


1I
0.03


I I
0.05


0 Centrifuge Data
+ Adjusted Data
- VLE Wilson
... VLE NRTL
VLE Margules













\
\





0.07 0.09


Mole Fraction, >

Figure 4-10 Chemical Potential Derivatives From Ultracentrifuge Data (as
Measured and Adjusted) and From VLE Data (Wolff and
Hoeppel, 1968, at 20 C) Fitted to Various Models, for the System
Carbon Tetrachloride(1) / Methanol(2).


c, /RT
dx1


25-

20-

15

10-

5--
0.01









NRTL equations with parameters regressed by Gmehling et al. (1977) and the data of

Wolff and Hoeppel (1968).

Table 4-9
Summary of Carbon Tetrachloride(1)/Methanol(2)
Ultracentrifuge Experimental Results
and Comparison Values Calculated from Literature VLE Data

x0 T [l'] ft1 fRT RT ( RT
-ax k J xaxi(J -axI J xt ax axi J (OxJ
0C Expt' Expt Wlsnt Wlsnt NRTLt NRTLt

0.01923 26.5 -10.41.2 41.3 -5.080 46.92 -6.900 45.10
0.01923 26.5 -9.71.6 41.9
0.01923 37.3 -6.25.8' 45.3
0.03438 26.5 -7.62.7 21.2 -4.833 24.26 -6.468 22.62
0.04913 26.5 -5.41.1 14.9 -4.611 15.74 -6.082 14.27
0.04913 26.5 -5.90.8 14.4
0.04913 37.3 -6.20.9 14.1
+ Standard deviations from various fringes.
t Calculated from VLE data of Wolff and Hoeppel (1968)
at 200 C.
Appears inconsistent.

In this case the experimental values are closer to the VLE data than in other

systems studied, but are still low by a factor of 1.1 which gives a reasonable fit to the

VLE results (Figure 4-10). The activity coefficient derivatives calculated from the

"corrected" chemical potential derivatives are shown in Figure 4-11.

One run was made at 37.30 C and 4.9% (mole) benzene. Statistically, its activity

coefficient derivative was the same as the corresponding 26.50 C runs. This is

consistent with the excess enthalpy which for this system is nearly zero (Abramov et

al., 1973), when the solution has less than 20% carbon tetrachloride. Also, the VLE

data of Wolff and Hoeppel (1968) show very little variation in the calculated activity

coefficients or its derivative from 200 to 40 C. The run at 37.30 C and 1.9% (mole)

benzene has a very high standard deviation and is obviously in error.











1-


0-


-1 -


-2-


-3-


I .02
0.02


I I
0.04


I I
0.06


0.08


Mole Fraction, x,


Activity Coefficient Derivatives From Ultracentrifuge Data (as
Measured and Adjusted) and From VLE Data (Wolff and
Hoeppel, 1968, at 20 C) Fitted to Various Models, for the System
Carbon Tetrachloride(1) / Methanol(2).


0
X .. .. '' ''
...... ...... ......
, o.* *'" --


dx1


-6-


-7-


-8-


-9


-10


-11 -


X Centrifuge Data
0 Adjusted Data
VLE Wilson
................ VLE NRTL
- VLE Margules


Figure 4-11









There were no mixture refractive index data for this system available in the

literature. The experimental data were measured at 300 and 350 C and extrapolated to

other temperatures. This data is summarized in Table 4-10. Again, pure component

refractive indexes were not used, so only the differential refractive index is given in

Table 4-10.

Table 4-10
Differential Refractive Index (An) Experimental
Results for the System Carbon Tetrachloride(1)/Methanol(2)

x, An An An
250 C 30 C 350 C
Extrapt Expt Expt

0.00000 0.00000 0.00000 0.00000
0.02389 0.00732 0.00720 0.00707
0.05500 0.01630 0.01606 0.01582
t Values at 250 C were extrapolated linearly from the
experimental measurements at 300 and 35 C.

Summary of Experimental Results

Chemical potential derivatives measured in the ultracentrifuge for each of the

five systems studied were lower than those calculated from VLE data. In each system

a multiplicative factor would bring the results to within experimental error of the

VLE curves. The factor was not the same for all systems, nor was it the same for all

systems using acetone as a solvent. Experiments were made at different temperatures

for all five systems. Generally the temperature effects on activity coefficient

derivatives could not be distinguished for experimental uncertainties.

The results obtained are surprising and disappointing. The error analysis made

for the method indicated more reliable results and a systematic deviation was

unexpected. An analysis was made to determine the source of such systematic

deviations. This included the effect of pressure and impurities.









Pressure Effects on the Results

Pressure in the cell during operation is a function of radius and speed. Thus,

during a run at constant speed there is a considerable pressure gradient in the sample,

and runs made at different speeds can have quite different average pressures.

However, it was found that the experimental runs made at different speeds with the

same sample were generally consistent (see Tables 4-1, 4-3, 4-5, 4-7 and 4-9). This

indicates that pressure, which increases greatly with speed, had little effect on the

results. However, to insure that pressure is not the cause of the experimental

chemical potential derivatives being low, a more complete analysis was made.

Pressure is related to speed in a centrifugal field by the following expression

dP = p wo2 r dr (3-5

Integrating equation (3-5) and using the values from Run 38-2 (see Appendix F), which

are typical of the experiments run, the pressure varies from 1 atmosphere at the top to

42 atmospheres at the bottom. It is conceivable that such a pressure could affect the

activity coefficient derivative calculation of equation 4-1 through the sedimentation

parameter and the refractive index profile.

To illustrate the affect of pressure on the sedimentation parameter, the

correlation of compressibility data of Winnick and Powers (1966) yields densities and

partial molar volumes at the bottom of the cell under 42 atmospheres of pressure.

The resulting sedimentation parameter is only 0.15% less than that at the surface,

clearly a negligible difference. Cullinan and co-workers also concluded this for their

studies (Sethy and Cullinan, 1972; Rau, 1975).

The pressure can also have an affect on the refractive index of fluids, causing the

refractive index to vary differently in a spinning cell from a static composition

variation. To determine the extent of the effect of pressure on the refractive index of

fluids, measurements of pure fluids were made in the ultracentrifuge following the

ideas of Richard (1980; Richard et al., 1979). The method entails loading a pure liquid









sample in one side of the ultracentrifuge interference cell and air on the other side as

a reference. When this cell is rotated in the ultracentrifuge the interference optics

show a fringe pattern from the changing refractive index with increasing pressure

through the cell. The pressure can be calculated at any depth in the cell using the

integrated form of equation 3-5. The refractive index is known at the surface (1

atmosphere) and the refractive index increment to each fringe is known from

equation 3-8. These yield the pure component refractive index as a function of

pressure.

As an example, the results for acetone are given in Figure 4-12. While the data

have some scatter caused by variations in microcomparator measurements, it is well

described by a quadratic equation in pressure. The other pure components studied

were adequately fit with a linear equation. The pressure coefficient of the refractive

index, n, dn/dP, for carbon tetrachloride was found by Richard to be 5.5x105 atm1;

here a value of 5.9x10-5 atm-1 was obtained. The agreement is probably within

experimental error, indicating proper technique for this measurement.

Measurements were made on three of the solvents used here to determine the

potential extent of pressure on refractive index profiles. Table 4-11 summarizes the

measured data.

If a solvent and solute have the same linear change in refractive index with

pressure (i.e., dn/dP constant), then the effect would not be observed since the change

in the solution side of the double sector cell would be cancelled out by a change in the

reference side of the cell. However, the different chemicals used do not have the

same pressure variations, and acetone (the solvent in three systems) has a quadratic

variation with pressure. The quadratic variation with pressure is particularly

important because the reference (pure solvent) side of the cell is filled to a higher

level than the sample (solution) side to insure that the interference pattern of the

sample is fully covered by solvent. However, this means that the pressure throughout














0.0015

0.0014


0.0013


0.0012


0.0011


0.001


0.0009
n
0.0008


0.0007


0.0006


0.0005


0.0004


0.0003

0.0002


0.0001

0


Figure 4-12


6 10 14 18 22 26 31

Pressure, atm

Pressure Effect on Refractive Index (n) as Measured in the
Ultracentrifuge for Acetone at 25.









the reference side is higher than the sample. Consequently, there can be no

cancellation of effects, even when the dn/dP for the solute and solvent are the same.

Table 4-11
Experimental dn/dP Values

Compound dn/dp Pressure Range
x 105 atm
atm-1

Acetone 5.0t 5 30
Chloroform 6.7 5 13
Chloroform 6.0 4 16
Carbon Disulfide 7.2 10 40
Carbon Tetrachloride 5.9 4 14

t Value given is at 20 atm, for acetone dn/dP is a function
of pressure, dn/dP[atm"1] = 5.2x10-5 P[atm]x10"7.

To quantitatively determine the magnitude of this effect, certain calculations

were performed for carbon disulfide in acetone, and chloroform in acetone. Using

the measured value of dn/dP, the change in refractive index due to pressure can be

determined,


Anp )P AP (4-3

This can be calculated as an average over the entire cell or for each fringe, as is

done in the computer program FRNGCNVP (see Appendix C). This calculation must

be performed for both the pure solvent and the solution. In the solution a mole

fraction average of the solvent and solute values of dn/dP is used. Using the relation

of fringes to refractive index difference (equation 3-8) the change in the number of

fringes due to pressure can be determined.


Ajpi = An. d / X (4-4

where i can be solvent or solute.

The net pressure effect on the fringes is found by subtracting the sample effect

from the solvent effect.









Ajp = Ajp(solvent) Aji(sample)


The result of this effect is summarized in Table 4-12 for the carbon disulfide in

acetone and chloroform in acetone systems. The measured number of fringes for

each run and the net effect of pressure on the number of fringes are given. In

addition, the number of fringes necessary to match the ultracentrifuge activity

coefficient derivatives with those derived from VLE data were computed and are

listed. It is apparent that the effect of pressure on the refractive index is much

smaller than the observed discrepancy.

Table 4-12
Fringe Adjustments Due to Pressure Effects
and Total Number of Fringes Required to Match
VLE Data In Two Typical Cases

x, Speed Fringes Fringes Percent Fringes Percent
RPM Actually with P Change to Match Change
Measured Correction VLE

Carbon Disulfide(1)/Acetone (2)
0.03559 21739 4.85 4.68 3.5 3.83 21
0.03559 29502 8.94 8.2 8.3 7.14 20
0.03559 37000 14.02 12.12 13.6 11.27 20
Chloroform(1)/Acetone(2)
0.03713 25969 9.78 9.32 4.7 6.43 34
0.03713 29501 12.56 11.89 5.3 8.26 34
0.03713 37019 19.55 18.21 6.9 13.25 32

The Effect of Impurities on Ultracentrifugation

Certified spectrographic grade chemicals from Fisher have been used here. The

acetone and acetonitrile were dried with Fisher molecular sieve 3 A. Acetone was

also tried without drying and no noticeable differences were found. Several factors

can be considered. Impurities in the solute could not cause the discrepancy, their

contribution would be too small. If an impurity only altered the solvent or solution

refractive index, its effect would be taken care of by the measurement of refractive

index in the differential refractometer, which used the same solvent samples as in the

ultracentrifuge. Further, the effect would have been the same for all of the acetone


(4-5









systems, but it was not. Finally, any effect of a solvent impurity would be nearly a

constant number of fringes at all concentrations of every solute, rather than a relative

number as it was.

The only impurity known for the present chemicals was one small amount (0.75

%) of ethanol used as a stabilizer for the chloroform solute. However, since ethanol

has a refractive index and density almost exactly the same as acetone (solvent), it

would appear as more acetone and thus have no effect.

Summary

Determination of activity coefficient derivatives using laser interference optics in

an analytical ultracentrifuge has been carried out on five systems. In all cases the

experimental results were consistently lower than those calculated from vapor-liquid

equilibrium data. In each system a single factor applied to the chemical potential

derivative over the entire range of compositions matched the ultracentrifuge

experimental quantities to the chemical potential derivatives calculated from VLE

measurements. Investigation of pressure effects on the physical properties of the

systems shows an insufficient corrective effect, while impurities in the chemicals also

do not appear to be the cause.

Accepting a constant multiplicative factor for the ultracentrifuge data gives

results which generally support the low concentration extrapolations of two equations

(Wilson and NRTL) applied to VLE measurements. It was not possible to

differentiate a best model for the VLE systems from the data measured here.















CHAPTER 5

DIRECT CORRELATION FUNCTION INTEGRAL
DATABASE AND MODELING

Introduction

In an effort to develop new thermodynamic property models, free of common

simplifying assumptions such as pairwise additivity of intermolecular forces and rigid

molecules, the fluctuation solution approach has been taken by several authors

(Kirkwood and Buff, 1951; O'Connell, 1971; 1981). Kirkwood and Buff (1951) formulated

the density derivatives of the chemical potential and total system pressure in terms of

integrals of the radial distribution function (gi). O'Connell (1971) used these

formulations and the Ornstein and Zernike (1914) equations to give the fluctuation

thermodynamic properties in terms of the direct correlation function integrals (DCFI

or Cij). These direct correlation functions appear relatively insensitive to the details

of intermolecular interactions and may be modeled with simple functions.

Additionally, because the thermodynamic properties are obtained by integration, the

results can be much less sensitive to their parameterization. For example, Mathias

and O'Connell (1979, 1981) found that this approach worked very well for gases

dissolved in liquids.

The direct correlation function is given as

cij = hij k Pk f Cik hjk drk (5-1

where hij -- gij 1, gij is the radial distribution function and Pk is the molar

concentration of component k.

The density derivatives of the pressure and chemical potential or activity

coefficient (shown here for component 1 of a binary system) are given as








P/RT (1 -(p C + P2 C12) / p (5-2
0p p KT RT

where Ciy = p f c drij, is the direct correlation function integral.

P/RT = T + 2=-( pp+2 C12C 2+p C2) / p2 (5-3
Bp J K RT
(a_../RT 1 Cn11 (5-4

01 T,p2 P1 P
(lny), = (1 -Cn)/p (5-5

If the direct correlation function integrals can be modeled as functions of

temperature and composition, integration of equations (5-2) and (5-4) can yield the

pressure, solution density and the chemical potential, essential in phase equilibrium

calculations. In particular, such a model may allow for accurate modeling of the

activity coefficient of highly non-ideal systems.

The DCFIs modeled by Mathias and O'Connell (1979, 1981) work well for liquids

containing supercritical components (dissolved gases) but apparently not as well for

condensed phase systems (Campanella, 1984). Campanella made an attempt to model

the DCFI for vapor-liquid equilibria (VLE) and liquid-liquid equilibria (LLE) systems

in a fashion similar to that of Mathias (1978),. using a hard sphere term and a

perturbation term. He found in VLE systems that the hard sphere term did not show

adequate compositional variation. Further, the data used to calculate DCFIs at low

concentrations were not accurate enough to properly determine the compositional

dependence in that region. His work was hindered by having to calculate activity

coefficient derivatives indirectly by differentiating either tabulated phase equilibria

data or data from a model, such as that of Wilson (1964). This has lead to uncertainty

in the actual compositional behavior of DCFIs.

The objective of the present work has been to thoroughly explore the

compositional behavior of the DCFI. The binary DCFIs can be written in terms of

mole fraction and the measurable binary quantities of; isothermal compressibility, KT,








partial molar volumes, Vi and V2, volume, V, and activity coefficient derivatives,
C dln__ (alny 'I
X ,and x,

(1 Cu)- + X2 Lny (5-6
VKT RT ax, )TP

(1 -C12)- V V2 I Lxn p (5-7
VKT RT 8x )TP

(1 -C2 -x + x1 (5-8
VKT RT I ax2 JTP

Binary data on non-ideal systems have been collected, aided by the recent compilation

of isothermal compressibilities by Huang (1986), and a database of DCFIs for a wide

variety of systems has been created.

To compensate for the lack of directly measured activity coefficient derivative

data, derivatives of the NRTL and Wilson models with parameters fit to VLE data

will be used to cover the range of values for a given system. With these activity

coefficient derivatives, a database of DCFIs can be developed. The sixteen non-ideal

systems selected by Campanella (1984) plus twelve other non-ideal systems were chosen

for study. For each of these systems, excess volume data, pure component

compressibility data and vapor-liquid equilibrium data was obtained. For nineteen of

the systems, mixture compressibility data was also available.

For modeling purposes, the DCFI can be conveniently divided into ideal and

excess terms. The ideal DCFI is calculated by assuming an ideal mixing rule for the

volume and isothermal compressibility. The activity coefficient derivative term is

zero in the ideal case. By subtracting the ideal DCFI from the real DCFI, determined
from experimental quantities, an excess quantity, resembling the activity coefficient

derivative, remains. The ideal and excess DCFIs in a binary system are given as

(1 C1)ID _____
( x1 V' K + x2 V' K42) RT (5-9


(1 C 2)ID = Vl V2
( x V KoI + x2 Vn zK) RT (5-10








(1 C22)ID = (V)2
( x1 V K + x2 V' K-) RT (5-11


(1 Cn)E = (1 ) ( C)ID = fn (vE, K) + x, l (5-12
( ax1 )TP

(1 C12)E = (1 C) (1 C12)ID = f21 (VE, KT) x1 x 2l(5-13

(1 C22)E (1 C22) (1 C22)ID = f22 (v KE) + x, 1 2 (5-14
V xa2 )TP
where the superscript 0 denotes pure component properties and fni, f12 and f22 are

functions of excess volume and excess compressibility. If this excess DCFI could be
easily modeled it would lead to a better understanding of the compositional variation
of the DCFI and aid future theoretical modeling efforts. Because the excess DCFI so
closely resemble the activity coefficient derivatives, a modeling approach that uses a

form similar to the derivatives of common activity coefficient models such as the
Wilson (1964) or NRTL (Renon and Prausnitz, 1968) might prove beneficial. Indeed,
Campanella (1984) pointed out that most of the activity coefficient models have a

common form when written in terms of the composition derivative. For component 1

of a binary, three common models, Van Laar (Van Ness and Abbott, 1982), Wilson and

NRTL, can be written as

Van Laar

lny (X2 + 2 (5-15
ax, )JTP (xl0 + X2F)2 (X10 + X2)3

where ( = 2 A12 Ai1 0 = A12 F = A21 T = 2 A21 A12 (A21 A2)


A12 and A21 are Van Laar model binary parameters.








Wilson


Oln__ 2 + 2 1X2
ax,1 P i=1 x1Oi + X2ri =1 (x1i, + X2Fi)2

where 1, = -1 0, = 1 17 = A12


(2 = A21


02= A21 F2 = I


(5-16


P = A12 (A12 1)


2 = A21 (A21 1)


A12 and A21 are Wilson model binary parameters.


NRTL
Lny 2 ix2 2 pix2
XTP i= X21 + + x2
ax, 1) i=1 (xI~i + x2,i)2 i=1 (xEie + X2 1)3


where (1 = 2 T21 G21


(5-16


0,= 1


02= G12 r2 = 1


=2 = 2 T12 G12 (G12 1)


T12, T21, G12 and G21 are NRTL model binary parameters.


The common term here,


S+ x2 may work well as a model for the excess
(xIE) + X20m


DCFI.

The activity coefficient derivative of component 1 in a binary system can be

written directly in terms of the three binary DCFIs and the reduced bulk modulus as


a =x2 [(1 CuX1 C22) (1 C2)2] p KT RT (5-18

Examination of this grouping of DCFIs calculated from the database may suggest a

new empirical relationship for the activity coefficient derivative.

DCFI Database
The database of DCFIs includes only significantly non-ideal systems. These types

of systems will have the largest and most complicated excess terms. Campanella (1984)

studied many systems, both ideal and non-ideal. He found the ideal systems to be


"2 = 2 T12 G12


11 = G21 1, = 2 T21 G21 (G,2 -1)




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