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The effect of single electrolytes on the vapor-liquid equilibrium of The effect of single electrolytes on the vapor-liquid equilibrium of

mixed solvents mixed solvents

Peggy Tomasula New Jersey Institute of Technology

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T o m asu la , P eg g y M ary

THE EFFECT OF SINGLE ELECTROLYTES ON THE VAPOR-LIQUID EQUILIBRIUM OF MIXED SOLVENTS

New Jersey Institu te o f T echno logy D.Eng.Sc.


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International

THE EFFECT OF SINGLE ELECTROLYTES ON THE VAPOR—LIQUID EQUILIBRIUM OF MIXED SOLVENTS

byPeggy Tomasula

Dissertation submitted to the Faculty of the Graduate School of the New Jersey institute of Technology in partial fulfillment of the requirements for the degree of

Doctor of Engineering Science 1985

APPROVAL SHEET

Title of Dissertation:

The Effect of Single Electrolytes on the Vapor-Liquid Equilibrium of Mixed Solvents

Name of Candidate:

Peggy Tomasula Doctor of Engineering Science, 1985

Dissertation and Abstract Approved:

Dr. Dimitrios Tassios Date Professor Department of Chemical Engineering & Chemistry

bate

Date

•

Date

Date

VITA

Name: Peggy Tomasula

Degree and date to be conferred: D.Eng.Sc., 1985

Secondary education: St. Mary's High School, Perth Amboy, NJ 1972

Collegiate institutions attended

Dates Degree Date of Degree

New Jersey Institute of Technology

9/80-5/85 D.Eng.Sc. 1985

New Jersey Institute of Technology

9/77-9/80 M.S.ChE 1980

Seton Hall University 9/72-5/76 B.S.Chem 1976

Major: Chemical Engineering

Publications:

Tomasula, P. and Tassios, D. "The Correlation and Prediction of the Vapor-Liquid Equilibrium of Multicomponent Electro-lytic Solutions," to be submitted for publication.

Tomasula, P., Czerwienski, G., and Tassios, D. "Vapor Pres-sures and Osmotic Coefficients: Electrolytic Solutions of Methanol," submitted for publication.

Czerwienski, G., Tomasula, P. and Tassios, D. "Vapor Pres-sures and Osmotic Coefficients: Electrolytic Solutions of Ethanol," to be submitted for publication.

Tomasula, P., Swanson, N., and Ander, P. "Co-Ion and Coun-ter-Ion Interactions with Sulfonated Polysaccharides," ACS Symposium Series (1978).

Kowblansky, M., Tomasula, P., and Ander, P. "Mean and Single-Ion Activity Coefficients of Sodium Halides in Aqu-eous Sodium Polyphosphate and Sodium Carrageenan Solu-tions," J. Phys. Chem. (1978), 82.

Positions Held:1984-85 Research Fellow in Chemical Engineering, New

Jersey Institute of Technology1980-84 Special Lecturer in Chemical Engineering, New

Jersey Institute of Technology1979-80 Adjunct Professor of Chemical Engineering, New

Jersey Institute of Technology1977-79 Teaching Assistant,

TechnologyNew Jersey Institute of

*

A B S T R A C T

A group-contribution model for the prediction of salt-effects on the vapor-liquid equilibria of multicomponent electrolytic solutions

containing a single electrolyte is presented. Coulombic interactions

are represented through a Pitzer term. Solvation effects and short- range interactions are represented through a UNIQUAC-type expression.

An ion-size, a solvation and three ion-solvent interaction parameters

per salt-solvent binary are required for multicomponent predictions.All parameters are obtained only through the correlation of

binary salt-solvent osmotic coefficient and vapor-pressure depression data at 25°C, in most cases, and binary solvent VLE data. The salt- solvent binary data were correlated with an average percent error in

c£> of 2.5 and an average percent error in P of 0.35 mm Hg up to a molality of 6 for 1-1 and 2-1 salts. The model is also useful in the prediction of aqueous binary salt data up to a molality of 6 and 2 0 0°C

and nonaqueous binary salt data up to a molality of 6 and 60°C.Methods are also presented for the estimation of the ion-solvent

interaction parameters needed for multicomponent prediction when the constituent binary data are not available.

25 data sets of isothermal and isobaric salt-alcohol-water and salt-alcohol mixtures were predicted using the binary interaction

parameters and gave an average absolute error in the vapor phase

composition of 0.019. The model predicts correctly the salting-in of

the appropriate component.

Vapor-pressure depression data of Nal, KCH^COO, NaSCN, and NH^SCN in methanol at temperatures of 25 and 40°C were measured in the

molality range of 0.1-5.0 m using a static method, where the vapor pressure of the electrolytic solution is compared to that of the

pure solvent.Osmotic coefficients were calculated from the vapor pressure data.

This data was used to obtain additional binary interaction parameters which could not be determined from the existing literature data.

PREFACE

The estimation of the effect of a single electrolyte on the

vapor-liquid equilibrium of mixed solvents is often necessary in the modeling of chemical reaction equilibria and separation

processes. While methods are available to predict nonelectrolytic solution phase-equi1 ibria from little or no experimental

data (Derr and Deal, 1969; Fredenslund, et al. , 1975), thosepreviously used for electrolytic solutions are usually limited

to correlation of existing data. In addition, the lack of salt- nonaqueous solvent data prevented the development of such models.

Recently, two models (Rastogi, 1981; Sander, et al., 1984) have been proposed which have some prediction potential of salt

effects on the VLE of mixed solvents. These models represent a significant advance in that the short-range interactions between all solvent species are accounted for through salt-solvent molecule (Rastogi) or ion-solvent molecule (Sander, et al.) parameters and solvent (A)-solvent (B) parameters. The long-

range ion-ion interactions are represented through a Coulombic

term. However, these models are basically useful for correlation purposes only.

Models in which the parameters are ion-solvent specific require a minimum of experimental data to effect prediction and it is the objective of this work to present such a model for the prediction of the VLE of mixed solvent systems consisting of one

ii

salt. This model combines a Pitzer (1977) expression to repre

sent Coulombic interactions, the Flory-Huggins expression (1 9 *1 1 , 1942) to account for differences in the sizes of the

solvent species and for the solvation of the ions by the

solvents, and the residual term of the UNIQUAC (Abrams and

Prausnitz, 1975) equation to account for the short-range inter

actions between all solvent species. The parameters, which are ion-specific, are evaluated from a binary data base of salts in

water and alcohols.

The validity of the model is shown for mixed alcohol-water and mixed-alcohol solutions consisting of one salt.

iii

TABLE OF CONTENTSChapter PagePREFACE................................................... ii

LIST OF TABLES........................................... viiLIST OF FIGURES........................................... xiINTRODUCTION............................................ . 1

1 THERMODYNAMICS OF VAPOR-LIQUID EQUILIBRIA............... 6

2 THE GIBBS FREE ENERGY, THE ACTIVITY, AND THEACTIVITY COEFFICIENT................................. 9

3 MODEL DEVELOPMENT................................... 124 EXPERIMENTAL........................................ 215 BINARY DATA REDUCTION

5.1 The Binary Equations the the Ternary and BinaryData Bases.......... 25

5.2 Parameter Estimation from Binary ElectrolyticSolution Data......................................35

5.3 Values of the Binary Ion-Solvent Interaction Parameters.................................... 53

5.4 Temperature Extrapolation of the Binary Parameters . . 675.5 Molality Range of the Model.................. 745.6 Values of the Binary Solvent (1) - Solvent(2)

Interaction Parameters ......................... 786 MULTICOMPONENT SYSTEMS

6.1 The Ternary Model for the Prediction of Salt-Mixed Solvent Data............................. 82

6.2 Case 1. Solvation Effects are Neglected in theModel.......................................... 85

6.3 Case 2. Solvation Effects are Considered . . . . . . 986.4 Case 3. Ihe Preferential Solvation of the Positive

Ions in Mixed Solvent Systems is Considered.(Equations (6-10a) and (6-10b) are Used.) .......... 105

iv

Chapter Page6.5 Case 4. The Preferential Solvation of the

Positive Ions in Mixed-Solvent Systems.(Equations (6-lla) and (6-llb) are used.) ........... 117

6 . 6 The Use of Preferential Solvation................... 121

6.7 Comparison of the Prediction Results of this Study with the Correlation and PredictionResults of Other Models .......................... 122

6 . 8 Use of the Ternary Model in Data Correlation......... 1306.9 Use of the Model in the Prediction of Salt-Mixed

Solvent Systems when a, or A has not been Established................................... 133

6.10 Use of the Model in the Prediction of Salt-Mixed Solvent Systems when more than one Value ofa^ or has not been established.................. 141

6.11 Use of the Model in the Prediction of Salt-Multicomponent Solvent Systems......................147

7 DISCUSSION OF RESULTS .................................. 150

8 CONCLUSIONS ........................................... 179

APPENDIX A1. Vapor Pressures of Solvents Used in This Study . . . . 1812. Calculation of the Fugacity Coefficients at

Saturation and of the Vapor-Phase Using theHayden-01Connell Correlation .. .................. 183

APPENDIX B1. Densities of Pure Solvents Used in This Study......... 1912. Estimation of the Densities of Pure and Mixed

Solvents......................................... 192

3. Estimation of the Change in Density of MixedSolvents with Composition ......................... 199

APPENDIX C1. Dielectric Constants of Pure and Mixed Solvents

Used in This Study . .............................2022A. Correlation of Dielectric Constant Data for Binary

Alcohol-Water Systems when Data are Available ....... 203

v

Chapter Page2B. Calculation of the Change in Dielectric

Constant with Composition of BinarySolvents when Data are Available................. 206

3A. Estimation of Dielectric Constant Data for Multicomponent Systems when Data as aFunction of Composition Data are Unavailable . . . . 208

APPENDIX D DERIVATION OF THE GIBBS FREE ENERGY EXPRESSIONOF THE PITZER (Coulombic) TERM USED TO OBTAIN THE ACTIVITY COEFFICIENTS OF THE SALT AND THE SOLVENTS............................... 209

APPENDIX E EXPERIMENTAL DATA RESULTS AND ERROR ANALYSIS . . 215APPENDIX F CALCULATION OF THE MEAN ACTIVITY COEFFICIENT

OF THE SALT IN SINGLE AND MIXED SOLVENTS . . . . 222APPENDIX G COMPUTER PROGRAMS........................ 238NOMENCLATURE............................................. 287BIBLIOGRAPHY............................................. 291

vi

LIST OF TABLES

Table Page

5.1 Data Base for Parameter-Estimation from BinaryElectrolytic Solution Data ........................... 29

5.2 Ternary.Data Base..................................... 31

5.3 UNIQUAC Volume (r ) and Surface Area (q ) Parameters . . . . 32

5.4 Pauling Crystallographic and Yatsimirskii ThermochemicalRadii for the Ions Considered in This Study............. 33

5.5 UNIQUAC Group Volume (P ) and Group Surface Area (0 )Parameters for the Ions of This Study......... ‘........34

5.6 Case 1. Test of the Binary Model (Eq. 5-6) with a = Srand h , . and h . Set Equal to Zero..................... 37o+j o-j ^

5.7 Case 2. Test of the Binary Model (Eq. 5-6) with a anAdjustable Parameter and h ,. and h .Set Equalto Zero.................... . . .°7J. ............... 39

5.8 Hydration Numbers for Some of the Ions Used in This Study . . 41

5.9 Case 3. Test of the Binary Model (Eq. 5-6) with a = Sr. for the Aqueous Systems and Adjustable for the Nonaqueous Systems.....................................43

c

5.10 Binary Interaction Parameters for the Aqueous 1-1Chlorides at 25° C ......................................54

5.11 Binary Interaction Parameters for the Aqueous 1-1Bromides and 1-1 Iodides at 25°C........................56

5.12 Binary Interaction Parameters for the Aqueous 1-1Nitrates at 25°C...................................... 57

5.13 Binary Interaction Parameters at 25°C. a is anAdjustable Parameter ................................. 59

5.14 Binary Interaction Parameters for Salt-Methanol Systemsat 25°C. a is an Adjustable Parameter................ JS3

5.15 Binary Interaction Parameters for Salt-Ethanol and Salt-Isopropanol Systems, a is an Adjustable Parameter . . . . .65

5.16 Average Percent Error in c£) and in P in Predictionfrom Parameter Values at 25°C in Aqueous Systems........ .68

vii

Table Page

5.17 A Comparison of the Average Percent Error in <£> in Prediction from Parameter Values at 25°C in Aqueous Systems for Four Models .................. . 70

5.18 Average Percent Error in <£> and in P in Predictionfrom Parameter Values at 25°C in Methanol Systems . . . . 71

5.19 A Comparison of the Average Percent Error in c£> inPrediction from Parameter Values at 25°C inMethanol Systems for Four Models . . . . . ........... 73

5.20 Prediction of the Osmotic Coefficients at 25°C ofSome Aqueous Systems at High Molalities.................75

5.21 Correlation of Binary Solvent Nonelectrolyte Data . . . . . 816.1 Case 1. Prediction of the Vapor-Fhase Compositions of

Ternary Data. (Solvation Effects are Neglected.) . . . . 8 6

6.2 Case 2. Prediction of the Vapor-Fhase Compositions ofTernary Data. (Solvation of the Positive Ionby the Solvents is Considered.)............ 99

6.3 Case 3. Prediction of the Vapor-Fhase Compositions ofTernary Data. (Preferential Solvation of thePositive Ion by the Solvents is Considered.) .......... -108

6.4 Case 4. Prediction of the Vapor-Fhase Compositionsof Ternary Data. (Preferential Solvation of the Positive Ion by the Solvents is Considered.Temperature Correction Applied.) ..................... 118

6.5 Comparison of the Results of This Study with Thoseof Rastogi (1981), Hala (1983), Mock, et.al. (1984),and Sander, et.al. (1984) ............................ 123

6 . 6 Demonstration of the Correlation Capability of theModel.............................................. 132

6.7 Prediction Results Using an Average Halide Value of a^ . .

6 . 8 Prediction Results Using Estimated Values of Amn anda4 ...............................................

6.9 Equations for the Estimation of A ^ ..................

6.10 Prediction Results Using Estimated Values of a andV . ...........................................

6.11 Prediction Results for Multicomponent Systems .........

134

139142

144

149

Table Page

A-l Antoine Constants for Solvents Used inThis Study...................................... 182

A-2 Constants Used to Evaluate the Pure-Component SecondVirial Coefficients, B, in Equation (A-6 ) ........... 187

A-3 Constants Used to Evaluate the Cross Second VirialCoefficients, B^, in Equation (A-6 ) .............. 189

B-l Densities of Pure Solvents Used in This Study....... 191B-2 Parameters for Equations (B-2) and (B-3) . .......... 193B-3 Values of Wg^, V", and T , ...........................195

B-4 Results of the Evaluation of Mixing Rules for theMethanol-Water System at 25 and 50°C.............. 197

B-5 Results of the Evaluation of Mixing Rules for theEthanol-Water, n-Propano1-Water, and Isopropanol-Water Systems.................................... 198

C—1 Dielectric Constants of Pure and Mixed SolventsUsed in This Study.............. 202

C-2 Values of A(l) through A(6 ) for Use in Equations(C-l) and (C-2)................................. 204

C-3 Values of (B-l) through B(6 ) for Use in Equations(C-l) and (C-3)................................. 205

E-l Experimental Values of AP, P, and c|> for theNal-MeOH System at 25 and 40°C................ 218

E-2 Experimental Values of AP, P, and <$> for theKCH3COO-MeOH System at 25 and 40°C .................. 219

E-3 Experimental Values of AP, P, and <3? for theNH^SCN-MeOH System at 25 and 40°C................ 220

E-4 Experimental Values of AP, P, and for theNaSCN-MeOH System at 25 and 40°C.................... 221

F-l Method 1. Average Percent Errors in y± inPrediction from Parameter Values at 25°C and Experimental Solvent Activities for Aqueous Systems . . 228

F-2 Method 2. Average Percent Errors in y± inPrediction Only from Parameter Values at 25°Cfor Aqueous Systems................................ 229

Table

F-3

F-4

Binary Interaction Parameters for the HCl-MeOH and HCl-EtOH Systems Obtained Through Regression of * Y± Data at 25°C...........................

Prediction of the Mean-Activity Coefficients of HC1 in MeOH-l^O and EtOH-t^O Mixtures at 25°C . . . . .

Page

231

236

LIST OF FIGURES

Figure Page

4.1 Differential Manometer ............................... 22

4.2 Test of the Differential Manometer. Vapor PressureDepression of Aqueous KC1 Solutions at 25°C......... 24

5.1 Case 1. The Contribution of the Pitzer, Flory-Huggins,and Residual Terms in the Calculation of In 'Y, n forthe LiCl-Water System at 25° C .............. 2 . . . . 4 4

5.2 Case 1. The Contribution of the Pitzer, Flory-Huggins,and Residual Terms in the Calculation of In TmpDH ôr the LiCl-Methanol System at 25°C..................... 45

5.3 Case 2. The Contribution of the Pitzer, Flory-Huggins,and Residual Terms in the Calculation of In 'V, Q forthe LiCl-Water System at 25°C.............. 2 . . . . 47

5.4 Case 2. The Contribution of the Pitzer, Flory-Huggins,and Residual Terms in the Calculation of In T mpDH or the LiCl-Water System at 25°C .............. 48

5.5 Case 3. The Contribution of the Pitzer, Flory-Huggins,and Residual Terms in the Calculation of In ''V, q for the LiCl-Water System at 25°C.............. 2........ 50

5.6 Case 3. The Contribution of the Pitzer, Flory-Huggins,and Residual Terms in the Calculation of In TmpOH or the LiCl-Methanol System at 25°C........... . . . 51

6.1 Predicted y-x Diagram for the LiCl-tLO-MeOH Systemat 25°C and m=l. (Solvation Effects areNeglected.)........................................ 39

6.2 Predicted y-x Diagram for the LiCl-Ô-MeOH Systemat 760mm Hg and Molality Range 0.1-3.8m.(Solvation Effects are Neglected.) .................. 90

6.3 Comparison of the Activity Coefficients of the Ô-MeOHSystem with and without Added Salt.Salt System: LiCl-Ô-MeOH at 25°C and m=l......... 91

6.4 Comparison of the Activity Coefficients of the tÔ-MeOHSystem with and without Added Salt.Salt System: LiCl-B?O-MeOH at 760mm Hg and Molality

Range 6 .1-3.8m..........................93

xi

Figure Page

6.5 Contribution of the Pitzer, Flory-Huggins, and ResidualTerms to In T n for the LiCl-HÔ-MeOH System at 25°C and 2 m=l. (Solvation Effects are Neglected.)......................................... 94

6 . 6 Contribution of the Pitzer, Flory-Huggins, and ResidualTerms to In "Y, 0 for the LiCl-lLO-MeOH System at 760mm Hg and 2 Molality Range 0.1-3.8m. (Solvation Effects are Neglected.) ............................. 96’

6.7 Contribution of the Pitzer, Flory-Huggins, and ResidualTerms to In Y , n for the LiCl-HÔ-MeOH System at 25°Cand m=l. 2 (Solvation of Li+ by H„0 andMeOH is Assumed.).................................. 102

6 . 8 Contribution of the Pitzer, Flory-Huggins, and ResidualTerms to In ' X , nrr for the CaC^-EtOH-MeOH Systemat 760mm and e m=1.806. (Solvation of Ca byEtOH and MeOH is Assumed.).......................... 103

6.9 Contribution of the Pitzer, Flory-Huggins, and Residual Terms to In 'Y. n„ for the CaCl^-EtOH-MeOH System „ at 760mm Hg e and m=1.806. (Solvation of Ca by EtOH and MeOH is Assumed. No T Correction Used.) . 110

6.10 Contribution of the Pitzer, Flory-Huggins, and ResidualTerms to In Yj n for the LiCl-HÔ-MeOH System at 25°C and 2 m=l. (Preferential Solvation of Li by H2O and MeOH is Assumed.)................ J.12

6.11 Contribution of the Pitzer, Flory-Huggins, and ResidualTerms to In ' Y , n for the CaCl?-H70-Me0H System at760mm Hg and 2 m=1.806. (With and withoutPreferential Solvation.)........................... .114

6.12 Contribution of the Pitzer, Flory-Huggins, and ResidualTerms to In Y, n for the LiBr-Isopropanol System at 75°C and 2 m=1.4 (With and Without Preferential Solvation.)........................................115

6.13 Contribution of the Pitzer, Flory-Huggins and ResidualTerms to In f°r the NaCl-Ho0-Et0H System at30°C . . . . . . ............. 2.....................136

7.1 A Comparison of the Solvation Numbers of the CalciumIon by Methanol and Ethanol with and without Preferential Solvation..............................166

7.2 Predicted y-x Diagram for the LiCl-H90-n-propanol Systemat 760 mm H g ...................z...................171

xii

Figure _ Page7.3 A Comparison of the Activities of n-propanol Calculated

Using the Model with the Experimental Values......... 173

7.4 Contribution of the Pitzer, Flory-Huggins, andResidual Terms to In ' Y ^ for the LiCl-^O-n-propanol System at 760 m m H g ........................ 175

7*5 Contribution of the Pitzer, Flory-Huggins, and Residual Terms to ln*% for the LiCl-^O-n- propanol System at 760 m m H g ........................ 176

xiii

I N T R O D U C T I O N

Thermodynamic data for solutions containing a single electro

lyte in a single solvent, and especially multiple solvents, are

often needed in the modeling of separation processes and chemical reaction equilibrium. Osmotic coefficient data at 25°C for binary

aqueous electrolytic solutions are available in the extensive compil

ations of Robinson and Stokes (1959). Vapor pressure depression data for binary aqueous electrolytic solutions at 100°C are tabulated in

Weast (1970). Data at temperatures other than 25 and 100°C are very

limited. (Snipes, et.al., 1975; Campbell and Bhatnagar, 1979; Holmes, and Mesmer, 1981) Osmotic coefficient and vapor pressure depression

data for solvents other than water are scarce. (Janz and Tomkins,1972; Bixon, et.al., 1979; Tomasula, 1980; Czerwienski, et.al., 1985)

Thermodynamic data for an electrolyte in mixed solvents are even

more limited (Ciparis, 1966; Sada, et.al., 1975; Boone, et.al., 1976)

and their prediction from binary data would be very useful for

industrial applications.Models for the correlation of nonelectrolytic solution data, such

as the Margules (1895) and NRTL (Renon and Prausnitz, 1968) equations,

have been applied to the correlation of ternary electrolytic solutions

consisting of a salt, water, and an alcohol. (Schuberth, 1974, 1977;

Schuberth and Nhu, 1976; Beckerman and Tassios, 1976; Mock, et.al., 1984) These models do not account for the long-range ionic forces, but

give recognition to the short-range forces. Binary interaction parameters are evaluated from binary and ternary data.

Recently, three models (Rastogi, 1981; Hala, 1983; Sander, et.al.,

1984) for the prediction of salt effects on the VLE of multicomponent

1

2

electrolytic solutions containing one salt have been developed.

Rastogi and Hala assume that the excess Gibbs free energy is the sum of two terms, a long-range Coulombic term to represent ion-ion

interactions and a short-range term to represent the interactions between all solution species. Sander, et.al., add an entropic term to the long-range Coulombic term and the short-range term.

The Hala model combines a semi-empirical electrostatic term and the Wilson (1964) equation. The LiCl-water-methanol system at 60°C was predicted from four binary salt-solvent parameters and two

solvent-solvent interaction parameters evaluated from binary data at 60°C. The model was not applied to the prediction of other ternary

systems.The Rastogi model combines a modified Debye-Huckel equation and

the NRTL equation. This model was used to predict salt-water-alcohol

systems from binary salt-solvent and solvent-solvent interaction parameters alone. Prediction of salt-binary mixed solvent data is only possible up to 2m and when the constituent binary data are available.

The model cannot be extended to more than two solvents.

Sander, et.al., combine the Debye-Huckel equation and the UNIQUAC equation. (Abrams and Prausnitz, 1975) The UNIQUAC parameters are functions of concentration and are ion-solvent specific. The ion-

solvent specific parameters were established through the correlation of electrolytic single and binary mixed solvent data. Prediction was

demonstrated for a few mixed solvent systems. However, prediction from ion-solvent parameters determined solely from binary data was not demonstrated.

All these models, however, are limited. The Rastogi model and

3

that of Sander, et.al. are basically applicable to correlation of mixed solvent-single electrolyte systems. The Hala model was applied

to one system only.The difficulty in modeling electrolytic solutions is due to the

phenomenon of salting-out. The addition of a salt to a binary or a

higher order solvent system increases the vapor phase mole-fraction of the solvent with the smallest dielectric constant. The solvent with the largest dielectric constant is salted-in. While this behavior is not general; for example, in the HgC^-methanol-water system, methanol is salted-in by HgC^ and water is salted-out, it is the one of most

interest in phase-equilibrium calculations. This effect is most

pronounced in electrolytic solutions consisting of water and can be explained by the concept of solvation, where it is assumed that solvent molecules are bound to the ions.

Due to the long-range nature of ion-ion interactions, the addition of a small amount of salt to a solvent results in an increase in the

solvent activity coefficient. As the concentration of the salt approaches an ionic strength of unity, short-range forces become important, and the solvent activity coefficient continues to increase until it reaches a maximum. The solvent activity coefficient then continues to decrease with increasing salt concentration.

The decrease in the solvent activity coefficient is a direct result of solvation. Increasing the concentration of the salt

increases the number of ions in solution which in turn remove solvent molecules from the bulk solution. As more and more solvent molecules are removed from the solution, the vapor pressure of the solvent

decreases.

4

The intermolecular forces which operate in binary electrolytic

solutions also operate in salt-mixed solvent systems. Even though

the constituent binary salt-solvent systems both exhibit negative

deviations from Raoult's law at concentrations above 1 molal, typically

only one of the solvents in a salt-mixed solvent system will exhibit

a decrease in the solvent activity coefficient relative to its value

in the salt-free solution. This phenomenon is often explained by the

concept of preferential solvation, where the probability of finding

solvent molecules with the higher dielectric constant in the vicinity

of the ions is greater than that for the solvent molecules with the

lower dielectric constant.

From the above discussion, it is apparent that a semi-empirical

model for the representation of salt-effects must not only account

for the long and short-range forces which operate in solution, but

should also explicitly account for the removal of solvent molecules

from the bulk solution by the ions.

While the Rastogi, Hala, and Sander models recognize this

phenomenon indirectly through modifications of their respective long-

range or short-range terms, solvation effects are not explicitly

accounted for in their expressions.

The model presented here combines a Pitzer (1977) expression to

account for the long-range and short-range ionic interactions, the

athermal Flory-Huggins (1941, 1942) expression to account for the

entropic effects of the solvated species, and the residual term of

the UNIQUAC equation to account for other short-range interactions

not described by the model.

The parameters of the mode], which are ion-solvent specific, are

evaluated from binary electrolytic solution data. The model is

applied to salt-mixed solvent systems to demonstrate its validity.

C H A P T E R 1

Thermodynamics of Vapor-Liquid Equilibria Electrolytes are generally classified into two groups. The

first group are known as the strong or non-associated electrolytes

and the second as the associated electrolytes. (Robinson and Stokes, 1959; Hamed and Owen, 1958)

The strong electrolytes are those which dissociate into their component ions when in solution. While the ions may interact with the solvent; i.e., associate with the solvent, there is no association between the ions of opposite sign. In addition, salts of this type do not vaporize at moderate temperatures and pressures. Salts which are termed strong in aqueous solutions are not necessarily strong in nonaqueous solvents, where the low dielectric constant leads to ion- pairing. Salts such as the alkali halides and the alkaline-earth halides are strong in water.

Associated electrolytes are termed either weak electrolytes or

ion-pairing electrolytes. Weak electrolytes exist as ions and molecular species in solution. Acids and bases, with the exception of the alkali metal and quaternary ammonium hydroxides, are weak electrolytes. The molecular electrolyte may enter the vapor phase but dissociates only at high temperatures. (Edwards, et.al., 1975)

Ion-pairing electrolytes are those in which the positive and negative ions associate. Bivalent metal sulphates in water and almost

all other salts in nonaqueous solvents at high concentrations form ion-pairs. Salts of this type are not present in the vapor phase.

All electrolytes in this study are assumed to be strong electrolytes; i.e., complete dissociation is assumed in both aqueous

6

7

and nonaqueous solvents. The electrolyte is not present in the vapor

phase.The condition for equilibrium for any solvent i in an electro

lytic solution of N solvents is then:

£.v = f.L i = 1, 2, ....N (1-1)

Xj vwhere f is the fugacity of solvent i in the liquid phase and f^ is

the fugacity of solvent i in the vapor.The fugacity coefficient, ($>., of solvent i is used to represent

1

the nonideality in the vapor phase.

- 4 ^ y, p d - 2 )

y^ is the mole fraction of solvent i in the vapor and P is the totalApressure. c£k is unity for the ideal vapor and is approximately unity

for systems at low pressures.The fugacity of solvent i in the liquid phase is given by

f.L = y . x, f,° i = 1, 2, ....N (1-3)

where f? is the pure-component reference fugacity of solvent i at the

temperature, T, and the pressure of the solution. *Xj_, is the liquid

phase solvent activity coefficient and will be discussed in Chapter 3. x , is the mole fraction of i calculated based on the total dissociation

of the electrolyte, where, for the solvent

x. = ni (1-4)1 ^ ns + Z k nk

2knk is the summation over all solvent species. ng, is the analytical

number of moles of electrolyte in the solution and V is the total

number of ions comprising the salt.

8

f/ is defined by the following expression:

fi =® i piS exp vi dP/RT (1"5)piS

where is the fugacity coefficient of pure solvent i evaluated at

T and the vapor pressure, P^ , of i. v , is the molar liquid volume of

pure i at T. It is not a function of pressure at low pressures.

The exponential term of equation (1-5), which is the Poynting

effect, reduces to equation (1-6) at low pressures

P

The Antoine equation is used to calculate the vapor pressures, P^s, for the solvents in this study. The Hayden-O'Connell (1975)

correlation for the prediction of second virial coefficients is usedAto calculate c£h and s. Ihe constants for the Antoine equation and

the Hayden-0'Connell correlation are in Appendix A.

The Hankinson and Thomson (1979) correlation is used to calculate

the pure-component liquid volumes, v . The method is discussed in

Appendix B.

C H A P T E R 2

The Gibbs Excess Free Energy, the Activity and the Activity Coefficient

The activity coefficient of solvent i is defined as the ratio of

the activity of i to the mole fraction of i.

y i - a1/xi (2-1 )

The activity of i is defined

(2-2)where f is the fugacity of i at T, P, and constant composition and f ° is the fugacity of i at T and a specified P and composition.

The excess Gibbs free energy is the difference between the actual total Gibbs free energy at T, P, and fixed composition and the ideal

total Gibbs free energy at the same T, P, and x.

GE = G real at T, P, x - G ideal at

T, P, x (2-3)

Differentiation of equation (2-3) with respect to the number of molesof solvent i, n , at constant T, P, and n gives

-Eg i = gi (real) - g ± (ideal) (2-4)

where

gt (real) = ^/dn.

and

g. (ideal) = ^/dn.

G (real) T, P, n. f^i + RT ln fi (real) J

(2-5)

(G(ideal))r, p, nj " Ai° + RT ln £i (ldeal)( 2- 6)

fJL^° is the chemical potential of the standard state. The chemical

potentials of the standard state in equations (2-5) and (2-6) are the

same since the ideal and real solutions are at the identical T, P, and

10

composition.

Substitution of equations (2-5) and (2-6) into equation (2-4)

gives

g^= RT ln £ ± (reaD/iu (ideal) (2-7)

The activity of solvent i in an ideal solution is equal to the mole

fraction of i. From equation (2-2), fp (ideal) is given byAfp (ideal) = fp0 x ± (2-8)

(real) is obtained directly from equation (2-2).

Substitution of equations (2-2) and (2-8) into equation (2-7)

gives

11 = RT ln a^x. (2-9)

From the definition of the activity coefficient of i given in equation (2-1), the partial molar excess Gibbs free energy is

g? = RT ln*X (2-10)

where-Eg i 6 ge / 6 n.x T, P, ^ (2-11)

Up is the total number of moles of species in the solution and

GE = n sE (2-12)EG is obtained from Euler s theorem, where

‘pg ■em

GE = n. gE (2-13)

orGE/RT = Z.n.lnTf (2-14)

1 1 lThe activity coefficient of component i is easily determined from

equations (2-10) and (2-11) given an expression for the excess Gibbs Efree energy, G . Conversely, the excess Gibbs free energy may be

11

calculated from equations (2-13) and (2-14) given an expression for

the activity coefficient, *X, of i.Any expression for the molar excess free energy of a binary

solution must obey the conditions:when x1 = 0 gE = 0

1 (2-15)x2 = 0 gE = 0

C H A P T E R 3

Model Development

The thermodynamics of an electrolytic solution as of all

solutions are determined by the forces which operate between the species of the solution. Assuming that the electrolyte dissociates into its constituent ions, interactions between the ions, between

the ions and the solvent molecules, and between the solvent molecules must be considered. These interactions may be loosely

categorized as physical or chemical in nature.The forces between the ions are the long-range Coulombic

interactions, which are important only in dilute solutions. The

short range ion-solvent molecule interactions, such as dispersion forces, ion-dipole, and ion-induced dipole forces, become important

as the concentration of the electrolyte is increased. (Bockris and Reddy, 1977). The molecule-molecule interactions are also short- range in nature. These interactions may be classified as induction

forces between a permanent dipole (or quadrupole) and an induced dipole, electrostatic forces between permanent dipoles and higher

poles, or dispersion forces between non-polar molecules. (Prausnitz,

1969)Strong physical forces can lead to the formation of loosely

bound species. They are referred to as chemical forces and lead to the phenomena of association and solvation. Association results from the formation of polymers, as in the case of water or methanol which are known to hydrogen bond. Solvation refers to the formation of complexes between unlike molecules. Solvation effects cause

negative deviations from Raoults's law.

12

13

Solvation effects are evident in electrolytic solutions. They

are typical in binary electrolytic solutions where the solvent

activity coefficients are less than unity and in multicomponent

systems such as, for example, the LiCl-water-methanol system. While

the activity coefficient of water is greater than unity in the

methanol-water system at 25<£ , it is less than unity in the LiCl-

water-methanol system at the same temperature.

In this study, it is assumed that the salt completely dissociates

into its constituent ions and that the ions are solvated by the solvent

molecules. Association effects are neglected. It is further assumed

that the apparent solvent activity coefficient, 7^> is the sum of

three terms:

ln 77 = ln 7^ (Coulombic) + ln 7^ (Combinatorial)

+ ln 7^ (Residual) (3-1)

The first term accounts for the long-range Coulombic forces, the

second for differences in the sizes of the molecules (combinatorial

term), and the third for the short range ion-solvent molecule and

molecule-molecule interactions, (residual term)

To account for ion-solvation, equation (3-1) is rewritten in

terms of 7^, the true activity coefficient.

' y ' ' Coulombic + ' C°mbinatorial + ' Residualn i i i i

(3-2)

For simplicity purposes, it is assumed that all solvation effects

are accounted for explicitly through the combinatorial term and

implicitly through the Coulombic and the residual terms, i.e.,

1<*

I n T 00111 ‘ + ln T R e s ' = l r 7.' Cou1' + ln T f ’ E e s '1 1 1 1

(3-3)

To relate the apparent solvent activity coefficients, *X, to the

true solvent activity coefficients, a relationship due to Bjerrum (1920)

and discussed by Guggenheim and Stokes (1969) is used. Bjerrum showed that the activity of the solvent is the same whether the solute is

solvated or not, i.e.,

a.1 = a. (3-4)1 1 v '

and

i i i i (3-5)Substitution of equations (3-1), (3-2), and (3-3) into (3-5)

yields

y . Comb- - Xj’V i ’ Co"ibVx1 (3-6)

Introduction of equation. (3-6) into (3-2) gives

l n 7 = l n X 0 o u l * + l ny R e s - + l n C y ’00* - x . ' / x . )1 1 1 ' 1 1 1

(3-7)iny°oui. , the Coulombic contribution to the solvent activity

coefficient is evaluated using the free energy expression developed

by Pitzer (1977) for binary electrolytic solutions. The expression

is extended in this study to electrolytic solutions consisting of an

electrolyte in mixed solvents. (See Appendix D)lnXCoul’ = 2A< ((M.W.)i/1000)I3/2/l+bI^ - A^bI2(M.W.)i/(1000(l+bI^)2)

- 2(M.W. )/1000 (I3/2/l+bI^ - I/b ln(l+bl^)) dA^/d^

+ 2 A ^ ( (M .W .) /1 0 0 0 ) ( ( I 2 / ( l + b l V ) + I / b 2 ln ( l+ b I ^ )

15

- I^/Kl+bl^))^ db/dni + v2(2TTa3/3)(N/10002)(m2(M.W.)nT d(d)/dni

- m2d(M.W.)i + m2d(M.W.)(3/a)nT da/dn^ (3-8)

(M.W.)^ is the molecular weight of solvent i. (M.W.) is the molecular weight of the mixed solvent system on a salt-free basis.(° indicates the salt-free basis)

(M.W.) = £. xj°(M.W.)j (3-9)

xj° ■ <>io)The molality is defined as the number of moles of salt per

kilogram of solvent.m = 1000ns/(M.W.)£knk° (3-11)

The ionic strength is given byI = IZ2z+z_|m/2 (3-12)

where z+ and z are the charges of the positive and the negative ions,

respectively. V is the total number of ions which constitute the salt.

t ie Debye-Huckel limiting coefficient, is given by = l/3(2TTNd/1000)^(e2/DkT)3/2 (3-13)

where d is the density of the mixed solvent on a salt-free basis and D is the dielectric constant.

b is a function of a, the ion-size parameter, which is the only adjustable parameter in the Coulombic term.

b = ( 8TTN/1000)^(e2d/DkT)^ a (3-14)

Differentiation of equation (3-13) and (3-14) yields

dA^/dn^ = A^/nr(l/2d d(d)/dni - 3/2D d(D)/dni)

(3-15)

16

db/dru = b/tXp(l/a da/dn^ + l/2d d(d)/dn^ - 1/2D dD/dru)

(3-16)

a, in a multiple solvent system is given by

(3-17)

where x^0 is given by equation (3-10) and a is the ion size parameter

for an electrolyte in solvent j.

The change in a with composition isda/dn^ = (da/dxi)(dxi/dni) (3-18)

The densities of the mixed solvents and the changes in density

with composition are evaluated using the Hankinson and Thomson (1977)

correlation. The method is discussed in Appendix B.

The dielectric constants of the mixed solvents and the changes in the dielectric constant with composition are evaluated using the

methods discussed in Appendix C.The residual contribution to the solvent activity coefficient

of equation (3-7) is that presented in the UNIQUAC (Abrams and

Prausnitz, 1975) development.

q^ is calculated using the procedure of the UNIFAC group contribution

method (Fredenslund, et.al., 1975) where

Q , the group area parameter, is calculated from the van der Waals

group surface area, Awk, given by Bondi (1968) and is normalized by

(3-19)

^i ' ^ k vk ^k wis defined as the number of groups of type k in molecule i.

17

gfactor 2.5 x 10 given by Abrams and Prausnitz.

Qk = Awk/2.5 x 109 (3-21)

Ihe values of Qk given by Fredenslund and coworkers are used for the solvents in this study.

The for the positive and negative ions are calculated using

the crystallographic radii, r, of Pauling. (1960) Awlc for the ions is given by

o2Awk = 4TTr N (3-22)0

where r is in centimeters. and C , for the positive and negative

ions, respectively, are calculated by equation (3-21).

®i> the area fraction, is given by

® i = qi xi/£j qj xj (3-23)Xj is the mole fraction of component j. For the positive (component 1)

and the negative (component 2) ions, respectively, x is given by

xi = ^ v V ^ s + 2= 3’ 4’— N 3-24^

x2 = 1^2 V ^ ns + ^jnj (3-25)

1/^ and 1^2 are numbers of positive and negative ions, comprisingthe salt.

The mole-fraction of the solvent is given by

xk ” V ( U n s + ’ jnj') k = 3» 4’ N (3-26)

The binary parameters, and m, are evaluated from the

experimental binary data. Prediction of ternary systems does not

require additional ternary parameters.

\ b = exp ( -(u -u /RT)) = exp -(A /T)rmn K ' rtm nn' v \ mn/ /

(3-27)

18

where the u represent the energy of interaction between species m

and n of the solution. The A |[(t[ are determined directly from the

experimental data. There are two binary.The combinatorial contribution to the solvent activity co

efficient of equation (3-7) is the activity coefficient expression

of Flory and Huggins. (1941, 1942)

l n ^ ,Comb- = Incjx ,/xi 1 + l-cfc-Vx^ (3-28)

The primes indicate the solvated basis. 1, the average segment

fraction, is given by

Oi' = r.' xi,/ 2 jrjx/ (3-29)

x/ is the true mole-fraction of component j. The true mole-fractions are related to the apparent mole fractions given by equations (3-24),

(3-25), and (3-26), by the following:

x^1 = x2/(l“ b"*"k ” x2^*kh"iP (3-30)

*2' = x2/^1- xl^*k h+k ” x2^kh”k (3_31)

Xj’ = (XJ " Xi h+j "X2 h-j>/(1" xl^k h+k 'x2^kh“k)k = 3, 4 ,N (3-32)

The h+j and h-j are the solvation numbers of the positive and negative ions, respectively, and are functions of the apparent solvent

mole-fractions.

h+i = ho+. x. j = 3, 4, ....,N (3-33)J J Jh-j = ho-j Xj (3-34)

ho+j and ho-j are the solvation numbers of the positive and negative

ions, respectively.

19

The parameter, r^', for the solvents used in this study, is

calculated using the procedure of the UNIFAC method (Fredunslund,

et.al., 1975) where

ri' - \ (1) \ <3-35>

R , the group volume parameter, is calculated from the van der

Waals group surface volume, Vwk, given by Bondi (1968). Vwk is

normalized by the factor 15.17 given by Abrams and Prausnitz (1975),

where

Rk = Vwk/15.7 (3-36)

The values of R^ given by Fredenslund, et.al., are used in this study.

The solvated positive and negative ions may be considered as

molecules consisting of a central ion surrounded by h+^ and h-^ solvent

molecules, respectively.

For the solvated positive ions, equation (3-35) is

r,' = R, + 2 . h+. r.' (3-37)1 1 J J J

and for the solvated negative ions

r ' = R9 + £. h- r.' (3-38)L J J J

0R^ and R2 are calculated using the crystallographic radii, r,

of Pauling (1960) for spherical ions or the Yatsimirskii thermochemical

radii for nonspherical ions. (Waddington, 1959) Vwk for the ions is

given by

Vwkion = (4/3)7T r3 N (3-39)

Substitution of equation (3-39) into (3-36) gives R- and R2 *

20

The apparent combinatorial activity coefficient is obtained by

introducing equation (3-28) into equation (3-6).

Substitution of equations (3-8), (3-19), and (3-28) into equation

(3-7) yields the final expression for the solvent activity coefficient.In7 = 2A(£((M.W.)./1000))(I3/2/(l+bI%))

- A^b I2((M.W.)i/1000)/(l+bI^)2 - 2((M.W.)/1000)(l3/2/(l+bI^)2

- I/b lnd+bl^))!^ dA^/dn^ + 2A^((M.W.)/1000)

(I2/(l+bI^ ) 2 + I/b2 ln(l+bl^) - I3/2/b(l+bI^))nT db/dni +v2(277 a3/3)

(N/10002 )(m2 (M.W.)ap d(d)/dn^ - m2 d(M.W.)^ + m2 d(M.W.)(3/a)Up

da/dn.) + q.(l“ ln( 2 © li/ . ) -T (© \ b . /T © \ b )) +

I n '/xi' + '/xi* + ln xi' " ln xi

(3-7)

The expressions for the mean activity coefficients of the salt

in single and mixed solvents are developed in Appendix F.

C H A P T E R 4

ExperimentalThe vapor pressure depression measurements were performed using

the differential manometer described by Oliver (1969) and further

modified by Tomasula (1980). Hie differential manometer is shown in Figure 4.1. Each flask, equipped with a magnetic stirrer to facilitate

stirring, has a capacity of 100 cc.The vapor pressure depression, AP, where

AP = Ps - P (4-1)

is determined directly by measuring the difference in vapor pressure between two flasks. One flask contains pure methanol (Ps) and the other contains the electrolytic solution (P). The direct determination of AP provides improved accuracy over measurements of the vapor pressure of the electrolytic solution alone since it eliminates the

effects of small temperature variations. These effects are more pronounced at low molalities where AP is very small.

The salts used in this study were reagent grade quality KCH COO, Nal, NaSCN, and NH^SCN, were used without further purification.

Each was dried under vacuum for 48 hours prior to use. J.T. Baker methanol of spectroquality (99.9 weight percent minimum purity) was used with no further purification. A Karl-Fischer titration indicated that the amount of water was less than 0.03 mole percent.

Solutions were prepared by adding the appropriate salt to a preweighed flask. The flasks were weighed again to determine the amount of salt added. Approximately 30 ml of methanol were then added to each flask and the flask was reweighed.

Briefly, the experiment consisted of degassing the pure methanol

21

22

10 m m QQ

uhon

— */ |r-15 m m O A

k

Differential ManometerFigure 4.1

83.8c m

23

and the methanol salt solutions before attaching to the differential

manometer. This was accomplished by immersing the flasks in a

methanol bath at -67°C while boiling at high vacuum. After the_2methanol solutions subcool and a residual pressure of 10 mm Hg was

indicated on a McLeod gauge, the flasks were removed from the bath and the contents warmed to room temperature. This procedure was

repeated until air bubbles no longer rose from the solvent.The entire differential manometer was immersed in a well stirred

constant temperature bath. The vapor pressure depression measurements were performed at 25 and 40°C ± 0.03°C. The vapor pressure depression, AP, was measured with the aid of a cathetometer to ±0.06 mm Hg. After a typical run, the manometer was removed from the bath and the flask containing the electrolytic solution was reweighed to determine any loss

of solvent. The molality, m, of the solution was then calculated.The performance of the experimental system is demonstrated in

Figure 4.2 where the results for aqueous solutions of KCL used as a test system are compared with the very accurate results reported by Robinson

and Stokes (1959).The results for the electrolytes used in this study are reported

in Appendix E. Values of AP, P, and (£>, the osmotic coefficient, are given. The vapor pressure data are obtained by subtracting the measured AP values from the vapor pressure of pure methanol reported

in each table under a molality of zero.

The osmotic coefficient is given by<£> = -1000 ln P/Ps/Vm(M.W.) (4-2)

where M.W. is the molecular weight of the solvent and IS is the total number of ions comprising the electrolyte.

2 . 7 -

2 . 4 -

2. 1-

1. 8-

1. 5 -

1. 2-

0 . 9 -

0. 6-

0 . 3 -

0 . 0 -H‘[ I I IT I 0 . 0

□ Robinson and Stokes(1959) + This study

S3

53

53

nri j i i i i i i i i i '| I'I'iti n ri |~T’» i m i i i r | i i j i i t i r » | » i ri’ f"i i » i » * i i i i | i i i i i i i i r |■

0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0

Molality

rigure 4.2 Vapor Pressure Depression of Aqueous KC1 Solutions at 25

C H A P T E R 5

Binary Data Reduction

5.1 The Binary Equations and the Ternary and Binary Data Bases For a binary electrolytic solution, equations (3-8), (3-19),

and (3-28), reduce to: (the subscripts 1, 2, and 3 refer to thepositive ion, the negative ion, and the solvent, respectively.)

l n T ^ 1* = 2Ac ((M.W.)3/1000)(I3/2/l+bI^))-Acj)bI2(M.W.)3/(1000(l+bI^)2)

-v2(27Ta3/3)(N/10002)m2 d(M.W.)3 (5-1)

b is given by equation (3-14).

ln)^ = q3(l- ln(01l/;i3 + © 2 ^ 2 3 + 3^33^

- © i / ( © i ^ n + © 2 ^ 2 1 + ®3^31^

- ©2^32^^® 1 * ^ 1 2 + ® 2 ^ 2 2 + ® 3*p32^

- © 3^ 3 3/ ^ 1 ^ 1 3 + ® 2 % 3 + ® 3% 3»

(5-2)From equation (3-27), and ^33 are equal to unity.

Also, \ j j 31 * l/>13 and * i/>23*

l^.Comb. = ln /x + 1-c5>3'/x3' (5-3)

where c£>3' and x3' are' given by equations (3-29) and (3-32),

respectively.

The activity coefficient of the solvent is obtained by

substituting equations (5-1), (5-2), and (5-3), into equation

(3-7).The solvent activity coefficient expression for a binary

electrolytic solution contains nine parameters.

25

26

The Pitzer (Coulombic) term has one adjustable parameter, a,

which is the ion-size parameter, a reflects the hard-core volume of

the ions and its value may be between the sum of the crystallographic

radii or the sum of the solvated radii.

The Flory-Huggins (Combinatorial term) has two fixed parameters,ho . and ho .. These values are obtained from Bockris and Reddy (1977) +1 "J

for various ions in water. They are estimated for ions in nonaqueous solvents and the method for estimation will be presented later in this

chapter.

The residual term has six adjustable parameters: A^> ,

A32’ A12’ arK* A21* Equati°n (5-2) is simplified further with the assumption

A12 = A21 (5-Aa)

that the interaction energy between two positive ions, , is the same

as the interaction energy between two negative ions, ^ 2 2 ' (See

equation 3-27)

and 23 are combined to one parameter, A+.

A± = Q1^13 + ( V1)Q2^23 ( 5 " 4 b )

With the assumptions of equations (5-4a) and (5-4b), equation (5-2)

is

lny3Res- = q3(l-ln®1 A ^ + ® 3)

- ®1'/ 31/(<S>1 + ®2^21 + ®3^31^

1 ^ 1 2 + ® 2 + ® 3 ^ 3 2 ^ “ ® 3/ ( ® i A±/Qi + @ 3 )

(5-5)

The parameters ri3

27

The number of parameters in the residual term is reduced to four.

The expression for the activity coefficient of the solvent is

then given by

In ~ equation (5-1) + equation (5-3) + equation (5-5) +

In x ' - In (5-6)and represent the interactions between the solvent and

the solvated positive and negative ions, repectively. A ^ represents

the short-range interactions between the solvated positive and negative

ions. A+ combines the interactions between the solvated positive and

negative ions and the solvent. To establish these parameters, an

extensive data base of salts in various solvents is needed.

The data base for parameter estimation is shown in Table 5.1 and

includes osmotic coefficient or vapor pressure data for salts in water,

methanol, ethanol, isopropanol, and n-propanol.

This data base was selected since it includes most of the salts

and solvents for which ternary salt-mixed solvent data are available.

The ternary data base is shown in Table 5.2. In addition, the binary

data base includes salts not covered in the ternary data base. The

parameters obtained for these systems are applied to the prediction

of binary systems at temperatures other than 25°C.

The UNIQUAC surface area (q ) and volume (r ) parameters of

equation (5-6) were calculated using equations (3-20) and (3-35),

respectively, and the group area (Q ) and group volume (R ) parameters

given by Fredenslund, et.al. (1975). They are tabulated in Table 5.3.

The UNIQUAC group surface area (Q and Q2 ) and group volume (R

and R2 ) parameters for the positive (1 ) and negative (2 ) ions are

calculated using the Pauling (1960) crystallographic radii in the case

28

of the spherical ions and the Yatsimirskii thermochemical radii (Waddington, 1959) in the case of the nonspherical ions. The ionic

radii (Table 5.4) are then substituted into equations (3-21), (3-22),

(3-36), and (3-39), to obtain the ionic and R . These values are

tabulated in Table 5.5.

29

TABLE 5.1Data Base for Parameter-Estimation from Binary Electrolytic

Solution Data

SaltWater

T°C or PmmHg

Maximum ,moles salt'> ' 1 # M l M M 1 V t M W1 4* ’

1-1 Chlorides 25°C 1-1 Bromides 1-1 Iodides 1-1 Chlorates 1-1 Acetates1-1 Flourides2-1 Chlorides 2-1 Bromides 2-1 Iodides 1-2 SulfatesMethanolLiClLiBrNalNaSCNNH4 SCNKCH3 COONaBrNaOHCaCl2

CuCl 2KI

EthanolLiClLiBrNalCaCl2

Molality kg solvent

6666

44666

3

25°C

24 .8 8 °C

15°C

35°C50°C

444, 35, 2 ,

1 ,

5. 2 ,

4. 0 .

4422

Reference

Robinson & Stokes (1959)

Tomasula ( 1985) This Study

Bixon et al. (1979)

Janz & Tomkins (1972)

Czerwienski et al. (1985)

30

SaltIsopropanolLiClLiBrCaCl2

n-propanolCaCl2

T°C orPmmHg Molalityvkg solvent

Maximum moles salt^

75.1°C 75°C 7 60mm

7 60mm

1.83.91 . 8

1 . 8

Reference

Sada et al. (1975)

Nishi (1975)

Nishi (1975)

31

TABLE 5-2 Ternary Data Base

SaltIsothermal Data T°C

Methanol-WaterLiCl 25 , 6 0

NaCINaBr 25,40NaFKC1CaCl2

Ethanol-WaterLiClNaClNalNaFKC1KINH4 CI CaCl 2

2530

30

Isopropanol-WaterLiCl 75LiBr 75Methanol-Ethanol

Isobaric Data mmHg

760

760760760760760

760755700700700700760760

References

Ciparis (19 6 6 ) Hala (1969) Boone (1976)CiparisCiparis; BooneBooneBooneNishi (1975)

Ciparis; Boone Ciparis

Nishi (1975)

Sada et al. (1975)

CaCl2 760 Kato (1 9 7 1 )

TABLE 5-3UNIQUAC Volume (r^) and Surface Area ( qj ) Parameters

h 2o MeOH EtOH nPrOH IsoProp

ri 0 . 9 2 1.4311 2.1055 2.7799 2.7791

9i 1 .40 1 . 4322 1.9720 2.5120 2 . 5 0 8 0

33

TABLE 5.4Pauling Crystallographic & Yatsimirskii Thermochemical

Radii for the Ions Considered in this StudyoIon r , Angstroms Reference

Li+ 0.60 Pauling (1960)

Na+ 0.95

K+ 1.33Rb+ 1.48

Cs+ 1.69

Mg+ 2 0.65

Ca+ 2 0.99

Sr+ 2 1.13

Ba+ 2 1.35

Co+ 2 0.74

F- 1.36

Cl" 1.81

Br" 1.95I- 2.16

SO4 - 2 2.30 Yatsimirskii (Waddington, 1959)

NO3 - 1.89

CIO4 - 2.36

CNS" 1.95

CH3 COO- 1.59NHi}+ 1.48

34-

TABLE 5.5UNIQUAC Group Volume (Rfc) and Group Surface Area (Qfc)

ParametersIon Rk Qk

Li + 0 . 0 3 5 9 0 0 .10893Na+ 0 . 14615 0.27771K+ 0.39195 0.53607Rb+ 0.54325 0.66639NHi|+ 0.54325 0.66639Cs + 0.75488 0 . 8 2 9 8 2

M g+ 2 0.03773 0 . 1 1 2 6 0

Ca+ 2 0 .16129 0.29657Sr+ 2 0.23984 0 . 3 8 6 3 8

Ba+ 2 0.40898 0.55148

Co+ 2 0.06736 0 . 1 6 5 7 0

F~ 0.39107 0.53526

Cl“ 0.97915 0 . 9 8 6 9 6

Br" 1.23253 1 . 1 5 0 6 2

I- 1 . 6 7 5 1 6 1.41179SO4 - 2 2.02246 1 .60073NO3 - 1 .12223 1 . 0 8 0 9 0

CIO4 - 2 .18490 1 .68534CNS" 1.23253 1 . 1 5 0 6 2

CH3 COO- 0.66817 0.76499

35

5.2 Parameter Estimation from Binary Electrolytic Solution Data

The 1-1 chlorides at 25°C of Table 5.1, which consist of LiCl,NaCl, KC1, RbCl, NH Cl, and CsCl, were chosen as the base systems

from which all aqueous ionic parameters of equation (5-6) are evaluated. This means that the values of and A3 2 obtained through regression of these systems are the values of A ^ and A ^ to be used for all

salts containing the same ions. For example, the value of A-water/Li+ is the same for LiCl as it is for LiBr. The value of Awater/Cl" the same for NaCl as it is for CaC^. (The values of a, and A±are discussed under the headings Case 1, Case 2, and Case 3,

respectively /)The 1-1 chlorides were selected over the other aqueous systems

of Table 5.1 since ion-pairing between a uni-valent cation and a chloride ion is not observed. (Robinson and Stokes, 1959) In addition, the osmotic coefficient data for these salts is available up to a molality of 6 , with the exception of the KC1 system which has a maximum molality of 4.5m. A maximum molality of 6 is desirable since the ternary salt-mixed solvent systems of Table 5.2 often extend to

this molality.It is equally valid to use the 1-1 bromides or 1-1 iodides as a

base system since these salts also do not ion-pair. However, the

osmotic coefficient data for the majority of these salts do not

extend to 6m.The LiCl and LiBr methanol systems at 25°C were chosen as the

base systems from which all methanol ionic parameters are evaluated.The data for these systems are available up to a molality of 4.3.

The binary model of equation (5-6) was tested considering the

36

three cases discussed below.

Case 1

a, the ion-size parameter of the Pitzer term, equation (5-1),

was set equal to the sum of the ionic radii given in Table 5.4.

Therefore, the Pitzer term has no adjustable parameters.

The values of hQ+j and hQ_j of the Flory-Huggins combinatorial term, equation (5-3), were set equal to zero. This means that the

ions are not considered 'to be solvated.

The residual term, equation (5-5), was used assuming that the

value of A^ 2 is the same for each salt and a particular solvent along

with the values of and given in Tables 5.4 and 5.5.

The 1-1 chlorides in water at 25°C and the LiCl and LiBr systems

in methanol at 25°C were used to obtain the model parameters. The

model parameters were calculated by minimization of the objective

function:

F (<*>exp - ^ c a l ^ e x p 2rb is the experimental osmotic coefficient from the data of Table ^exp c

5.1. ^Pca]_ i-s calculated osmotic coefficient and is obtained from

equation (5-6) using the relationshipC£>cal = -1000In( y 3 x3 )/^(M.W.)m (5-8)

The results for Case 1 are shown in Table 5.6. The results for

the 1 - 1 chlorides were obtained by simultaneously solving for

A3 1 , A3 2 > and A+. The value of A ^ > which represents the interaction

between a water molecule and a chloride ion, is the same for each of

the chloride salts. The value of A^, which represents the interaction

between a water molecule and a positive ion, differs from salt to salt,

since it depends on the type of positive ion. A ^ > which represents

37

TABLE 5-6Case 1. Test of the Binary Model (Eq. 5-6) with

a = 2rc and ho+j and ho_j Set Equal to Zero

SystemWaterLiCl

NaCl

KC1

RbCl

CsCl

MethanolLiCl

LiBr

*12

-3624 . 1

15820.6

-31 32

-12224.1-5925- 1

-5643 . 8

-2824 .7

- 2670.6

-2505 . 6Average % error cj) =

-4633.0 -14788.1-9077.4

Average % error $ =

Avg. % Error $

22.7

7-3 1 .5 2.4

5 ■ 1 7.8

56.0

52.5

54.3

38

the interaction between a positive and negative ion, was assumed to

be the same for each salt. The value of A+, which was assumed to be

the same for all salts, was found to be approximately.zero.

The overall average percent error in cj> for the 1-1 chlorides is

7.8. The performance of the model is especially poor for the LiCl

system where the average percent error in cj> is 22.7.

The LiCl and LiBr methanol systems could not be correlated by the

model. The overall average percent error in c£> is 54.3. The reasons

for the poor performance of Case 1 will be discussed after the

presentation of Case 3,. The other salt-methanol systems of Table 5.1

also could not be correlated by the model.

Although not shown, the value of A± was also found to be zero in

the regression of the 1-1 bromides and the 1-1 iodides. The other

aqueous systems of Table 5.1 were not tested using Case 1.

Case 2

Case 2 is the same as Case 1 with the exception that a, the ion-

size parameter is now an adjustable parameter. The 1-1 chlorides and

the LiCl and LiBr methanol systems were again used to obtain the model

parameters. The regression procedure used in Case 1 to obtain the

model parameters was also used in Case 2.

The results for Case 2 are shown in Table 5.7. The overall average

percent error in for the 1-1 chlorides is 2.1 while that for the LiCl

and LiBr methanol systems is 1.9. These results are a significant

improvement over those in Table 5.6. Again, A+ was found to be

approximately zero for the aqueous and the methanol systems.

The values of A ^ and A.^ obtained from the regression of the 1-1

chlorides were used to obtain the values of a and A3 2 for the 1 - 1

39

TABLE 5-7Case 2. Test of the Binary Model (Eq. 5-6) with

a an Adjustable Parameter and ho+j and ho_j Set Equal to ZeroAvg. %

System a A -|2 A3 -] A32 Error in <t>WaterLiCl 3.98 -591.7 -341.6 -2907.8 0.9NaCl 3-24 -302.7 0.6

KC1 3-14 188.1 1 . 5

RbCl 3.29 337.1 2.4CsCl 3-46 494.8 5.1

LiBr 4.24 -341.6 -232.5 1.8

NaBr 3-55 -302.7 0.4KBr 3-28 188.1 2.4

RbBr 3.44 337.1 3-0CsBr 3.61 494.8 4.7MethanolLiCl 5-53 -364.7 -166.1 762.1 2.2

LiBr 5.88 -166.1 919.7 1.5

NaBr 5-32 -353-1 919.7 3-7

bo

bromides, where A^2 is the water-bromide ion interaction parameter.The values of A^2 and A ^ obtained from the regression of the LiCl and LiBr methanol systems were used to obtain the values of A^, the

methanol-sodium ion interaction parameter, and a for the NaBr- methanol system. A± was assumed to be zero. These results will be utilitized in Chapter 6.

Case 3a, the ion-size parameter of the Pitzer term of equation (5-6)

was set equal to the sum of the ionic radii given in Table 5.4. Thiswas done in Case 1 also.

The solvation numbers of the positive, hQ+^, and the negativeions, hQ j, in water were set equal to the values of the primary

hydration numbers given by Bockris and Reddy (1977). These values

are listed in the second column of Table 5.8. The wide variation inthe values of h ,. and h . for each ion arose since the hydration o+j o-j 3

numbers were determined by five different methods. Since the valuesof h . are 1±1 for the Cl , Br , and I ions, h . was set equal to o-j ’ ’ ’ o-j ^zero for these ions, h . was also set equal to zero for the othero-j nanions encountered in this study for simplicity purposes.

Since no solvation numbers are available for ions in nonaqueousmedia, it was assumed that h0+j for a positive ion in a nonaqueous

solvent is given by, /nonaqueous\ _ n/nonaqueous\9ron/no+j solvent ' o+j water ' solvent J ' H2C>25oc

(5-9)

where D ( - ~ s) is the dielectric constant of the nonaqueoussolvent at 25°C and Du is the dielectric constant of water at 25°C.

2

4-i

TABLE 5.8Hydration Numbers for Some of the Ions Used in This Study

ho+ j or ho_j h°+i or ho_-jIon (Bockris & Reddy, 1977) (This StudyJLi + 5— 1 5Na + 4 + 1 3K+ 3+2 2Rb + 2+1 1Cs + 0 0Ca+2 7.5-10.5 9Mg+2 13-16 10F“ 4 + 1 0

ci- 1 + 1 0Br” 1+1 01“ 1 + 1 ' o

42

h , . is the hydration number of the ion from the values ofoJ-j water 1

Table 5.8.

Again, the aqueous 1-1 chlorides and the LiCl and LiBr methanol

systems were used to evaluate the model parameters using the

procedure outlined under Case 1. A preliminary run indicated that

a should be an adjustable parameter for the methanol systems and that

the hydration numbers of the aqueous positive ions should be adjusted

to the values shown in the third column of Table 5.8. These values

are within the experimental error indicated by Boclcris and Reddy.

The regression results for the aqueous 1-1 chlorides and the

LiCl and LiBr methanol systems are shown in Table 5.9. The average

percent error in for the 1-1 chlorides is 1.4. This is an im

provement over the value of 7.8 obtained in Case 1 and the value of

2.1 obtained in Case 2.

The average percent error in cjb for the methanol systems is 1.9.

This is an improvement over the value of 54.3 obtained in Case 1 and

is the same as the value obtained in Case 2.

A comparison of the results of Cases 1, 2, and 3 indicates that

the model fails overall for the choice of parameters of Case 1. (a

was set equal to the sum of the crystallographic radii and liQ+j and

hQ j were set equal to zero). The results for the KC1, RbCl, and

CsCl systems are comparable to those of Cases 2 and 3.

The Pitzer, Flory-Huggins, and residual terms for the LiCl-water

and the LiCl-methanol systems of Table 5.6 are plotted in Figures 5.1 and 5.2, respectively.

Figure 5.1 indicates that the contributions of the Pitzer, Flory-

Huggins, and residual terms to the calculated solvent activity

4.3

TABLE 5-9Case 3. Test of the Binary Model (Eq. 5-6) with a = ZrG

for the Aqueous Systems and Adjustable for the Nonaqueous SystemsAvg. %

System a ho+j *12 *31 CMon<

\ Error inWaterLiCl 2.41 5. -828.7 -368.2 -87.5 1.6

NaCl 2.76 3. -442.3 0.9KC1 3.14 2. -13.7 1 .7

RbCl 3-29 1 . -142.0 1 .4

CsCl 3-47 0. -89.7 1 .5Average % Error <j) = 1 .4

MethanolLiCl 5.35 2. -253- 1 620.8 782. 3 2.3LiBr 5.70 2. 938.4 1 .5

Average % Error <j) = 1.9

44

Figure 5.1 Case 1. The Contribution of the Pitzer, Flory-Huggins, and Residual Terms in the Calculation of ln'V, Q tor the LiCl-Water System at 25 C. 2(The parameter values are found in Table 5.6)

0

- 0.1experimentalcalculatedPitzerFlory-HusginsResidual

- 0.2

molality

^5

Figure 5.2 Case 1. The Contribution of the Pitzer, Flory-Huggins, and Residual Terms in the Calculation of forthe LiCl-Methanol System at 25 C.(The parameter values are found in Table 5.6)

Me OH

- 0.1

- 0.2experimentalcalculatedPitzerFlory-HugginsResidual

5.02.0 3.01.00.0molality

46

coefficients of water are negligible as the molality increases above

2m. The 1-1 chloride data were regressed again varying the starting values of A ^ , and , in equation (5-6) to check that the

parameter values of Table 5.6 are the optimum ones. A± was set equal to zero. No changes in the parameter values or the value of the

residual term were noted indicating no multiplicity of roots in the term.

Figure 5.2 shows that the contributions of the Flory-Huggins and the residual terms to the calculated solvent activity coefficients of methanol are negligible even at high molalities. The major contri

bution is due to the Pitzer term. However, this contribution is

opposite in sign to that needed for the correlation of the experimental activity coefficients. Again, the residual term was checked for multiplicity of roots by varying the starting values of A^> A^, and A ^in equation (5-6). Again, A± was set equal to zero. No changes in

the parameter values or the value of the residual term were noted.

In Case 2 (Table 5.7), the values of hQ+j and hQ_j were set equalto zero and a, A^j A^ , and A ^ were adjustable parameters. A± wasfound to be zero. The results for the LiCl-water and the LiCl-methanol

systems at 25°C are plotted in Figures 5.3 and 5.4, respectively.

Figure 5.3 shows that the contributions of the Flory-Huggins and

the residual terms to the calculated activity coefficients of water are negligible over the entire molality range. The major contribution to

the calculated activity coefficients is due to the Pitzer term which decreases with increasing molality. This contrasts with Case 1 (Figure

5.1) in which the contribution of the Pitzer term is small.Figure 5.4 indicates that the major contributions to the calculated

^7

Figure 5.3 Case 2. The Contribution of the Pitzer, Flory-Huggins, and Residual Terms in the Calculation of ln'V. „ for the LiCl-Water System at 25 C. 2(The parameter values are found in Table 5.7)

0.0

l n %


- 0.23.02.01.00.0

molality

4-8

Figure 5.4 Case 2. The Contribution of the Pitzer, Flory-Huggins, and Residual Terms in the Calculation of ln'V. p... for the LiCl-MeOH System at 25 C.(The parameter values are found in Table 5.7)

§■

Me OH

- 0.1


5.03.02.01.0molality

-4*9

activity coefficients of methanol are due to the Pitzer and the

residual terms. The Pitzer term decreases with increasing molality while the residual term increases with increasing molality. This is in contrast to Case 1 (Figure 5.2) where the Pitzer contribution is

positive and the residual contribution negligible. The Flory-Huggins contribution in Cases 1 and 2 are identical.

Solvation of the positive ions was assumed in Case 3. a is the sum of the crystallographic radii for the aqueous systems and is an adjustable parameter for the methanol systems. (The case where a is the sum of the crystallographic radii and solvation is assumed for

the methanol systems is not presented here since the methanol data are correlated as they are in Case 1 with this option.)

The results for the LiCl-water and the LiCl-methanol systems are

plotted in Figures 5.5 and 5.6, respectively.Figure 5.5 shows that the major contribution to the calculated

solvent activity coefficients of water is due to the Flory-Huggins

term. The Pitzer contribution is the same as it is in Case 1. The

residual contribution is now positive where in Cases 1 and 2 it is

negative.The Flory-Huggins term is always negative. The magnitude of

this term increases with increasing solvation number; i.e., thecontribution of this term is greatest for the LiCl system which hasthe largest value of hQ+j and smallest for the CsCl system which has

a value of h ,. of zero o+jFigure 5.6 presents the results of Case 3 for the LiCl-methanol

system at 25°C. Hie major contribution to the calculated solvent

activity coefficients of methanol is due to the Pitzer term; however,

50

Figure 5.5 Case 3. The Contribution of the Pitzer, Flory-Huggins, and Residual Terms in the Calculation of ln'V., Q for the LiCl-Water System at 25 C. 2(The parameter values are found in Table 5.9.)


- 0.2

*1 .02.01.0 5.00.0molality

51

Figure 5.6 Case 3. The Contribution of the Pitzer, Flory-Huggins, and Residual Terms in the Calculation of 1 0 % Q„ for the LiCl-Methanol System at 25 C.(The parameter values are found in Table 5.9)

- 0.1


-0.3

5.02.0 3.01.0o.omolality

52

this contribution is not as great as it is in Case 2. The contri

bution of the Pitzer term is less in Case 3 than in Case 2 since the Flory-Huggins contribution is greater in Case 3 than in Case 2. The

assumption of solvation decreases the value of this term. The residual contribution in Cases 2 and 3 is approximately the same.

Although the performance of the binary model is similar for Cases

2 and 3, it will be demonstrated in Chapter 6 that only the parameters of Case 3 allow the prediction of the properties of single salt-mixed

solvent systems. It is for this reason that only the parameters

determined using the assumption of Case 3 will be presented in the next section for the binary data base of Table 5.1.

53

5.3 Values of the Binary Ion-Solvent Interaction ParametersThe approach of Case 3 with a, the ion-size parameter of the

Pitzer term set equal to the sum of the crystallographic radii, and hQ+j of the Flory-Huggins term set equal to the values of the third column of Table 5.8, is used to obtain the binary ion-solvent parameters of the residual term. The hQ_j, the solvation numbers of the negative ion, are set equal to zero. The calculation scheme used to obtain A^j A ^ , and A ^ for the 1-1 chlorides has already been described in Section 5.2. A± was found to be zero and is assumed to be zero for all salts in all solvents.

The parameters for the aqueous 1-1 chlorides at 25°C are shown in Table 5.10. The average percent error in (f> and the average percent error in the vapor pressure, P, is shown for each salt. The overall average percent errors and P are also reported. Since no value of the crystallographic radii is available for the hydrogen ion, it was assumed that the radius of the hydrogen ion is zero. The value of hQ+j for the hydrogen ion was obtained by regression since no value is reported in the literature. The ammonium ion is assumed to have a solvation number of one since this ion is the same size as the rubidium ion which has a solvation number of one.

It is assumed that the value of A ^ given in Table 5.10 is the samefor all salts in water. The values of A ^ from Table 5.10, the water-positive ion interaction parameters, and A^> were used to determineA^2 > the water-bromide ion interaction parameter and the water-iodideion interaction parameter. Again, a is the sum of the crystallographicradii and the h ,. are obtained from Table 5.8. The HBr, NaBr, KBr, o+jRbBr and CsBr systems were regressed together to obtain A^ o/Br” an<

TABLE 5.10Binary Interaction Parameters for the Aqueous 1-1 Chlorides at 25°C

Avg. % Avg. %Salt a= %rn h°+j ^12 ^31 ^32 Error, $ Error,

HC1 1 . 8 1 6 . 0 -828.7 -365 .5 -87.53 3 - 0 0 . 44

LiCl 2.41 5.0 -3 6 8 . 2 1 . 6 0 . 1 8

NaCl 2 . 7 6 3.0 -442.3 0.9 0. 05KC1 3. 14 2 . 0 -13.7 1.7 0 . 1 2

RbCl CMon 1 . 0 -142.0 1 .4 0 . 1 0

NH4 CI 3.29 1 . 0 -116.3 2.9 0.27CsCl 3.47 0 . 0 -89.7 1 .5 0 . 1 7

Average % Errors 1.9 0. 19

55

the HI, Nal, KI, Rbl, and Csl systems were regressed together to

obtain q/j“* The LiBr and Lil systems were predicted from the resulting parameters.

The parameter values, the average percent error in cjp, and the average percent error in P, for the 1-1 bromides and 1-1 iodides are presented in Table 5.11. The average percent errors in c£> and P for

these systems are also tabulated.The average percent errors in c£> for the 1-1 bromide and iodide

systems are 2.6 and 2.9, respectively, compared to the value of 1.9 obtained for the 1-1 chlorides. This is to be expected since only

one parameter was used to correlate these systems versus the three parameters used to correlate a chloride system.

An attempt was made to correlate the 1-1 nitrates, perchlorates, acetates, and flourides, the 2-1 chlorides, bromides, and iodides, and the 1-2 sulfates of the binary data base of Table 5.1 using the parameters of Tables 5.10 and 5.11, the values of hQ+j given in Table5.8, and the assumption that a is the sum of the crystallographic radii. In the case of the 1-1 and 1-2 salts, the data were regressed for A^2 and in the case of the 2-1 salts, the data were regressed for A^. While the results for the 1-1 flourides, perchlorates, and 2-1

chlorides are not as good as those of Table 5.10 and 5.11, poor correlation of the data was obtained for the 1-1 nitrates, acetates, and the 1-2 sulfates.

Typical results are shown in Table 5.12 for the 1-1 nitrates at 25°C. The data were regressed for Awater/N0 the water-nitrate ion interaction parameter. The average percent error in is 8.5 and the

average percent error in P is 0.55. These results are poor compared

56

TABLE 5-11Binary Interaction Parameters for the Aqueous 1-1 Bromides

and 1-1 Iodides at 25°C

A 12 = -828.7Avg. % Avg. %

Salt a=Erc ho+j A 3 1 A 3 2 Error, <j> Error,

HBr 1 .95 6.0 -365.5 -49.9 3-7 0.08

*LiBr 2.55 5.0 -368.2 3.5 0.70

NaBr 2.90 3-0 -442.3 2. 1 0.13FCBr 3.28 2.0 -13.7 2.5 0.19RbBr 3.43 1 .0 -142.0 2. 1 0 . 22

CsBr 3.61 0.0 -89.7 1 .5 0.14

Average % Errors = 2.6 0.24

HI 2.16 6 .0 -365.5 -13.1 3.6 0.16

Li I 2.76 5.0 . -368.2 3-2 0.16

Nal 3-11 3-0 -442.3 3.4 0 . 20

KI 3-49 2.0 -13.7 3-4 0.07Rbl 3.64 1 .0 -142.0 2.6 0.31

Csl 3-82 0.0 -89-7 1 .4 0.19Average % Errors = 2.9 0.18

^Predicted from the binary parameters

57

TABLE 5-12Binary Interaction Parameters for the Aqueous 1-1 Nitrates

at 25°C

a = 2rc and A 1 2 = -828.7

Salt a=£re ho+J A31 A32Avg. % Error,

Avg.Error

LiN03 2 .49 5.0 -368.2 555.0 5.5 0.46

NaN03 2 .85 3-0 -442.3 8.8 0. 85

k n o 3 3.22 2.0 -13.7 7.8 0.29

RbN03 3-37 1 .0 -142.0 11.9 0.55Average % Errors = 8.5 0.55

58

to the fit of the data obtained for the 1-1 chlorides, bromides, and

iodides.

For these systems, the Flory-Huggins term may be regarded as

fixed since it depends on literature values of the solvation number.

Likewise, the residual term may be considered fixed since two out

of three parameters, and either or A ^ , have already been

established. (Tables 5.10 and 5.11) The only other parameter that

may be modified is a, which is assumed to be the sum of the crystal

lographic radii for the 1-1 chlorides, bromides, and iodides, and is

adjustable for the methanol systems.

The 1-1 nitrates, perchlorates, acetates, and flourides, the 2-1

chlorides, bromides, and iodides, and the 1-2 sulfates, were regressed

again, with a as an adjustable parameter, and for either A ^ and A ^ •

The results are shown in Table 5.13.

The average percent error in for the 1-1 nitrates is reduced

to 3.1 with a an adjustable parameter compared to a value of 8.5

where a is assumed to be the sum of the crystallographic radii. The

results for the other systems listed in Table 5.13 have also improved

dramatically.

It should be noted that the values of h ,. for the Ba , Co ,o+j ’++and Sr ions, were obtained regarding hQ+j as an adjustable parameter

since these values are not available in Bockris and Reddy (1977).

Even though Bockris and Reddy indicate that the value of the hydrationf_ [

number of the Mg ion is between 13 and 16, it was found that a value

of 10 gives better correlation results.

Robinson and Stokes (1959) indicate that the so called strong

electrolytes, those which completely dissociate in solution, are

59

TABLE 5-13 Binary Interaction Parameters at 25°C

a is an adjustable parameter

A 12 = -828.7

Avg. % Avg. %Salt a ho+j A31 A32 Error, $ Error,

LiN03 1 .79 5.0 - 3 6 8 . 2 -13- 3 4 . 1 0.42NaN03 1 .45 3-0 -442. 3 3-9 0.24kno3 1 . 6 9 2 . 0 -13.7 2.3 0 . 0 8

RbN03 1 . 9 8 1 . 0 -142.0 1.7 0 . 0 1

hno3 1 .05 6 . 0 -365 .5 3-6 0 . 1 6

Average % Errors = 3.1 0 . 1 8

LiC10i| 2.72 5.0 - 3 6 8 . 2 -3 3 -8 1 .4 0 . 0 8

NaClOii 2 . 2 6 3-0 -442. 3 2.3 0.15HCIO4 1 .42 6 . 0 -365 .5 1 .7 0.13

Average % Errors = 1 . 8 0 . 1 2

LiAc 1 . 6 0 5.0 - 3 6 8 . 2 -2 1 8 .5 3.5 0 . 2 1

NaAc 3 . 2 8 3.0 -442.3 1 .5 0 . 0 8

KAc 3.93 2 . 0 -13.7 1 .5 0 . 0 8

RbAc 4.14 1 . 0 -142.0 1 . 2 0.07CsAc 4 . 36 0 . 0

001 1 . 6 0 . 1 0

Average % Errors = 1.9 0 . 1 1

NaF 2.23 3.0 -442.3 -147. 3 0 . 8 0 . 0 1

KF 3.27 2 . 0 -13.7 0.4 0 . 0 2

Average % Errors = 0 . 6 0.015

Li 2 S0ij 1 . 1 7 5.0 -3 6 8 . 2 147. 6 14.4 0.85Na2SOij 2 . 1 6 3-0 -442. 3 0 . 8 0 . 0 1

K2 S0 4

CMmon 2 . 0 -13.7 2.5 0.07Rb2 S0 4 3-63 1 . 0 -142.0 3-0 0 . 1 0

Cs2 SOi| 4.18 0 . 0 -87 .7 4.9 0 . 1 1

Average % Errors = 5.1 0.23

60

Avg. % Avg. %Salt a ho+j A 3 1 A 3 2 Error, $ Error,

CaCl2 2.93 9-0 -220.2 -87 .5 2.8 0.67CaBr2 3.52 -45.9 2.8 1.14Cal2 3-72 • -13.1 1.6 0 . 0 7

BaCl2 3-19 7.0 -261 .1 -87 .5 3-3 0.14BaBr2 3-57 -45.9 2.3 0.11Bal2 4.21 -13.1 1 .4 0.08CoCl2 2.89 9.0 -235.0 inir-co1 4.9 0 .69CoBr2 3-56 -4 5 . 9 2.3 0 .65CoI2 3-98 - 1 3 . 1 3-0 1 .86SrCl2 3-19 8.0 -227 .7 -8 7 . 5 2.2 0.13SrBr2 3-54 -4 5 . 9 1 .7 0 . 0 7

Srl2 3-94 - 1 3 . 1 1.3 0.05MgCl2 2.82 10.0 -193.9 inc—COI 2.8 0 . 5 0

MgBr2 3-33 -45-9 1 .6 0 . 2 3

Mgl2 3-70 -13.1 2.2 0.67Average % Errors = 2.4 0.47

61

comprised of the alkali halides, the alkaline-earth halides and

perchlorates, and some transition-metal halides. The flouride, perchlorate, and the 2-1 salts of Table 5.13 are in this category.

The nitrate, acetate, and sulfate salts are weak electrolytes since it is believed that association occurs between the oppositely charged

ions as a result of electrostatic attraction.The Pitzer term, if used alone in the correlation of 1-1 and

2-1 strong electrolytes in water, has been shown to be applicable up to a molality of 6 for the 1-1 salts and a molality of 2 for the 2-1 salts, when a is an adjustable parameter. (Tomasula, et.al., 1985)

It does not correlate these systems if a is set equal to the sum of

the crystallographic radii. The addition of the Flory-Huggins and the residual terms extends the molality range of the Pitzer term in

the case of the 2-1 salts of this study since the data are correlated up to a molality of 6 when a is adjustable. If a is set equal to the sum of the crystallographic radii, the average percent errors in (£>

for the CaC^j CaB^, and Ca^ systems are 4.5, 7.1, and 7.9, respect

ively, up to a molality of 2. Tne addition of these terms allows the correlation of the 2-1 aqueous salt systems up to a molality of 6.

The Bromley (1973) model which contains four fixed water specific

parameters and one salt-specific parameter, correlates these systems up to a molality of 1. The Pitzer (1973) model, which contains two

fixed water specific parameters and three salt-specific adjustable

parameters, correlates the MgCd^ system up to a molality of 4.5, the

SrCl2 system up to a molality of 4, and the other 2-1 salts of Table

5.13 up to a molality of 2.

The addition of the Flory-Huggins and the residual terms also

62

allows the value of a to be set equal to the sum of the crystallographic radii for the 1-1 chlorides, bromides, and iodides. This is apparently coincidental since a must be an adjustable parameter for the strong electrolytes listed in Table 5.13.

For the ion-pairing electrolytes of Table 5.13 (1-1 nitrates and 1-2 sulfates), the value of a obtained through regression is generally less than the sum of the crystallographic radii. Even though equation (5-6) was derived assuming complete dissociation of the electrolyte, the model correlates the 1-1 nitrates with an average percent error in (jp of 3.1 and an average percent error in c£> for the 1-2 sulfates of 5.1. However, the I^SO^ system is correlated with an average percent error in (jp of 14.4.

The binary interaction parameters for the salt-methanol systems listed in the binary data base of Table 5.1 are shown in Table 5.14.

a is an adjustable parameter and hQ+j i-s calculated using equation (5-9). The LiCl and LiBr methanol systems at 25°C were regressed together to establish the individual values of a for LiCl and LiBr,Af2, A^, and A^* A± was found to be equal to zero. The value of A^ 2 obtained for these systems is assumed to be the value of A ^ for

all the other salts in methanol. The values of ^methanol/Br- an< ^1 2

were used to establish the value of a of the NaBr-methanol system and

the Amethanol/Na+ ^ Anethanol/Na+ Parameter and A,2allowed the determination of the values of a for the Nal and NaSCNsystems as well as the values of Amethanol/I- and Amethanol/SCN-.The parameters for the other salts listed in Table 5.14 are determined as above.

The overall average percent error in (J) for the systems listed

63

Table 5.14 Binary Interaction Parameters for Salt-Methanol Systems at 25°C. a is an Adjustable Parameter. = -253.1

Salt a h .o+i A31 A32Avg. % Error ,<£>

Avg. ' Error

LiCl 5.35 2.00 620.8 782.3 2.3 0.50

LiBr 5.70 2.00 620.8 938.4 1.5 0.43

Nal 5.35 1.25 -281.6 219.8 0.8 0.20

KI 4.73 0.88 -302.1 219.8 2.0 0.03NaBr 5.19 1.25 -281.6 938.4 2.0 0.14

NaSCN 4.73 1.25 -281.6 -42.6 2.4 0.35kch3coo 4.00 0.88 -302.1 213.0 3.3 0.30

CUCI2 3.65 4.17 -2362.8 782.3 8.0 0.93

CaCl2 5.11 3.75 -118.2 782.3 2.7 0.42

NH^SCN 3.93 0.42 -410.7 -42.6 7.9 0.99

AVG. = 4.77 AVG. = 3.3

64

in Table 5.14 is 3.3. Most systems are correlated well with the exception of the CuCl2-methanol system which has an average percent error in cp of 8.0 and the NH^SCN system which has an average

percent error in <p of 7.9.The binary interaction parameters for the salt-ethanol systems

listed in the binary data base of Table 5.1 are shown in Table 5.15. The LiCl and CaCl2 systems were chosen as the base systems from which

the other salt-ethanol interaction parameters are determined. The value of A^ 2 was not regarded as an adjustable parameter for these systems but was calculated based on the values of obtained for the water and methanol systems. This was done to decrease the number of parameters in the residual term which need to be established for

solvent systems in which few salt-solvent data exist.The values of f o r the water and methanol systems were assumed

to be linear functions of the dielectric constants of these solvents at 25°C. The relationship between A^ 2 ar|d the dielectric constant is

A12 = 156.8 - 12.57D25oc (5-10)

The values of A^ 2 an<5 the values of hQ+ calculated using

equation (5-9) were used to determine the value of Ag2, theethanol/Cl interaction parameter, and the values of A^, the

+ , +2 ethanol/Li and ethanol/Ca interaction parameters. The values ofa of the Pitzer term for LiCl and CaCl2 were also calculated.

The ethanol/Li parameter allowed the calculation of the values of a and A^ 2 for the LiBr-ethanol system. The parameters for the Nal-ethanol system, with the exception of A^2, were determined independently since no other salt-ethanol data are available which have ions in common with this system.

65

Table 5.15 Binary Interaction Parameters for Salt-Ethanol andSalt-Isopropanol Systems, a is an Adjustable Parameter.

A1 2 ethanol

r—I 1il 9 A = - 1 2 isopropanol ■6 8 . 8

(The parameters were established at the indicatedtemperatures.)

EthanolSalt T°C a A31 A32

Avg. 7> Error, cj>

Avg. % Error, P

LiCl 35.0 4.84 4634.1 926.9 3.3 1.33

LiBr 50.0 6.60 4634.1 1349.5 2.9 2.29

CaCl2 50.0 5.27 90.3 926.9 3.7 0.50

Nal 50.0 6.08 -279.2 1098.3 1 . 6 0.34

AVG. = 5.70 AVG. == 2.9

IsopropanolSalt T°C a A31 A32

Avg. 7o Error,

Avg. 7a Error,

LiCl 75.1 6 . 1 0 2398.9 9377.9 4.3 0.51

LiBr 75 6.50 2398.9 2160.0 3.2 0.80

AVG. = 6.30 AVG. = 3.8

66

The binary interaction parameters for the salt-isopropanol

systems are presented in Table 5.15. A.^ was calculated from+equation (5-10) and the value of hQ+ for the Li ion was calculated

by equation (5-9). The LiCl and LiBr-isopropanol systems were regressed simultaneously to obtain the values of a for the two salts,

Ai sopropanol/Li+ ’ Ai sopropano 1/Cl anc* Aisopropanol/Br ' '^a e va uesof a and A. , , for the CaCl0-isopropanol system couldisopropanol/Ca-H- I f t - j

not be established since this system could not be correlated by the binary model.

The parameters for the CaC^-propanol system are not presented, since this system could not be correlated by the binary model.

The possible reasons for the failure of the model to correlate the CaCl 2 - i s opr opano 1 and the CaC^-propanol systems are discussed

in Chapter 7.

6 7

5.4 Temperature Extrapolation of the Binary ParametersIt is important to examine the effect of temperature on the

parameter values of the model since in the isobaric salt-mixed solvent systems, temperature varies with composition.

Table 5.16 presents the prediction results, based on the parameter values at 25°C, for some aqueous systems. These results can be considered good for the temperature range involved. The

average percent error in (J> for all systems is 4.2. An average percent error in surpassing 10% is observed only for the LiCl- water system at 150°C.

The parameters of the Bromley (1973) model and the Pitzer

(1973, 1977) models, which were obtained at 25°C, were used to predict the osmotic coefficients and vapor pressures of some salts in water up to an ionic strength of 6m. (Tomasula, et.al., 1985)

The Bromley model, termed B-l, contains four fixed water specific parameters and one adjustable parameter. The Pitzer (1973) model,

termed P-3, has two fixed water specific parameters and three salt- specific adjustable parameters. The Pitzer (1977) model, termed P-l, is given by equation (5-1) and has one adjustable parameter.

The models are compared in Table 5.17. The binary model of this study allows the prediction of salt-water systems with temper

ature independent parameters as well as the B-l, P-l, and P-3 models, Table 5.18 presents the prediction results, based on the

parameter values at 25°C, for some methanol systems. The average percent error in for all systems is 3.2 and it may be concluded that this model provides reliable extrapolations with respect to

temperature using temperature independent parameters up to a molality

68

Table 5.16 Average % Error in cp and in P in Prediction from Parameter Values at 25°C in Aqueous Systems

Avg. %Salt T°C Error ,(i£>

LiCl 50 3.8

75 5.7

1 0 0 4.4

125 6.7

150 1 0 . 2

CsCl 1 1 0 2.5

140 3.4

170 4.3

2 0 0 5.4

KC1 40 1.750 1 . 6

60 1.5

70 1.5

80 1.5KI 1 0 0 8.5

Li I 1 0 0 3.1

Nal 1 0 0 7.7KI 1 0 0 3.7Li I 1 0 0 2.3

Nal 1 0 0 8 . 0

LiBr 1 0 0 1.3NaBr 1 0 0 5.1

LiBr 1 0 0 1.5

Avg. 7o Maximum Error, P Molality0.35 3.2

0.570.430.65

0.930.43 6 . 0

0.560.67

0.770 . 1 1 4.5

0.080.090.09

0.092 . 6 8 1 0 . 0

1.28

2.090.59 6 . 0

0.851.60

0.29 1 0 . 0

0.84 8 . 0

0.24 6 . 0

ReferenceCampbell and Bhatnagar (1979)

Holmes and Mesmer (1981)

Snipes, et.al. (1975)

Weast (1970)

Table 5.16 continued

SaltNaBr

Avg. % Avg. % MaximumT°C Error Error, P Molality References100 5.8 0.80 6.0 Weast (1970)

70

Table 5.17 A Comparison of the Average % Error in c]b in Prediction from Parameter Values at 25°C in Aqueous Systems for Four Models

Avg. % Error,This Maximum

Salt T°C B-l P-l P-3 Study MolalityLiCl 50 3.4 3.6 3.0 3.8 3.2

75 6 . 2 6.4 5.1 5.71 0 0 4.9 4.8 3.5 4.4125 7.6 7.3 5.7 6.7150 1 1 . 6 10.9 9.1 1 0 . 2

KC1 40 0.9 1.9 1 . 1 1.7 4.5

50 1.4 2.4 1.7 1 . 6

60 1 . 8 2 . 8 2 . 2 1.570 2 . 0 3.1 2 . 6 1.5

80 2.3 3.4 2.9 1.5CsCl 1 1 0 4.9 6.5 7.0 2.5 6 . 0

140 5.1 7.3 8 . 1 3.4170 4.9 7.9 9.1 4.32 0 0 4.5 8 . 2 1 0 . 6 5.4

71

Table 5 1- oo Average % Error in d£> and in P in Prediction fromParameter Values at 25°C in Methanol Systems

Avg. % Avg. % MaximumSalt T°C Error ,(J> Error, P Molality ReferenceLiCl 35 1.9 0.46 4.6 Tomasula (1980)

45 1 . 6 0.2960 3.6 1.30 5.9 Hala (1969)

LiBr 35 1.9 0.44 4.345 2 . 6 0.46

NaSCN 40 3.1 0.53 3.4 This Studynh4scn 40 7.0 0.89 5.2kch3coo 40 4.2 0.40 2.5

Nal 40 2.9 0.76 4.3

Average % Error; 3.2

72

of 6.0 and up to 60°C.The results of this study are compared to those of the B-l, P-l,

and P-3 models in Table 5.19 and are comparable in all cases to those

of the P-3 model.The temperature dependency of the ethanol and isopropanol salt-

solvent parameters could not be investigated since the data for these

systems (Table 5.1) which are at temperatures other than 25°C, were

used to establish the model parameters.

73

Table 5.19 A Comparison of the Average % Error in in Prediction from Parameter Values at 25°C in Methanol Systems for Four ModelsAverage % Error in the Predicted Osmotic Coefficients

Salt T°C B-l P-l P-3 This :

LiCl 35 2.4 4.1 1 . 1 1.9

45 4.7 6 . 6 1.4 1 . 6

60 5.4 8.5 3.5 3.6

LiBr 35 5.5 7.0 1 . 2 1.9

45 4.6 7.0 2 . 0 2 . 6

7^

5.5 Molality Range of the Binary ModelThe binary parameters of Table 5.13 were used to predict the

osmotic coefficients at 25°C of some 1-1 electrolytes in water at molalities greater than 6m. The results are shown in Table 5.20 and

are reported as the relative percent error in at each molality.

The results indicate that relative percent errors in <x> greater than 1 0% are to be expected if the parameters obtained from regression of the data up to a molality of 6 are used to extrapolate to molalities greater that 6 . The only exception occurs with the LiCl-water system. Reliable predictions are obtained up to m = 13.

The dashed lines for some of the salts in Table 5.20 indicate

the region in which the model breaks down entirely. This occurs when the model indicates there are no longer any solvent molecules with

which to solvate the ions, or x ', given by equation (3-32) becomes

zero.x ', the mole-fraction of the free-solvent molecules in a binary

electrolytic solution, is given by

x3* " <x3 -xlho+lx3) / ( 1 ^ l W (5'n) where hQ+ is the solvation number of the positive ion, x , is the

apparent mole-fraction of the positive ion. x and x^ are given byequations (3-24) and (3-26), respectively, x^ 1 becomes zero when the

numerator of equation (5-11) is zero, of when

x = l/hQ+ (5-12)For the LiCl-water system in which the solvation number of the

Li+ ion is 5, x ' from equation (5-11) is zero and x from equation

(5-12) is 0.2. This corresponds to a molality of 18.5. At molalities

greater than 18.5, x ' becomes negative.

75

Table 5.20 Prediction of the Osmotic Coefficents at 25°.C of Some Aqueous Systems at High Molalities. The Data are from Robinson and Stokes (1959).

Salt

HC1

HC10,

Rel. % Rel. %m Error, c£> Salt m Error ,(£)

7.0 -1 0 . 8 LiN03 7.0 -1 2 . 6

8 . 0 -14.8 8 . 0 -17.3

9.0 -19.9 9.0 -2 2 . 6

1 0 . 0 -27.8 1 0 . 0 -28.8

1 1 . 0 -39.2 1 1 . 0 -36.1

1 2 . 0 -56.5 1 2 . 0 -44.4

13.0 -87.2 13.0 -54.5

14.0 -186.2

15.0 --- LiBr 7.0 1 0 . 8

16.0 --- 8 . 0 1 2 . 2

9.0 13.1

7.0 -1 0 . 8 1 0 . 0 14.2

8 . 0 -1 0 . 6 1 1 . 0 14.8

9.0 -11.3 1 2 . 0 13.6

1 0 . 0 -13.7 13.0 12.3

1 1 . 0 -18.3 14.0 9.4

1 2 . 0 -27.1 15.0 3.7

13.0 -44.9 16.0 -4.9

14.0 -109.3 17.0 -19.5

15.0 --- 18.0 -54.3

16.0 — — — — 19.0

76

Table

SaltLiCl

.20 continued

Rel. % Rel. %m Error ,cj) Salt m Error,<£>

7.0 3.78 . 0 4.3 CsCl 7.0 -1 0 . 6

9.0 4.0 8 . 0 -15.0

1 0 . 0 2 . 8 9.0 -2 0 . 6

1 1 . 0 0.9 1 0 . 0 -27.1

1 2 . 0 -2.3 1 1 . 0 -34.3

13.0 -6.9

14.0 -13.3

15.0 -2 2 . 2

16.0 -35.317.0 -56.8

18.0 -105.919.02 0 . 0

77

It is recommended, therefore, that the binary model parameters not be used to predict the thermodynamic properties of electrolytic

solutions beyond a molality of 6 in water.The molality range for the salt-nonaqueous systems could not be

investigated since data greater than 6m are not available.

78

5.6 Values of the Binary Solvent (1) - Solvent (2) Interaction

Parameters

The solvent activity coefficient of component i in a multi-

component nonelectrolytic solution is:

ln7i ~ I n T i ^ r . ) + In 7. (Residual) (5-13)where

■ ln<W +1 - <*>iV (5-1A)and In' Y ^ (Residual) is given by equation (3-19). The volume fraction,

is defined<£. = r. x. °/T .r .x. 0 (5-15)l i ' j j

and the area fraction, ©^, is defined

® i * V i v 2 jqjxj° (=-16)where the x^ 0 are the mole-fractions of the solvents. The UNIQUAC

volume (r ) and surface area (q ) parameters for the solvents in this

study are found in Table 5.3.

The solvent activity coefficient of solvent 1(3), in a binary

nonelectrolytic solution of 3 and solvent 2(4), is obtained from

equations (5-13), (5-14), and (3-19).

In = ln<$>3/x3° + 1 - C|>3/x30

+ q3 (l- ln( © 3 + © 4 ^ 3 4 ) " © 3 / ( 0 3 +© 4 ^ 3 4 <)

“ © 4 3 4 / ( © 3 3 4 + © 4 )) (5-17)The solvent activity coefficient of 4 is given by

In = ln<£>4/x4 ° + 1- c£4/x4°

+ q4 (l- ln( © 3 ^ 3 4 + © 4 ) - + ®4^43>- © 4 /(© 3 ^ 4 3 + ® 4)) (5-18)

The group interaction parameter, V'mn- is defined

79

\ j j = exp - (A /T) (5-19)rrnn K ranThe binary nonelectrolyte solvent activity coefficient

expressions contain two adjustable parameters, A and A/n.'34The mixed solvent data of this study were correlated using the

objective functionNP 9 NP 9

-F = X((P i - P )/P ) + S ((Yq i - Yq )xlO)v cal exp' exp s 3cal 3exp' s(5-20)

where pexp and Y^exp are the experimental values of the total pressures of the mixed solvent system and the vapor-phase compositions

of solvent 1. A weighing factor of 10 is used for the deviation in

vapor-phase composition in order to make the magnitude of this term

equal to that of the first term of equation (5-20).

For isothermal P-x-y data, Pca is obtained from the value of the

temperature, the values of the solvent mole-fractions, equations (5-17)

and (5-18), and the relationships developed in Chapter 1.

P cal = x 3 3Ct )3Sp3S(exP^Pcal “ P3S)v3//RT)/($ )3

+ x4 ^ 0 4Sp4S<exP(Pcal - P4S)v4/RT)<i)4 (5”21)Y3cal = X3 ^ 3 ^ 3 Sp3S(exp Pcal " ^ ^ / ^ ^ V c a l

(5-22)The fugacity coefficients are calculated using the Hayden-0'Connell

(1975) correlation which is presented in Appendix A. The Hankinson

(1979) correlation is used to calculate the pure-component liquid

volumes. This method is discussed in Appendix B.Equations (5-21) and (5-22) are solved by using a bubble point

calculation.

For isobaric T-x-y data, Tca and Y3cal are obtained from the

80

value of the experimental total pressure of the system, the values

of the solvent mole-fractions, equations (5-17) and (5-18), and the relationships of Chapter 1. P -.is set equal to P in equationsC31 6Xp(5-21) and (5-22). A bubble point calculation is used.

The regression results for the mixed-solvent systems of thisstudy are presented in Table 5.21. The quality of the correlation

of the data is shown by comparing the experimental and calculated

quantities through the following expressions:NP

AT = (1/NP)S | Tcal - Texp |°K (5-23a)NP

AP = (1/NP)S |Pcal - Pexp[™Hg (5-23b)NP 2

Ay = (1/2NP)2 2 | y c a i - y e x p | <5-23c)In the case of the isothermal systems, AP and Ay are reported.

AP, AT, and Ay are reported for the isobaric systems.

No Ay is reported for the l^O-EtOH system at 30°C since

experimental values of y and y^ were not available.

Correlation

of Binary

Solvent

Nonelectrolyte

Data

81

—. --H0- O-0- to- NCO a>t— '-N t— T-

___ n ■—• 'cP'i>-• CO • •

i—1 .— rH rH0) cdp . p -p<1J i—i QJ *— 0)

0 cd c—o bO bO ac bOa G p SC '— C0) •r—1 O

m in in ■=j- ■=r co co C- ■=r ■=3“ cr>CTc CO on cr> o co OO on CO o

on m .— .=r CM cO in CO p - n ID ■=r ■=3">J o o o o o 1 o o o o O o o< o o o o o 1 o o o o O o o■ • . . . . . » . . . .

o o o o o o o o o o o obOEGQ oo t— CO <— o- in ^r <— o r— CM I- r—5 • . . . . . . ■ . . . • .Oh o 00 in o o o VO o on o o o o<

OO 1 1 I CM 1 i I on 1 .=r on T“ onH 1 1 i • 1 i i • I • . • •< o o o o o o

o on o .=r o ^r C— DC o onon • • • • . . • . . . . .

LTt OJ ;=r f- .=r CM on ID o in CM in< <- CM cO •=3" on LO CO CD n 0- OJ OJ

1 1 r— *— T— '— r~ 1

T_ OJ t_ CO ID o CM ID o \— CD co• • • • . ■ ■ • • • • •

on co CO 0- CM on o in o CO =r cD in CD«! cO CM on p- oo o on CM n co CM on H-

OJ T- OJ r“ CM CM on on on ^r 1

bO o O o c_> O oEC o O o E o O o S E E E so co o E o O o s o E E E s. . . o . . . o o o o o o§ in ai o x> in o o XI n ID CD CD COO h oo on cO r— OJ on n 0- c— t- c— t—

■=r■'_/CMP EC ECG O EC oQJ U O P> EC EC O h Sh EC O ho O o O h O O0 CD +3 m 1 P nw s H M c w MH—\onT—

HOG EC0) O> EC ShrH o O o o O Ql,0 OJ OJ CM OJ a) 1CO EC EC EC EC s SC

-PrOH

EtOH

760mm

-79-5

98.9

0.5

0.2

0.00

817

C H A P T E R 6

6.1 The Ternary Model for the Prediction of Salt-Mixed Solvent Data

Equations (3-8), (3-19), and (3-28) are used to obtain the solvent

activity coefficients for a ternary electrolytic solution containing a

single salt. (The subscripts 1,2,3, and 4 refer to the positive ion,

the negative ion, solvent 1(3), and solvent 2(4), respectively, in the

following discussion.)

The Coulombic contributions to the solvent activity coefficients

are calculated directly from equation (3-8). The terms in equation

(3-8) are defined by equations (3-9) - (3-18).

The residual contributions to the solvent activity coefficients

are calculated from equation (3-19). The terms of equation (3-19) are

defined by equations (3-19) - (3-27).

The Flory-Huggins contributions to the solvent activity

coefficients are given by equation (3-28). The terms of equation (3-28)

are defined by equations (3-29) - (3-38).

The solvent activity coefficient for component 3 in a salt-mixed

solvent solution is therefore given by

In 7 ^ = In 7 ^Coulombic^equation 3-8 ) + lncj^'/x^' + 1 “

+ q3 (l- ^(©.jAt/C^ + © 3 + © 4 t//43)

- <S>1 V-r31/(<5>1 + © 2 ^ 2 1 + ® 3 ^ 3 1 +®4*feP

-©2^32/(®i^ 12 + ® 2 + ® 3T/;32 +®4 ;42')- <S> 3/(<3> 1 A±/Q1 + © 3 + ® 4t/743)

“ © 4 ^ 3 4 / ( © iB±/Qi + ® 3^ 34 + ® 4 ))

+ lnx3' - lnx^ (6-1)

The solvent activity coefficient for component 4 in a salt-mixed

solvent solution is given by

82

83

In 7^ = In 7 ^ ulombicXequation 3-8) + lnc^'/x^ 1 + l- (J^Vx^'

+ q4 (l- bi^Bi/C^ + ® 3 3 4 + © 4 )

- © i S M ® ! + ® 2 ^ 21 + @3^31 * © * ^ )- © 2V'42/<©lV'l2 + © 2 + ® 3 % 2 + ©4Vi2)

- © 3 ’ l3/(© 1A±/Q1 + © 3 + © 4 ^ 4 3 )- © 4 / ( ® 1B±/Q1 + ® 3 <//34 + ® 4))

+ lnx/+' - lnx^ (6 -2 )

A± is defined by equation (5-^b). B± is defined

B± " «lV>14 + (v2/ v l )C*2'p24 (6'2a>(Note that the solvent was referred to as component 3 in Chapter 5

for both aqueous and nonaqueous electrolytic solutions. In this

Chapter, component 3 is generally water and component 4 is the

nonaqueous solvent.)

The values of A± and B± were found to be zero from the

regression of binary aqueous and nonaqueous data.

It is also to be noted here that the solvation numbers of the

negative ions are assumed to be zero.

For isothermal P-x-y data, the experimental values of the liquid-

phase mole fractions for each component, the temperature of the

system, and equations (6-1 ) and (6 -2 ) allow the calculation of the

pressure and vapor-phase mole fraction at each data point. From the

relationships of Chapter 1,3

Pcal - ’‘j7 JrjSCt>jS<exp(Pcal - FpvyRT)/^.(6-3)

yical = Xi^iPi W (exp(Pcal “ PiS> V RT)/^ i Pcal

(6-4)

The vapor pressures, P^s, of the solvents are determined using the Antoine equation. The constants are given in Appendix A. The Hayden-O*Connell (1975) correlation is used to calculate the fugacity coefficients. The procedure is discussed in Appendix A. The Hankinson and Thomson (1979) correlation is used to calculate the pure-component liquid volumes, v . (See Appendix B) Equations (6-1) (6-4) must be solved using a bubble point calculation.

For isobaric T-x-y data, the experimental values of the system pressure and the liquid-phase mole fractions for each component are used to generate the bubble-point temperatures and the vapor-phase mole-fractions. Pca]_ i-n equations (6-3) and (6-4) are replaced by Pexpj the experimental system pressure. A bubble point calculation

is used to solve the equations.

The ternary model of equations (6-1) - (6-2) was used in the prediction of the vapor-phase compositions given by equations (6-3)

and (6-4) considering the cases discussed below.

85

6.2 Case 1. Solvation Effects are Neglected in the Model

The binary solvent-ion parameters of Table 5.7 and the binary solvent-solvent parameters of Table 5.21 are utilized. The binary

solvent-ion parameters of Table 5.7 were obtained with a, the ion-

size parameter of the Coulombic (Pitzer) term, an adjustable

parameter. The solvation numbers of the positive ion were set to

zero. The prediction results for the corresponding data of the

ternary data base (Table 5.2) are presented in Table 6.1. The

quality of the prediction of the data is indicated by AP, AT, and

Ay, defined by equations (5-23a), (5-23b), and (5-23c). The average percent errors in 7 3 and 'Y ^ given by

Average 7. error 7. - (1/NP) Xj K T j ^ - 7 ^ ) |x 100

(6-5)are also reported.

The average percent ratio is defined V/ - y a - salt-freeAverage/. , ( 1 ) ^ --- ) x 100

Ratio v NP ' y y t , - Ya. salt-free'^exp j

(6-6)and is a measure of how well the calculated values of y^ agree with

the experimental values of y . The calculation is based on y^ since

this component is enriched in the vapor-phase upon the addition of a

salt to a mixed-solvent system. Component 4 is referred to as the

salted-out component and component 3 is referred to as the salted-in

component. This will be discussed in a later section.

Table

6.1

Case

1. Prediction of

the Vapor

Phase

Compositions of

Ternary

Data

Using

the Binary Parameters

of Tables 5.7 and 5.21.

(All

Solvation

Numbers

are Set Equal

to Zero.)

86

B-SCDOO o O o OO 00cd o • • • • • • •

-H tH r. r* 00 Nl* tHCD 4-) CO rH rH rH CN cn rH> cd < Pi 1 1 1 1

CDoo - r^ LO m o CN tH ON• • « • • • •-<1* in m CN CO COCN rH rH rH rH rHc w

CD 00 ~ Cd M M O CD M > M < Wco

CO cnOJ

ooin

'tfCO

CN G> o CN• • • •O 00 1 OrH

CN OJ CO in CMCN <1* •o- CN inO o o o o o o

• • • • • • •o o o o o o o

87

A value of Average % Ratio equal to zero indicates that the

predicted vapor-phase compositions of component 4 in the salt- mixed solvent solutions are the same as the vapor-phase compositions

of component 4 in the salt-free mixed-solvent solutions of the same salt-free liquid phase composition. This indicates, poor prediction.A value of Average % Ratio equal to 100 indicates that the predicted

vapor-phase compositions of component 4 equal the experimental values, or perfect prediction. An Average % Ratio between 0 and 100 shows that the predicted vapor-phase compositions of component 4 lie

between the experimental values of for the salt-mixed solvent solutions and the values of y ^ for the salt-free mixed solvent

solutions.Negative Average % Ratios indicate that the model predicts that

component 3 is enriched in the vapor phase instead of component 4, a sign of very poor prediction. Average % Ratios greater than 100 mean that the model overpredicts the vapor phase composition of

component 4 in the salt-mixed solvent system relative to the

experimental values.

The results presented in Table 6.1 show that the Average % Ratios for the LiCl-IÔ-MeOH system at 25°C and the NaBr-tÔ-MeOH systems at

25°C and 40°C are between 0 and 100 indicating that the model predicts

correctly the salting-out of methanol. The negative values of the Average % Ratios for the LiCl-tÔ-MeOH systems at 60°C and 760mm Hg and for the NaBr-Ô-MeOH system at 760mm Hg indicate that the model

predicts that water is salted-out instead of the methanol, clearly

88

a failure of the model.

Figures 6.1 and 6.2 show the y-x diagrams for the LiCl-Ô-

MeOH systems at 25°C and 760 mm. Figure 6.1 shows that even

though the Average X Ratio for the LiCl-tÔ-MeOH system is 31.0,

the relative X ratios range from approximately unity at low mole-

fractions of methanol x ° = .152 to over 100 at high methanol mole-

fractions x ° = 0.958. This indicates that the model, where the

solvation numbers of water and methanol are assumed to be zero, is

only applicable at mole fractions of methanol greater that 0.95.

The impact of molality can not be ascertained for this system since

the data are available only at m = 1.

Figure 6.2 shows that the model, where the solvation numbers of

water and methanol are assumed to be zero, predicts that methanol is

salted-in at mole fractions of methanol less than 0.5. The data for

this system are available at molalities ranging from 0.1 - 3.8.

Molality has no impact on the prediction of the vapor-phase

compositions of component 4.

Figure 6.3 compares the solvent activity coefficients for the

tÔ-MeOH system at 25°G with and without LiCl. In the absence of

the salt, the solvent activity coefficients of water and MeOH exhibit

positive deviations from Raoult's law. The addition of LiCl increases the activity coefficient of MeOH, indicating that MeOH is salted-out,

but causes the activity coefficient of ^ 0 to exhibit negative

deviations from Raoult's law. Negative deviations from Raoult's law

are indicative of solvation effects, in this case, between the salt and the water. (An example of solvation effects in nonelectrolytic

solutions is observed for the chloroform-acetone system.)

89

Figure 6.1 Predicted y-x Diagram for the LiCl-t^O-MeOH System at 25°C and m=l. (Solvation Effects are Neglected.)

0.9

0.8

0.7

0.6

0.3

0.2# - experimental calculatedO - salt-free

0.1

0.00.6 0 . 8 1.00.20.0

x rteCH

90

Figure 6.2 Predicted y-x Diagram for the LiCl-^O-MeOH System at 760mm Hg and Molality Range 0.1-3.8m. (Solvation Effects are Neglected.)

O - salt-free • - experimental^lm calculated <lm^ - exp. 1 m 2 ■ — calculated l<m<2 ■ - exp. 3< m<^ calculated 3<ra*i|'

0.6 0.8 1.00.20.0XMe0H

91

Figure 6.3 Comparison of the Activity Coefficients of the t^O-MeOH System With and Without Added Salt. Salt System: LiCl-HpO-MeOH at 25UC and m=l.

1. 50..

O - 7^ q(salt-free)

\ o ( s a l t )

A " ^MeOH(salt_free) A ’ ^IeCH(salt)

1. 30..

r

AO

O

1.10.,

0.90-

MeOH

92

Figure 6.4 compares the solvent activity coefficients for the H20-MeOH system at 760mm with and without LiCl. The activity coefficients of H20 in the presence of salt are less than their values

in the salt-free solution, but negative deviations from Raoult's law are not always observed. Activity coefficients less than unity for

this system are only observed at molalities greater than 1 and at

salt-free mole fractions of water greater that 0.45. This is to be

expected since the greater the molality, the greater the number of

ions in the solution which can remove water from the bulk solvent to

enter into solvation. At salt-free mole fractions of water less than 0.45 and any molality the activity coefficient of H20 is less than it

is in the salt-free mixture but is not negative. This indicates that there are not enough water molecules available to solvate the ions of

the salt.

The addition of LiCl increases the activity coefficients of MeOH

relative to those in the salt-free solution with one exception. At a

salt-free mole fraction of water of 0.06, and m = 0.1 the activity

coefficient of MeOH in the absence of salt is 1.00194 and that in the

presence of salt is 0.99875. While it can be argued that the lowered activity coefficient is due to experimental error, it will be shown

that negative deviations from Raoult's law are observed at low (around

X2 °= .1) mole-fractions of water indicating that methanol, as well as

the other solvents of this study, is involved in the solvation of the

ions of the salt. Unfortunately, this is observed only at one data- point for this system.

Figure 6.5 shows the contributions of the Pitzer, Flory-Huggins,

and the residual terms to the calculated activity coefficients of

Figure 6.4 Comparison of the Activity Coefficients of theMeOH System With and Without Added Salt.Salt System: LiCl-^O-MeOH at 760mm Hg and Molality

Range 0.1-3.8m.

3 . 00-

r

O - A -

□ - • -

A -

^ h 20 ( <lm)

^ H 20 (1<m<2>r „ 2 o O ' h k'o

^ H e O H ( <lm)

^/MeOH^1<m<2

^ M e O H ^ <m<^

2.00_.

1.00_

q (salt-free)'THeOH^sal't"free

0.0MeOH

94

Figure 6.5 Contribution of the Pitzer,Flory-Huggins, and Residual Terms to In 7 Q for the LiCl-MeOH-HgO System at 25®C2and 1 molal.

in 7

in 7,exp.calc.

PitzerFlory-Huggins Residual In 7, salt-free

0.0

- 0.2

ivieun

o *

95

water for the LiCl-^O-MeOH system at 25°C. (Solvation is not

considered.) It is seen that that contribution of the Pitzer term

is positive while that of the Flory-Huggins term is negative. The

sum of these terms gives a negligible contribution to the calculated

activity coefficients. The major contribution to the calculated

activity coefficients is due to the residual term which is positive

and predicts water activity coefficients greater that unity up to a

salt-free mole fraction of water of 0.3. The residual term is

negative at a salt-free mole fraction of water of 0.042. Although

not shown, the model predicts solvent activity coefficients of MeOH

which are essentially the same as those in the salt-free solution.

With the exception of the data point at x^°= 0.042, it is evident

that solvation should be considered in the model. At 0.042,

the solvation of the ions by water is negligible due to the low

concentration of water.

Figure 6.6 shows the contributions of the Pitzer, Flory-Huggins,

and the residual terms to the calculated activity coefficients of

water for the LiCl-^O-MeOH system at 760 mm Hg. Again, solvation

is not considered.

The contribution of the Pitzer term is negligible in most cases

except at molalities greater than 3.0 and salt-free mole fractions of

water greater than 0.75. The contribution of the Flory-Huggins term

is negligible at all molalities and water compositions. The

contribution of the residual term is negligible at molalities greater

than 1 and salt-free mole-fractions of water greater than 0.8. The

contribution is significant for the other compositions. In most cases,

the calculated activity coefficients agree closely with those in the

96

Figure 6.6 Contribution of the Pitzer, Flory-Huggins, and ResidualTerms to In % n for the LiCl-FLO-MeOH System at 760mm Hg .... .... --- — ..—1 -- - iL

and Molality Range 0.1-3.8m. (Solvation Effects are Neglected.)

0 . V '

InXh 2o

0.2

0.0

- 0.10

1 - salt-free2 - calculated(m< 1;3 - experimental(m <1)

O A D ” Pitzer( m<l, m<2 , m<^}# Flory-Huggins(m<l,m<2,m<^)© A D - Residual(m<'l,m<2'-,in<^)

0.0 1.0MeOH

97

salt-free solution indicating the failure of this model. The

calculated activity coefficients of MeOH in the presence of the

salt (not shown) also agree with those of the salt-free solution and

not the experimental values.The trends observed in Figures 6.1 - 6.6 for the LiCl-Ô-

MeOH systems at 25°C and 760 mm Hg are also observed for the LiCl-

Ô-MeOH system at 60°C and the NaBr-Ô-MeOH systems at 25 and 40°C

and at 760 mm Hg.

It is to be noted that the results for the LiCl-Ô-MeOH at 60°C

are reported for two molality ranges. While Hala (1969) reports the

experimental ternary data up to a molality of 14.1, the binary inter

action parameters used to predict the data were established from binary

data that are available up to a molality of 6.0m. Also, as discussed

in Section 5.5, the model should be used up to a molality of approximately 6.0 since all the binary interaction parameters were determined

up to this molality.

Figures 6.3 and 6.4 indicate that solvation effects are important

in salt-mixed solvent systems and must be considered in a model which

predicts the properties of such solutions. (The trends observed in

these figures also apply to the other systems of the ternary data base

of Table 5.2.) In the next section, solvation effects will be

considered.

6.3 Case 2. Solvation Effects are Considered

98

The binary solvent-ion parameters of Tables 5.10-5.15 and the

binary solvent-solvent parameters of Table 5.21 are utilitized in equations (6-1) and (6-2). The binary solvent-ion parameters of Tables 5.10 and 5.11 for the aqueous 1-1 chlorides, bromides, and iodides were obtained with a, the ion-size parameter of the Pitzer term, set equal to the sum of the crystallographic radii of Table 5.4. a is an adjustable parameter for the other aqueous 1-1 electrolytes,

the 2-1 aqueous electrolytes, and for all binary nonaqueous electrolytes. The solvation numbers of Table 5.8 for the solvation of the positive ion by water at infinite dilution were used. The solvation numbers at infinite dilution for the solvation of the positive ion by nonaqueous solvents are given by equation (5-9). The positive ion in a mixed solvent solution is assumed to be solvated by both solvents. The solvation numbers are assumed to vary linearly with the mole-fractions of the solvents. (See equation (3-33))

h+3 ho+3x3 ,;6-8)

h+4 ‘ ho+4x4The prediction results for the corresponding data of the ternary

data base of Table 5.2 are presented in Table 6.2. AP, AT, and Ay, which indicate the quality of the prediction, are defined by equations (5-23a), (5-23b), and (5-23c). The average percent errors in *7 and

defined by equation (6-5), and the average percent ratio, defined

by equation (6-6) and discussed in Section 6.2, are also presented in the table.

The results for the LiCl-l^O-MeOH and the NaBr-t^O-MeOH systems

are significantly improved. (See Table 6.1 for the case where

Table

6.2

Case

2. Prediction of

the Vap

or Phase

Compositions of

Ternary

Data

Using

the Binary Parameters

of Tables 5.8 - 5.15

and 5.21.

(Solvation is

Considered)

99

a) oo ts oM .Ha) •<->> CO< PS

in o o 00 O o O' r-. O o CN• « • • • • • • • • • •<1- CN rH CO cn vO o r^

CD m OO oo O co C^ cn o cD NfrH rH tH rH rH

ar&o - co M ^ oOJ M > !-< < W

r- 00 m 00 CN in o rH O CO• • • • • • • • • • •rH <1*CN orH CO 00 r * CT> CNrH

O''cnCN

COcu>bOCO M 8 CN

00 CTNco

mcn

CN mr*-

o<r

00 orH vOT—!

CNoCN

ooHo

mo

00

oCN m

£

<300 CN iD CN Nt- CN CN <1* CN CN• • • • • • • • • • •rH OO m O ■<T o o CN O OrH rH

00CNCO

Table

6.2

Case

2. continued

100

 CO C Pi

oo OO 0O• • •m CO O]<1* c- O'

I

CO

M O CO M > 5-1 < W

r-O'

1-CT> UO OCn]

CO "00 »■ CO 5-1 ^ OSi h1 c w

oMl* cn ox—!

<1*HCM

OoH< o

£1 m CM Ml* co• • • •O' CM t—I ot-H CM rH

<1

101

solvation is not considered.) The salting-out of methanol is now

observed in all cases. The LiCl-^O-MeOH system at 60°C is now overpredic ted.

The results for some salts in the mixed solvents, water-ethanol, water-isopropanol, and ethanol-methanol are also presented. The results are good for the water-ethanol systems but are poor for the water-isopropanol systems. (The results for the water-isopropanol systems are shown for two molality ranges.) Salting-in is observed for the CaCl^-MeOH-EtOH system.

Figure 6.7 shows the contributions of the Pitzer, Flory-Huggins, and residual terms to the calculated solvent activity coefficients of water for the LiCl-^O-MeOH system at 25°C and a molality of 1. A comparison of Figure 6.7 with 6.5 shows that with solvation of the positive ion by water and methanol, the magnitudes of the Pitzer and Flory-Huggins terms decrease. Where in Figure 6.5 the sum of these contributions is approximately zero, in Figure 6.7, there is a net negative, although small contribution to the calculated activity coefficients at high methanol concentrations. In addition, the residual contribution does not decrease as drastically above a mole fraction of methanol of 0.7 as it does when solvation effects are neglected.

Figure 6.8 shows the contributions of the Pitzer, Flory-Huggins, and residual terms to the calculated solvent activity coefficients of methanol for the CaC^-MeOH-EtOH system at 760 mm and 1.806m. Ethanol is the salted-out component for this system. At this molality, the activity coefficients of methanol show negative deviations from Raoult's law at salt-free mole fractions of ethanol from 0.1 to 0.9. Below a

102

Figure 6.7 Contribution of the Pitzer, Flory-Huggins, and Residual Terms of In for the LiCl-H^O-MeQH System at 25°C

and m=l. (Solvation of Li+ by Ho0 and MeOH is Assumed.)

l n ~Ki2 c

ln\ o

Pitzer

Flory-Hue;gins Residual

ln K. c (salt-free)

exp.

calc.

0.2

0o Ao

MeCH

- 0.2

103

Figure 6.8 Contribution of the Pitzer, Flory-Hug- gins, and Residual Terms to ln)jJIe0H for the CaClg-KeOH-EtOH System at 7 6 0 mm Hg and 1.806 molal,(The positive ion is sol-

. vated by x IeOH and EtOH.)

0 - experimental • - calculatedA - salt-free1 - Pitzer2 - Flory-Huggins3 - Residual

0.2

0.0

- 0.20.60.2 0.80.0 1.0

XEt0H

104-

mole-fraction of 0.1, positive deviations from Raoult's law are

observed.The Pitzer term is the major contribution to the calculated

activity coefficients of MeOH at salt-free mole fractions of EtOH less than 0.4. The sum of the Flory-Huggins and residual contributions is approximately zero in the region. Above a mole-fraction of 0.4

the residual term becomes important.The results indicate that salting-out is always predicted when the

solvation number of the salted-in component is much greater than that of the salted-out component. For example, the solvation number of the Li+ ion in water is 5 while that of the Li+ ion in methanol is 2. However,in the case of the CaC^-MeOH-EtOH system, the solvation number of the

+2 +2 Ca ion in methanol is 3.8 while it is 2.8 for the Ca ion in ethanol.Since the model breaks down for the CaC^-MeOH-EtOH system, it is

apparent that the solvation numbers calculated by equation (6-7) and (6-8) should be modified to give some recognition to the properties of

the mixed solvent system. This will be done in the next section utilizing the concept of preferential solvation.

105

6.4 Case 3. The Preferential Solvation of the Positive Ions inMixed-Solvent Systems is Considered.Debye (1927), who introduced the concept of preferential

solvation, showed that the solvent with the higher dielectric

constant will preferentially solvate the ions in a mixed-solvent

system. He assumed that the solvent activity coefficients are unity.

(Ideal solution)He developed an expression which relates the salt-free mole-

fractions of the components of the bulk mixed-solvent system to the mole-fractions of the solvent species in the vicinity of an ion. v^br^A^0 - V2lnx^/x^° = - V2(z^e^/87TkT)(l/D^r^) dD/c3n^

(6-9)r is the average distance between the central positive ion and the nearest solvent molecules. x ° and X r f are the mole fractions of each of the solvents in the bulk solution when r = °°. x^ and X2 are the

mole-fractions of the solvents in the vicinity of the positive ion; i.e., the solvated compositions, v^ and V2 are the molar volumes of the solvents and D is the dielectric constant of the salt-free mixed-

solvent.Equation (6-9) cannot be used readily since the value of r is

unknown. If it were to be used, an additional parameter would be added

to the ternary model which could only be evaluated through regression

of the ternary data.While equation (6-9) cannot be adapted for use in equations (6-1)

and (6-2), it does suggest which solvent properties and system conditions are important if one is to account for preferential

solvation; namely, the change in the dielectric constants of the

106

mixed-solvent systems with composition, the temperature of the system, and the molar volumes of the solvents. Hie effect of the molar volumes of the solvents was not considered since the solvents of this study have nearly identical molar volumes.

To maintain the predictive capability of the model, only the CaC^-MeOH-EtOH system at 760 mm and m = 1.806 was used to develop the

expressions which account for preferential solvation. (If the other systems listed in Table 6.2 were used, the model would reduce to a correlation method.)

Several sets of expressions were developed and were tested using the CaC^-MeOH-EtOH system. The expressions for h+3 and h+4 given by equations (6-7) and (6-8) were replaced by the equations for preferential solvation. Only two of the sets of expressions caused equations (6-1) and (6-2) to predict the salting-out of EtOH and the salting-in of MeOH. The first set of expressions are given by

h _,_3 = h0-f-3x3 exP(z+z_ (Dit/D3t)x4) (6-10a)

= V 4 X4 exp(" ZjrZ- ^Dri/D4T^x3 (6-10b)The second set of expressions are given by

h+ 3 = ho+3x3 exp(z+z_(Dm/D3T)x4)(298.15/T) (6-lla)

h+4 = h0+4 x 4 exp(-z+z_(Dm/D4T)x3)(298.15/T) (6-llb)and will be discussed in the next section. In equations (6-10) and (6-11), component 3 is the salted-in component and component 4 is the

salted-out component.Equations (6-10a) and (6-10b) recognize the temperature of the

system indirectly through the dielectric constants of the mixedsolvents, Dm, and the pure solvents, and D^. Dm, D^, and

are evaluated at the system temperature. Both equations reduce to

107

aquation (3-33), which gives and h+4 as functions of composition

for the binary solutions, when the appropriate limits are taken.

The results for the CaC^-MeOH-EtOH system using equations (6-10a)

and (6-10b) in equations (6-1) and (6-2) are shown in Table 6.3. The

expressions were also used in the predictions of the other systems of

Table 6.3.The CaC^-MeOH-EtOH system now has an Average % Ratio of 152

compared to the value of -49.6, obtained when the concept of

preferential solvation was not utilized. The average value of Ay is

now 0.034, which is a significant improvement over the value of

0.070 obtained previously.

The contributions of the Pitzer, Flory-Huggins, and residual terms

to the calculated activity coefficients of MeOH for the CaC^-MeOH-EtOH

system are shown in Figure 6.9. The Pitzer and residual terms are the same as those in Figure 6.8 since the same binary interactions are

utilized. However, the contribution of the Flory-Huggins term is now

more negative than it is when preferential solvation is not considered.

While the average percent error in ^ is approximately the same in

both cases, the calculated activity coefficients with the assumption

of preferential solvation correctly indicate negative deviations from

Raoult's law.

The contributions of the Pitzer and residual terms to the

calculated activity coefficients of EtOH for the CaCT^-MeOH-EtOH

systems are the same as the case when preferential solvation is not considered. The Flory-Huggins contribution is negative when

preferential solvation is not taken into account and positive when it

is. The Flory-Huggins term, when used in nonelectrolytic systems,

Table

6.3

Case

3. Prediction of

the Vapor-Phase

Compositions of

Ternary

Data

Using, the Bainary

Parameters

of Tables 5.8 - 5.15

and 5.21

(Preferential

Solvation

is Considered;

No Temperature

Correction

108

S'S

bO cO H ■I a

CO O rH 00 o o O O o CF' LO• • • • • • • • • • •m CO o r*-. 00 o r-- in o m 3 in CMo m Mf Ml* tH m rH v£> CM CM 00 r*r—I CM CM T-i tH tH CM t—1 rH tH

§P MM O a) n> M < W

mMl*

CMrHin cm

oo r*-CO

CM CM O'* oo CM CO M3 M3• t • • • • •M) CO CM CO m r-* in

t-H tH tH

or oo - co u u oQJ U

COr-»r-*

in CM tH CM tH tH o p-• • • • • • • • •o 00 CD O M0 CM CO CO CMT— 1 tH CO tH CM CM

LOCM CO

aoH<]00 M3 CM M3

O

£

<1

CM CM m CM Ml* <rn C\J o CM 00• • • • • • • • * • •»CO COCO — i CO O CM <o o o CO o o

M3

M3

M3COtH

Table

6.3

Case

3. continued

109

S'SCL)60 oCO O •V -H CT>a) 4-j O> cO — 1< PCS

COv£><T> CMLO

tH

<DM O <D Vi > V < W

co kD<1* oCM

a)60 r cd M Vi O CL) Vl > Vi < W

m o\D cr>

aoH<]

mCsl

Figure 6.9 Contribution of the Pitzer, Flory-Huggins, and Residual Terms to ln< ]eoH ^or ’the CaClg-MeOH-EtOH System at 760 mm Hg and 1,806 molal. (Preferential Solvation assumed.)

0.4 -

0.2

ln MeOH

0.0

- 0.2 -

experimental calculated salt-free Pitzer Flory-Huggins Residual

1.0EtOH

110

Ill

should never predict positive deviations from Raoult's law since its

development is based on the assumption of an athermal solution. The

Flory-Huggins solvent activity coefficient for EtOH on a solvated

basis, m 7 - EtOH’ ne§ative> w ic.h is correct since this term is identical to the Flory-Huggins equation for nonelectrolytic solutions.

This contribution only becomes positive when the term lnx1gtOH^xEtOH

is added to lr/X'^Q^. (See equation (3-6)) If In^'jrtOH were positive, the model would be incorrect.

The introduction of preferential solvation increases the

magnitude of the Flory-Huggins term for the salted-in component; i.e.,

causes it to become more negative. The magnitude of the Flory-Huggins

term for the salted-out component becomes more positive.


to In *7 q for the LiCl-^O-MeOH system at 25°C are shown in Figure

6.10. A comparison with Figure 6.7 shows the impact of preferential

solvation. The Flory-Huggins contribution to the solvent activity

coefficient has become more negative. Although not shown, the Flory-

Huggins contribution to the coefficients of MeOH have become more

positive. However, this term is negligible compared to the residual

term.

The introduction of preferential solvation given by equations

(6-10a) and (6-10b) in equations (6-1) and (6-2) worsened the results;

i.e., the Average % Ratio, for many of the systems of Table 6.3.

(Compare with Table 6.2, where preferential solvation is not taken

into account). Poor results are indicated when the Average percent

ratio is greater than 150% (overprediction) and Ay is greater than

0.03. A value of Ay of 0.03 was chosen since predictions of the

112

Figure 6.10 Contribution of the Pitzer. Flory-Huggins. and Residual Terms to I n ' Y „ n .rl LTThe positive ion is solvated by water andmethanol., Preferential Solvation assumed)

0.5

l n 7 * 2

Pitzer Flory-Huggins ResidualIn 'Y ,r n (salt-free)

exp.calc.

0.3

0.1

0

- 0.2

113

vapor phase compositions by the UNIFAC model of nonelectrolytic solutions are usually considered poor if Ay is 0.03.

For example, when preferential solvation is not taken into account, the average error in Ay and the Average percent ratio for the CaC^-tÔ-MeOH system at 760mm Hg and m = 1.806, are 0.010 and 96.7, respectively. When preferential solvation is used, Ay is

0.055 and the Average % Ratio is 215.0.The contributions of the Pitzer, Flory-Huggins, and residual

terms are plotted in Figure 6.11 for the CaC^-Ô-MeOH system. The

results with and without preferential solvation are indicated. ThePitzer and residual contributions are the same in both cases. Itis obvious that the Flory-Huggins term overcompensates for the

+2preferential solvation of the Ca ion by water. In this case, no preferential solvation term is needed.

The results of Table 6.3 for the LiCl-Isopropanol-Ô and LiBr- Isopropanol-Ô systems at 75°C improved with the introduction of

the preferential solvation term. The results for the LiCl-Isopropanol- H2O system up to a molality of 11 are poor since this system was predicted with binary interaction parameters obtained only up to a molality of 1.8.


are plotted in Figure 6.12 for the LiBr-Isopropanol-lÔ system at 75°C. The results with and without preferential solvation are indicated.Since the Pitzer and residual terms are the same, the improvement in the results is due solely to the Flory-Huggins term. Again, the Flory- Huggins term with preferential solvation is more negative that it is without preferential solvation. The results would be further improved

114

Figure 6.11 Contribution of the Pitzer, Flory-Huggins, and Residual Terms to ln% r,n o wComparison of Preferential Solvation Assumption and Solvation.

2

0

O - experimental • - calc. No Pref. Sol © - calc. Pref. Sol.A - salt-free

- Pitzer

2PS- FH( Pref.sol.) 3 - Residual

- 0.2

2 PS

0.8 1.00.20.0

115

Figure 6.12 Contribution of the Pitzer. Flory-Huggins.and Residual Terms to lnO^ Q for the LiBr- HgO-Isopropanol System at ^6°C and m - !.*<•. (With and Without Preferential Solvation.)P.S. = Preferential Solvation

1.0 -

ln-)H20

0.0

- 0.1

- experimental- calculated with P.S.- calculated with no P.S.- Flory-Huggins with P.S.- Flory-Huggins with no P.S.- Residual- Pitzer A

0.0 0.2x

0 . 8 1.0Isopropanol

116

if the Flory-Huggins term with preferential solvation was more

negative at salt-free mole fractions of isopropanol of 0.48.

In the next section, the results using the preferential solvation

results of equations (5-lla) and (6-llb) will be presented.

117

5.5 Case 4. The Preferential Solvation of the Positive Ions in Mixed-Solvent Systems. The Temperature Correction Term is Applied..(Equations (6-lla) and (6-llb) are Used.)

Equations (6-lla) and (6-llb) recognize the temperature of the system implicitly through the dielectric constants of the solvents,

and explicitly, through the factor 298.15/T. Tnese equations reduce

to equation (3-33) for the binary systems when the appropriate limits are taken.and only if the temperature of the ternary system is

298.15°K. Their use is not strictly valid unless the solvation

numbers in the binary are temperature dependent in the same manner;

i.e., equation (3-33) is multiplied by the factor 298.15/T.

The temperature correction term serves to lower the values of

h_,_2 and h+ given by equations (6-10a) and (6-10b). This correction

in turn decreases the impact of the Flory-Huggins term for the

salted-in component; i.e., makes it less negative, depending on the

values of h and h ,, , and decreases the values of the Flory- o-i-o 0+4Huggins terms for the salted-out component; i.e., makes them less

positive.

The temperature correction term will not affect the results

already shown in Table 6.3 for the systems at 25°C. The results

utilizing equations (6-lla) and (6-llb) are shown in Table 6.4.

As expected, the Ay and Average % Ratios for the systems which are

overpredicted (See Table 6.3) have decreased.

While this is desirable for most of the systems of Table 6.3,

it is not desirable for the Isopropanol-I^O systems. (In Section 6.4,

it was pointed out that the preferential solvation numbers calculated

by equations (6-10a) and (6-10b), should be larger at salt-free mole-

fractions of isopropanol of 0.47.) In addition, the Average % Ratio

Table

6.4

Case

4. Prediction of

the Vapor-Phase

Compositions of

Ternary

Data

Using

the Binary Parame

of Tables 5.8 - 5.15.

(Preferential

Solvation

is Considered.

Temperature

Correction Term)

118

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120

for the NaBr-t^O-MeOH systems at 760mm Hg is lowered from 67.9 with

no temperature correction term to 56.7 with the temperature

correction term.

It is recommended that the temperature correction term be used

to improve the results of systems that can only be predicted using preferential solvation. The CaC^-I^O-EtOH system at 760mm Hg and

the CaC^-MeOH-EtOH system at 760mm Hg are examples of systems that

can be predicted well using preferential solvation.

121

6.6 The Use of Preferential Solvation

It is apparent from Tables 6.2 and 6.3 that the prediction

results for some systems are improved when preferential solvation

is used and worsened in other cases. The results of Table 6.3

indicate that the prediction results are inproved for the NaBr-tÔ- MeOH system at 760mm, the CaC^-fÔ-EtOH system at 760mm, the

Isopropanol-I-Ô systems, and the CaCl9-MeOH-EtOH system at 760mm.

The prediction results for the LiCl-lÔ-MeOH at 60°C deviated

considerably. The results for the other systems, which deteriorated

with the introduction of preferential solvation, are acceptable and

represent good prediction of the experimental data.

It is recommended that the preferential solvation equations,

(6-10a) and (6-10b), be used for all nonaqueous mixed-solvent systems

that have dielectric constants less than that of water. As shown in

Table 6.2 and Figure 6.8 salting-in is predicted for the CaC^-MeOH-

EtOH system if preferential solvation is not used.

It is also recommended that preferential solvation be used for all salts in tÔ-MeOH mixtures except for the lithium and calcium

salts, and for all other salts in IÔ-nonaqueous solvent mixtures.

122

6.7 Comparison of the Prediction Results of this Study with theCorrelation and Prediction Results of Other Models.

The prediction results of this study are compared to the

correlation and prediction results of the Rastogi (1981), Hala (1983),

Mock, et.al. (1984), and Sander, et.al. (1984) models. The comparisons are shown in Table 6.5.

The Rastogi model combines a modified Debye-Huckel equation and

the NRTL (Renon and'Prausnitz, 1968) equation. The interaction

parameters are salt-solvent specific as opposed to the ion-solvent

specific parameters used in this work. Prediction is only possible when the constituent binary data are available. The maximum molality at which data can be predicted is 2m.

The prediction results of this study are much better than those

of the Rastogi model. In addition, the model allows the prediction of the data up to a higher molality range.

The Mock, et.al., model can only correlate electrolytic solution

data. The model is based solely on the NRTL model and contains salt-

solvent and solvent-solvent interaction parameters. The model, which

contains 9 adjustable parameters, does not represent the long-range coulombic forces through an additional term. For a salt-water-alcohol

system, three of the adjustable parameters represent the water-alcohol

interactions, three represent the salt-water interactions, and the last three represent the salt-alcohol interactions. Salt-water data and

water-alcohol data were correlated to obtain the needed interaction

parameters for the ternary model. This reduces the number of

parameters in the ternary model to three salt-alcohol parameters.These salt-alcohol parameters were determined through the regression

Table

6.5

Comparison of

the Results

of Thi

s Stu

dy wit

h Those

of Rastogi

(1981),

Hala

(1983),

Mock,

et.al.

(1984);

and Sander,

et.al, (1984).

123

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(Sander)

Table

6.5 continued

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125

of the ternary salt-water alcohol data. The authors did not regress

these systems with the available binary salt-alcohol data meaning

that a unique set of salt-alcohol interaction parameters which

correlate the ternary as well as the constituent binary data were

not obtained. The authors obtained a different set of binary salt-

alcohol interaction parameters through the regression of salt-alcohol

systems. Substitution of these parameters, along with the 3 solvent-

solvent and 3 salt-water parameters, do not allow the prediction of

the ternary system.

The prediction results of this study compare remarkably well with

the correlation results of Mock, et.al. The only exception noted is

for the LiCl-tÔ-Isopropanol system up to a molality of 11. However,

this system was predicted with ion-water parameters established from

data available up to a molality of 6 and ion-isopropanol parameters

established from data available up to a molality of 1.8.

The Hala model, which contains six adjustable parameters, combines

a semi-empirical electrostatic term and the Wilson (1964) equation.

The model was used only in the prediction of the LiCl-HÔ-MeOH system

at 60°C. Two of the adjustable parameters, which represent the

interactions between water and methanol, were established through the

regression of water-methanol data at 60°C. Two of the parameters were established through the correlation of LiCl-water data at 60°C

and the other two through correlation of LiCl-methanol data at 60°C.

The results of this study compare well with those of Hala. Apparently, the Hala model performs better at high molalities of salt.

It should be pointed out that in this study, the LiCl-Ô-MeOH system

at 60°C was predicted from binary solvent-ion interaction parameters

126

determined from data at 25°C and water-methanol parameters determined

at 60°C. As shown in Section 5.4, the binary interaction parameters

obtained at 25°C allow the prediction of <£) and P for binary data up

to 200°C in the case of aqueous systems and up to 60°C in the case of

methanol. (See Tables 5.16 and 5.18)

The Sander, et.al., model combines the Debye-Huckel and UNIQUAC

(Abrams and Prausnitz, 1975) equations. The UNIQUAC parameters are

functions of concentration and are ion-solvent specific. Parameters

were established for salt-water-alcohol and salt-mixed alcohol systems

through the simultaneous, regression of tlie ternary data and the

constituent binary data. Since the binary and ternary data bases were

used to establish the parameters, (the data base of Sander, et.al. is

similar to that given in Tables 5.1 and 5.2) few systems were left with

which to test the predictive capabilities of the model.

The results of this study compare well with those of Sander, et.

al., especially when the temperature correction term is applied. The

results for the CaC^-MeOH-EtOH system of this study are not true

prediction since this system was used to establish the preferential

solvation equations. However, the ion-solvent parameters were obtained

only from binary data at 25°C whereas the ion-solvent parameters of

Sander, et.al., were determined from isothermal and isobaric binary

and ternary data. It is unknown if the Sander model allows the

prediction of ternary and higher-order systems from binary data alone,

since the concentration dependency of the parameters can only be

established in mixed solvent systems and not in single solvent systems.

This would be a true test of the model.

127

The models presented here, with the exception of the Mock, et.al.

model, were developed assuming that the excess free energy of solution

is the sum of two terms. The first term is a Coulombic term, which

represents the long range ion-ion interactions. The second term

represents the short-range interactions between the various species in

the solution. The Mock model neglects the Coulombic term.

The activity coefficients of the solvent in a binary electrolytic

solution exhibit negative deviations from Raoult's law except for a

region at low salt concentrations (typically, for 1-1 salts, at m<l),

where positive deviations from Raoult's law are observed. In this,

study, these deviations are interpreted through a solvation model of

the electrolytic solution.

If a solvation model is used to describe the properties of a

binary electrolytic solution, it is assumed that the ions of the

solution are solvated by the solvent. At extremely low concentrations

of the electrolyte (m< 0.001m) where positive deviations from Raoult's

law are observed, the long-range forces between the solvated ions

predominate. These forces, which are inversely proportional to the

distance between the species squared (r ), are the most important

since there are too few ions in the solution available to interact

with each other at close distances, or to affect the properties of the

bulk solvent. As the concentration of the electrolyte increases to

approximately 1 molal, the short-range interactions between the ions

become important and remain important over the entire concentration

range. These short-range solute-solute interactions are of the

128

charge-induced dipole type; i.e., the solvated ions induce dipole

moments in the other ions. These forces are inversely proportional£

to r . In addition, there are dispersion (London) interactions

which occur when two ions are attracted to each other. These

dispersion forces are inversely proportional to r .

Increasing the salt concentration causes more and more solvent

molecules to be removed from the bulk solution. At molalities above

1 molal, sufficient numbers of solvent molecules are removed from the

bulk solution to cause a decrease in the solvent activity coefficient.There are also short-range interactions between the solvated-

ions and the solvent molecules of the bulk solution. These are of

the same type as those which operate between the solvated-ions.

The solvent-solvent interactions are also short range in nature

and have been described in Chapter 3.

The forces between molecules which operate in binary electrolytic

solutions also operate in salt-mixed solvent solutions. However, in

these systems, the ions of the salt are preferentially solvated by the solvent with the larger dielectric constant in most cases. The

activity coefficents of the solvent which preferentially solvates the ions are lower than their values in the salt-free solution and most

times exhibit negative deviations from Raoult's law. The activity

coefficients of the salted-out solvent exhibit positive deviations

from Raoult's law. This is in contrast to its behavior in the pure solvent.

The quality of the prediction of salt-mixed solvent systems

from binary data alone indicates that the assumptions used to

129

develop the model of this study are valid.

The model of this study differs from those previously mentioned in that solvation effects are explicitly accounted for. In addition,

the short-range interactions between ions are accounted for in the

Pitzer term. These interactions are neglected in the Rastogi and

Sander models.

130

6.8 Use of the Ternary Model in Data Correlation

The ability of the ternary model, given by equations (6-1) and

(6-2), to represent salt effects is important in the event that the

experimental salt-mixed solvent data are already available.

Tne correlation capability of the model was tested using the

LiCl-Ô-MeOH systems at 25 and 60°C, and the LiCl-HÔ-Isopropanol

systems at 75°C. In the examples to follow, it is assumed that the

ion-nonaqueous solvent parameters are unknown.

The number of parameters in equation (6-1) and (6-2) are reduced

to five immediately since data for the tÔ-nonaqueous solvent and

LiCl-I- O systems are readily available. Tnese systems were regressed

to obtain the binary interaction parameters given in Tables 5.10 and

5.21, respectively.

Two of the five parameters in equation (6-1) and (6-2) are the

solvation number of the nonaqueous solvent and A^* The solvation

number is calculated from equation (5-9) and A ^ may be estimated

from equation (5-10). Tne parameters that need to be established

are a , A41 and A^.

The ternary data can now be regressed for a , A^, and A^- To

show that meaningful parameters are obtained when the model is used

in correlation, the ternary systems were regressed for a^, A^, and

A^ 2 along with the binary salt-nonaqueous data. This is not possible

using the Mock, et.al., (1984) correlation. (See Section 6.7)

The LiCl-Ô-MeOH and LiCl-MeOH systems at 25°C were regressed

to obtain a , A41» and A^2 * Tnese parameters were used to predict the LiCl-I-Ô-MeOH systems at 760mm Hg. (The preferential solvation

terms given by equations (6-10a) and (6-10b) were not utilized.)

131

The LiCl-IÔ and LiCl-MeOH systems at 60°C were regressed

together and the resulting parameters were used to predict the LiCl-HÔ-MeOH systems at 25°C and 760 mm Hg.

Finally, the LiCl-BÔ-isopropanol and LiCl-Isopropanol systems at 75°C were regressed to obtain a , A^, and •

The results for all systems are shown in Table 6.6. The error in Ay obtained through regression of the ternary system and the

average percent error in <$> for the binary data are indicated, as well as the Ay for the predicted systems.

The Ay obtained from the regression of the ternary systems

agrees well with those of Table 6.2 and 6.3. This is to be expected since the data were correlated with the same binary data used to

predict them. The only exception to this is for the LiCl-tÔ-MeOH

system at 60°C which was correlated with the binary LiCl-MeOH data

at 60°C. This would explain the difference in the values of the

parameters obtained in Table 6.6 compared to the parameters of Table 5.14.

The results of Table 6.6 indicate that the model can be used in

the correlation of ternary data. The parameters are meaningful since

the same ternary systems at different temperatures or isobaric

conditions can be predicted from the parameters. Also, the model

parameters allow the prediction of the osmotic coefficient data of the

LiCl-MeOH system at 25°C using the parameters obtained from the 60°C

data and vice versa. This is not possible with the Mock model.

Table

6.6

Demonstration

of the Correlation

Capability of

the Model

132

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133

6.9 Use of the fcodel in the Prediction of Salt-MixedSolvent Systems when a,, or A has not been^ mn~~" i .... .Established.For the NaCl and KCl-HgO-MeOH systems and the NaCl-

HgC-EtOH systems of the ternary data base of Table 5.2, the values of a^, the ion-size parameter of the Pitzer term in equations (6-1) and (6-2), have not been established since the constituent salt-alcohol data do not exist. (The data are not available since sodium chloride is generally soluble in alcohols up to a molality of approximately 1.0.)The values of A have been established for these systems.mn ^(See Tables 5*1^ and 5.15) In order to predict these systems, the average values of a^ obtained from the regression of the halide salt-alcohol data are assumed to be the values of for these salt systems.

The prediction results (preferential solvation terms of equations (6-10a) and (6-10b) are used) are shown in Table 6.7. The average value of a^ for the halide salt- methanol systems is 5.23. The average value of a^ for the halide salt-ethanol systems is 5.70.

The prediction results for the HgO-EtOH systems are reported up to two maximum molalities. Reducing the molality ranges improves A y but does not improve the Average % Ratio significantly.

Examination of Table 6.7 for the HgO-EtOH systems, indicates that at the upper molality limit reported for each system, the average percent error in 7^ is generally large compared to that in 7^. Reducing the molality range

ible

6.7

Prediction Results

Using

an Average

Halide Value

of

13^

B-8

CU00 c p CN m co 00CO O • • • • •S-l *rH vO CN CN co in<\) 4-J > CO M< w

CO CN vr> cD rH• • • • • •CP CO CN CN CP CO

t“H T—1 r~H

"dCO

s-e-vma>C| &o - co s-j J-l o a) j-i > W< w

o -d" -d-co

cn-d CN

r-.

OoH<1

iD c oo

cnCO o

0 r--i niig4-1w1 CO

cnCN

eu< 1

in•rH Ctf X r—H

CNO

OCOo

ocD

CNO

e'eno

c

«CO

CO CNO

CNo

CN CO ON o<t tH in ,rHo o o oo d d o

tH o co mCN d t-H

gcu s: I co U 03

o IOE-* puCN O

r-*CJoOCO

mm

g<usIoCN

4-J r*Hzco

r— CJ r—H wcC CJ 1CO Z o-M

CJcO CJcO

135

reduces the average percent error in 7^ but has no affect on the average percent error in 'Yy

To see why this occurs, the contributions of the Pitzer, Flory-Huggins, and residual terms to the solvent activity coefficients of ethanol for the NaCl-HgO-EtOH system at 30°C are plotted in Figure 6.13. A.t low ethanol concentrations and high salt molalities the residual term provides the major contribution to the calculated solvent activity coefficients. The data would be predicted well in this region if the Pitzer and Flory-Huggins terms were positive. However, the Flory-Huggins term is correct at this low concentration of ethanol, where solvation of the positive ion by ethanol is negligible. Therefore, it is the Pitzer term which is causing the poor results at high molalities.

The Pitzer term which accounts for the long-range ion-ion and short-range ion-ion interactions, incorrectly predicts the salting-in of ethanol instead of the salting- out of ethanol at high molalities and low ethanol concentrations. The reason for this is that the Pitzer equation assumes that the dielectric constant of the solvent is constant. It does not allow for changes in the dielectric constant with salt concentration.

The NaCl-HgO-EtOH system at 30° C was regressed for a^ to see if the assumption that a^ is the average value

aij. for the halide salts is a valid one. The regression results yielded a value of a^ of 11.0, A y of 0.01^, and an

136

Figure 6.13 Contribution of the Pitzer. Flory- Huggins. and Residual Terms to for the NaCl-H^O^tOH System at 30°C. Molality range» 0. 56-*l-. 10m.

2.0

1.5

• • • - exp. 3*0 < m< *K1B O - exp. 2.0 < m< 3.0

- A • © - exp. 0 . 5 < m < 2 . 0B A - calculateda h A - Pitzer

fi O o□ - Flory-HugginsB - Residual

1.0

xn EtOH

0.5

n o ▲ B o

8

£

80.3 0.5

EtOH

137

Average # Ratio of 112. The average percent error in 7^ is 6.2 and that in 7^ is 3*8. '.'/hen a^ is 11.0, the Pitzer term is positive. However, this value of a^ is unreasonable in light of the values of a^ obtained for theethanol systems of Table 5.15*

There is also the possibility that the experimental data are inexact, but this is unlikely since the trends observed for the NaCl-HgO-EtOH system at 30°C are also observed for the other systems of Table 6.?.

There does appear to be a link between the solubility of the salt in ethanol and the molality range to whichthe ternary data can be predicted. The salts of Table6.2 are soluble in the nonaqueous solvents over the entire molality range at which the ternary data are available. The prediction of the ternary data from the binary is also excellent. (See Tables 6.2 and 6 .3 .) Good predictions are obtained up to twice the saturated values for the NaCl-HgO-EtOH systems. The results for the NaCl and KCl-HgO-MeOH systems are not significantly improved by reducing the molality range.

The values of a^ and AMeoH/F~ ■f,°r "the NaF“H 2°“Me0H system, a^ and ^ t O H / P ” ^or NaF-H20-Et0H system, and a^ and for the KI-HgO-EtOH system have not beenestablished since the constituent salt-alcohol data are not available. These salts are not very soluble in the

138

nonaqueous solvents.As in the case discussed above, is assumed to

equal the average halide values of a^ obtained from the regression of the salt-alcohol data. The average value of a^ for methanol is 5*23. The average value of a^ for the ethanol systems is 5 *7 0 .

The A]yieQH/F" interaction parameter is assumed to have the same value as the AMeoH/Cl” itt^raction parameter. This was done since the Cl” ion is the only negative ion which is close in size to the F~ ion. The

ocrystallographic radii of the Cl” ion is 1.81 A while that

oof the F” ion is 1.^0 A. The A-tr-toH/F- interaction parameter is assumed to have the same value as the Ag-toH/Cl“ interaction parameter.

It is further assumed that the A-Tr-fcOH/K+ dnteraction parameter may be calculated from a linear relationship between Asoiven^/^+ and the dielectric constant of the solvent at 25 °C. The A^ q/K+ an<* AKeOH/K+ dn^erac’tdon parameters of Tables 5-10 and 5*1^ are used to establish this linear relationship.

Asolvent/K+ = ”^07.5 + 6 '3D 25°C (6-12)The value of ■A2-tQH/K+ calcula‘ted from equation (6-12) is -353.8.

The average halide values of a^ and the estimated parameters were used to predict the vapor phase compositions for the systems of Table 6 .8 . All of the salts listed in the table are barely soluble in the nonaqueous

Table

6.8

Prediction Results

Using

Estimated

Values of

A and a

tnn

>

139

S'S0100cfl O co O' cn cnU -H • • • •CU J-) O CN CN o> cu cn c n P" cr>C Pi

<u' oo -cfl f-iu o cu n > Hc w

ea

rnCO

-d-uo CN

S facnCU'*00 - CO J-i H O CU M > J-i < fa

m

C N

1*0I"--

Oi n

ooH< 1C O

oi n

o

OC N

dC N

OCNo

CN

o

<| oo

O c n n -■*—i c n oO o o

• • •o o o

3 •f'J B rH •H Ctf XI i— 143 &o CO

«

oUOvO

CM

CN

EC!-)O g sO p <5 so IT) oHf-U n~

co

or-*

gQ)T3 n

4-» grH (-H fa faCO w 43 t— 1CO 1o fa-4

140

solvent alone.The results for the KI-HgO-EtOH system are improved

when the maximum molality is reduced to 2.2. As in the NaCl-HgO-EtOH system, it is the Pitzer contribution to the solvent activity coefficient of ethanol which lowers the contribution of the residual term. The residual term predominates at high molalities and low ethanol concentrations.

The NaF-HgO-alcohol system^ are predicted with A y values of 0.01 but low Average fo Ratios. No conclusions can be drawn as to why the Average % Ratio of the NaF- HgO-EtOH system is low since there are too few data points for this system with which to analyze the data.

The contributions of the Pitzer and Flory-Huggins terms to the calculated activity coefficients of MeOH for the NaF-HgO-MeOH system are negligible compared to the contribution of the residual term. Although negative deviations from Raoult's law are observed for this system at salt-free mole-fractions of water of 0.8, the model predicts positive deviations from Raoult's law. Increasing the solvation number of the Na+ ion in water would decrease the Flory-Huggins contribution and increase the Average fo Ratio. However, this option is not possible since the solvation numbers are fixed in this study. The residual term can be readjusted through regression of the data to find an optimum value of

141

6.10 Use of the Model in the Prediction of Salt-Mixed Solvent Systems When More Than one Value of a^ or A has not been Established.

In Section 6.9, the model was used to predict salt-mixed solvent systems in the case where a^ or A ^ has notbeen established. The value of a^ in the salt-mixed solventsystem was assumed to equal the average halide value ofa^ obtained from the values of a^ for the halide salt-nonaqueous systems already established. The value ofA , was assumed to equal the value of the ion of the mnsame charge and nearest in size if it is unavailable in a particular solvent. For example, the values of AjyjeoH/F” and ^g-toH/F” were assumed to have the same values asAMeOH/Cl"’ and AEtOH/Cl"" * respectively. If the value ofA was established in two of the solvents but unavailablemnin a third, a linear relationship between the values of A ^ and the dielectric constants was assumed. (See equation (6-12).) This relationship was used to obtain the value of AEtOH/K+ froin values of A^ o/k+ and AMeOH/K+ *

Since these methods of predicting the unknown parameters gave reasonably good prediction results, they are utilized in this section.

The values of Asoiven-t/ton estimated usingthe equations shown in Table 6.9• All the equations, with the exception of that for the AS//j+ parameter, were established using the values of the interaction parameters

Table

6.9

Equations

for the Estimation of

.S = solvent

D is

equal

to value

at 25°C

1^2

GO•rHH->cG

4->srHo—! *H

CL) 4-1 M 4-1 S-l CU

■Q c3

mo

ot-H o

cn CO CO CO CO COa (Q a a a a acr1 o o o o o ovX) 00 CN o vO o oCN co NO tH co m» • • • • •00 in 00 on NOrH CN rH NO£o 1 1 1 1 i 1 i•H4-> o CO 00 Ni- 00 m CNcO * •r~H m O'' CO no rH CNCU CO CN CN rH O' O CNM co 00 rH r-* rH in rH

tH rH rH -<}■ I iOa II II II II 11 II II

o •Hhj !SOJ+ cUCJ

1^3

obtained for water, methanol, and ethanol. The equationcorrelates the A ^ values with a correlation coeffi-s/ioncient of 1.0 for three of the ions. Poor correlation is obtained for the iodide, lithium, and calcium ions. (Logarithmic and exponential equations were also tried, but similar results were obtained.)

It is recommended that the equations of Table 6.9 he used only when the values of As^ on are not available.

The systems of Table 6.10 were predicted using the equations of Table 6.9 for the missing parameters. Preferential solvation is assumed. The values of a^ were assumed to equal the average halide value of obtained from the values of for the halide salt-nonaqueous systems already established. Since no values of a^ for salts in n-propanol are available at all, it was assumed that a^ for LiCl in n-propanol has the same value as the average a^ of the salt-isopropanol systems. (At 25°C, the dielectric constants of isopropanol and n-propanol are 18.0 and 20,2, respectively. This indicates that the Pitzer contribution is approximately the same for both systems.

It should be noted that the binary data for the CaClg-Isopropanol system could not be correlated by the binary model given by equation (5-6). The binary data

were regressed for ak and A-iEOpropanoi/ca+2 - The value

of A isopropanol/Cl" was established from regression of the LiCl-Isopropanol system at 75-l°C. A.^ was calcula-

Table

6.10

Prediction Results

Using,

Estimated Values of

a_ and A.

Hl4

'*—1£

vr>

cu6

LO1at

G"Scoto

toe'er.<uGooCQ

cuW) o O oG O • • •5-1 -H to LO Cba) -u r-- CN> Gc a tH

Oc o

aoS-JG0 CO M1oCMai

a)txOG H M O CU S-i > Vj < W

6'SsooQJ'"bOCO M 54 o a) m^ £

oHO

aO

* 1

M tlfOO Io9H P-J

COC/D

mco

vO

CNo

O

CO

CO00CO

ocr*

soVO

a

CNCO

CO

tH CNtH 00 tHo CN o

• • •o o o

-cf

oco m

CN CN rH t“Ho a<3 <3

COr»*

gAJ■oa?

oV—1 >»• rHCN (U• CN 4->

CN T“1 gun •rHXoG uo &*r—t Cuin 4-1 cd• Go G MH3 " o

G 4->G •iHto i— I

cdr— 1CM • oBO G

a)G cdG 4->8

cdG CO<uT“H rH COCM G cd

O G3 -Sq Cuo e IoG ££

40oo tH 4—5

to o P• •HCN G00 COG 4JG •rH5-4 r— 1Cu4^ CO

1 •HrH §cd -po rH COVO O >s

s■5c

CO

o

145

ted from equation (5-10). Since the data were correlated poorly using the approach above, the CaClg-Isopropanol system was then regressed for all the parameters of equation (5-6). The data were still correlated poorly.Since the LiCl and LiBr-Isopropanol systems were correlated well using the model, (see Table 5.15)» it can only be concluded that the binary data for the CaClg-Isopropanol system are poor and that the poor results are not due to a failure of the model.

The data for the CaClg-HgO-Isopropanol system at 75.1°C are available up to a molality of 6.0. The prediction results up t© this molality indicate a value of A y of 0.281, which is extremely poor. The reason for this is that the range of the model, given by equation (5-12), has been exceeded. This occurs when either x^' or x ^ ' , the mole-fractions of components 3 and 4, respectively, on a solvated basis, become zero. (For the CaC^-HgO-Iso- propanol system, Xjj q * becomes zero when the molality is greater than 4.0) The values of A y are also very poor at molalities greater than 0.44m and up to salt-free mole-fractions of isopropanol of 0.33» the maximum value reported by the workers. The value of A y is 0.012 at a molality of 0.44.

Mock,et al.,(1984), correlated this system only at a molality of 0.44 and report a value of A y of 0.013.

This system cannot be predicted beyond a molality of 0.44 since it splits into two liquid phases at a molality

1A6

of approximately unity. No data are reported between a molality of 0 . ^ and 1.0.

The prediction results for the LiCl-HgO-n-propanol system are good. This system was predicted with estima-

ted values of Vpropanol/l,i+ > and An_propanol/01-.A y is 0.021.

34-7

6.11 Use of the Model in the Prediction of Salt-Multi- component Solvent Systems .

In Sections 6.9 and 6.10, the model was used topredict salt-mixed solvent systems in the cases wherea., or A have not been established. The value of ah h> mn *rwas assumed to equal the average halide value of a^ obtained from the values of a^ for the halide salt-nonaqueous systems already established. may be calculated from the equations of Table 6.9.

In this section, the LiCl-water(3)"methanol(4)-n-propanol(5) and the LiCl-water(3)-ethanol(4-)-n-propanol(5) systems at 760 mm Hg are predicted using the methods described in Sections 6.9 and 6.10. The data are from Boone(1976).

The Pitzer term contains one parameter, a. a is calculated using equation (3-17). The values of a^ and a^ have been established. (See Tables 5*10, 5.1^# and 5.15.) The value of a^, the ion-size parameter of LiCl in n-propanol, must be estimated. As in Section 6,10, it is assumed that a^ for LiCl has the same value as the average a^ of the salt-isopropanol systems.

The values of the solvation numbers for the Li+ ion in water, methanol or ethanol, and n-propanol for use in the Flory-Huggins terms are given by the following expres- sionst (Preferential solvation is assumed.)

148

= h0+ifx4exP (”z+z-(DM//D4T ^ x3 + x5 ^ (6-12b)

h+^ * h0+^x5exp(-z+z-(DM/ D ^ ) ( x 3 + x^)) (6-12c)

The exponents of equations (6-12b) - (6-12c) are negative since the alcohols are the salted-out components.

The values of A ^ , A^2 » Azn» and A^2 of* the residual term are found in Tables 5.10, 5*1^* 5.15. Thevalues of A ^ , A ^ , A ^ , A ^ » A ,- and A ^ are found inTable 5.21. The values of A ^ and A^2 are calculated from the equations of Table 6.9.

The prediction results for the two systems are shown in Table 6.11. The A y are compared to those of Boone who used the Wilson and UNIQUAC equations and a pseudobinary approach.

The results of this study compare well with those of Boone. They are surprisingly good considering that only binary parameters and estimated binary parameters were used to effect the multicomponent predictions. The Boone approach utilizes ternary data to effect multicomponent predictions.

Table

6.11

Pred

icti

on

Results

for

Mult

icom

pone

nt

Syst

ems

149

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CM•

CMVO

cn•

od

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oCM

l>-00

d© HbO i cd Pi, Pi oi © Pi,> Pij < Wl

© O bD *i cd Pi Pi O © Pi> Pi <! W

°I o |e-ti

ftl<

< 1

vo OOrH

Oo6

_ >jE P3 *H -—

vr> 6 |H i—1 V O d•H Cd • '—- •X I*! rH cn X cno cd o OPc s e P<ft ft1 1C 3I w 1

X r— >d d

6 o oX E vo X voo Cv- o Cv-© ft ps w1 1

P rH O' rHrH O O

o © H o •HCM C/3 X cm X

XI XI

co•HP©3O*©COwrH•rl©Si+>bDC•H013

CO•HPcd3O*©oofMXX©XCO o p• • —>

o CM vO bDCv- COv •Hi—1 CO—- 3cn cn• • © ©o /— . o r—% c £rH CM rH CM o o■- VwX" __ o oo o CO vo vo d X Xcn CM CM cn cn cno O O o o O• • • • • . o oo o o o o o X0

01 o Pi A A. cd>.PicdC•HAO*03©wftII

SiCJcdoPiftftcd>5a.5A0T33©01 ftIICM

C H A P T E R 7 DISCUSSION OF RESULTS

The objective of this study was to develop a model for the prediction of mixed solvent-single electrolyte mixtures from binary data alone. The following approach was takeni 1) Establish a phase equilibrium data base of systems containing mixed solvents with one electrolyte, 2) Develop the corresponding binary data base from the literature data, 3) Provide the missing data through experimental measurements of salt-methanol systems, 4) Develop a model containing binary parameters only, 5) Evaluate the model with binary data and determine the corresponding binary parameters, and 6 ) Test the model with multicomponent data.

The phase equilibrium data base for salt-mixed solvent systems is shown in Table 5*2 and contains data for 1 6 salt-water-alcohol systems and one salt-mixed alcohol solvent. These systems were chosen since the construction of a group contribution model for the prediction of salt-mixed solvent data from binary data alone requires that the ion-solvent interaction parameters evaluated from the constituent binary data be firmly established. This is only possible if the binary data are reliable} i.e., verified through a variety of measurements.

The only binary data meeting these requirements are those for the salt-water systems. The compilation of Robinson and Stokes (1959) reports osmotic coefficient

150

151

and mean activity coefficient data for these systems. The aqueous data used in this study are listed in Table 5.1.

The data for the salt-nonaqueous systems (also shown in Table 5.1) comprising the ternary systems of Table 5.2, were obtained through vapor pressure measurements only.The methanol data base was extended in this study. (See Chapter 4 and Appendix E . ) The data base is almost complete for the methanol systems} i.e., the values of a^, the ion-size parameters of the Pitzer term, and the interaction parameters of the residual term can be obtained directly from the regression of the data in the binary model. The values of a^ cannot be established for the NaCl, NaF, and KCl-water-methanol systems, nor can the value of *They were estimated in this study. The method is presented in Chapter 6 and will be discussed later in this section.

The values of a^ for the NaCl, NaF, KC1, and Kl-water-ethanol systems as well as the values of -^EtOH/K+ anc*AEtOH/F” ’ also had be estimated since the data forthese systems are not available. These salts, as well asthose listed above for the methanol systems, are not very soluble in their respective solvents. This means that if the data were available, they would extend to a molality of no more than unity. Meaningful ion-solvent parameters are not obtained if data below this molality are regressed since each parameter is multiplied by a concentration term. If the concentration is below unity, the ion-

152

solvent parameters of the residual term will assume any value. For example, consider one of the terms of the binary residual expression of equation (5-5) given below.

If © ^ and © 2 , the area fractions of the solvated positive and negative ions, respectively, are very small,

then © 1V 12 + compared to © 3^ 2 * Evenif assumes a large value in regression, it is minimized through multiplication by ©^. The term then reduces to

“ ®2%2 / ®3%2 and any value of ^ 2 will give the same value for this term.

The only way to obtain meaningful parameters for these systems is to regress them with a system that has a common ion and is soluble in the solvent up to a high molality. For example, the maximum molality of the KI- MeOH system is 0.8 meaning that this system cannot be regressed in the binary model to obtain a and the ion- solvent parameters. However, it can be regressed with either the KCH^COO-MeOH system which has a maximum molality of 2,5 or the Nal system which has a maximum molality of ^.3*

The binary data base for the prediction of the salt- isopropanol-water and methanol-ethanol systems is complete.

The. binary model presented in Section 5*1 (equation (5-6)) was developed assuming complete dissociation of

153

the salt. Ion-association effects are neglected. The dissociation of a salt in a solvent depends on the charge densities of the ions comprising the salt, the dielectric constant of the solvent, and the temperature of the system. An increase in temperature results in a lowering of the dielectric constant.

Most of the salts in water at 25°C of this study are completely dissociated, with the exception of the 1-1 nitrates and 1-2 sulfates. The dielectric constant of water at this temperature is 78.33* In solvents other than water, incomplete dissociation of the salt usually occurs. Dissociation constant data are reported for some of the salts in this study including the NaSCN and KC1 salts in methanol and the LiCl, NaCl, KC1, and KI salts in ethanol at 25°C. (Kratochvil and Yeager, 1972) It is reported that LiCl is completely dissociated in methanol. Waddington (1969) gives the rule of thumb that an electrolyte can be considered completely dissociated up to a moderate concentration range in a solvent with a dielectric constant greater than 30. The dielectric constant of methanol at 25°C is 32.6 and that of ethanol is 24.3. The incomplete dissociation of LiCl and LiBr in isopropanol can be assumed based on Waddington's rule and the fact that LiCl is incompletely dissociated in ethanol which has a higher dielectric constant.

The assumption of complete dissociation appears to be valid for the aqueous 1-1 and 2-1 halides, the 1-1

15k

chlorates, and the 1-1 acetates of this study since the average percent error in cj) for these systems is 2 ,2k. However, the average percent error in c£> for the 1-1 nitrates and 1-2 sulfates is k.l indicating a decline in the correlation ability of the model for these systems.In systems in which there is incomplete dissociation, the decrease in the solvent activity coefficient curves with increasing molality is not as great as it is for systems in which there is no ion-pairing. This effect can be explained in terms of the hydration model. As the concentration of the electrolyte is increased, the incidence of ion-pairing increases^ i.e., since there are more ions in solution, the probability that they will come into contact increases. This contact is likely to reduce the ion-solvent interactions, so that the paired ions will have less complete solvation sheaths. (Robinson and Stokes, 1959) The net effect, then, is to return solvent molecules to the bulk solution which results in solvent activity coefficients which are higher than those noted for completely dissociated systems having the same positive ion or lowered osmotic coefficients. At a molality of 3.0, the LiCl-water system has an osmotic coefficient of 1,286, while the LiNO^-water system has an osmotic coefficient of 1.181.

Two terms in the model can be modified to account for ion-pairing, the first being the Flory-Huggins contribution. For the LiCl-water system of Figure 5*5, the

155

Flory-Huggins term represents the largest contribution to the calculation of the activity coefficients of water.This contribution is approximately the same for the LiNO^ system since the crystallographic radii of the nitrate ion is I .89 & while that of the chloride ion is 1.81 5.

The values of h+ in this study are assumed to be a linear function of the solvent mole fractions. (See equation (3-33)•) For a particular negative ion, decreasing the value of hQ+ decreases the contribution of the Flory- Huggins term, (making it less negative). The value of hQ+ should not be adjusted to improve the results for the LiNO^ system since at low molalities, the osmotic coefficients of LiNO^ and LiCl are of nearly the same magnitude. At a molality of 0.1, the osmotic coefficient of LiCl is 0.939 while that of LiNO^ is 0.938. This indicates that the "amount of solvent removal" by the Li+ ion is about the same in both cases. The values of the osmotic coefficients diverge at a molality of 0.7. Since hQ+ cannot be adjusted, the concentration dependence of the solvation number would have to be investigated for ion-pairing systems. A power law model, where is raised to the ( n ) power in equation (3~33), would decrease the value of h+ and therefore increase the Flory-Huggins contribution, (making it less negative.) This would increase the calculated values of the activity coefficients of the LiNO^ system compared to those of the LiCl system.

The Pitzer term could also be modified, but any modification would involve the addition of a parameter

156

to the equation. This parameter would have to include an equilibrium constant to account for the fraction of ions removed from the solution.

The values of a obtained from regression of the nitrate and sulfate salts are less than the sum of the crystallographic radii of these salts. (See Table 5«13») This indicates a "failure" of the Pitzer term since the minimum value a can have is the hard-core distance between the ions which is assumed to be the sum of the crystallographic radii.

In a sense, the model has already been modified since it has been found that a should be less than the sum of the crystallographic radii to improve the correlation of the nitrate systems. The average percent error in when a is set equal to the sum of the crystallographic radii is 8.5 and 3»1 when a is an adjustable parameter.

It is most likely that both of the modifications suggested above would have to be incorporated into the present binary model to improve the correlation of incompletely dissociated salts, as well as the physical reality of the model.

The data of Kratochvil and Yeager (1972) indicate that while LiCl completely dissociates in methanol, NaSCN and KC1 do not. The value of a for the LiCl system is-5.7 while that of NaSCN is ^.73. The values of a for LiBr, Nal, NaBr, and CaCl2 are also above 5*0* Since

157

the value of a gives some indication of whether or nota salt undergoes ion-pairing, it can be deduced that KI,KCH^COO, and CaClg form ion-pairs since their values of a are less than 4-. 73* In addition, the fit of the CuClg and NH^SCN systems is poor compared to that of the other systems. With the exception of these systems, the model correlates the data well with an overall average percenterror in Of 3.3.

Even though ion-pairing is observed for the LiCl- ethanol system (Kratochvil and Yeager, 1972), and most likely occurs in the other systems of Table 5*15» the fit of the data is quite good. The average percent error in c£> for the ethanol systems is 2 .9 . The average percent error in c£> for the isopropanol systems is 3*8.

It should be noted that the CaClg-Isopropanol and the CaClg-n-propanol systems of the binary data base could not be correlated by the model. The average percent errors in <£> for these systems were approximately 60.The LiCl and LiBr-Isopropanol systems were correlated well. It can only be concluded that the binary data for the CaClg-Isopropanol system are poor and that the poor results are not due to a failure of the model. Since no other salt-n-propanol data are available, it is difficult to conclude if the CaClg-n-propanol data are poor by comparison.

Even though the model neglects the effects of ion- pairing, it is evident that in the majority of cases the

158

model gives a good representation of the experimental data. Since this representation is good, the assumption of complete dissociation of the salt is a valid one.

The Pitzer(1973) term accounts for the long and short-range ion interactions. It gives some recognition to ion-solvent interactions since it is a function of the dielectric constant. The contribution of this term is smallest in water systems and largest i n ‘the isopropanol systems, (greatest negative contribution)

1" r H20 < 1"r L o H < ln^/EtOH< ln^ Is opr op This is to be expected since the forces between ions are inversely proportional to the dielectric constant of the solvent. Decreasing the dielectric constant of the solvent, increases the magnitude of the forces between the ions.

The residual term accounts for short-range interactions between the species of the solution not defined in the model. The contribution of the residual term is negligible for the aqueous systems indicating that the model adequately accounts for the intermolecular forces operating in the solution.

It would be expected that the contribution of the residual term would be greatest for the ion-pairing systems. However, the method of parameter estimation in this study prevented this observation. The values of

159

A12 and A ^ were established from data for the 1-1 chlorides, where the residual term gives a negligible contribution to the calculated solvent activity coefficients. These values of A 12 and A ^ were then used to estimate A ^2 for ^he nitrate and sulfate interaction parameters, respectively. However, since the contribution of the term is essentially "set" by A ^2 an<* - 31* the value of A ^2 affects the residual term by at most 5%, It can be argued that the 1-1 nitrates or the 1-1 sulfates should have been selected as the base system for parameter estimation. However, these aqueous ion-pairing systems are not included in the ternary data base of Table 5*2. All of the salts of the ternary data base are completely dissociated in water.

The Flory-Huggins term accounts for deviations from ideality due to the sizes of the molecules. (En- tropic Effects) In other words, it is assumed that a solution of solvated ions would not show ideal behavior if the interionic forces were absent. In addition, the term accounts for the lowering of the solvent activity due to the removal of solvent molecules by the ions. (See equations (3~1)“ (3~7)) ■ This was done to simplify the model. The impact of hQ+ and h+ on the Flory-Huggins term has already been discussed.

V/hile literature values of h . are available for the0 +

aqueous systems, none were found for the nonaqueous systems. They were estimated from equation (5-9). This

160

equation was chosen because it relates the values of h Q+ of water to the best values of hQ+ which fit the binary LiCl and LiBr-methanol data at 25°C.

The model for the prediction of the properties of single electrolyte-mixed solvent systems is presented in Section 6.1. The Pitzer term contains the parameter, a, which is the mole-fraction average of a^ and a^ obtained from the constituent binary data. The Flory-Huggins term contains the parameter, hQ+^, where the j refers to each of the solvent species. The residual term contains seven parameters. Ion-solvent interactions are represented through four of the parameters. Two of the parameters represent the interactions between the solvent molecules, and one represents the short-range interactions between the positive and negative ions. The ion-solvent, solvent- solvent, and positive ion-negative ion interaction parameters are obtained directly through regression of the constituent binary data. Since the ternary expression contains only one positive ion-negative ion interaction parameter, it is assumed that A ^2 is the mole-fraction average of the A ^2 obtained from the constituent binary data.

The performance of the model was first evaluated by neglecting the solvation effects; i.e., ^0+3 and ^o-H^ were set equal to zero. The performance of the model

161

is extremely poor as indicated by negative values of the Average % Ratio for some of the systems. (See Table 6.1.) The model predicts the salting-in of methanol instead of the salting-out of methanol.

The contributions of the Pitzer, Flory-Huggins, and residual terms to the calculated activity coefficients of water, are shown in Figure 6.5 for the LiCl-HgO-MeOH system at 25°C. (Water is the salted-in component.) The Pitzer term in this case, predicts the salting-out of water (indicated by its positive contribution at high salt-free methanol mole-fractions) instead of the salting- in of water. The Flory-Huggins term is in the right direction but its magnitude is approximately equal to that of the Pitzer term canceling the effects of both terms.The residual contribution is positive at salt-free mole fractions of methanol less than 0.7, and becomes negative at salt-free mole-fractions of methanol of approximately0.9. Since the Flory-Huggins term is in the right direction, it can only be concluded that the poor results are due to the Pitzer and residual terms.

Regression of the binary LiCl-water system at 25°C, where solvation effects are neglected and a is an adjustable parameter in the Pitzer term, shows that the major contribution to the calculated solvent activity coefficients of water is due to the Pitzer term. (See Figure 5.3.)Since the solvation effects are neglected, the Flory-

162

Huggins contribution only accounts for entropic effects and their effects are negligible. The residual contribution is also negligible. This indicates that the Pitzer term alone adequately describes the system. At a molality of unity, which corresponds to the molality of the LiCl-HgO-MeOH system for which data are available, the Pitzer contribution is very small. In the ternary system, the Pitzer term initially decreases and then increases with increasing salt-free methanol, mole-fraction. As the mole-fraction of methanol increases, the dielectric constant of the mixture changes from that of water to that of methanol. Apparently, the Pitzer contribution to the calculated solvent activity coefficients of water, behaves like the solvent activity coefficients of methanol in the LiCl-MeOH binary. (See Figure 5t^.) Although not shown, the Pitzer contribution to the calculated solvent activity coefficients of methanol is negative, where it should be positive. The Pitzer contribution to the activity coefficients of methanol is similar to the contribution for the LiCl-HgO system.

The Pitzer (1977) term was derived through consideration of the long and short-range ionic interactions only. It does not account for interactions between different solvent molecules or explicitly for ion-solvent interactions.

The behavior of the residual term is dictated solely by the parameter values obtained from binary regression.

163

It corrects for any short-range interactions not accounted for by the Pitzer and Flory-Huggins terms. Figure 6.5 shows that at salt-free methanol mole-fractions less than0.6, the residual term represents the salt-free activity coefficients of water in the water-methanol system. Above this concentration, the residual term gives the value of the infinite dilution activity coefficient of water.

Solvation effects were introduced into the model to correct for the inadequacies of the Pitzer, Flory- Huggins, and residual terms in predicting ternary electrolytic solutions. (See Section 6.2.)

Rastogi (1981), whose model is based on the Debye- Huckel and NRTL equations, noted that the Debye-Huckel equation calculates the salting-out of water in a salt- water-alcohol system, instead of the salting-in of water. He modified the Debye-Huckel term using a semi-empirical expression. The expression cannot be extended to systems consisting of more than two mixed solvents.Sander, et al.,(198k), developed a model based on the Debye-Huckel and the UNIQUAC equations. The UNIQUAC equation corrected for the deficiencies in the Debye- Huckel term through the introduction of concentration dependent parameters.

The results when solvation is introduced into the model are shown in Table 6.2. Salting-out of the correct

16^

component is predicted in every case with the exception of the CaClg-IVIeGH-EtOH system at 760 mm Hg.

The contribution of the Pitzer, Plory-Huggins, and residual terms for the LiCl-HgO-MeOH system at 25°C to the calculated values of the activity coefficients of water when solvation of the lithium ion by water and methanol is assumed, is shown in Figure 6.7. The introduction of solvation increases the negative contribution of the Flory-Huggins term. The larger the value of h0+* 'the larger the negative contribution. A comparison of Figures 6.5 and 6.7 indicates this. The Flory-Huggins term also corrects for the residual term. The residual term does not become negative at mole fractions of methanol greater than 0.9* but its contribution is approximate ly the same at mole fractions of methanol less than 0.7.

The contribution of the Pitzer, Flory-Huggins, and residual terms to the calculated activity coefficients of methanol for the CaClg-MeOH-EtOH system are shown in Figure 6.8. The effects of introducing solvation to the model are canceled by the magnitudes of the Pitzer and residual terms. The contribution of the Pitzer, Flory- Huggins, and residual terms to the calculated activity coefficients of ethanol are almost equal in magnitude but opposite in sign to those shown in Figure 6.8. The solvation number of the calcium ion by methanol is 3.7* while that of the calcium ion by ethanol is 2.7.

To improve the results of the CaClg-MeOH-EtOH system

165

at 760 mm Hg, the concept of preferential solvation was introduced in Section 6.^. The contributions of the Pitzer, Flory-Huggins, and residual terms to the calculated activity coefficients of methanol are shown in Figure 6.9. hQ+ for methanol and ethanol are calculated by equations (6-10a) and (6-10b).

The Flory-Huggins term cancels the effects of the Pitzer term; however, its effects are extreme. The Flory-Huggins term decreases rapidly with concentration as the salt-free mole fraction of ethanol increases.

Many expressions for hQ+ and were testedfor their effectiveness in causing the salting-in of methanol, but only those of equations (6-10a) and (6- 10b) gave the desired results. According to Debye (192?), the expressions for hQ+ and must be functions ofthe dielectric constant of the mixed solvent and the temperature of the system. The expressions were used in the prediction of the systems shown in Table 6.3. A comparison of these results with those of Table 6.2 indicates a worsening of the predicted A y and Average % Ratio for many of the systems; i.e., they are overpredicted.

The preferential solvation term works to increase the solvation number of the salted-in component and to suppress the solvation number of the salted-out component. The solvation numbers of the calcium ion by methanol and ethanol, with and without preferential solvation are plotted in Figure 7*1* The use of preferential solvation

166

Figure 7*1 A Comparison of the Solvation Numbers of the Calcium Ion by Methanol and Sthanol With and '.vithout Preferential Solvation.

O - Pref.Sol. by MeOH No Pref.Sol. by WeOH• - Pref. Sol. by StOH •— - No Pref.Sol. bv StOH

1.0

0.6 0.8 1.00.20.0x°Me OH

167

enhances the solvation number of methanol over the entire concentration range while suppressing that of ethanol.As already indicated, the preferential solvation term overcalculates the solvation number of methanol.

The results of Table 6.3 suggest the possibility that equation (5“9)» which was developed to relate the solvation number of the lithium ion in methanol to that of water, should be modified. (This relationship is assumed to be valid for all systems.) The form of this relationship can only be established using the systems of Table 6,2. This study would then reduce to a correlation scheme.

To reduce the impact of the preferential solvation term if the contribution of the Flory-Huggins term is too large; e.g., for the CaClg-MeOH-EtOH system, the preferential solvation term was multiplied by the factor 298.15/T. (See equations (6-lla) and (6-llb) and Table 6.^.) Even though this correction term is useful for some systems, it was found to be inapplicable for the LiCl-HgO- Isopropanol system at 75.1°C which would be predicted better if the Flory-Huggins term were more negative; i.e. if the value of the solvation number of water were increased and that of isopropanol decreased. In addition, the introduction of this term prevents the ternary expression from reducing to the binary expression.

168

In Sections 6.9 and 6.10, the model was used in the prediction of salt-mixed solvent systems when a^ or have not been established because the constituent binary salt-alcohol data do not exist. In order to predict these systems, the average halide values of a^ obtained from the regression of halide salt-alcohol data are assumed to be the value of a^ in the ternary system. If an ion- solvent interaction parameter is not available, it is assumed to have the same value as the ion nearest in size and of the same charge. For example, the AjyteoH/F“ interaction parameter, which is unavailable, is assumed to have the same value as the AiyieQH/ci“ interaction parameter. The prediction results are shown in Tables 6.7 and 6.8.

The results indicate that good prediction of the ternary data are achieved for salts in HgO-MeOH mixtures and for HgO-EtOH mixtures up to approximately twice the solubility of the salt in the nonaqueous solvent. It is also important to note that the salts of Tables6.7 and 6.8 are not completely dissociated in the nonaqueous solvents of the ternary systems. However, this does not seem to affect the quality of the predictions. If it did, the predictions would probably be valid only up to molalities of 0.5 where the properties of incompletely dissociated electrolytes diverge from those of completely dissociated solutions.

If A /. is available for at least two of the s/ion

169

solvents, the value of A-g/^on ^or another solvent may be estimated by assuming a linear relationship between the already established and the dielectric constant of the solvent at 25°C. This method was used to develop the equations of Table 6.9 in Section 6.10.For example, the An »pr0pan0i/iJi+ parameter and the ^n-propanol/Cl” parameter were estimated from the equations for and A.g^ci- of Table 6.9. Sven thoughthe correlation coefficient of the equation for is 0.5, indicating a poor correlation of the with dielectric constant, the prediction results for the LiCl-HgOOJ-n-PrOH^) system of Table 6.10 are quite good. The correlation coefficient for the equation is 1.0. a^ was also estimated.

It is not recommended that the estimated values of a^ and or the equations of Table 6.9 be used to predict the osmotic coefficients or vapor pressures of binary nonaqueous electrolytic solutions. As shown in Chapter 5» the major contribution to the calculated solvent activity coefficients of the nonaqueous solvent is the Pitzer term. The wide range in the values of a^ within each nonaqueous system shown in Tables 5*1^ and 5.15» indicate that it would be coincidental if the average halide value of a^ was able to predict the binary data.

It is possible to use an average halide value of

170

in the prediction of the ternary systems since in most cases, the Flory-Huggins and residual terms provide the largest contributions to the calculated solvent activity coefficients.

In Section 6.11, the model was used to predict the salt-effects on the vapor-liquid equilibrium of the LiCl- water( 3 )-niethanol(4)-n-propanol (5) and the LiCl-water(3)“

ethanol^)-n-propanol (5) systems at 760 mm Hg. The data are from Boone (1976). The parameters of Tables 5»10»5.1^» and 5*15 and the estimation techniques of Sections 6.9 and 6.10 were used to effect the predictions.

The results are shown in Table 6.11 and compare well with the results of Boone who used a pseudobinary approach and ternary data to obtain the parameters for the multicomponent predictions.

The results of this study would of course be improved if binary data for the LiCl-n-propanol system were available. Even though the LiCl-HgO-n-propanol system at 760 mm Hg of Table 6.10 was predicted with a A y of 0.021, it should be noted that a plot of the predicted vapor phase compositions of n-propanol as a function of the salt-free mole-fractions of n-propanol indicates that the model possibly predicts an immiscible region around a mole-fraction of 0.2 for the molalities indicated. (See Figure 7.2. The results for a molality of 2.0 are not indicated. The molal ities are on a propanol-free basis.)

To investigate this possibility, a plot of the exper-

171

Figure 7.2 Predicted v-x Diagram for the LiCl- H^Q-n-PropQH System at 760 mm Hg. (Indicated molalities are on a pro- •panol-free basis.)

▲ - salt-free O - experimental m = 1.0 # - calculated m = 1.0 □ - experimental m = ^.0 ■ - calculated m = ^.0

6

n-PropOH

2

00.6 0.80.0 0.2 1.0

xn-PropOH

172

imental and predicted activities of n-propanol as a function of the salt-free mole fractions of n-propanol at molalities of 1 .0 , 2 .0 , and ^.0 , respectively, was prepared. (See Figure 7.3.) The figure indicates that the experimental data exhibit a point of incipient instability at a mole-fraction of n-propanol of approximately 0.2 and a propanol-free molality of ^.0 .

A point of incipient instability is indicated when ( dlna^ / = 0 (7-1)(&2lna^ / dx£)T>p = 0 (7-2)

Graphically, a point of incipient instability is indicated by a horizontal inflection point.

An unstable system is indicated by a maximum on a plot of activity as a function of liquid-phase mole fraction.

Equations 7.1 and 7.2 require that isothermal data be available to evaluate the derivatives. However, as Boone (1976) indicates, the boiling point range for the LiCl-HgO-n-propanol system is l ^ C , but for the n- propanol composition range (0 .03-0.8 mole-fraction propanol ) for which data are available, the boiling point varies only 3°C. Therefore, the system can be assumed to be isothermal.

The prediction results of Figure 7.3 indicate that at

173

1.0

0. 8

0.6

an-Prop0H

0.U

0.2

0.0

Figure 7.3 A Comparison of the Activities of n - orooanol Calculated Using the P/lodel With the Experimental Values.Svstemi LiCl-H20-n-Propanol at 760 mm Hg. (Indicated molalities are on a propanol- free basis.)

- salt-free- experimental m*1.0- calculated m=1.0- experimental m=2,0- calculated m=2.0- experimental m ^ . O- calculated m=4-.0

“O

0.6 0 . 80.2 1.0

xn-Prop0H

17**-

a propanol-free molality of 2.0, a point of incipient instability exists at a mole-fraction of propanol of approximately 0.2. At the same mole-fraction and a molality of ^.0, the model predicts that the system is unstable. Boone's model gave similar results. Boone's experimental data indicate that the LiCl-^O-n-propanol system at 760 mm Hg splits into two liquid phases at a propanol-free molality of and a mole fraction of propanol of 0.2.

Even though the model predicts a point of incipient instability at a molality of 2.0 and an unstable system at a molality of ^.0, it does predict correctly the mole fraction of n-propanol at which the point of incipient instability is observed for the experimental data.

Figures 7«^ and 7.5 show the contributions of the Pitzer, Flory-Huggins, and residual terms to the calculation of the solvent activity coefficients of n-propanol as a function of the salt-free mole fractions of propanol at molalities of 2.0 and ^.0, respectively. As Figure 7*^ shows, the agreement between the experimental and calculated activity coefficients of n-propanol is good. The figure gives no information as to why a point of incipient instability is predicted. The contribution of the Pitzer term is negligible over the entire concen

tration range of n-propanol. Of course, an adjustment in the value of the solvation number of the Li+ ion in n-propanol to lower the contribution of the Flory-

175

Figure 7 A Contribution of the Pitzer. Florv-Huggins.and Residual Terms to lnT^ for the LiCl- H^O-n-ProuOH System at 760 mm Hg. (m = 2 on a n-propanol-free basis.)

lnX,

) - experimental I - calculated1 - Pitzer2 - Flory-Huggins3 - Residual

.0

0

.0

0.6 o.8 i.o0.20.0x.

176Figure 7.5 Contribution of the Pitzer. Florv-Huggins.

and Residual Terms to ln}^ for the LiCl- H^O-n-ProoOH System at 760 mm H g . (m = b on a n-propanol-free basis.)

experimentalcalculatedPitzerFlory-HugginsResidual

SB

0.0

0.6 0.8 1.00.20.0

177

Huggins term (make it more negative), would decrease the contribution of the residual term and improve the resultsti.e., a point of incipient instability would not be predicted. Adjustments in the values of An_pr0panoi//Lj[+ and“Si-propanol/Cl” could also be made but the binary data for the LiCl-n-propanol system are not available.

As shown in Figure 7.5 for a propanol-free molality of ^.0 where an unstable system is predicted at a mole- fraction of propanol of approximately 0.2, the agreement between the calculated and experimental activity coefficients of n-propanol is not good. The relative percent error in the activity coefficient at this point is -5.0.(At a mole fraction of propanol of 0 .38, the relative percent error in the activity coefficients is -8.8) A p parently, at a mole fraction of 0.2, the contribution of the Pitzer term is too large. (negative contribution)The contribution of the Flory-Huggins term is negligible. However, at a mole fraction of O.3 8 , the contributions of the Pitzer and Flory-Huggins terms are negligible and the residual term, is the predominate contribution. As in the case where the molality is 2.0, the results would be improved through an adjustment of the solvation number of the Li+ ion in propanol. However, at a molality of ^.0, the Flory-Huggins term should be increased in a positive direction where for a molality of 2.0, this term must be more negative. In both cases, adjustments in the value of a^ of the Pitzer term would also improve the results but of

178

course, this would require the experimental data of the LiCl-n-propanol binary.

It is generally difficult to predict liquid-liquid equilibria using parameters obtained from vapor-liquid equilibrium data. In the case of the LiCl-HgO-n-propanol system and the systems of Table 6,11, the value of alf. or a^ and the values of the ion-solvent interaction parameters for the constituent LiCl-n-propanol binary were estimated. Even though these parameters were estimated, the correct mole-fraction at which the point of incipient instability occurs for the experimental data of Boone is predicted although the molality at which this occurs is incorrect. The results of Table 6,11 can also be considered good.

179

C H A P T E R 8 CONCLUSIONS

A group-contribution model for the prediction of salt- effects on the vapor-liquid equilibria of multicomponent electrolytic solutions containing a single electrolyte has been presented. The model uses only binary parameters obtained from the regression of binary salt-solvent osmotic coefficient and vapor-pressure depression data at 25°C and binary solvent VLE data.

Methods are presented for the estimation of the ion- solvent and ion-size parameters needed for multicomponent prediction when the constituent binary data are not available. However, these parameters should not be used in the prediction of binary electrolytic solutions.

The prediction of liquid phase activity coefficients and vapor phase compositions was demonstrated for 25 data sets of isothermal and isobaric salt-water-alcohol and salt-alcohol mixtures and gave an average absolute error in the vapor phase compositions of 0.019.

The ability of the ternary model to represent salt- effects was also shown. The results are superior to those of Mock, et.al., (198^),

The liquid phase immiscibility of the LiCl-water-n- propanol system was also predicted. Although the model predicted that phase separation occurs above a molality of 2.0, the correct mole-fraction of n-propanol at which phase separation occurs was predicted.

APPENDIX A

180

181

1. Vapor Pressures of Solvents Used in this Study

The Antoine equation

log Pisat = A — (A-i )T°C+C

where p^sat 1S saturated vapor pressure in mmHg and T is the

temperature of the system in °C, is used to calculate vapor pressures. This equation is not used in the following cases: 1)

for binary aqueous systems at all temperatures, where the vapor pressures are given by Weast (1970); 2) in the calculation of osmotic coefficients from the vapor pressure depression data of

salts in methanol measured in this study since the vapor pres

sures of methanol at m=0 are given; and 3) in the calculation of osmotic coefficients of salts in various solvents measured by other workers where the vapor pressure at m-0 is given.

The constants of equation (A-l) are given in Table (A-l).

182

TABLE A-lAntoine Constants for Solvents Used in this Study

All Values are from Gmehling and Onken (1977)

Solvent A B C

Water 8.07131 1730.630 233.426Methanol 7.76879 1408.360 223.600Methanol 7.97010 1521.230 233.970Ethanol 8.11220 1592.864 226.184n-Propanol 7.61785 1374.890 193.00Isopropanol 8.11676 1580.630 219.610

TemperatureRange

1-100°C25-56°C

65-100°C20-93°C

1-100°C1-100°C

183

2. Calculation of the Fugacity Coefficients at Saturationand of the Vapor Phase Using the Hayden-O*ConnellCorrelation

The generalized method of Hayden and O'Connell (1975) for the prediction of pure component and cross second virial co

efficients for simple and complex systems is used to calculate

the fugacity coefficients at saturation and of the vapor phase. This method was chosen since it does not require experimental

data to obtain parameters. The correlation requires only the critical temperatures and pressures, the dipole moments, the mean radii of gyration, and association and solvation parameters

for the components of the mixture.

The fugacity coefficient of component i in the vapor is given by

ln = It ]-( i ~ P )dP (A_2)

where T is the system temperature and V- is the partial molar volume of i in the mixture.

The density-explicit virial equation which is valid at low pressures is

■7 _ p v - 1 J. B P ^RT RT (A3)

where Z is the compressability factor and V is the molar volume of the mixture. B is the second virial coefficient for the mixture and is given by

184N N

Bmix = 2 2 YiYjBij (A-4)i=l j=l

for i, j = 1, 2, 3, . ..., N components.

The B^f and Bjj represent interactions between like components and are termed the pure-component second virial co

efficients. Bij/ the cross second virial coefficient, represents interactions between unlike molecules i and j.

Combining equations (A-2)-(A-4) gives

N pIn cj) i - [2 YjBij Bmix ] — (A-5)

Hayden and O'Connell assume that the B are the sums of various types of molecular interactions.

B = Bfree + Bme^as .aj;) e + Bj-,oun3 + (A-6)

Bfree accounts for unbound pairs of molecules; Bme^astaj:) e , for metastably bound pairs of molecules; Bkoun(j, for physically bound pairs of molecules; and Bcjiem, for chemically bound pairs of molecules.

Bfree in equation (A-6) is the difference between

Bfree-nonpolar anc Bfree-polar- For interactions between like molecules, Bfree-nonpolar is given by

Bfree-nonpolar = bQ (.94-1.47/T*'-.85/T*’2+1.015/T*’3

(A-7)

T*', the reduced temperature, is

= e/T - 1.6 w (A-8)T

185

where w, the nonpolar acentric factor is a function of R , the mean radius of gyration.

w = .006 r ' + .02087 r '2 - .00136 r ’3 (A-9)

e , the energy parameter for polar pairs of molecules is

e 2e = e 1 - .n + n(n+l) (A-10)

where

£ 1 = TC (.74 8 + .91w - .4n/(2+20w)) (A-ll)

Tq is the critical temperature and n the association parameter.

n = (16 + 400w)/(10 + 400w) (A-12)

C = y 4/(C.ei.o6.Tc .5.723xl0-8) (A-13)

where y is the dipole moment and

C = 2.882 - 1. 882w/'(0.03 + w) (A-14)

a = (2.44 - w)(Tc/Pc )i/3 (A-15)

a is the molecular-size parameter for non-polar pairs. bQ , the equivalent hard-sphere volume of molecules is given by

bQ = 1.2618 a' (A-16)

where a', the molecular-size parameter for pure polar andassociating pairs is

a'3 = a 3(l + 3 c/(10 + 400w)) (A-17)

Bfree-polar' i°r interactions between like molecules, is given by equation (A-18).

Bf ree-polar =’ boU* ’ ( . 75-3/T* '+2.1/T* ' 2 + 2 .1 /T*'3)(A-18)

186

where bQ and T*' are given by equations (A-16) and (A-8),

respectively, y*'* the polar-reduced dipole moment is related to the reduced dipole moment by the following

y*' = y* - -25 y* ^= 0 .25>U*>.04= y* .04>y*>0 (A-19)

y* is given by

y* = 7243.8 y2/ea' (A-20)

e and cr' are given by equations (A-10) and (A-17), respectively. The sum of Bm e t a s t a b l e and Bb ou nd is given by

Bmetastable + Bbound = ^o^ exp[AHc/T] (A—21)where

A = .3 - .05 y* (A-22)AH = 1.99 + .2 y*2 (A-23)

y* is given by equation (A-20).

Bchem is given by

Bchem = b0exp(n (D-4.27))(l-exp(1500n/T)) (A-24)where

D = 650/(e + 300) (A-2 5)

Values of T^f PC' n ' R ' • an(i V or solvents used in this study are given in Table (A-2).

To evaluate B^j lor interactions between unlike polar molecules, the following mixing rules must be used in equations (A-

6), (A-7), (A-18), (A-21), and (A-24):

18?

TABLE A-2Constants Used to Evaluate the Pure-Component Second Virial

Coefficients B in Equation (A-6)

Ref: (Fredensluna, et al. , 1977)

Solvent TC°K Pc(atm) R'(&) y (D) JQ

Water 647. 3 218.3 0. 615 1.83 1.70

Methanol 512.6 78.5 1.536 1.66 1.63Ethanol 516.2 63.0 2.250 1.69 1.40n-Propanol 536. 7 51.0 2.736 1.68 1.40

Isopropanol 508. 3 47.0 2.726 1.66 1.32

188

e i j - 0 . 7 ( e i e j ) + . 6 ( l / e i + 1 / e j ) (A-2 6)

(A- 2 7)

w i j = . 5 ( w i + W j )

V*ij* - 7243.8 yiyj/teijo'ij)

(A-2 8)

(A-29)

The ej / ai/ °j f wj_, and Wj, are calculated from equations (A-10), (A-l7) and (A-9), respectively, n, in equation (A-24)

is replaced by the solvation parameter for unlike inter

actions. The values of riij are found in Table (A-3).

Once the pure-component and the cross second virial coefficients are evaluated from equation (A-6) , they are substituted into equation (A-4) to obtain Bm ix . The virial co

efficients and Bmix are then substituted into equation (A-5) to obtain the vapor phase fugacity coefficients for each conden

sable component of the mixture.

The pure component virial coefficients calculated above are

also used to evaluate the fugacity coefficients, $is, at saturation. 4>is is evaluated at the system temperature and saturated

vapor pressure, P^sat, of component i and is given by

Pj_sat given by equation (A-l).

189

TABLE A-3Constants Used to Evaluate the Cross Second Virial

Coefficients, in Equation (A-6)

Ref: (Fredenslund, et al., 1977)

Solvent Mixture Uii

Water-Methanol 1.4Water-Ethanol 1.7Water-Isopropanol 1.55

Methanol-Ethanol 1.63Isopropanol-n-Propanol 1.50Water-n-Propanol 1.55

APPENDIX B

190

191

Densities of Pure Solvents Used in this Study

TABLE B—1

SolventWaterMethanolEthanol

n-PropanolIsopropanol

Temperature Range0-100°C

25-50°C

0-40°C0-30°C0-30°C

ReferencePerry (1973)Mikhail & Kimel (1961)

Perry (1973)Perry (1973)Perry (1973)

192

2. Estimation of the Densities of Pure & Mixed Solvents

The correlation for the prediction of saturated densities of liquids and their mixtures developed by Hankinson and Thomson

(1979) is used in this study. The correlation is a corresponding states equation which requires the reduced temperature, acen

tric factor, and a characteristic volume for each pure compound comprising the mixed solvent. Six combinations of mixing rules

are presented to evaluate the pseudocritical constants of the mixture. The model is applicable over the reduced temperature

range 0.25<TR<0.98.

For a pure or mixed solvent, the saturated liquid volume, Vg, is given by

Vg = V*VR (°) U-WgRKVR ( 6> ] ( B-l)

where

VR (°) = 1 + a(l-TR )1/3 + b (1-Tr )2/3 + c(l-TR )

+ d(l-TR )4/3 (B-2)

and

VH <«> = [e + f.TR + g.TR2 + h.TR3]/(TR-l.00001) (B-3)

The parameters for equations (B-2) and (B-3) are given in Tabie B-2.

V*, the only adjustable parameter, is the characteristic volume specific for each pure compound and WgRR is the acentric

factor determined from the Soave equation of state.

TABLE B-2 Parameters for Equations (B

a. -1.52816b. 1.43907c. -0.81446

d. 0.190454e. -0.296123f. 0.386914

g- -0.0427258h. -0.0480645

2) and (B-3)

19^

Table B-3 presents the values of Wsrk' • an< f°r *-he solvents used in this study.

For the estimation of the densities of mixed solvents, V*

and W srk of equation (B-l) are replaced by Vm*, the characteristic volume of the mixed solvent, and Wm , the acentric factor

for the mixed solvent. Six combinations of mixing rules may be used to evaluate Tcm, the pseudocritical temperature of the

mixed solvent, and Wm . Vm* and the six combinations of mixing

rules, Aa, Ab, Ba, Bb, Ca, and Cb, are given by the following

equations:

vm* = T< SxiVi* + 3(ZxiVi*2/3)(2xiV i*1/3)) (B-4)i i i

A. Tcm = ZEx-[X jV-[ j*Tci j/Vm* (B-5)i j

Vij*Tcij = (V-[*TciV* jTc j )1/2 ( B-6 )

Tcm — ^xiVi Tcj^/2^x^Vj* (B 7)i

C. Tcm = [ExiV i*(Tci)1/2]2/(ZxiV i*)2 (B-8)i i

a. Wm = 2xjWSRKi (B-9)i

b. Wm = 2xiV i*WSRKi/ZxiV i* (B-10)i i

The six combinations of mixing rules were evaluated for the

methanol-water system at 25 and 50°C, the ethanol-water system

at 25°C, the n-propanol-water system at 30°C, and the isopropanol-water system at 30°C, using equations (B-l) to (B-10). The reduced temperature in equations (B-2) and (B-3) is given by

195

Solvent

WaterMethanolEthanoln-PropanolIsopropanol

TABLE B-3 Values of Wgj^, V*r and Tc

WSRK v*(li^er)mole 1 TC°K Reference

-0.65445 0.04357 647. 30.5536 0.1198 512.6

0.6378 0.1752 516. 2

0.6249 0.2305 536. 7

0.6637 0.2313 508. 3

196

Tr = T/Tcm (B—11)

The liquid volumes were converted to density, d, using the

relationship

d = E^x^MWi/(Vs•1000) grams/cc (B-12)

where MWj[ is the pure component molecular weight. The results

are shown in Tables B-4 and B-5.

Combination Aa is used in this study although combinations Ba and Ca give comparable results.

197

TABLE B-4Results of the Evaluation of Mixing Rules for the

Methanol-Water System at 25 and 50°C(The data are from Mikhail and Kimel (1961))

Mixing Rule Combination Average Percent Error in Density

25°C 50°C

Aa 2.16 2. 50

Ab 2.48 1.59Ba 2.23 2. 58Bb 2. 40 1. 50Ca 2. 28 2.64Cb 2. 33 1. 43

198

TABLE B-5Results of the Evaluation of Mixing Rules for the

Ethanol-Water, n-Propanol-Water, and Isopropanol-Water Systems(The data are from Perry (1973))

Mixing Rule Average Percent Error in DensityCombination EtOH/H2O f 25°C n-Prop/H20, 30°C iso-Prop/H2Q

Aa 1.77 1.12 1.24

Ab 5.15 6.93 6.61

Ba 1. 72 0.729 1.034

Bb 5.27 7.45 6. 92

Ca 1.76 0.763 1.093

Cb 5.21 7.40 6. 84

199

3. Estimation of the Change in Density of Mixed Solvents with Composition

Differentiation of equation (B-12) with respect to the mole fraction of each component of the mixture gives the change in density of the mixed solvent system with composition. All

derivatives presented here are with respect to component 1 for a k component system.

From equation (B-12)

entiation of equations (B-2)-(B-4) and (B-10) with respect to

xi.

1000 V (B-13)

S-g— is obtained from equation (B-l)

3 Xy3VS

(B-l4)

9VR (0) 3Vr (6) 3Vm* 3Wmand -r— are obtained through differ-

1 .00001)

1.00001) (B-16)

(B-15)

200

1 1 ^ - = i ( V l * + + 3 ( V i * 2 / 3 + . ^ V j * 273 ^ M Z i X i V i * 1 7 3 )

+ 3( EiXiVi*2/3) (Vi* 1 / 3 + 2 Vi* 1 / 3 (B-17)j_ l J dX^

awm axi= WSRKx + WsRK-j 3^ ( B—18)

3Tr is determined from equations (B-8 ) and (B-ll).

3TR Tr 3Tcin3X1 ^cm ^X 1

(B-19)

3 Tcm 3x-i 3V *^ = l/vm.t2ZIx1V lj*Tolj - Tcm 3^ - ] (B-20)

The change in density with respect to the number of moles of component j in the mixed solvent is

3d _ 3(d) 3X13nj 3x^ 3nj

where

(B-21)

3xl 1TZT ~ TT (1 - xl) (B-22a)3 n^ nij x

and3xl Xl3n j n<p for j^l (B-22b)

APPENDIX C

201

202

1. Dielectric Constants of Pure and Mixed Solvents Used in this Study

TABLE C—1

Solvent Temperature Range ReferenceWater 0-350°C Bradley & Pitzer (1979)Methanol/Water 5-55°C Albright & Gosting (1946)Ethanol/Water 20-80°C Akerlof (1932)n-Propanol/Water 20-80°C

Isopropanol/Water 20-80°C

203

2A. Correlation of Dielectric Constant Data for Binary Alcohol-Water Systems When Data are Available

The constants of the equation,

D = AeBt (C-l)

where t is in °C, were determined from the data of Table C-l at a fixed alcohol composition. The values of A and B thus obtained were fit to a fifth-order polynomial where

A = A(1) + A(2)x 4 ' + A(3)x4 '2 + A(4)x4 '3 + A(5)x4 '4 + A(6)x4 '5(C-2)

and

B = B (1) + B(2)x4 ' + B(3)x4 '2 + B(4)x4 '3 + B(5)x4 '4 + B(6)x4 '5

(C-3)

x4 ' is the mole fraction of the alcohol on a salt-free basis.

The values of A(l) through A(6) and B(l) through B(6) are

shown in Tables C-2 and C-3 for the alcohol-water systems of this s tudy.

Values

of A(1)

through A

(6) for

Use

in Equations

(C—1) a

nd (C-2)

VO oovO \— x C'-N CM CO in in

VO • • • •N_X LO •=r 43- X< on cr> C\J cn

i i

onon CTn on X/-N o- CO x X

LH • • • •’W VO oo -=r CO.< LO *— CM

1CMI

X

C— on X)VO ^r X X<■> LO co o X• • • •o CO X

< on 0- on T—i l X1

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oo“N vO CM CM XCM oo CT\ on CMw • • • •< r C*— o •=r

C— on Xi CM

iCM1

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oo CMCM DCo DC S

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Q) 4-3 1 ms w G M

00CJV 0- vO 1—O Cr> O o

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o o o o

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PQ • • • •o o o o

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o O o o-aPQ<H

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1 1

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o o o oN PQ • •T™* o o o o'w' 1 1 1 1PQ<HO o

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206

2B. Calculation of the Change in Dielectric Constant with Composition of Binary Solvents When Data are Available

Differentiation of equation (C-l) with respect to the mole fraction of each component of the mixture gives the change in

dielectric constant of the binary solvent system with respect to composition.

3Dr = eBt[ 9A, + At 9Bv] (C-4)8x4 3x4 8x4

where from equation (C-2)

— ~ r = A(2) + 2 »A(3)x4 ' + 3.A(4)x4 ' 2 + 4.A(5)x4 '3 + 5.A(6)x4 '4 8x4

(C-5)and from equation (C-3)

— = B(2) + 2.B(3)x4 ' + 3 »B(4)x4 '2 + 4.B(5)x4 '3 + 5.B(6)x4 '4 8x4

(C-6)

The change in dielectric constant with respect to the number of moles of alcohol or water is given by

3D 3D 3X4 '3n4 8x4 3n4 (C-7a)

and3D _ 3D 3*4 (c_7b)3n3' 8x4' dnj'

where3x '3n44r = x3 ' /nip ' ( C-7c)

20 7

3x 4 ' x 4 ‘r— V = - -2-t (C-7d)d n3 nip

X 3 ' is the mole-fraction of water on a salt-free basis.

208

3A. Estimation of Dielectric Constant Data for Multicomponent Systems When Data as a Function of Composition are Unavailable

The mixture dielectric constant data for the methanol-

ethanol and isopropanol-n-propanol systems were estimated as

suming that the dielectric constant of the mixture is given by

D 3 and D 4 are the dielectric constant of the pure solvents at a

fixed temperature.

D 3 and D 4 are calculated from equations (C-l), (C-2), and

(C-3) with X 4 ' set equal to unity. The values of A(l) through

A ( 6 ) and B(l) through B ( 6 ) are given in Tables (C-2) and (C-3).

The change in dielectric constant with composition is ob

tained through differentiation of equation (C-8 ) with respect to

X 3 ' and X 4 '.

D = D 3 X 3 + D 4 X 4 (C-8 )

3D , = D 3 + D 4 8x 4 7 8 x 3 (C - 9 a )

^ — D 3 8 x 3 7 8 x 4 + D 4 (C-9b)

where

d x 3 (C-9c)

The change in the dielectric constant with respect to the number

of moles of components 3 and 4 are given by equations (C-7a)

through (C-7d).

APPENDIX D

209

210

Derivation of the Gibbs Free Energy Expression of the Pitzer (Coulombic) Term Psed to Obtain the Activity Coefficients of the Salt and the Solvents

The coulombic contribution to the activity coefficient of

the salt or of the solvent is evaluated using the equation for

the osmotic coefficient developed by Pitzer (1977).

j. -WLK , ,2TTa3 , 7TaW2L 2 ix

w = ZiZi2 Ci/Vc (D-3)

k 2 =■. 4ttlwVc (D-4)

a is the ion-size parameter; e, the electronic charge; D, the

solvent dielectric constant; k, the Boltzmann constant; T, the

temperature of the system; V/ the total number of ions; zj_, the

ionic charge; ej_, the concentration of ion i; and c, the total

ionic concentration, c is converted to molarity in this study

since the units of c given by Pitzer are ions/cc.

c = cN/1000 (D - 5 )

Expansion of equation (D-3) gives

z+ 2v+ + z- 2v- xw = ----- (D-ba)

where v+ and v_ are the number of positive and negative ions,

respectively. From the principle of electroneutrality

211

|v+z+| = iv-z_| (D-6b)

equation (D-6a) becomes

w = ( |v_z_|z+ + |v+ z+ |z_)/v

| Z _ |_ Z _ | ( v+ + v _ )

= |z+ z_|v (D - 6 c )

since v+ + v_ = v.

The ionic strength on a molar basis is defined

Ic = cwv/2 (D - 7 )

Substitution of equations (D-5), (D-6c), and (D-7) into equations (D-4) and (D-l) gives the following expressions for «2 an(j

K2 = 87TL J qI q I c (D - 8 )

and. i _ 1 z+ z_ | k ycN , 2ira8 -naw2!,2^ " 1 " 6 (1+Ka) 1(100 ( 3 3 ( 1 + K a ) 2

Equation (D — 9) must be converted to a molal basis since the

experimental osmotic coefficient data are in terms of molality.

With the assumption that concentration is proportional to mo

lality

c = mdD (D-10)

where m is the molality and a0 the density of the pure solvent,

equations (D-7), (D-8), and (D-9) become

Im = l-c/ o (D—11)

212

K2 -• 8 irL(N d 0/ 1000) Im (D-12)

and|z+ z _ | k l . vmd0N 7raw2L 2

♦ - 1 - - 6(1 + <a) + T O O O (2”a3/3 + 3 (1+K a)2 (° - 13)

The coulombic contribution to the activity coefficient of

the solvent for a binary salt-solvent system is easily obtained

from the following relationship

lnys = -(<t>-l) vm(M.W. )/1000 (D-14)

where (M.W.) is the molecular weight of the solvent.

, Pitzer ... .. v,n n n w KlL v2m^d0N OfTa2 , iTaw2L 2 ..inYs (M-W -/1000) (3( 1+ica) 1000 ( 3 3 ( 1 + K a ) ^

(D-15)

Pitzer gives the following expression for the excess Gibbs

free energy of a binary electrolytic solution

_,E Coulombic Lw ^ 1 o(£— ) = “ + i ln( 1+Ka) ) + 2TTa2c/3 (D-16)ckT ^ cl

G e has units of ergs/cc solution.

To convert GE to calories, let

G e = nL gE (D-17a)

where nE has units of total number of moles of solution/cc

solution and gE of calories/total number of moles of solution.

The total number of moles of solution is given by

nT = n solvent + Vnsalt (D-17b)

Multiplying equation (D-16) by cnT/nL gives

,Ge .Coulombic Lwcnip k Lwcnrp 2ira 2c 2nijikT ” 6nL ( 1+Ka) " 6anL ln(1+ a)+ 3 ^

(D-17c)

213

Replacing k by R/N and letting

c = x saltnL (D— 17d)and

n salt = xsaltnT (D-17e)

equation (D-17c) becomes

/-•E w L n sai *. it i§- = --------------------+ - In (1 + Ka))RT 6 1+ka a

+ v 2 (2na3/3) ( ° 0 oo^m n salt (D-17f)From the definition of molality

m = 1000 n sa i■(-/ (M. W. ) nso^vent (D — 1 7 g )

and equation (D-17f) , the excess Gibbs free energy on a molality

basis for binary electrolytic solutions is:

(GE/RT)Coulombic= - ( (M,W;ooo0 lve n t )[<— + i ln(l-HCa))

+v 2 ( 2tt a3/ 3 ) ( J q^ q-) m 2 ] (D-18)

Equation (D-18) may be written in terms of the Debye-Huckel

parameters, and b, given by equations (3-13) and (3-14).

, Ge * Coulombic ) ns olvent, r— 1 { > + r ln(l+bll/2 )](RT> “ ~ 2V ----- 1000--------H l + b l l / 2 b

+ )( 2 ^ d0 (M.W.)n s o xvent m ^ ) (D-19)J 1000^

Equations (D-18) and (D-19) are easily extended to multi-

component solutions containing one salt by replacing (M.W.), the

21k

molecular weight of the solvent, by the molecular weight of the

mixed solvent, given by equation (3-9). a, the ion-size para

meter, is evaluated using equation (3-17). The density and

dielectric constant are given by their values in the salt-free

mixed solvent system. ^solvent is replaced by the total number

of moles of mixed solvent.

Differentiation of equation (D-18) or (D-19) with respect

to the number of moles of solvent i gives the expression for the

Coulombic contribution to.the activity coefficient of the sol

vent (see equation 3-8).

The mean activity coefficient of the salt is obtained using

the following relationship:

i v _ ,aGE/RT vl" Y± - (3T55iT^ (D-20)

Differentiation of equation (D-19) with respect to the

number of moles of salt and substitution of this result into

equation (D-20) gives

v In Y± = ~ v | z+ z_ | + E ln(1 + bll/2) ]

+ (mdoN v 2/'1000) (4Tra3/3 + iraw2L2 } (D-21)° 3(l+bll/2)2

Equation (D-21) is applicable to single and mixed-solvent sys

tems containing one salt.

APPENDIX E

215

216

ERROR ANALYSISDetermination of the Maximum Error in

The osmotic coefficient where

1000 In P/PS* = " v m M.W. (E_1)

and

P = P s - A P (E-2 )

is a function of Ps , ApS and m. M.W. and v are constant. m is

a function of the number of moles of salt, n s , and the number of

moles of solvent, n3 . m is defined by the following:

m = 1000 n s/ ( n 3M.W.) (E-3)

Differentiating equation (E-3) yields

dm = (am/3ns )dns + (am/8n3 )dn3 (E-4)

ordm d w s dw 3 (E- 5 )m w s w 3

w s is the weight of the salt and w 3 is the weight of the solvent.

It is assumed that the errors in weighing the salt, d w s , and the

solvent, d w 3 , are jlO.OOOlg.

Taking the absolute value of equation (E-5) gives

dm = m( 0. 0001) [— + — ] (E-6)w s w 3dm was calculated for each electrolytic solution in this study

and was found to be ±.0. 0001m.

Differentiation or equation (E-l) gives

wherea = P/Ps (E-9)

Differentiation of (E-9) gives

dP dPsda = a[^| - (E-10 )

and taking the absolute value of (E-10) gives

da = a[£| - (E-ll)

P is the error in the measurement of the vapor pressure of the electrolytic solution. ApS is the error in the measurement of the pure solvent. APS is ±0.06 mmHg. AP is ±0.1 mmHg since it is calculated by subtracting Ap from Ps.

Combination of equations (E-8) and (E-ll) and taking theabsolute value gives

d) . 1000 r APS . AP •, ■, n,Acb = rr Am + ■ [— - + — r-J (E-12)T m vmM.W. ps P

For the methanol systems of this study

Ac() = ^( 0.0001) + 15, 60452 [^4 ^ + (E-13)m m ps p

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rH •—i H CN CN CN CO c o r o r o r o

CD 00 r-~ p- P- VD VD VDCN rH rH O o o O O O o O

1 O O O O o o O O O o O1 • • • • • • • • • • •

o O O o o o o o o o o

ro r" o «3< p- CD oin in in CO r-' r~ p* 00 r- 00

1 oo 00 00 00 oo oo 00 oo oo 00 001 • • • • • • • • • • •

o o o o o o o o o o o

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♦d o o rH H rH rH CN CN CN CN CN CN

cn cn r**- vo vo in ro ro CM CM CMo o o o o o o o o o o O O Oo o o o o o o o o o o O O O■ ■ • • • • • • • » • • • •

o o o o o o o o o o o o o o

ou0 <0*■snjin01

o <n CM cn i— 1 in rH cn o O ["*• rH r-o rH 00 o in 00 in cn to ro ro rH r*• • • • • • • • • • • • • • •o o O ro r- o m 00 CM CO vo CM cn1—1 rH i—I rH •—I rH CM CM CM ro in vo

«—I cn r** CM rH vo ■*p CM vO CM in r- 00 O ro• • • • • • • • • • • • • • •rH 00 o o r- CM <T\ in rH VO 00• vo in in in in in in ro ro CM CM rH O'*OV ,

CM CM CM CM CM CM CM CM CM CM CM CM CM CM 1— 1

ro CMin

VD i—I vo CN CM 00 CM VD ro 00O rH CM rH ^P ^p 00 r- 00 o o rH o00 00 00 00 00 00 CO 00 00 00 o> cn cn cn• • • • • • • • • • f • • •

o o o O o o o o o o o o o o

cnIH33

*0c(0

%

W0)3H<0>(04JceM0)aXW

-©4

A<

O'* 00 r- in in <5P roCM CM CM CM rH rH rH O O o o o O o

1 O O O O O O o O O o o o O o1 • • • ■ • • • • • • • • • •

O o O O O O o O O o o o O o

^P CO CM LO CM cn oro ro VD 00 VD cn cn cn O rH rH CM ro

1 00 00 00 00 00 00 00 00 00 <n <n cn cn cn1 • • • • • • • • • • • • • •

O o o o o o o o o o o o o o

o1— 1 o o CM <y\ VD in m • CM CM 00 00 i— i• • • • • • • • • 1— 1 • • • • •

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o cn r*' 00 CM ro in rH VD rH O M 1 r O inVD o o o 00 CM in VD VO CO rH cn cn CM ro oo • • • • • • • • • • • • • • *• o o in VD r- 00 00 o CM P in o CM ino rH rH rH i—I CM CM CM ro

rHO 1— 1 rH VD

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220

Expe

rime

ntal

Values

of Ap

} p?

and

<j> for

the

NaSCN-MeOH

System

at 25

and

40°C

221-&■ cn C*“— m =r on on on on oo*—• o o o o o o o o o oT3 1 o o o o o o o o o o o•fl 1 • • • • • • • • • • •o o o o o o o o o o o

m on LO c— o CM cnLO CM in 30 CM m on on on <o 00

1 00 cn cn O o r— <— CM CM CM•e- 1 • • • • • • • •o o o o r- T— t— T— t—

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bOSBO Cn in t>- =r ■=T ■=r CM CM <o n CO

vO O r— *— cn •=T f*- in <o VO cn on cnO • * • • • • • • • • • •

■ o CM VO O in o O O in cn -=ro c— v— CM CM on in in vo+1Oh<

cn te- te~ vo in LOon CM T— \— o Ct O o o oT3 1 o O o O o o o O o o o+ 1 1 • • • • • • • • • • •o o o o o o o o o o o

on vo VO oo on onLO on cn oo 0— o CO OO cn T- on

1 OO cn cn o o t— r~ s— 00 on on-©1 1 • • ♦ • • • • • • • •o o o 'r_ * T—* ' ' '

bOB3S T— o- on c— r— on cn cn cn oo t— r“a • • • • • • • • • • •T— CO h=3" CM cn CO in CM i>- o co vO• oo C\J CM T— T- t— V— o o o cn cno - r— x— T— \— r— 1— \— *— V—+ 1Oh

hOSB§ j=T m te CM cn in CO CO CM vo vo

lO 1 co rn o CM r— t— cn on cno 1 • • • • • t • • • • •• on in o o o CM in o o c— cn T~o r— *— T- CM CM CM CM m+ 1Oh<

>- -=r ino - r 00 CM m =r on -=T =T n •=r t—O t— *— cn ^r o 0- in -=r oO cn CO vO in ^r c o in in in cn vO• o c— O CM in C*- CM oo cn o ono • • • • • • • • • • • ••y

o o o i— OJ CM CM on on

A P P E N D I X F

222

223

Calculation o£ the Mean Activity Coefficient of the Salt in Singleand Mixed Solvents

Stokes and Robinson (1948) present the following relation

between the observed molal mean activity coefficient, y ± , and the rational (mole-fraction basis) activity coefficient, f±', of the

solvated solute for binary electrolytic solutions. y± and f±' observe the unsymmetric convention.

mlr7± = In f±! - J h+/Z>dln a3 - ln(l-((M.W.)/1000)(h+ - V )m )

0(F-l)

Kj. is the solvation number of the solvated positive ion and is

discussed in Chapters 3 and 5. a3 is the activity of the solvent.In f±' is separated into three component terms:

FLORY-lnf±' = lnf±'(Coulombic) + lnf±'(HUGGINS) + Inf±'(Residual)

(F-2)The Coulombic contribution is evaluated using the equation for

the mean molal activity coefficient presented by Pitzer (1977).

In7±' = - lz+z_fA(j)(2lV(l+bI^) + I/b ln(l+bl%))+ (mdQN 27/1000)(47Ta3/3 + TTaw2L2/3(l+bI^)2)

(D-21)

Equation (D-21) is derived in Appendix D.Since the value of a in equation (D-21) may have a value between

the sum of the crystallographic radii and the sum of the solvated radii, it is assumed that equation (D-21) predicts the rational coefficient of the solvated ions or f±'. This assumption was made

by Stokes and Robinson where lnf±' in equation (F-l) was replaced by

224

the Debye-Huckel equation in the development of their hydration

model.

lnf±' (hUGGINs) i-s readily obtained from equation (3-28) and

the following relationship:

I'lnfi’ = ^ln^' + i"2 lnf 2 ' (F“3)

where f ' is the rational activity coefficient of the positive ion,

(component 1), and f2' is the rational activity coefficient of the

negative ion, (component 2).

V lnf±' ( [toxins ) = ^(IhCJ^'/x-l' + l-C^'/x^)

+ ^(lnC^'/x^ + 1-<&2 ' / k 2 ' (F-4)

Equation (F-4) is normalized to the unsymmetric convention by taking the limit of equation (F-4) as the mole fractions of the positive and

negative ions approach zero (Equation (F-5)) and subtracting this

result from equation (F-4). (Equation (F-6))

lim Vmtt'(^-HUGGINS) _ +x-,'— » 0 4- v. i °# a. 4.C1 ~ * u + 1 2 (In r2' /r3 + 1- r2'*°/r3)x9’-^ 0

(F-5),.FLORY-HUGGINS. , , _ , , , _ t/

V Inf±'(UNSYMMETRIC ) = ^(lnC^'/x^ + 1- C^'/x^)

+ ^(lnCj^'/x-?1 + 1- c£>2*/x21) - ^(ln r^'” /r + 1- r1l~Vr3)

- Z 2(ln r2,0°/r3 + 1- * 2 ' " ° /r3)(F-6)

,oois given by

rl'” rl + V S3 (F‘6a)and

r2,0= r2 (F-6b)

225

since the negative ion is not solvated.

The residual contribution to equation (F-2) is obtained from

equation (3-19) and the relationship of equation (F-3).

i^lnf±(Residual) = Z2 Q (1- ln(®^ + © 2 ^ 2 1 + ® 3 ^ 3 1 ?

®-^/(©^ + ® 2 ^ 2 1 + ®3'tp31^ ~ ® 2 ^ 1 2 ^ ® l ^ P l 2 + ® 2 +® 3 ^ 3 2 ^ ~

@ 3 ^ 1 3/ ® iA±/Qi + © 3 )) + ^ u ( ® i 4 * i 2 + ® 2 +® 3 ^ 3 2 ^ ~

® l ^ p 2 i f ® l + ® 2 ^ 2 1 + ® 3 Jft31* “ ® 2 ^ ® l * P l 2 + ® 2 + ® 3 ^ 3 2 } “

® 3l/;23^®lA±/Ql + ®3> (F~?)

A± is defined by equation (5-4b) and is set equal to zero. The

fifth and tenth terms of equation (F-7) may be combined using

equation (5-4b). Equation (F-7) then reduces to the following:

Z^lnf±(Residual) = Z/ Q^(l- ln(®^ + ® 2 ^ p 2 1 + ® 3 ^ P 31

- ®-^/(®^ ®2^p21 ®3^J^31^ ~ ® 2 ^ 1 2 ^ ® lV^12 ® 2 ® 3 ^ 3 2 ^

+ U2Q2 ( 1 - l n ^ l p ^ + ® 2 + ® 3^ 32)

“ ®iV^21^®l ®2}p21 ®3^3l! ~ ®2^®l}Pl2 ® 2 "*"®3V 32(F-8)

Equation (F-8) is normalized to the unsymmetric convention by

taking the limit of equation (F-8) as the mole fractions of the

positive and negative ions approach zero (equation (F-9)) and subtracting this result from equation (F-8). (Equation (F-10))

limit Z^lnf±(Residual) = Q^(l“

226

^^Unsymmetric) = e<luation O 8) “ equation (F-9)(F-10)

Substitution of equations (D-21), (F-6), and (F-10) into

equation (F-2) allows the calculation of lnf±' in equation (F-l).

lnf±' is therefore given by:

lnf±' = equation (D-21) + (1/I S)(equation (F-6))

+ (l/Z'O(equation (F-10)) (F-ll)

Substitution of equation (F-ll) into (F-l) allows the calculation of

lnY±.The mean activity coefficient of the salt in a single solvent can

be calculated in two ways from equation (F-l). In both methods, ft1

is given by equation (F-ll).

The first method, (method 1), utilizes the experimental values of

the solvent activities, a,, and the parameter values obtained through

the regression of the osmotic coefficient data shown in Tables 5.10,

5.11, 5.13, 5.14, and 5.15.

To evaluate the integral of equation (F-l), the values of h+,

the solvation number of the positive ion given by equation (3-33),

are plotted against lna~. The area under the curve up to a molality

of m is then determined. For computational purposes, h may be fit

to a polynomial in lna3*

h,_ = A(l) + A(2)lna^ + A(3)lna^ + A(4)lna~^ + A(5)lna^

A(6)lna35 + A(7)lna36 (F-12)

Once the constants are determined, equation (F-12) may be integrated to

give:

J " h_,_dlna3 = A(l)lna0 + A(2)lna3 /2 + A(3)lna,^/3 +A(4)lna3V 4 +0 '

227

A(5)lna35/5 + A(6)lna36/6 + A(7)lna37/7 (F-13)

This method was used to calculate the mean activity coefficients

of the LiCl-water and the NaCl-water systems at 25°C. The parameter

values given in Table 5.10 were used to evaluate lnf±'. The solvent

activities were calculated from the osmotic coefficients of these

systems (Robinson and Stokes, 1959) using equation (4-2). The results

are presented in Table F-l and are compared to the values of the mean

activity coefficients given by Robinson and Stokes.In method 2, the values of a3 in equation (F-l) are calculated

from equations (2-1) and(5-6), orlna3 = equation (5-6) + lnx3 (F-14)

where x3 is the mole-fraction of the solvent given by equation (3-26).

f±' is evaluated as in Method 1. Method 2 represents true

prediction since only the parameters obtained through regression of

the osmotic coefficient data are used to predict the mean activity

coefficients of the salt.To evaluate the integral of equation (F-l), h+ is fit to a poly

nomial in lna3, given by equation (F-14). The polynomial expression

of equation (F-12) is used. The constants thus obtained are then used in equation (F-13).

The results for this method are shown in Table F-2 and are

comparable to those of Method 1.

Equations (F-l) and (F-12) through (F-14) can also be used to

obtain the model parameters if mean activity coefficient data only are

available. This was done for the HCl-MeOH and HCl-EtQH systems at 25°C.

No osmotic coefficient or vapor pressure data are available for these

228

TABLE F-1Method 1. Average Percent Errors in y± in Prediction

from Parameter Values at 25°C and ExperimentalSolvent Activities for Aqueous Systems

Average % Maximum RelativeSalt Error, y± % Error, y—LiCl 5.0 -13-1 at 6m

NaCl M .1 -11.3 at 6m

229

MethodTABLE F-2

2. Average Percent Errors in y± in PredictionOnly from Parameter Values at 25°C

for Aqueous SystemsAverage % Maximum Relative

Salt Error, y± % Error, y±LiCl 4.8 -10.4 at 6m

NaCl 4.2 -11.2 at 6m

230

systems.

The values of for HC1 in methanol and HC1 in ethanol as well

as the values of AtvfeoH//H+ anc* AEtOH'/H+ were determined. For the HC1- MeOH system, the values of and the methanol-chloride ion inter

action parameter, A ^ j from Table 5.14 were used since they have already

been established from the methanol data base. The values of A ^ and

the ethanol-chloride ion interaction parameter, A^? from Table 5.15 are used for the HCl-EtOH system.

The results are shown in Table F-3. The values of A ^ obtained through the regression of these systems are meaningless since the

maximum molality range for the HCl-MeOH system is 0.56m and that for the HCl-EtOH system is 0.10m. The Flory-Huggins and residual

contributions are negligible at such low molalities. Any value of

A ^ would give the results of Table F-3. The values of A ^ obtained

here should be used with caution in extrapolation to higher molalities.

In addition, the poor performance of the model in the correlation of

these systems at such low molalities suggests that extrapolation is

not possible.

The expression which relates the rational activity coefficient,

f±', of the solvated solute for a salt in two solvents to the observ

ed molal mean activity coefficient, y±, is developed below.If solvation is not considered, the activity of the salt, a , isO

related to the activities of solvent 1 (a ) and' solvent 2(a ) by

ngdlnag = - n2dlna2 - n^dlna^ (F-15)where ng, n , and n^ are the numbers of moles of the salt, solvent 1,

and solvent 2, respectively.

If solvation is considered, the activity of the solvated salt,

231

TABLE F-3Binary Interaction Parameters for the HCl-MeOH and HCl-EtOH

Systems Obtained Through Regression of y ± Data at 2 5 ° C

(Reference: Harned and Owen (1958))

SystemHCl-MeOH

HCl-EtOH

a

4.03 1 . 2 7

MolalityRange0.560 . 10

A12-253-1 -147.9

A 3 1

0.00.0

Avg. % Max. Rel. A32 Error, y— % Error,

782. 3 926.9

3.9

10.7

7.4 at

-15.7 at

56m

01m

232

ag', is related to the activities of solvent 1 and solvent 2 by ngdlnag' = - n^'dlna^ - n^'dlna^ (F-16)

where n01 and n,’ are the numbers of moles of solvent 1 and solvent 3 42 not involved in solvation of the salt, n ' and n ' are related to

and n^ by the following:

n3* " n3 ‘ h+ h I1s (F'17a)"4 ' “ n4 ' d+^2ns (F-17b)

In equations (F-17a) and (F-17b) solvation of the positive ions alone

is assumed. h+ represents the number of solvent 1 molecules involved

in solvation and d+ the number of solvent 2 molecules involved in

solvation, h, and d are given by equation (3-33).T *TSubstitution of equations (F-15), (F-17a) and (F-17b) into (F-16)

givesdlna 1 = dlna + h M dlnaQ + d Mdlna, (F-18) s s + 1 3 + 1 4

By definition

a = a±^= (fix )^ (F-18a)s sand

a ' = a,±Z/= (f±'x ' ) U (F-18b)s swhere x and x ' are the mole-fractions of the salt on the two bases, s s

Substitution of equations (F-18a) and (F-18b) into equation (F-18)

yieldsdlnfi' = dlnft + h+(l^/V)dlna3 + d+(^/Z^)dlna^ + dln(xg/xg')

(F-19)

dln(x /x ') may be written in the following manner, s sdln(x /x ') = dln(l+ {V- h, -d,)(m(M.W.)/1000))

O S *

- dln(l+^m(M.W.)/1000) (F-19a)where M.W. is the molecular weight of the mixed solvent.

233

If solvation is not assumed, the observed mean activity

coefficient is related to the rational activity coefficent by the

following expression

dlnf± = dlnTi + dln(l+Mn(M.W.)/1000) (F-19b)

Substitution of equations (F-19a) and (F-19b) into equation (F-19)

givesm m m mJ din/ ± = f dlnf±' - ( J / . /U ) f h dlna, - ( U / U ) j d dlna.0 0 1 0 J i 0 4

m- / dln(l+ ( y - h+ -d+)(m(M.W.)/1000)) (F-20)0

Integration of equation (F-20) gives the following expression which

allows the calculation of the observed mean activity coefficient of

a salt in a mixed solvent.m m

In/ ± = lnf±* - { V J v ) / h.dlna, - ( U / U ) J d dlna,X 0 J 1 0 4

- ln(l+ ( y - h+ -d+)(m(M.W.)/1000)) (F-21)

Equation (F-2) is used to evaluate f±'. Equation (D-21) is used

to evaluate the Coulombic contribution to equation (F-2). However,

the density and dielectric constant of the pure solvent are replaced

by the densities and dielectric constants of the mixed solvent. The

molecular weight of the pure solvent is replaced by that of the mixed solvent. The value of a in equation (D-21) is replaced by equation

(3-17).

ft'(F£ £ S S> is giveri by eHuation (F“6)* andcp2' mustbe replaced by their values in the ternary solution, is given by

i R, + d R, o+3 o+4’ - r, + h R, + d R, (F-21a)

and ^2 '"° is given by equation (F-6b).

The residual contribution to equation (F-2) is obtained from

equation (3-19) and the relationship of equation (F-3).

I''Inf ±( Residual) = ln( © 1 + ® 2 ^ 2 1 + ®3^31 + "

- ©-[/(©-l + ©2^21 + ® 3 XI;31 + ® 4V ;41>

- ® 2 l//l2/(©i^i 2 + © 2 + ®3*fe2 + ® k XP k l )

- ® 3 ^ 1 3/(® 1A±/Q1 + ® 3 + © 4 ^ 4 3 )

- © 4 ^ 1 4/ (® j_B±/Q1 + ® 3 34 + ® 4 »

+ ^2^2^" -*-n^®l^i2 + ® 2 + ® 3 ^ 3 2 + ®4$42^

- @ { ^ 2 1 ^ ® ! + ® 2 XtJ21 + ® 3 lp31 + ©4^41

- ®2/(®lV;i2 + ® 2 + ® 3 XI*32 + ©4^42^- @ 2 ^ 2 3 ^ ® + © 3 © 4 ^ 4 3 )

‘ ® 4 V ;24// ®1B±//( 1 + ©3 ^ 3 4 + © 4 ^ (F-22)

A± and B± are equal to zero in this study. The sum of the fifth,

sixth, eleventh, and twelfth.terms of equation (F-22) are equal to

zero from the definitions of A± and B±. Equation (F-22) then reduces

to

V Inf ±( Residual) = Z^Q1 (1- ( ® 1 + © 2 l/ / 2 1 + © 3 ^ 31 + ©4^41^

- <3>1/(®1 + © 2 ^ 2 1 + ©3^31 + ® 4 I/,4p

- @ 2 ^ 1 2 ^ ® 1 ^ 1 2 + ® 2 + ® 3 Xp 3 2 + ®4*/,42^

+ 2^2 ^ ” © 2 ® 3 ^ P 32 ©4^42^

- @ { ^ 2 1 ^ ® ! + © 2 ^ 2 1 + ® 3 XP31 + ® b XlJA l )

- ® 2/(®i^12 + © 2 + ®3*p32 + © 4 ^ 4 2 ^ (F-23)The limit of equation (F-23) as the mole-fractions of the positive

and negative ions approach zero is given by

limit l^lnfi'(Residual) = Z^Q^(1- ^ ( © 3 ^ 3 + © 4 ^ 4 1 )xit-*2

235

+ 2Q2(1- ln(©3tp 32 + © 4^ 42)) (F-24)© 3 = Q3x3" /(Q3x3 + Q4xp (F-24a)® 4 = Q4V 7(Q3x3* + V ? (F"24b)

x3 and x^ are the mole-fractions of solvent 1 and solvent 2 on a

salt-free basis.Subtracting equations (F-24) from equations (F-23) gives the

unsynmetric residual contribution to equation (F-2).

V Inf ±' (unsymmetric^ = eciuation (F“32) “ equation (F-24)(F-25)

Substitution of the ternary forms of equation (D-21), (F-6)

and equation (F-25) into equation (F-2) allows the calculation of

f±' in equation (F-21). lnf±' is given by the following:

lnf±' = equation (D-21) + (1/j/) equation (F-6)

+ (1/V) equation (F-25) (F-26)

Equation (F-21) was used to calculate the mean activity

coefficients of HC1 for the HCl-MeOH-lÔ and HCl-EtOH-tÔ systems

at 25°C. The solvent-ion parameters of Tables 5.10 and F-3 and the

solvent-solvent interaction parameters of Table 5.21 were used in

the prediction of the mean activity coefficients.

The results are presented in Table F-4. The maximum molalities

and the salt-free mole fractions of the nonaqueous solvents at which

the data were available are also indicated.For the HCl-MeOH-FÔ system, the average percent errors in "Y±

increase with increasing methanol mole fraction. The calculation of y± does not appear to be affected by changes in molality.

For the HCl-EtOH-Ô systems, increasing the salt-free mole

fraction of ethanol from 0.0427 to 0.0891 does not affect the

236

TABLE F - H

Prediction of the Mean Activity Coefficients of HC1 in Me0H-H20and Et0H-H20 Mixtures at 25°C

(Reference: Harned and Owen (1958))

System in'Maximum

MolalityAverage % Error, y±

Maximum % Error, y±

HCl-MeOH-H20 0.0588 2.0 3-6 -4.7 at m=0.001

0. 1233 2.0 20.2 -24.9 at m=2.0

0.0588+0.1233 2.0 11.9 16.6 at ra=2.0

HCl-EtOH-H20 0.0417 2.0 8.4 16.6 at m=2.0

0.0891 2. 1 6.0 11.2 at m=2.1

0.50 2.5 464.0

0.0417 2. 1 7.2 11.2 at m=2.1

237

calculated values of the average percent errors in r± . The

calculation of y± is affected by molality. The relative percent errors in y± increase with increasing molality and are greatest at m = 2.0.

The model fails to predict the mean activity coefficients for the HCl-EtOH-t^O at a salt-free mole fraction of 0.5 and over the

available molalilty range. This is to be expected since the binary model, which was used to obtain the ethanol interaction parameters,

poorly correlated the HCl-EtOH data.It is recommended that the methanol-H+ and ethanol-H+ inter

action parameters be used to predict ternary systems only at low mole-fractions of methanol or ethanol.

A P P E N D I X G

238

239

This appendix contains the following programsi

G .1 Binary Program for Salt-Solvent Systems

This program allows the correlation or prediction of binary electrolytic solution data. This program calls subroutine LSQ2 which in turn calls subroutine PN. The data may be in the form of osmotic coefficient or vapor pressure depression data as a function of molality.

G.2 Binary Salt-Free ProgramThis program allows the correlation or pre

diction of mixed-solvent binary data. Subroutines LSQ2, HANK, and PHIB are called. The salted-in component is termed solvent (1 ).

G.3 Ternary Program for Salt-Binary Mixed-Solvent Systems.

This program allows the prediction of ternary electrolytic solutions containing a single salt using the binary parameters obtained from programs G.l and G.2, The values of for the two solvents must be supplied to the program.

Subroutines HANK, PHIB, and HANKI are called by the program. The parameters AB(1)-AB(1*0 were determined from a fit of the mixed solvent dielectric constant data to a polynomial using FITIT.(See Appendix C for the constants.)

The salted-in component is Solvent (1) and is referred to by subscript 3 .

240

G . Subroutine LSQ2This subroutine uses a search technique to find

the optimum values of the variables which minimize the objective function. Subroutine. FN of the binary programs is called.

G .5 Subroutine HANKThis subroutine calculates the pure-component

liquid molar volumes using the Hankinson and Thomson (1979) correlation. (See Appendix B.)

G .6 Subroutine PHIBThis subroutine calculates the fugacity coeffi

cients at saturation and of the vapor phase using the Hayden-O-Connell correlation (1975)« Subroutine SVIR is called for the prediction of the pure component and cross second virial coefficients. (See Appendix A.)

G .7 Subroutine HANKIThis subroutine calculates the densities of

the salt-free mixed solvents and the changes in density with respect to composition. (See Appendix B.)

G.8 Program FITITThis program calls subroutine POLIFI which

makes a least-squares fit of the data. POLIFI calls subroutine DETERM which performs the error analysis for POLIFI. A polynomial of order 6 can be described by FITIT.

G.1 Binary Program for Salt-Solvent Systems

l, OHM ON A P H I P ( H r ' i ( ' ) r A P H I r f it.: f <> ) 7 P C A I.. ( 9 ^ 1 0 ) ; H O N v X H .1. ( 0 / <> 0 > 7 X H 2 ( 9 7 <1') ) r N O P T ( B ? .'< )

R X P v B ) r R ( B r 4 0 ) t (I ( H i - ' i i 1) f ( B r ' K O r ) ! B ( B 7 ' i ( • ) ? GP. XP ( B r 4 0 ) r VD C O MMO N X P ( B 7 - ) 0 ) » : < N ( 9 7 « ) ' ) ) i N I M D •> H B Y S 7 R J. ( 0 ) i P 2 P < {■)} 7 HOP ( .1.0)CO MMO N N P P 7 NP ( B ) r F 7. P ( 9 ) 7 F Z N ( B ) 7 P NP ( B ) r F MH ( 9 ) 7 V K ( t<) j T ( B ) r I A ( B ) v

C 0 I E I..!;. C ( B ) 7 0 0 ( 9 ) 7 A MW ( 9 ) y P S M ( 0 ) 7 jr. J! 7 A A A ( 0 > :■ 0 B B ( 8 ? < ) ' i ) 7 A P H C 1 P P ( B ) r P N ( 9 ) 7 (!.) ( B ) r H Ji ( B ) : R 2 7 f t'Ax ( 9 ) 7 l i J C M B )

D I M E N S I UN AN B ( B ) 7 IK PH I ( 9 7 <) ' ) ) ; E E R O O ( B 7 - ) ' ) ) 7 E P R O P ( B 7 <>')) 7 P O P ( 9 7 <10 )d i m k n b :r 0 n ).i >: ( :i 2 ) 7 y ( x 2 > 7 >: ( y 7 ,i. c ) ? :>: I ( x 2 ) ? d p p: (9 7 4 <> > r p ■< % t # %. % £ % % % % Z % £ % % % % %%%%%%%%

B I N A R Y P R O O R A M

T H E B I N A R Y BA I A A R P '►Ji ) R E G R E S S E D H O P I M P B I N A R Y P A R A M E T E R S W I T H N I N B ■■■ X 2 ) P R I K B I C T l K B U S I N G T H E B I N A R Y P A R A M E T E R S W I )' H M l MB ^ 2

.*. (: -j: X Y >' Y Y Y T Y (< Y Y Y (< Y Y Y Y Y Y Y Y Y YY Y Y >X Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y % % Y I N T E G E R F I P . • F ENN 0 P I v X F O R M 0 1. A I.. I I Y V B , B P. I.. I A Pn o p 1 - 2 p i he n b a r a a r e h oi . .ai . . i r r v s > p i: h iNO r i f A I B N O , OP BA I A S E T 1 0 BP UBK.1)F 7. P I B I'lllK C H A R OF ON 1'HlK P O B I I I OF I O NF / . N I B 1 HP. ( . HA ROM ON I IIP. NH. OAI I BP . I ON ( A B B , OAI . . IJP. )F K I B I M F BUM OF VHlK P O B I )' I OF A N B N F O A l ' I B F I O N B A N D

P I A P N P + F HMF N P I B I MP BOM OP T H U NI JHBK. R OP P O B I T I M H I O N BF Mi i I B VHPJ BOM O F I N F N F O A I ' I V F I O N Sr I B I HP. I P.HP P. RA' ( OR P. OP T PIP. B Y B ' I M HAHW I B T H F MOI. .F.C.JI . .AR O F I OH I' OF ) H F BO I.. OF N TPB. I i I B IMP. P O R P. BO I.. OH. NT O A P O R P R P B B U R EI H B P'" I. B A ' I A ARP HOI.. AP. I f Y OB : BP. P R MBS I ON I N O A P O R P R E S S U R E I M B P‘ 2 BA P A A R E HOI . A I.. I T Y O S . O A P O R P R F B B U R E X MI ) P. I B I HP. HO LAP. I ( Y OP f H H B A I . I B O U J f l O N X MOL J. I S T H E M O L A L I T Y OF 1 H E B 0 I . . U 1 I 0 N A C C O U N V I N O F U R

BO P. OAT I O NX S I B I H P. Pi 0 P. P. P R A C f I 0 N 0 P I I I P. B 0 1.. 0 P N 7 A B BOH I N 0

C O M P L E T E D I S S O C I A T I O N OF T HE BAI . .T XF' I B 7 HP. Pi 01.. P. P R A C I I O N OP I I IP P O S I T I V E . (Oi l ABBl JPi l NO

C O M P P . E l ' E D I S S O C I A T I O N OF T HF S A L T XN I B ( HP H O L E P R A C I I O N OP ' IHK. NE O A' I I V E I O N A . BB UP i l N B

C O M P P . E l ' E D I S S O C I A T I O N OF T HE S A L TX HX I B ( H E H O L E P R A C I I O N OP I I I P BOP. VA' I EH P O S I T I V E I O NX PI 2 I B (PIE H O P E P R A C I I O N OP I HP B O I . . V A I E D N E O A ' I I VP. I O NX T ' ) I S T HE MOI. .E F R A C T I O N OF T HE S O I . . V E N T ON A S O L V A T E D B A S I SA P P I I I B T H E E X P E R I M E N T A L O S M O T I C C O E F F I C I E N T A P I I I P I I B T H E C A L C U L A T E D O S M O T O C C O E F F I C I E N T P I B T H E E X P E R I M E N T A L V A P O R P R P B B U R E P C AP. I B T HE CAP. CUP. A T E D V A P O R P R E S S U R E :G B E X P I B ( H E E X P E R I M E N T A L S O L V E N T A C T I V I T Y C O E F F I C I E N T GBR I B T H E CAP. CUP AT E D S O L V E N T A C T I V I T Y C O E F F I C I E N T

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24-3

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GO TO 9 AO 9 0 3 DO a 3 9 2 X - J. ? N P P

R E A D 2 j XPiOI.. ( J J r X ) : A P H X ( J J r X )1 3 V 2 C O N T I N U E 9 1 ,i. DU 9 ::J2 X X r N P P

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9 3 2 C O N T I N U E 9 1 (> DO '■;('<'<• X". i . r N P P

A P - F N P ( J J ) » : < r i O I . . < J J ? I )AM I Ni l ( J J ) fc XPiOI . ( J J ? X )XB ( J J ? X ) - A N B ( J J ) / ( AMO C J J ) P A P P A N )X P < J J f X ) - A P / ( A N B ( J J ) + A P + A N )X H ( J J ? X ) ‘" A M / C A N B ( J J ) P A R I A N )1F ( N O P ( < J J r X ..I) . H 0 , 3 ) CO I 0 4 ( ' ( ' 0G B E X P ( J J ? X ) -I" ' ( J J ? X ) / ( XB ( J J ? X ) t P B i'1 ( J J ) >X F ( NOP' f < J J r X J ) . I! 0 t 2 ) CO 1 0 4 0 0 0A P H X ( J J ? X ) =:-• .1. 0 0 0 . #AI . . OO ( P ( J J ? X > / P B M C J J ) ) / ( F K ( J J ) * X H O L < J J ? I )

( ; * A M W ( J J ) )- 4 0 0 0 C O N T I N U E5 ( ) ( ' ( ' CON ( X NO P.4 1 . 0 C O N T I N U E

i: r - 2J! P < N X ND < E C i 2 ) CO ( 0 X 6 3 I K ( N I N D . E U > A ) 0 0 TO A 0 1

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6 0 A H = 7M l - P i H M 3 - H P 3

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P R Y N T 5 2 2 f- O R M A Y ( •' - ' )DO 7.1 .5 K J " 1 7 M H Y 8

N P P : rN P ( K J )I F ( H Y M D , EU , 2 > Y j: - 2I. F ( N Y H D , K 0 , 1 , A H » , H 0 P I ( K . I 7 j Y ) , H. 0 < 2 ) P R Y H I i 4 6

I F ( N Y N J) , E U , Y , A M D , H 0 P T ( K J 7 Y Y ) , E U , Y ) P R Y N Y Y 4 8Y OR MAY OB HO I Y C RH.C ' )F O R M A D E L T A P R F C ' )P R Y H I 5 <-3 ('7 A A A ( K J )F O RMA V ( ' - ' 7 ' A A A ( K J ) » ' 7 F Y 0 , 5 )P R Y N T 5 0 9 9I- O R M A Y ( / / 7 1 X r 2 H R P 7 Y OX 7 7 H R M 7 Y OX 7 2 H R 3 )P R Y M T 5 9 9 0 7 R P ( K J ) 7 R N ( K J ) ? R 3 ( K J )

O R M A I ( / 7 3H ! (■« 5 )P R Y M I' 5 8 9 1f: 0 RM A I ( / / 7 Y >: 7 2 HR 1 7 1 0 X 7 2 H R 2 7 1 0 X 7 2 H 0 1 7 YOX 7 2 H 0 2 7 YOX 7 2 H 0 3 )P R Y M I 5 H 9 2 ? R Y ( K J ) 7 R 2 I K J ) y 0 J. ( K J ) 7 U 2 ( K J ) 7 U 3 ( K J ) h O R M A I ( / 7 5H A <•. 5 )P R Y N T 3 8 0F O R M A T ( / / 7 Y X 7 3 HA i 2 7 9.X 7 3 H H ( - P 7 9.X 7 3 H A P M 7 9 X 7 3 H A 3 1 ? 9 X 7 3 H H 0 M 7

C 9 X 7 3 M A 3 2 )P R Y H I 3 5 i 7 A 1 2 < K , J ) r H O P < K .1) 7 AP M 7 A 3 i ( K J ) 7 H 0 H 7 A 3 2 < K .J )F O R M A T ( / 7 6 F Y 0 , 2 )COH I YHOTn - ( h y n d i e » . 2 ) 0 0 t o 2 0F O R M A T ( 7 F Y 0 , 5 )I F ( H O P I ( K J 7 Y Y ) , I..h. . 2 ) 0 0 I 0 2 2 4

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245

.1 F ( I I , E d , 3 , A (•! U , N (J F‘ I ( K J 7 I I ) , h'.U < 3 ) (5 0 I 0 3 :i.I F < N O P T < K J 7 I I + 1 > . I!!0 , 0 , A N N , N O P ) ( K J 7 I I ) » F U , 3 ) (50 VO J. 9 2

5 1 B U H 2 " 0 < 0HO 2 9 I - I 7 N P PU t » " ( A B B < ( P C A I < K J 7 I ) - P ( K J f 1 ) ) / P ( K J 7 I ) ) ) * * 2 .

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4 0 2 5 F O R M A T < ' - • ' 7 IO . X 7 ' A V R B E 2 - ' 7 E 1 b ,. 8 )P R I N T 5 0 2 3

5 0 7 3 F O R M A T ( ' - ' » ' X H D P E U P C I P R U P ' )P R I M ) 1 0 0 7 ( X i TOL ( K J v I ) 7 D P R ( K J 7 1 ) 7 D P C ( K J 7 1 ) 7 O P < K J 7 I ) 7 I - I 7 M P P )SUM 1 2 " 0 < ( - 0DO 5 0 7 4 I ■■I 7 M P P» P " A B B ( NPK. < K.J r I ) - U P C < K J r I ) )

5 0 7 4 SUM I. 2 - 9 Oi l 1 2 1 DPA O B P " H U H 1 2 / N P P P R I M ) ' 5 0 7 5 j A U I i P

5 0 7 5 F ORMA' I ( ' - ' 7 l O X v ' A V U P - ' ? K J . 5 . 8 >1 9 2 C O N T I N U E

P R I N T 1 3 21 3 2 F O R M A T < 7 ' XM P H I E X P PF I I CA I . . E R R O R ' )

P R I N ) 1 0 0 7 ( X H O E ( K J 7 t ) 7 A F’H 1 ( K J 7 I ) / AF‘FI I F ’ 1 ( K J 7 I ) 7 F P IT I ( K J 7 I ) 7 I - I. 7 MF*F’ ) I F ( 1 1 . H. 0 , 3 , A H ).T , 0 0 F' 7 ( K J 7 1 1 ) , H. 0 , 3 ) 0 0 7 0 15 2 I F ( NO P T ( i< J y 1 1 + J. ) , F U , 0 , AMU , N O P 1 ( K J 7 I I ) , F U > 3 ) 0 0 TO 2 0 9

5 2 S U i ’l - 0 . 0DO 5 3 1 N P PD E E " ( A B B < A P H 1 P I ( K J 7 I ) - A P H I ( K J 7 I ) ) )D F P - U F P : * * 2 ,

5 6 S O M - B O F i + A B B ( E P H I ( K J 7 I ) )A O R O F - 9 U N / N P P

P R I N T 4 0 2 0 7 A V R U E 4 0 2 0 F O R M A 7 ( ' • • ■ ' r .1 OX 7 ' A 0 R H E RH I..H E R R O R - ' E 1 5 , 9 )2 0 9 C O N T I N U E

I F ( N O P ( ( K J v 1 1 + 1 ) . E C , 0 , ) CO 7 0 5 0 3 3 5 0 3 3 C O N T I N U E

P R I N T '3 0 ( i85 0 0 8 F O R M A T ( ' - ' * ' XT A C OB E X P O BO I . . UCA L E R R O R ' )5 0 0 7 F O R H A T ( ' - ' 7 J . O X 7 ' f W R O E R F C / E R R O R - ' F J. 5 > 8 >

P R I N T i ( ' < * 7 ( X B ( K J j I ) 7 OB E X P ( K J 7 I ) 7 O B B ( K J r I ) 7 EF.'ROO ( K J 7 1 ) 7 : i > l 7 N P P )I F ( 1 1 , F U > 3 , A M U , N O P T ( K J 7 1 1 ) > F 0 > 3 ) 0 0 TO 2 9 1

I F ( N O P T ( K J 7 1 1 + 1 ) < E d < ( 1 , A N D , N O P T ( K J * I I ) , E U , 3 ) 8 0 7 0 5 0 2 02 9 1 S U H O - 0 , 0

DU 5 0 1 0 I - 1 * N P P5 0 1 0 9 1J M 0 - B U M 0 F A B B ( F R R 0 0 ( K J 7 I ) )

A U B - B U F i O / N P PP R I N T 5 0 1 1 7 A V O

5 0 1 1 F O R M A T ( ' - ' 7 l .OX'7 ' A 0 8 FT K 1.1C FLR RO R- ' E 115, 8 )5 0 2 0 C O N T I N U E

P R I N T 5 0 0 55 0 0 5 F O R M A T ( ' - ' 7 ' X H 1 X H 2 X F I3 ' )

21+6

P R A N T A <> 4 r A X H A A K J r A ) r X H 2 A K J r 1 ) f X X A' ( K J r A ) 7 A '■■■■ X r H P P )7 1 ii COM I ' ANtJE

S I OP EM ElS U B R O U l A NR H N A V Y 7 X T )C O M M O N f t . PHAP ( B 7 A A' ) t OP H I A O r 4 A' ) r f t CT A Or <10 ) r U P C A 0 r A ( • ) r ft P H I P li. ( H r ft (>)

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C O M M O N X P A B r 4 0 ) 7 X M A B y 4 0 ) ■> M 1 M )) 7 M B Y B 7 KM. ( B ) 7 R 2 P A B ) 7 H O P A A O )C O M M O N N P P r N P A B ) r K X P ( B ) r l: X H A B ) r P HP ( B ) r F NH 7 h o ( b ) ■> a m w a b > y p b m a b ) 7 x x ? a a a a b )7 o » b a b 7 4 0 > 7 a p mC » R P r x i (.12> r b a i b a a :><•) r s f t r x 2 ( o a - ) s b o a 1 2 a o a m r b o a x a < s o )D A M E M O A O N F H E T A ( B ? 4 0 ) ? V H E T 2 A B ? 4 0 ) , T H E XA ( B j - ' I O ) 7 P H 3 A B 7 4 0 )

C r S U M H ( B r ' i ( ' ) r BUMM'.i ( B r ' l O ) r SUM M'S ( 0 r A ( • ) r I f t U X A B r A 0 ) r OB < B r A <•)C 7 OU A ( By 4 0 ) y 0 0 2 C B 7 4 0 ) 7 X N O U . ( B y 4 0 ) 7 H P ( B 7 4 0 ) 7 HM ( B 7 4 0 ) 7 A A A B 7 4 0 )C r 0 0 A B r 4 A1) r 0 » H B A ( B r 4 0 ) r BRH.B < B 7 A A-)

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256G.3 Ternary Program for Salt-Binary Mixed Solvent Systems.

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I N I E C E P F i. P 7 F Y. N M 5 Y B : : 1

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257

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258

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i oo :< (x ? a > •=:C A L I . !• H ( V ( A > :• T >D O A O A J - ' 2 < H 1XT < J~ A > ->: 'f < J - A ) +D>: < j - A )D O . L O T X •' J. ? H

. l o t >:< x !■ j ) -•>."( < x >

c a l . I . , f m ( y i. j :■ y y. i ;■X T < J - A > " • > : < J " A .- A >

1 0 6 C O N I ' I H U E L . 2 ( . ' = : ( 'F L. (3 - A > 0 ( 3 0 ' f ' 0 l i 0

1 0 9 I.. X 0 - I.. X C f 1I K 0 0 T O T O O

50 YL." A » OF 38Y H " ' - Y L YLO-YH Y X - Y LD O A A < 1 J - A r i'i II I ' - ( Y ( J ) , I . . T > YI I > 0 0 V O A 0 9 1Y 2 - Y Hi a - ' X HY H ' - Y ( J )I H - J( 3 0 T O A 0 V

10 9 A X F ( Y ( J ) > L V •.. Y 9 ) (30 VO A 0 9Y 2 " ’ Y < J )1 2 * : J

1 0 9 XH < Y U ) , 0 1 , Y L ) B 0 T O A A 0:1Y X •••- Y L I X :; X I.1 1 . . - ' J

271

1 .i CM

11.0

6 0

61

6 2

65

.1.1 1

:l a 2J. 1 5 1 2 A

135

.L3B

V I . . - V ( . J )G O T O 1 1 0

1 H ( Y C 2 ) i O f i Y 3 ) G O I 0 1 1 0 Y 5 ‘=Y ( J )1 '.■■■ .)

COM 1 ii M11E I... 2 0 1. 2 C M1 F C L 2 C ( I., f : I.. 2 > CO T O 1 1 1 L 2 C ■ • 0 j J c i ) -• :i. i J J ( 2 ) -• .1. 2

J J C 3 M X 3 D 0 6 K J. 1 7 K 3 J J "-..CM K A )DO 6 1 K 2 ■ :K J. 7 K 3 J 2 " \ . U < K 2 )S « 0 * 0D i.) 5 5 X 7- .1. 7 Mh ” • b i ( ( x v j i ) ~ >; c x ;• ;c h ) ) t c x c x ? j 2 > ••• >: c 1 :■ 1 h ) >

0 C K 1 7 K 2 ) " : S D A C 1 1 1 ) {: A c 2 v 2 ) - 0 ( 1 :• 2 ) if if. 2 G O I ' O ( A 2 7 6 J. ) y X 4

1< 1 :;i A ( A r 1 ) if. A ( 2 r 3 ) — A ( A r 2 ) if 0 ( A r A )1 1 C A ( A 7 A ) >£■! U > > ) , ' ) ) A C A 7 A ) - .1. , . ) £ - 5.0 = ( C A ( A r A ) if A ( ' , !•: 3 - A ( A. :• 3 ) if Y 2 ) >!< D - D A Y D A ) / C <

A F C D . F ( . ! > ' ) > • ) > 0 0 T O 6 5 11 < D < I..H. , 0 , 0 ) D SB ( Vi)D- - ' C D / A , > ) ) if T (3 I H C D ■ I . . T < K. A G O T O 6 5 F I . O ' " A > 0 G O 1 0 A A A

X F ( F I . . G , I . . T > 0 > 0 ) 0 0 V O 1 0 0 F I . . O - A , 0

D O A. I . 5 X " A ? M X I C .1. ) " < > { 0 D O A 1 2 . 1 - 1 7 i-i 1I F C .1 , N H . . A H I ) X T ( A ) ' ' X T C I M X < X s . J )COM 1' X NO E

X T C I ( 3 < O Y X ' f ( X. ) + X C X v 1 2 ) - X C X 7 X L ) ) X H . M - ) C A I . . I . . F M ( Y T v X 1 >

I F C X T , CL : Y2 ) GO 10 16 /I R C • M A H ' lI F C YT , CH. . VI . ) CO TO A 6 0Y ) 1 :- YTD O 1 3 5 X : r A r H

:< r c x > 1. , 5 * : < v c o - o , 5 * : < c x ? x h :•C A I . I . H M C Y M X M I F C Y V > l . . t : . Y L . > 0 0 1 0 1 6 0 D O A 3 B X ; - A r IT

x c x r x h ) *••• c 2 ,0 a. x t •: x ) + >: c x 7 x h > > / 3.0Y C X H ) ' " Y I T

C A r A ) if 7 ( C ' )

; c x 7.1 hi :>

2?2

80 10 10 81 2 0 DO 12 2 I ■■ J. 7 l-i.1 2 2 >: ( i j x 11) «■

y< x h > ■■■< tGO TO 1 0 8

XT ( I

i 4 / :«h i : ho - 1i h ( : u - ic , i:.u , 0 > GOI K ( Y V :.0 t : , YKDGO

d o I a h :c 1. :• i'ix s -=:X T ( I ) -X C . r h .)

1 AH :< ( X 7 X H > •=:<s.1.73 DO 17 2 X "A r M:L 7 2 :< r ( x > •=•••> >7 '5 % :< ( X

CAI. I. KN< YT r X T )i k ( y r > o r >Y l i > 0 0Y :GO 10 A 0 8

■f ( I. >

.1 8 0 DO 1 0 5 2 '- I K < J i FO ■ DO 1 0 2 :o-

1 7 H 1 1 ’>00 1. 7 i-i

r

x i ( x >=-<>:< X 7.1) i.X1 8 2 :< ( X 7 J ) HI T ( I >

Cnl. 1. F I K Y ( J > 7XT185 CONTIN UE

GO TO 1 0 03 0') X H O 7. Y-il 1

IK ( I ' i , OH , 3 ) B 0 ( 0S--0 >0DO 3 0 2 X - 1 r MX( X 7 H i 2 > ": :< < 1 7 I HX ( 1 r I'i i 3 ) >; <; ;i; y I H

3 0 2 8 -B i . ( ( X 7 i'i H2 ) # **>3 0 3 B - B O R T ( B )

I F ( B < K.(l , 0 , (■ ) B : r 13 0 2 I J ( 2 7 H i 2 ) / S

X ( 2 r li f 7 ) X ( 1 7 Hi IX ( 1 7 i'i 4-2 ) •” IJB = >; ( 1 i I'i i 2 >#>:< 1 7DU 3-) 8 I 1 7 h

3 0 5 >:< x r i'i i 2 > "•>: ( x .- H i3 0 A DO 3 0 / I ” 1 7 H3 0 7 x t ( x ) =••>: ( x 7 I H ) I X

COI.. I. F M ( Y V 7 X V ).DO 3 ( '9 I :" 1 7 M

3 09 :< r ( x ) -x< ( i 7 I Kl ) ... /

Col.. L. H I ! (Y T T 7 >: T )I I - ( YT r . 1 K > Y T ) 0 0DO 3 1 1 1 “ ■i. 7 H

( 0 ;■». (»o1 0 .1 .7 3

? j::i> •!•'>, 28* :< r ( i >

r o .i.«o

1) . 1 . 8 5

(I r I. I. i .) /;.> t o

)

3 5 0

; X ( 1 i X L .*) - 7 ( I r it 3 )

< O H . - 5

2 > / S

H i 3 ) i X ( . ? ? H i 2 > S X < 2 r M i 3 )

2) * B

( I r H i 2 )

( A 1 H )

VO 320

273

3i:i >:t <:o o: < :c r xn>+>: < x * h + 2 >Y T '!' Y T

320 Y O i H O V nIH.) 32J. i " h H

32.1 >;:i (;ogo n.) xoe

350 IK) 3 1) 2 ;o :o hX l ' ( I ) C< ( I 7 I I I ) - X ( I 7 11- >x < 1 1 h+2 :«-• v { r v x i i ) - >: (I x 2 >

3 : < c o m + 3 ) c< ( i ? i h ) •••:< ( i 1 3>S: (1 { (1 S 1 -o > 0DO 35 5 X O r M S : B + X T ( I ) * * 2

353 BO-B.i+X( I 7 M + 3 > **2S ■" B U R I ( B ) s:l ::B 11R f ( B O s , oDO 33/ X o U H I K ( B , 0.1 , 0 = 0 ) B - I , 0 6. ••••5XT ( I ) ■■■y.l < 1 ) / B S 2 “:R 2 + X 1(1) *:< ( I ? i'i Y 2 ):lf ":!< **2B J. " B 0 R I ( B J. )DO 363 I d r M11-' ( B J. >6:0, 0,0) B I O 0 0 - 5

3 63 >! ( I 7 i-i 4- 2 ) «•>: ( X y h + 2 )/Bl5 J. K:0 , 0 52-(' : 0DO 36/ 1 0 . 7 ii B 1 B I 4-X I ( I ) * X ( I :• M + 3 )

36/ B 2 " B 2 + :< ( I 7 i'i4-2 > * X ( 1 1 ri4-3 )I K) 3 7 ( 1 X O O M

37(1 >: ( X 7 h + 2 > " B* ( B.i. *Xi ( X ) + B 2 ¥ X ( X 7 H + 2 ) - X < X r H + 3 ) )GO TO 3-)&

4 0') B-'YO. >Y ( X > "■ Y ( XI..)Y ( Il.)-SDO 4 ( ' 2 X •' .17 Mx r ( x ) -=:< (x 7 x i.. >X ( X 7 x !.. ) =• >: ( X 7 .i. )

4 02 :< (x 7 .i.) o<r < x )P R X M l 7 7 2 7 1. X (.:

7/ 2 FORiir'i V < ' 7 'I.. I (O' 715)REi URN E i ID

27^

G.5 Subroutine HANK

1 ;■!

BUBROI J ( X l-iH. H n M K < f H.i'iP r i-i COMP r ! C r B r 0 r WBRK )Ei HP!MB XDM ORl) ( '50 ) ? ORD ( BO) i VC ( B ) i TR ( B > ? OB i '1.1. > ? 0 ( '10 ) ? MBRK ( .U; A , 3 2 B . i 6 B^.l. > 0BV0 7 C ~ - . H I V : r i D - ■ , J. 9 0 OB 4eF - , 3B6 9J. 4g - - <H-:- > 0 -190A45BO . 1 J J" ' i. •• N C O H P

) :' .1)

TR < .J '.'RO (

c c * ( j. > - 1ORB ( J J :> OB < J . ) )RE I URN

END

(I-:, h p ) / f c •; j j >). . •{ r , > : • - . 1 . . ( r ( . i j ) ) * i>. ( .i. > / A , ) ) -f » >,< ( < .i. , •• ( r <; j , . i ) ) ■:<% < ; > , / , ) ) +

( J .) > ) B :* ( .1 > • • TT: ( J J > ) * < -I . / 3 > >< E + E * ' ( R ( J J ) - l - O M R ( J J ) m 2 : •MI 'KT R ( J J > * * 3 . > / C! R ( J J > - J. . <«('■:• 0 5 )0 0 0 , >5:0 ( J . J ) * O R U ( J J > >5. < J. , - W B R K ( J J > T O R D ( J J ) )

G. 6 Subroutine PHIB

C C A L I . AY I O N OH H I J G A C I f V COK F H I C I li M Y C H OR P U R L B U B S T A H C L B AND C B X M A R Y M I X T U R E S W I T H COM!- ' OB X T X ON ( 5X0 PM BY V H H-! i’i OI..F. HR A C T X ONS

V A'f PRH. 0 0 U R L P < .1 N AY H ) A N N I I! Hi P H R A Y 11R fc. I Ll - iP < XM M B K>

M - * OH COHPONH M l 0 <0 OR 2 )C P R 0( 5 RA M C A L L S U B R O U T I N E : 5 0 1 f t W H E R E T H E COM HEM I B OF O I R D A T C X B H' X P I.. A X M HI nC MACXA REPRESENTS YHP NUKBH'R OH CARBON A fO H B XM C AM OR(5AHXC A C I D , FOR A C E T IC A C ID MACD) XB 2 .CC f H H B U B R O U l X MH W I L L V A L U E S OH H U B AC I ' I V C O E F F I C I E N T S FU(5 C t * t :!< * * * * # * t * * B t * >Y >5t >X H X B * T * * >'/■ >Y 1 * .X % * T t >!■ * * >'/■ * * B * * * * * * * * S'- *

$ U B k 0 IJ I X NF PHI X B D X i’i F NB X 0 M A ' A ) ■> B ( • ) ) ? Y X < A. ) F IJ(5 (('5> ? F V A ( 5 ) ? B B < 5 > ? B F < .1.0 >

CEG '• 0 2 , 0 6P R " P R / ( R( 5 * HEMP >

PURL OOi iPONPNl OR B IN A R Y i' iX>:TURF iIt) AC X BB X MO01..OFD E ( ! ' B ( 2 , 6 ) AMU ( 2 , / )

CC CAI.!. BOXRDO ■ > .1. X •: J. ? I'I X F ( H Y A ( X ) , L 0 , 6 , 0 > COY 0 X ( - 2

lo. l . COM IXMUEI F <N . K (51 2 ) COfO XOX FOB C D “ P R *B B (J . )H U C C l ) =" L X P ( H U C C D )R E T U R N

X 0 2 BH X X Y X < X ) * * 2 * B B ( X ) •}• Y X ( 2 ) * * 2 * B B ( 2 ) 5-2 , * Y X ( X ) * Y X ( 2 ) * B B < 2 )FIJ 0 ( I. ) »: P ft # ( 2 , * Y X ( X ) * B B ( X ) •!■• 2 > * Y X < 2 ) * B B ( Y ) - B i’i X X )F U 0 ( 2 ) PR * ( 2 , *Y X < X ) * B B C D H2, * VX (.2 ) * B B ( 2 ) - B H X X )DO I - M X " .1.7 2

x <m f u c ( X ) " i:>:p ( h o c ( x >)RFIU RM

c.C PURF COMPONENT OR B IN AR Y M IX TU R E C OHF COMPOHFMT XS AM OROAMXC AC tD C E C ' S ( 2 , 1 6 ) r ( 2 , 1 0 ) AND ( 2 , 1 0 )C.1.0.2 IH ( X , EO ■ X ) GOTO 1 0 6

NA •■= 2 MB "• 1 GOTO 1 0 7

106 MA •" 1 MB =•• 7

1 0 7 XH ( MAC I D , C T , 6 ) COfO 1 0 5CC E X P F R . I H F N Y A L V A L U E OP K C

A l\ A " A ( N A (.' X B ) - B ( N A (51D ) / T P M P

276

cc !■;

c t:u c1 015 109

.toy

i’i K (•; - F >: P ( - A K i’i >!< 2 . 3 0 2 !5 9 ) Y 7 A 0 G O T O .1.08

p r k. j:i :c o t k. » f r o m r i-_ c n n jj v r a i.. c o f: f f i o :c i: h -i h ( > y )

A K A ■■■■■- * P R )S 0; ^ B 0 R I < t , •! A , * A K T * Y X < K A ) * ( 2 ■ - Y X < N A ) ) )z a ■•••• (:•> o •■•!,)/( :•>» t a k r >:< c y , - y :< ( m a > > >F U (•; ( N A ) < 7. A / Y X < W A,> ) * H.X P ; ( m p > # < ? . . + a « y a i ; ' i y < :■? < ••• y x ( n a ) ) - b o )

2 A ■: 7 . B / ( 2 . YAK f * C 2 , - Y X <NA) ) )F U(> < HB ) < .7.B / Y X < W B ) ) # H .XP ( BF ( NB ) * P R )C O M r 1 MOE RE I URN E N D

27 7

Subroutine SVIR

C * Y 1 1 % & # % ft * t t # >j{ * % ft t * t * # (f-11 # # 1 1 & % # & % 1 1 % t % # * 1 1 % t K 1 1 % *C CA1. .CUI . .A 1' XON O F P U R E C O M P O N E N T AM) ) ( ' R O B S V I R X A L C O E F F I C I E N T S C H OF: I N O C O M P O N K M l B (VI I E H P E R A T U R E I I "HiPC F ROM H A Y D E N AM).' 0 ' C O N N E L L C X EC P R O C . ).iFTB < D E V , X 4 ( 2 > 2 0 9 ( X 9 7 1 5 ) (.:C N ;; NUMBER OF COMPONENTS (X OR 2 )C BP :■ BF RF E C BP •■: B l 'OVAl .C; N(; " (>■ OH COMF'DNF.M'I SC PC =: CRT D C AX. PRESHORE; ATMC RD =- Ml I. AM RAD.'. OB OH OYRA'f X 01! r AC DMU - DXPOLE MOMENT I N DEBYEC E T A ( X ) AMD E T A ( 2 ) = ABBOCXA I ION PARAMETERS (PORE COMPONENTS )C E T A ( S ) *: S O LV A T IO N PARAMETER (CROSS I N T E R A C T I O N )

C R I ! I C A I. ( E Hi P E K A 1 1) R E r .D h" C K( ' r x r x i : a i.. c. o m p r e s b x b x i. . x r y f a c t o f:

OHMOM VALUES OP V I R D A f (HE SUBROU I I EE OXI..L.C RETURN VAI..OEH OK BE REE AMD C(TO HAL.C t « >K $< (' # t >!: * t f 4:41 4:4! * t % t it• 4: # * * * * % * >£ * >!<* # * 4: % # % * % %■ C * % %■ * 4

SU BRO UTIN E SVXRC O M M O N 0 3 K ( T O ) f X P 4 < AIX > r N C H O X C r I C ( X 0 ) r AM r AH r M AC X D r BMX X f BH ( 4 5 )C O M M O N X MO l . ( 414) ; P C A L C 4 3 > ; P ( 4 l ' 4 ) ; P P ; MP ; K K ; B S C ( 4 0 ) ; 0 4 C ( 3 ; 4 0 )C O M M O N X 3 ( 4 0 > r 0 4 E ( 414) r B B T ( 4 15) f 0 4 f ( 415) >• D ( ' P r D ( ' N f T C A L ( 4 14 >C O M M O N Y E S C ( 4 3 ) ; Y F 4 C ( 4 0 ) ; Y E S ( 4 0 ) ; D D E N X 4 ( 4 0 ) ; MOP V ; H O P ; HON r

C A AA ( 4 0 ) r P (.' ( 5 ) r DHUJ ( 14) r H '('A ( 15) r E E R 0 R 4 ( 41 4) f RX r R 2 f X P 3 ( 4 0 ) f V ( 4 '.5) C O M M O N AX ( 4 O f WS RK< 0 ) •; Y F 4 ( 4 1 5) ; E R R O R P ( 415 > r N C O M P ; T E M P ; VB< 4 5 ) C O M M O N E R R O R S ( 4 0 ) ; X P ( 414) ; XM ( 4 0 ) ; X 4 ( 4l '4) * AS ; A 4 ? B S ; B4

C TCC Z CC F O R G I V E N

COMMON C O M M O N COMMON C O M M O N

! y V X I-' F F; 7. N ; H M E f I' M H f H M f I ( 4 1 4 ) r D 0 ( 415 ) r A M W ( 4 ) f A S 2 f A 4 2

P 7 R N t R S j R 4 ; U J. 7 U 2 ; Q S ; U 4 ; A M V X ; A N V 2 ; A N V S ; A B ( 2 0 )P M r A 2 :l r B P M f A 4 X f A S 4 r A 4 2 f A M T 4 f A M T 15 y A M f 6 t P T f R D ( 6 )S f : i . P ( 4 ! 4 > f (44 T .1 P ( 415 > r PCAI . . P ( 414) f Y P S C P ( 4 1 5 ) f Y F 4 C P ( 4 5 ) f

CF-'OO ( 414) # B B ( A > ; Y !< ( 6 )C

D :i M E M B X ON 0 ( 2 ) f E P S 3 ( 2 ) r B X CMS ( 2 ) r RDi ' iO ( 2 ) f RDMM ( 2 ) f A ( 2 ) >1 DEI . . H ( S ) 7 D ( S ) 7 BO ( 2 )

CC CAI . . COI . . A I XOM OH C O M P O N E N T P A R A M E T E R SC E 0 ' B X O ; 2 0 / J.7 ; 2 4 ; 214; 7 S ; 2 J. ; 7 2 ; XOc:

DO X ( ' X X - X f MW ( X ) ■ 0, ( ' ( ' A # k D ( I ) T O < ( ' 2 ( 1B 7 # RD ( X ) # # 2 - 0 . ( ’ OX 2 A # R D ( X ) # * 3E P S X ( X ) - T C ( X ) t ( 0 > 7 4 0 + 0 > 9 X :1<Xl ( X ) ~ 0 , 4 1 E T A ( X ) / ( 2 . + 2 0 , StW ( X ) ) )B X 0 M 2 ( X ) =• ( 2 < 4 4 - N ( X ) ) * * 2 * ( I C ( X ) / P C ( X > )I F ( 0 M 0 ( X ) ••• X , 4 14) X 0 X ; X 0 X ; X 0 3

.1.03 PM X 3 « + 4 ( ' ( ’ < StW ( X )C •• 2 . S B 7 - X > H 0 2 YO ( X )/(•), OS + O ( X ) >X I =• D M 0 ( X ) « 4 / < C # E P S X < X ) * y ( x , ■■■>: x * p p m++■ p m * ( p e m+ x . ) s< (>: x # # 2 ) / 2 <)S X 0 MS ( X ) :••• S X (7 M'S ( .r. ) * ( X > + 3 , H C ( X / ( P N - A > ) )

,i. 0 X R D M11 ( .1 ) < ) ) Hi 1.1 ( X > * * 2 ) * 7 2 4 3 . 0 / ( If P B X ( X > * B X B Hi 2 ( X ) )I F ( N - X ) S 0 0 7 S 0 0 7 4 0 0

278

30 0 J :• iGOTO 30.1

4 0 0 J - 3GOTO ')0:l

CC N N P 0 L A R -!•!(3HP01..AR r EO ' r 32 s 3 3 r 3 4C P A R A if IB T E R 0 FOR HCC fU R F CALOUL.O I S ONC

o o i epb:c <3> - o , ? * b o r t < h : f - b *epb:c ■; 2 ) > i o< 60/ c i , / e p b : c c s ) + 1 < /k.pb:c ( .?>)3 :i: (5 N 3 ( 3 > •••• 8 U R 1' < 8 113 N 3 ( 1 ) * 3 X (3 rt 3 ( 2 ) >W <3> <•. 5*<N< 1 ) + W< 2 ) )I F ( Di'HJ ( J.) DMU ( 2 ) ) 8 0 0 ? 00 J. ? BOO

CC POLAR- N()NP0 1. AR r HO' B 30 r 24 r 36 r 37C

5 0 1 I F < 33 HO < 1 ) }• 31 Hi I.I ( 2 ) ~ 2 , ) 50 0 t 5 0 0 r 1 ?19 :<:c3 8 =•= < < j. )*:&;•>#<e p s x <2 > xgm*?<:•>>>#*< 1 » / 3 > )#s:cgm3<:>> +» h u <2 >*

J. * 2 * (\i H' B1 < 1 ) :f t 'A * B X (3 H '3 ( ;i. ) ) * * < 1 , / 3 < ) * B 3 (3 M 3 ( 1 ) ) / ( F P B X ( 3 ) % BIB H 3 ( 3 ) * #)PH =: J. 6 . F 0 0 0 * 0 ( 3 )EPS X < 3 > «- K.PBl ( 3 ) * ( 1 < T > : : ( 3 B * P H / ( P H - 6 < ) ) s :i: o h 3 ( 3 ) - b y o m a < 3 > * < i. > - 3 . * : >

cC P 0 1..AR-POLAR f k0 ' B 3 5 r 37r:

5('<- R3.IH0 ( 3 ) «» 72 4 3 . B* UMU ( 1 > *3.1 H0 < 2 ) / ( K.PS X ( 3 > # 8 XGM3 < 3 ) )3 OJ. DO 6 0 0 1 J. i J

I F < R>:iHO ( . ( ) - 0 < 04 > 1 4 r :i. 3 r i. 510 R OHM ( 1) == ROMO ( X. )

GOfO 6 0 0 1 5 y F ( R31H0 CC ) - 0 . 0 2 5 ) 1 6 r 1 7 ? 1 716 r n h i'i < i:) - o.

GO'fO 6 ( '017 RYJi'f rl ( X. ) Rl.ii'HJ ( X ) - 0 > 256 0 0 C O K ' i m . ' F

cC LABI PARAMETERSf E O ' S 7 r B r 9 ? 2 9C

DO 6 0 9 I " i . r JBO C O ■" I > 26 .1. B* B X. OH3 ( I )ACC ) . 3 - 0 ■ 0 5 * R .0 M U ( X )d f l h c c ) - i > 99 + 0 , 2 * r d h o c o **2IF( in ACC) “6 , ) 60 4 V 60 0 r 605

6 0 4 D Cl ) 65 0 < / ( K. P B .( ( X ) + 3 , 0 0 . )GOTO 6 09

6 0 5 D ( :c ) • • 4 2H00 , / ( K P B I ( X ) A22 4 00 , )6 0 9 OOMT X HOE

( .:

C CAI..COI..A I :COH OH '.LCRIAI COHFF iCIHH' l B r H. 0 ' B .14 r :i 3 t 26 r 6 r 29C

DO 6 5 1 1 :• I r . l T B T R :: FP B :c ( :c ) / r F i-iP - J. . 6 * W ( I )BFI-! 0 , 94~x . 67*1 B T R- 0 . B5* l B l RT.S2 + 1 < 01 5 * T B T R * * 3

2?9

B H R -• 3 , * I B T R -!• 2 < A * T S T R *#2 + 2 « .1 * T B 1 R * * 3 ) * R ).lN M < X )B F ( I ) " ( B F'N - B F P ) :Jt }5> () ( X )BE ( X ) « BF < X ) FBI ) < X ) *Yi (1 ) * h . X P ) t ( A , - P X P ( A . #K. Tf i ( 3 B B ( X > " 3 B < X ) F B C H E M

6 0 A P O N T X H O P

R E T U R N

END

c > / th:h p ) >

280

G .7 Subroutine HANKIB U B R O U T XMH I lf';NIC IC O M M O N (■: A h ( 4 3 > 7 X P 4 ( 4 3 > 7 N O H O Y 0 ; 11) ( :i. (< ) 7 A H 7 AH 7 N A O Y B ? BH Y X r B F ( 4 3 ) C O M M O N XMOI. . ( 4 3 ) 7 F'C A L ( <13) ? F ( ■‘I 3 ) 7 PR 7 NP 7 KK 7 0 3 0 ( 4 3 ) 7 0 4 0 ( X 7 4 3 ) C O M M O N X 3 ( 4 B ) 7 B 4 H : ( 4 3 ) 7 0 3 1 < -Vo ) r 0 4 T < 4 3 ) 7 ) . 1 < ' P r ) . K ' H r T C A I . < V . i )C O M M O N YF 3 0 ( 4 0 ) 7 Y F 4 0 < 4 3 ) r YF 3 ( 4 0 ) 7 B B F N X 4 ( 4 CO 7 N O P f 7 H O P 7 H O N j

c a a a < 4 o ) 7 1 1: < 0 > 71:1 mu < > 7 f : t a < 0 > 7 f: f : r o r 4 < 4 0 > 7 r y ? r 2 ? : 7 0 < 4 5 )C O M M O N AY ( 4 3 ) r N B R K O O 7 Y F 4 ( 4 3 0 7 H R R O R P ( 4 3 ) 7 N O C M P ? Th' MP 7 OS ( 415) C O M M O N F R R 0 R 3 ( 4 3 ) 7 X P ( 4 3 ) 7 XH ( 4 3 ) 7 X 4 ( 4 3 ) 7 A 3 7 A 4 7 S 3 7 B 4 C 0 MMON N 7 F X P 7 F 7. N 7 F N P 7 F NM 7 F K 7 T ( 4 3 ) 7 B 0 < 4 3 ) 7 A M W ( 4 ) 7 A 3 2 7 A 4 2C 0 M M (J N h: F' 7 R N 7 R 3 7 R 4 7 NY 7 (12 r 0 3 7 0 4 7 A N T Y 7 A N I 2 7 A N 'I 3 7 A B ( 2 0 )C O M H 0 N A P M 7 A 3 .1.7 BP M 7 A 4 I. 7 A 3 4 / A 4 3 7 AM )' 4 7 A N T 3 7 AM T A 7 P I 7 R )) ( 6 >C 0 M M 0 N 0 3 T .1 P ( 4 3 ) 7 0 4 T :i. P ( 4 3 ) r F' C A I.. P ( 4 3 ) 7 V F 3 C P ( 4 3 ) 7 Y F 4 C F' ( 4 3 > 7

CF I JO ( 4 3 ) , B B ( A ) 7 Y X ( A )i 1 :i. m h n b y t) n f o m ( 4 3 ) 7 0 m b ( 4 3 0 7 0 r 0 ( 4 3 ) 7 w m ( 4 3 ) 7 v b b < 4 3 )D I i - i FNB Y ON f R ( 4 3. > 7 0 R .0 ( 4 3..) 7 BM ( 4 3 ) 7 B WM :i. ( 4 3 ) 7 B '3HB Y ( 4 3..) 7 B T O H .1 ( 4 3 >D X M F : N B I ON B )'R J. < 4 3 ) 7 B 0 R 0 R ( 4 3 ) 7 B 0 R J. ( 4 3 ) 7 B 0 R B ( 4 3 ) 7 B 0 B ( 4 3 )

C C O M P Y Y F A L C O H O LC C O M B , A A N B F L F C - : 1

A = Y , 3 2 BY 6 B- J. , 4 3 9 0 7 (4 - 1 B Y 4 4 6B ::: , Y 9 0 4 3 4 I."■ - ~ , 2 9 4 Y 2 3 F > 3 0 A 9 .1.4 (3a - . <<4 2 7 2 3 8 H - - , 0 4 BO A 4 5 K J - KK

1 0 0 M K X ) : >.' P 4 < K I ) ft W B R K < 2 ) T X P 3 ( K X ) ft W B R K ( 1 >Li !<J M Y ( K :i: ) 7 < W 3 R ;< ( 2 ) - >,1 B R K ( Y )

Y Y A A " ( 0 ( 2 ) # • »<; < 2 ) ft '3 ( 2 ) * T C< 2.> ) S f t < 3H H •• ( U ( Y ) ft T O ( Y ) * 0 ( Y ) f t T 0 < Y ) ) ftft > 5 B B :: < ( 2 ) f t I 0 ( 2 > f t M ( Y > f t T 0 < * ) > « « < 3 B A ” ( 0 ( Y > # T C ( Y ) * 0 ( 2 ) * T C ( 2 ) ) f t f t . 50 M B ( K Y ) : ( J , / 4 1 > ft (>: P 4 ( K X >f t V ( 2 ) + X P 3 ( K X ) ft 0 ( Y ) + 3 . * < X P 4 ( K X >

Cf t O ( 7 ) ftft ( 2 , / 3 .

0 ) T X F' 3 ( K X > ft 9 ( .1 > ft ft < 2 . / 3 . ) ) * ( X P 4 ( K X ) ft V < 2 ) ft ft ( Y . / 3 . ) + X P 3 ( K I )C K O ( Y ) f t f t ( Y > / 3 > ) ) )

D M !:• Y ( K X ) ■ ■■< Y < / 4 ♦ ) t ( 0 ( 2 ) - 0 ( Y ) 3 2 ) % * ( 2 < / 3 , Y ) Z %{ 2. , / 3 ,c ) ) % ( :<P4 < k .1: ) >:<o < 7 ) %% ( 1. . / 3 , ) + : >C + 3 « < >:P 4 ( KX ) 4 : V ( 2 ) « ; * <0 2 ■ /'A . ) + X P 3 ( KX ) «'Y ( Y ) tt ( 2 . / A « > ) * < < 2 ) f t # ( Y , / 3 < ) - > . ' ( .1 ) f t # ( Y , / A . ) > )

T O M < K X ') ■■■ ■ (( X P 4 ( K I ) ft ft 7 , ) ft A A I- X P 4 ( K X > ft X P 3 < K X ) ft 0 B0 + >:p a ( k Y ) ftx p 4 < k :c ) ft b a + < >:p a < k y ) ft ft 2 < > ft.HH > m b < iv.t >

D VOMY (KX ) ( 2 , f tXP-4 (KX ) ft A A T 7. , f t X P 3 < K X ) ft O B - 2 , f t X P 4 ( KX ) ft O B - 2 , f t X P 3 ( i ( I )C ft FI H ) / 0 M B ( K X ) - I 0 M < K X ) ft B 0 M B Y ( K X ) / 0 M B ( K X )

:L2 v r < k x ) ■■■■( r o A L ( id: > > / i -o m < k : oIi I R Y ( K X ) • " ■- ( ( I O A L ( N X ) ) / ( T Ol-i ( Id ! ) f t f t 2 ■ ) ) ft AT O M Y ( K X >O RO ( K Y ) ■ -I i T A f t ( ( Y , - I R ( KY ) ) ft ft ( Y , / A < ) ) YBf t ( ( Y , - ( R < KY ) ) ft ft ( 2 « / ' 3 . > >T

CO ft ( I. » - V R ( K C ) ) Y B ft ( Y , - TRY KY ) ) ft ft ( 4 >/A > )B O R OR ( l< Y )< -■ < Y < /A , > ft Af t ( Y . - I R ( K Y ) > ft ft ( - 2 . /A, ) - ( 2 , / ' 3 , ) f tBf t ( Y , -

C T R < KY > ) ft ft ( • • Y , / 3 , ) - 0 - ( 4 , / 3 , ) f tBft ( Y , - VR< KY ) ) ft ft ( Y , /A , >B 0 R J ( K J ) B 0 R 0 R ( K Y ) ft B T R Y ( K Y )

281

V R J . K K X ) " ( H + F X ’( I’; < K ) + B * l R ( I U ) * * 2 . + H * T K ( KX ) * X B , ) / ( T !:; < KX ) - .1 , 0 0 0 0 J. )g a a - = t r ( k x )XX2 .DA)!!"' U R ( K X ) - - . i < < ' 0 0 0 1 )B V R D < K I ) "JJ' I R 1 < K X > # < !•' -I 2 « * 7 R < KX ) S iB +B , # / < T R < K I ) J. , ' ) ' ) • ) 0 1 >'J B B < K I )■■ \> H B ( K X m 1 R () < K X ) X < 1 . ~ W M < K X ) X 0 R )3 < K I ) )DOB(KX ) ":l/' H B < K I ) :*)3Vrt J. < KX >+VRt)( KX ) X 13V MB J. < KX ) - V M B < K X > *yR0< KX ) *WM < KI )

C * ).i 0 K H ( K J ) - 0 Fi B ( K X ) X 0 R .0 ( K .1 ) X 0 R 0 ( K X ) X D W M1 ( K X ) - V t'i B < K X ) X W i'l ( K X ) * 0 K ) i < K 3 ) C*D*,'RJ. ( KX ) - yRO (KX ) X id M ( K X ) KOND ( KX ) SOX VMS J. ( KX )

A N W T " X P - X K X > X A K W ( 2 ) 1 X R B ( K X ) * A i-i W ( 1 )DO ( K X ) " A M \'.l T / ( O' B B ( K X ) X J. 0 0 0 : >I.i H H A H W ( 2 ) A i i W ( J. ).0n F .M >: 1 ( KX ) • ( X , / :i. (-(■<•, ) X ( D H k ' / O B B ( KX ) - ( AKW 1 SUXOB ( KX > / ( V B B ( K!l )XX2 < ) ) >

R E T U R N END

282

G .8 Program FITIT

d i h l n b i o n y c k o f > : o k o c :< - j: m m p m ):i m y v a r i a b l eC Y - .11H.Pr: H D F i l l AF; A I Y;B I. P.C HP - 4 OF PO IN TS

Rlr.AIK I > * ) M P v Iu r i t e u u ^o h f o tDO I P I X NF‘r e a d 9 9 9 , ( ;i:) 7 v ! i :>

99 9 F ORHAT (2F .19 , 6 )1 0 C O N T I N U E .

C n i. l . f : n : n < n p >: > v >2 0 C O N T I N U E

S I OPe n n

SIJDT’OIJ I I i !H. I- I I I T (N PO IN T v'/.i >■ AOAHA)CC T I H B P Ti 0 (■; P A l-i T I T B A I-' 0 1. V IT 0 H I A I. 0 F 0 R D IE P 6C

I! .I'M FITS I UN B I (•■MAY <IK 1) > X C K O i A O AM AC K O s A C K O 7.DH.I. TAY ( 0 0 ) >Y C A L C K OI F < N P OIN T > I..F , ) 0 0 TO 99

CDU 2 .C C U IT P O I N T S fOMAY ( I > ■■■■') ,

2 C O N T I N U EM CO Die <•M A X O P D " N P O I N 1 / 2I F ( MPOINT < I..E , 'i ) HA):OK'D-' 2I F ( M A X 0 R D , 0 T , A ) H A :< 0 R D -■= 6N N IT H A >: 0 K‘ .0DO A K :: J. i H N KK l - K + i

CC A I . L P O L I T I ( >! I 7 A B M H A 7 B I C M A Y j N P O I N T r K X 7 (' ? A i OH I BOR )

UR I TE ( 2 7 A ' ) ' ) > K6 0 0 TOR M A I ( / / / ? 7 >! r ' POL. Y N 0 I i I A I.. T I T T I-." D I B 0 F I H K. D H. 0 R F. FI - ’ 7 I \\ )

UR I TE ( 2 7 J. OO).1.0(> F O R H A T ( / / / 7 B> U ' YH. XP ' 7 BX 7 ■' Y C A L ' 7 i 3>: >• ' D Y ' )C

E R R U R " 0 , 0DO <1 J "J. 7 N P O I N TS U M - A ( 1 )DO 0 I " 2 7 K 1S U H ”• B U M + (Y ( I ) * >! ;l ( ..I) % % ( I • .1 )

5 C O N T I N U EYCAI . ( J ) - BUM

D EI..T A Y ( J )■■■{{ YCAL. ( J ) - A B A H A ( J ) ) / AO AM A ( ..I) ) % 1 (■(>,.U R I TI I ( 2 7 2 0 0 ) A 0 A IT A ( J ) 7 Y C A I.. ( J ) 7 D E I . . )' A Y ( J )

2 0 0 F O R M A T ( / y OX 7 2 0 1 2 . 5 )4 ERR 0 R ■■E R R 0 R F D E I.. I A Y ( ,J) * Y 2

E KR O IA -LR R O R /N PO IN T

283

M R I I H. ( 2 > l i <• <•)5 0 0 F O R H A T < / / / r !•.>.' r •' P O L Y M O M I A I. COM H I A M I H ' )

DO 2 0 I - . I . ? K1 WRIT K. < 2 7 2UC ) h/:( it )

2:io format (// 7 •;■;:< 7' a (7 ? 1:2 u.i.2 > 5)2 0 C O M ! I H U E

UR I I t! ( 2 7 3 ' ) ' ) ) E R R O R 3 0 0 F O R H A T < / / / 7 5 X r ' BlJi'i OF DE . EL T A Y H O O A R F D D I O I Jit H. D DV I H K.'

* OP P O I N T S ■■■' 7(U2.>Vi)C3 CO M 'I I MO FIC99 R E T U R N

E N DS U B R O U T IM F! P O L 11- I ( X 7 V 7 B I O H A Y j MPT B 7 NT LI': l-i'.' 7 M O D E > A 7 C H I BOR )

CC F XT P A C T L D F R O M ! BE V I MOT OM 7 P < R , 7 ' D A T A R E D U C T I O N AMDC E R R O R A N A L Y S I S E U R H I E P H Y H 1 0 A I.. H C I E M C EH 1 7 MCORAW H I I.. I.. 7 J. V & 9CC B U D R OUT I M L P 0 1. I F I I P(J RP 0 B ECC M A K K A L L A B I - B (M IA R H B F I I TO D A I A M I I I I A. P 01.. Y M 0 M I A I.. C U R 0 LC Y - A ( .1. ) }• A ( 2 ) Y X F A ( 3 ) * : < * * 2 -F A C 4 ) * X * Y 3 FCC D E S C R I P T I O N OF P A R A M H I E E SC X - A R R A Y O F D A T A P O I N T S F O R I M D E P E M D E N V V A R I A B L EC Y - A R R A Y OF D A T A P O I N T B F O R D L F ' L H D L M T V A R I A B L EC B I O M A Y - A R R A Y OF B T A N D A R D D i : V I A T 1 0 MB F O R Y D A T A P O I N T SC N E T S - MU I T B F . R OF P A I R B OF D A T A P O I N T SC N T E R M S - N U H B E R O F O O E F F I C I EM I B ( D E G R E E OF P O L Y N O M I A L -F 1 >C MODE - D E T E R M IN A N T S METHOD OF WEIOHTIM O I..E AST -BOUAREB F I TC T J. ( IM H T R11M E N T A I... > W E 1 0 H T - J. > / S 1 0 M A Y ( I ) *• * 2C 0 < MO W E IO H T IM O ) MEI CUT ” 1 ,C - I ( B T A T I B T I C A L ) MF I OI - I T ( I > =• I . / Y < I )C A - A R R A Y O F C O E F F I C I E N T S OF P O L Y N O M I A LC C H I HOF: - R E D U C E D C H I BOO A R E F O R F I TCC S U B R O U T I M E B AMD F U N C T I O N B U B P R O O R A H B R E I M J I R E DC D E I.. I I ■: R M ( A R R A Y 7 i ! () R D E R )C E V A L U A T E S T H E D E T E R M I N A N T S OF A S Y M M E T R I C T W O - D I M E N S I O N A LC M A T R I X OF M U R D E RC

D O U B L E P R F f. I H I OM BUM) : 7 S U M Y • XT I- RM 7 YT L E M 7 A R R A Y 7 CF I I BO D I M E N B 1 0 M X ( 5 • ) ) 7 Y ( 5 <)) 7 B 1 0 i'i A Y ( Vi >)) ? .A < 'A 0 )D I M E N S I O N B U M X <!)<■> 7 B U M Y < S O ) 7 A R R A Y ( 0 7 B )

CC A C C U M U L A T E ME I B i l l I N C B U M SC1 1 N M A >: «• ; >*MT E R M B - 1

DO 13 N - .! 7 MM AX 13 SOH>:<M) « (■<

DO 1 5 ,L-.i. 7 N T E R M S

2 m

1 5 SUMY < J ) - 0 .c h i s u =” •>.

2.1 DO 3 0 X =• A r HR'(S >: i: o (.()y l v (:i.)

31 I F ( M O M ) 32 7 37 7 3932 1 F ( Y I ) 3 1) r 3 7 r 3 333 w t: i (j h r - j. , / y i

GO TO 4 J.35 WFIGHV - .1. , / ( - Y . O

GO TO 4 1 37 WFIOHV ” 1 .

GO TO 4.i3 9 W H I () H I "■ i , / 5 I tv H A Y ( I ) * Z 241 x t k r i o w f i g h t

DU 44 NO. VIM AX SUi ' lX(M) •” t>DM:<(?•!) + XVFRM

44 XYK.RFi ” XYFRi i Z X I4 5 Y X f RH W O OH 1* Y 1

DO 4 0 N O V NYK. RMB S U M Y < M ) i i U i t Y ( H > Y l lKK'M

4 B Y I H . R M - V I K R H * X I4 9 OH I HO OH 1 0 0 I- 1-JO (il l 1 * Y I Z Z 25 0 CUNY IF!UKC0 CUN BY HU (71 MAI R 1 01: B AMD OAI. CUI..AI K OOKF K 1 0 11! M I BC5.1 DU 34 J O 7 NYKRMB

DO 5 4 KO . 7 NFORMSN " .J + K -■ J.

54 A R R A Y ( J 7 K ) •” BUMX(M)D E I. T A - D li I K R M < A , R R A V 7 M Y C. R M B )I O DKL TA ) A.I. 7 57 7 *1

5 7 C H I BOR " 0 ,DO 59 J O 7 NI lIRMS

5 9 A ( J ) - 0 ,GO VO BO

61 DO 70 I. O f NYKRMB6 2 DO 66 J O . 7 M 1 0 RMS

DU 65 K" 1 vNIKRHEi N -■ J F K - 1

65 ARRA Y ( J t K ) ” BIJHX ( M )66 ARRAY ( J 71..) ” BUHY ( J )70 A C L ) ” D K. I K R M < A R R A Y r H 11- I'J H B ) / .D I! I.. I A,CC CALC UI. AY K.B C H I BOUARKC71 DO 73 J :: 1 v MYKRHB

CIHIBU == OH I BO - 7 > * A ( J ) YSUHY ( .1)DO 7 5 K ” 1 f MYKRHB N": J FK- 1

7 5 CHI BN” OFII BN + A < J ) t A ( K ) *BUMX ( N )

285

7 6 K R H I'." II I'' I B - M ' ( E R l-i S7 7 C M ) ! B U R - - C H I B U / F R E E

W R I I K ( 2 > I O C ) C I I I B U R 1 0 ' ) F OR HA ) ' ( / / / ? OX •/ ' C H I B U R ” O 0 >. 2 > 0 / / / / / )BO RK.I URN

ENDFUMf/ l I ON UK. I H Rt'i ( ARRAY r HOEBEE )

CC EXT RAC I K I) K ROM i Mil 0 1 MCI CM r R , R . 5 ' BA'I A EK I'lIJC'f I OM AMUc e r r o r a n a l y b i b f o r t h e -: ph y s i c a l bc i f i n b f b ■ ? m c b e a w m i l

c;C F IIM C I I ON Bh' IKEMCC PURPOSEC CAI..(;ULAIi : .B IMF BE I K R H I RAMI OK A BOO ARK. M A ' IE IXCC 1.1 BACKC BK I IIK.TK.RH ( A, RRA Y r MOR.0K.R )CC: DE BCR I P I I OM OK PARAHEl K.RBC A R R A Y ~ M A 1' R I XC NOEUER -ORDER OK BE'I l.'.Ri-iIMAM I ( BE. BE Eli. OK M A T R IX )CC SUBROUTINE AM)) KIJMCIIOM BUB PR OCR A MB RK.UO I REDC NOMECC COMMENTSC I H I B BlJBPROBRAl' i BKC IR O YB I HE I MR I.II M AT R IX ARRAYC

DOUBI. K. PR EC I B I OM ARRAY r SAVE D I KUHNS I OM ARRA Y ( 0 / 0 )

10 D E I ERA ' - I .11 BO 0 0 KO . 7 MORBFRCC INTEECHANCK COLUMNS IK BIACMOI.. ELEMENT I B 7KR0C

I F (ARRAY ( K r K ) ) A X r 2 A r 4 1 21 CK) 2 3 J - K 7 MORBFR

I F < ARRAY CKr J ) ) 31 0 3 > 3:1 23 COM VI NOE

BE TERM "• 0*GO VO 60

31 BO 3 4 I K - MROBERSAUF " ARRAY < 1 7 J >ARR A Y ( I v J ) ARR A Y ( I r K >

3 A ARRAY ( I i i\ ) " 'BAREI )ETERM ”■ - BE I K RH

CC BUB IR A C I ROM K l-ROM LOWER ROUB 10 BE'I BIAOOMAI. MATRIXC41 BE TERM B E ' I E R H # A R R A Y < K tK )

. 7 J. V 6 ?

286

X F ( K ~ HO R D H R ) 4 3 y !:><'r 5 0 *13 K l - ' K - K

DO 4 6 X " K .1 > R' ORDKR DO 4 6 J " K J. » NOR D E R

4 6 A R R A Y < X r J ) " A R R A Y < X y J ) - A R R A Y ( X r K ) * A R R A Y ( K y J ) / A R R A Y ( K y K )5 0 COM t ' XMl JE6 0 R R ' i U R N

E M D

N O M E N C L A T U R E A = Antoine constant, equation (A-l)A ^ = UNIQUAC binary interaction parameter, equation (3-27)A\vk = van ^er aa -s grouP surface areaA± = parameter of equations (5-5) and (6-1).A = see equation (3-13)a = ion-size parameter of equation (3-8).

calculated using equation (3-17) for mixed solvents.a^ = solvent activity, equation (2-1)B = virial coefficient. See Appendix A.B = Antoine constant, equation (A-l)b = see equation (3-14)bQ = equivalent hard-sphere volume of molecules.C = Antoine constant, equation (A-l)c = total ionic concentration, (ions/cc). equation (D-l)c = molar concentration, (g - mol/liter)D = dielectric constant of pure or mixed solvent

on a salt-free basis.d = density of mixed solvent on a salt-free basis.dQ = density of pure solvente = electronic charge, 4.8029 x 10 esuF = minimization function given by equations (5-7) and (5-20)f = fugacityf± = rational activity coefficientEG = excess Gibbs free energyEg = excess Gibbs free energy per mole-Eg^ = partial molar excess Gibbs free energyA H = see equation (A-23)h , v = solvation number at infinite dilution of the positive ion in a° single solvent. See Table 5.8

h _K = solvation number at infinite dilution of the negative ion in a0 single solvent. See Table 5.8

h+k = solvation number of the positive ion as a function of solventmole-fraction. (See equation (3-33)) for binary electrolytic solutions and equations (6-8) and (6-lla) and (6-llb) for ternary electrolytic solutions.

h , = solvation number of the negative ion as a function of solventmole-fraction. Set equal to zero for binary and ternary electrolytic solutions.

287

= ionic strength, equation (3-12)*16= Boltzmann constant, 1.38045 x 10 erg/deg

= see equation (D-2)= molecular weight of mixed solvent on a salt-free basis, equation (3-9)

= molality (g - mol/kg solvent)23 -1= Avogadro's number, 6.0232 x 10 mole

= number of moles = total vapor pressure, mm Hg = pure-component vapor pressure = group area parameter, equation (3-20)= pure-component area parameter= gas constant, 1.98726 cal/deg mole or 82.0597 cc atm/deg mole = group volume parameter, equation (3-36)= mean radius of gyration= pure-component volume parameter, equation (3-35)= crystallographic radii. Table 5.4 = intermolecular distance = temperature, °K or °C = reduced temperature= UNIQUAC binary interaction parameter, equation (3-27)= molar volume of the salt-free mixed solvent equation (A-3)

= saturated liquid volume, equation (B-l)= van der Waals group surface volume, equation (3-39)= partial molar volume of component i= characteristic volume, equation (B-l)= molar volume of pure solvent= number of groups of type k in molecule i.= weight of salt or solvent in equation (E-l)= nonpolar acentric factor= parameter in equation (0-1)= acentric factor determined from the Soave equation of state.= liquid-phase mole fraction equations (3-24) - (3-26)= liquid-phase mole fraction on a salt-free basis, equation (3-10)

= liquid-phase mole fraction on a solvated basis, equations (3-30) - (3-32)

289

y = vapor-phase mole fractionZ = compressability factor, equation (A-3)z = ionic chargeGREEK LETTERS7 . = solvent activity coefficient% = mean activity coefficient of the saltA = indicates difference between an experimental and

a calculated value€ = energy parameter for polar pairs of molecules,

equation (A-10)7) = association parameter. Table (A-2)

= area fraction of group k. equation (3-23)K = parameter in equation (D-l)fJL = dipole moment. Table (A-2)

jU,! = chemical potential of the standard state of component iV = number of ions

TT = 3.14159CT = molecular-size parameter for non-polar pairs, equation (A-15)

G~' = molecular-size parameter for pure polar and associating pairs,equation (A-17)

= osmotic coefficient, equation (4-2)<E>[ = segment fraction of component i. equation (3-29)cf). = fugacity coefficient of component i. equation (A-2)\ f jmu = UNIQUAC binary interaction parameter, equation (3-27)SUBSCRIPTS1 = positive ion2 = negative ion3 = solvent 14 = solvent 2+ = positive ion

= negative ion c = critical property c = molar basis cal = calculated valuecm = indicates pseudocritical mixing rule used.E = excess property indicated

exp = experimental valuei = component iij = interaction between molecules i andj = component jk = component k or group km = molal basism = group mmix = mixturen = group ns = saltT = totalSUPERSCRIPTS0 = pure component or standard state' = solvated basis*' = reduced property00 = infinite dilutionL = liquid phases = saturatedv = vapor phase

291

BIBLIOGRAPHY

Abrams, D.S. and Prausnitz, J.M., AICHE J., 1975, 21, 116.

Akerlof, G., J. Am. Chem. Soc., 1932, 11, 54.Albright, P.S. and Gosting, L.J., J. Am. Chem. Soc., 1946, 68, 1061.Bixon, E.; Guerry R.; Tassios, D., J. Chem. Eng. Data, 1979, 24, 9.Beckermann, E. and Tassios, D., ACS Advances in Chem. Ser., 1975, 155, 3.

Bockris, J. O'M. and Reddy, A.K.N., Modern Electrochemistry Vol. 1., A Plenum/Rosetta Ed., 1977.

Bondi, A., Physical Properties of Molecular Crystals, Liquids, and Glasses. Wiley, New York, 1968.

Boone, J., The Correlation and Prediction of Vapor-Liquid Equilibrium Behavior. PhD. dissertation, North Carolina State University at Raleigh, 1976.

Bjerrum, N., Z. anorg. Chem., 1920, 109, 275.

Bradley, D.J. and Pitzer, K.S., J. Fhys. Chem., 1979. 83, 1599.

Bromley, L.A., AICHE J., 1973, 19, 313.Campbell, A.N. and Bhatnagar, O.N., Can. J. Chem., 1979, 57, 2542.Ciparis, J.N., Data of Salt Effect in Vapor-Liquid Equilibrium,

Edition of Lithuanian Agricultural Academy, USSR, Kaunas, 1966.

Czerwienski, G.J.; Tomasula, P.; and Tassios, D., Vapor Pressures and Osmotic Coefficients: Electrolytic Solutions of Ethanol, to besubmitted for publication to Phase-Equilibrium.

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