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MlDBB@@K

@FA UEOUS

THERMODYNAMICS ELEC 9 ^ROLYTE

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BFAQUEOUS ELECTROLYTE

Theory & Application Joseph E Zemaitis,Jr.

Chem Solve, Inc.

Diane M. Clark

OLI Systems, Inc.

Marshall Rafal

OLI

Svstems.

Inc.

Noel C. Scrivner

E.I.

dubnt de Nemours & Co.,

Inc.

WILEY-

INTERSCIENCE

A

JOHN WILEY & SONS, INC., PUBLICATION

A publication of ^the

Design Institute for Physical Property Data (DIPPR) Sponsored by ^the

American Institute of Chemical Engineers

345 East 47th Street New York, New York 10017

No part. of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otheiwise, except as permitted under Scctions 107 or 108 of thc 1976 United Statcs Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923. (978) 750-8400, fax (978) 750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., I 1 1 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008.

0 Copyright ¹⁹⁸⁶

American Institute of Chemical Engineers, Inc.

345 East Forty-Seventh Street New York, New York 10017

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN 978-0-8 169-0350-4

AlChE shall not be responsible for statements or opinions advanced in papers or printed in its publications.

Dedication: 1 his book is dedicated to the mem- ory of Dr. Joseph F. Zemaitis, Jr. Dr. Zemaitis, a colleague and friend for many years, was responsible for the outline of this book and the writing of the first three chapters. In a larger sense, he provided for the synthesis of a very diverse body of work into a coherent frame- work for problem solving. We dearly hope that our dedication to and respect for his memory is reflected in the content of this work.

Diane M. Clark

Marshall Rafal

Noel C. Scrivner

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Acknowledgment: T h e authors wish to express their gratitude to Lisa Perkalis. This book could not have been completed without her word processing skills, patience and dedication in the face of never-ending “small” changes to the manuscript.

vi

Sponsors: T h e DlPPR sponsors of this project, and the technical representatives who served on the steering com- mittee are listed below. Stars indicate those companies which sup- ported the project throughout its three year duration.

Sponsors of DIPPR Project 811 (1 981 -1 983)

Partlclpant Technical Representative

*Air Products & Chemicals, Inc.

Allied Corporation

*Amoco Chemicals Corporation Chevron Research Company

*Chiyoda Chemical Engineering

& Construction Co., Ltd.

*E.I. du Pont de Nemours

& Company, Inc.

'El Paso Products Company

'Hatcon SD Group, Inc. Exxon Research and Engineering Co.

Hoff mann-LaRoche, Inc.

Hooker Chemical Company lnstitut Francais du Petrole

*Institution of Chemical Engineers M.W. Kellogg Company

Kennecott Copper Corporation

*Kerf--McGee Chemical Corporation

*Olin Chemicals Group 'Phillips Petroleum Company

*Shell Development Company

*Simulation Sciences, Inc.

*The Standard Oil Company (SOHIO)

*Texaco, Inc.

'Texasgulf, Inc.

Union Carbide Corporation

Dr. M.S. Benson Mr. W.B. Fisher Dr. D.A. Palmer Dr. W.M. Bollen Mr. T. Maejima Dr. N.C. Scrivner Mr. K. Claiborne Dr. C. Tsonopoulos Dr. C.EChueh Dr. R. Schefflan Dr. W.L. Sutor Dr. J. Vidal Dr. 6. Edmonds Mr. H. Ozkardesh Mr. D.S. Davies Dr. D.S Arnold Mr. W.M. Clarke Mr. M.A. Albright Dr. R.R. Wood Dr. C. Black Mr. TB. Selover Dr. C. Wu

Mr. S.H. DeYoung Mr. E. Buck

Federal Grantor

*National Bureau of Standards Dr. H.J. White, Jr.

'Supported Project for 3

year

life

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Foreword: A n international conference on the ‘Ther- modynamics of Aqueous Systems” sponsored by the American Institute of Chemical Engineers (AIChE), the National Science Foundation (NSF), and the National Bureau of Standards (NBS), was held in Warrenton, Virginia, on October 22-25,1979. The papers presented reflected a great deal of research on electrolyte solutions. However, it was apparent that there was no fundamental document to tie all of the different information together and so to form a framework for solving real problems.

Therefore, AIChE’s Design Institute for Physical Property Data (DIPPR) decided to publish this book to meet such a need. Through a cooperative effort by participating corporations, different correlations have been compiled and objectively compared to experimental data, in regions of industrial interest. Effective methods of finding and using data are also described. The Handbook incorporates and extends previous work in a well-organized, easy to understand format, with a focus on applications to serious industrial problems. It will become a cornerstone in the study of aqueous electrolyte thermodynamics.

Electrolyte mixtures come in various forms and add another dimension to the normal complexities of nonelectrolyte solutions: entirely new species can form in water, some of which are not obvious; components can precipitate; soluble components can affect the vapor pressure of the solution very significantly. In industrial applications, the solutions are often highly concentrated and encounter high pressures and temperatures. There- fore expertise in electrolyte systems has become increasingly critical in oil and gas exploration and production, as well as in the more traditional chemical industry opera- tions. A variety of correlations are available that can solve the problems that are encountered in industry, but which ones work best? This comprehensive handbook not only provides easy access to available data but also presents comparative studies of various correlations up to extreme conditions.

As the Chairman of the Technical Committee of AIChE’s Design Institute for Physical Property Data, I conceived this cooperative research project and chose as its leader, Dr.

Noel C. Scrivner of the E.I. duPont de Nemours Company, one of the leading practi- tioners of electrolyte thermodynamics. With his expertise and enthusiasm, Dr. Scrivner defined the work that needed to be accomplished, promoted the project until it was funded, and directed its completion. He was elected to head the steering committee of representatives from the supporting companies.

The initial work was carried out in 1981 by the Electrolyte Data Center of the National Bureau of Standards, under the direction of Dr. B.R. Staples. That work continued throughout the project, generating the bibliographies for electrolyte systems. One of these bibliographies, developed by R.N. Goldberg, is included in this volume.

A significant portion of the project funding came from the Office of Standard Refer- ence Data of the NBS (Dr. David R. Lide, Jr., Director). The NBS liaison to DIPPR was Dr.

Howard J. White, Jr. DIPPR provided additional funding along with administrative and technical assistance.

The task of creating the actual handbook was given to the late Dr. Joseph F. Zemaitis,

Jr., owner of Chem Solve, Inc. His intellectual contribution to this project will remain as his legacy. He wrote the first few chapters and carefully outlined the remainder of the book. His principal colleague in this work, Ms. Diane M. Clark, dedicated several years of creative effort to take the outline and complete the project. Another colleague of Or.

Zemaitis, Dr. Marshall Rafal, owner of OLI Systems, Inc., assumed the contractor’s responsibility for execution of the handbook and contributed some of the writing. Dr.

Scrivner gave technical direction and technical contributions to assure that the work would meet the standards set by the steering committee.

This book is dedicated to Dr. J.F.Zemaitis. Dr. David W.H. Roth, Jr., the administrative committee chairman of DIPPR, and I would like to express sincere appreciation to the other authors, the steering committee members, the corporate sponsors, and the National Bureau of Standards.

David A. Palmer, Chairman

DIPPR Technical Executive Committee and DIPPR Technical Committee

X

TABLE OF CONTENTS

I I1

111

IV

INTRODUCTION

THERMODYNAMICS OF SOLUTIONS Basic Thermodynamic Functions

Solutions

-

Basic Definitions and Concepts Equilibrium

-

Necessary Conditions

Activities, Activity Coefficients and Standard States EQUILIBRIUM CONSTANTS

Ionic andlor Reaction Equilibrium in Aqueous Solutions Solubility Equilibria Between Crystals and Saturated Solutions Vapor-Liquid Equilibria in Aqueous Solutions

Temperature Effects on the Equilibrium Constant

Estimating Temperature Effects on Heat Capacity and Other Equilibrium Constants from Tabulated Data

Pressure Effects on the Equilibrium Constant Appendix 3.1

-

C r i s s and Cobble Parameters

ACTIVITY COEFFICIENTS OF SINGLE STRONG ELECTROLYTES Thermodynamic Properties

History

Limitations and Improvements to the Debye-Huckel Limiting Law Further Refinements

Bromley's Method Meissner's Method Pitzer's Method Chen's Method

Short Range Interaction Model Long Range Interaction Model Temperature Effects

Bromley's Method Meissner's Method Pitzer's Method Chen's Method Bromley's Method Meissner's Method Pitzer's Method Chen's Method

NBS Smoothed Experimental Data Test Cases:

Application

HC1 K C1 KOH NaCl NaOH CaCl2 Na2SO4 MgSOO,

Bromley's Extended Equation MgS04 Test Case

1 11 13 14 16 17 25 27 30 31 32 33 36 36 39 45 48 55 56 64 67 71 76 76 82 84 84 84 86 89 90 91 93 94 96 98 99 103 106 110 113 117 121 124 127 128

Comparison of Temperature Effect Methods B romley

Meiss ner

Pitzer and Chen Experimental Data Test Cases :

HC1 at 50' Celsius KC1 at 80' Celsius KOH at 80° Celsius

NaCl at 100 and 300' Celsius NaOH at 35O Celsius

CaClz at 108.85 and 201.85' Celsius Na2S04 a t ^80° Celsius

MgS04 at 80' Celsius

Appendix 4.1

-

Values for Guggenheim's 0 Parameter Table 1 : B Values for Uni-univalent Electrolytes

Table 2: 0 and B Values of Ri-univalent and Uni-bivalent Methods for Calculating 8

Table 1 : B Values at 25'C Determined by the Method of Least Squares on Log Y to 1=6.0 (or less if limited data) Table 2: Individual Ion Values of B and 6 in Aqueous Solutions

at

^25OC

Table 3: Bivalent Metal Sulfates at 25OC

Table: Average Values of Parameter q in Equation (4.46) Electrolytes from Freezing Points

Appendix 4.2

-

Bromley Interaction Parameters

Appendix 4.3

-

Meissner Parameters for Selected Electrolytes

Table 1: Inorganic Acids. Bases and Salts of 1-1 Type Table 2: Salts of Carboxylic Acids (1-1 Type)

Table 3 : Tetraalkylammonium Halides

Table 4: Sulfonic Acids and Salts (1-1 Type) Table 5: Additional 1-1 Type Organic Salts Table 6: Inorganic Compounds of 2-1 Type Table 7:

Table 8: 3-1 Electrolytes Table 9: 4-1 Electrolytes Table 10: 5-1 Electrolytes Table 11: 2-2 Electrolytes

Table 1: Temperature Derivatives of Parameters for 1-1 Electrolytes Evaluated from Calorimetric Data

Table 2: Temperature Derivatives of Parameters for 2-1 and 1-2 Electrolytes Evaluated from Calorimetric Data

Table 3: Temperature Derivatives of Parameters for 3-1 and 2-2 Electrolytes from Calorimetric Parameters

T Values Fit for 5:olality Mean Ionic Activity Coefficient Data of Aqueous Electrolytes at 298.15 K

Appendix 4.4

-

Pitzer Parameters

Organic Electrolytes of 2-1 Type

Appendix 4.5

-

Pitzer Parameter Derivatives

Appendix 4.6

-

Chen Parameters Table:

V A CT IV IT Y CO E FF ICIE NTS 0 F MU L T 1 CO hlP ON E N T ST RO N G ELECTROLYTES

Guggenheim's Method for Multicomponent Solutions

130 130 131 131 132 133 136 140 144 150 153 159 162 165 165 166 167 170 170 173 174 175 175 179 179 131 181 182 183 183 185 185 18t1 18 7 187 188

188 189 190 19 1 191

205 209

xii

Bromley's Method for Multicornponent Solutions Activity Coefficients of Trace Components Meissner's Method for hlulticoinponent Solutions Pitzer's Method for Multicomponent Solutions Chen's Method f o r Multicomponent Solutions Application

Guggenheiin's Method Broinley's Method Meissner's Method Pitzer's Method Water Activities

B romley's Water Activity Pdeissner's Water Activity Pitzer's Water Activity Phase Diagram Calculations

Basic flow of t h e testing program Program block descriptions H:O

-

NaCl

-

KC1

H 2 0

-

NaCl - HC1 H 2 0

-

NaCl

-

NaOH Hz0

-

KC1

-

HC1 H z 0 - NaCl

-

CaC12 H$I

-

NaCl

-

MgC12 I 1 2 0

-

NaCl

-

NaeSOs H z O

-

KC1

-

CaC12 I 1 2 0

-

NaOH

-

Na2S04 H z 0

-

NaCl

-

CaSOs H 2 0

-

HCI

-

CaS04 H 2 0

-

CaC12

-

CaS04 H 2 0

-

Na2S04-CaSOk I i 2 0

-

MgS04

-

CaSOb

Appendix 5.1

-

Values for Pitzer's 0 and J, Parameters Table 1: Parameters f o r mixed electrolytes with virial Table 2:

Effects of Higher-order Electrostatic Terms

Table 3: Parameters for binary mixtures with a common coefficient equations ( a t 25OC:)

Parameters for the virial coefficient equations a t 25OC

ion at 25OC

V I ACTIVITY COEFFICIENTS OF S'TKONGLY COMPLEXING COMPOUNDS Identification of Complexing Electrolytes

Literature Review

U s e of Meissner's Curves

Comparison of Osmotic Coefficients Phosphoric A c i d

Sulfuric Acid Zinc Chloride Ferric Chloride Cuprous Chloride Calcium Sulfate Sodium Sulfate

Other Chloride Complexes Manganous Chloride Cobalt Chloride

211 212 214 2 19 223 23 1 232 234 235 236 238 238 240 24 1 242 243 245 250 261 268 278 29 0 299 315 326 335 347 363 357 369 377 384 394 385 386 389 39 9 406 407 407 407 409 415 419 424 428 43 1 436 440 440 443

xiii

Nickel Chloride Cupric Chloride

Activity Coefficient Met hods Summary

Appendix 6.1

-

Cuprous Chloride Table la: Interaction Parameters Table l b : Three Parameter Set

Table 2 : Equilibrium Constants and Heats of Reaction

Table 3a: Equilibrium Constants and Changes in Thermodynamic Properties for Formatiog-of CuC1; and c'uC1;- from CuCl(s) + nC1- = CuC1,+1

Table 3b: Equilibrium Constants and Changes in ThermodynRmic Properties for Formtttio of CuCI; and CuC1:- from

cu+

^t nC1- = c u c l y - l f

V I I ACTIVITY COEFFICIENTS OF WEAK ELECTROLYTES A N D MOLECULAR SPECIES

Setschihow Equation

Salting Out Parameter Determination by Randall and Pailey Salting Out Parameter Determination by Long and McDevit Salting Out Parameter Determination by Other Authors

Edwards, Maurer, Newman and Prausnitz Pitzer Based Method Beutier and Renon's Pitzer Based Method

Chen's Pitzer Based Method

Predictions Based upon Theoretical Equations Ammonia

-

Water

Carbon Dioxide

-

Water

Ammonia

-

Carbon Dioxide

-

Water Sulfur Dioxide

-

Water

Oxygen

-

Sodium Chloride

-

Water Conclusions

Pitzer Based Equations

Appendix 7.1

-

Salting Out Parameters for Phenol in Aqueous Salt Appendix 7.2

-

Solutions at Salting Out Parameters from Pawlikowski and Prausnitz 2 5 O Celsius

for Nonpolar Gases in Common Salt Solutions at Moderate Temperatures

Lennard

-

Jones Parameters for Nonpolar Gases as Reported by Liabastre (S14)

Salting Out Parameters for Strong Electrolytes in Equation (7.18) at 25OC

Temperature Dependence of the Salting Out Parameters for Equation (7.19)

Salting Out Parameters for Individual

Ions

for Equation (7.20)

Temperature Dependence of the Salting Out Constants for Individual Ions

Table 1:

Table 2:

Table 3:

Table 4:

Table 5 :

VIII THERMODYNAMIC FUNCTIONS DERIVED FROM ACTIVITY C 0 E F F I C I E

N

T S

Density

Binary Density

Multicomponent Density

Strong Electrolytes Which Complex Weak Electrolytes

447 449 453 455 457 457 457 458

458

459 479 485 486 496 491 5 03 503 505 512 517 517 524 524 524 524 524

538

540 540 540 541 542 542 551 554 554 558 558 5 59

xiv

Temperature Effects Illustrative Example

Heat of Formation at Infinite Dilution Enthalpy

Molecular Species Ionic Species

Range of Applicability Excess Enthalpy

Example

IX WORKED EXAMPLES Model Formulation Obtaining Coefficients Model Solution

Specific Examples

Sodium Chloride Solubility Water

-

Chlorine

Water

-

Ammonia

-

Carbon Dioxide Water

-

Sulfur Dioxide

Chrome Hydroxides Gypsum Solubility Water

-

Phosphoric Acid Table -1 :

Table 2 : Table 3 : Table 4 : Table 5 :

Appendix 9 . 1

-

Parameters for Beutier and Renon's Method

Temperature fit parameters for equilibrium constants Temperature fit parameters for Henry's constants Pitzer ion-ion interaction parameters

Temperature fit molecule self interaction parameters Dielectric effect parameters

Appendix 9 . 2

-

Parameters for Edwards, Maurer, Newman and Prausnitz Method

Table 1 : Table 2 : Table 3 : Table 4 :

Table 5 : Molecule-ion interaction parameters Table 1 : Pure component parameters

Table 2:

Table 3 : Interaction parameter ^a92 for polar-nonpolar mixtures Table ^{4 :}

Table 5 :

Temperature fit parameters for equilibrium constants Temperature fit parameters for Henry's constants Ion-ion interaction parameters

Temperature fit molecule self interaction parameters Appendix 9.3

-

Fugacity Coefficient Calculation

Nonpolar and polar contribution to parameters a and

B

for four polar gases

Parameter a12 for binary mixtures of nonpolar gases Interaction parameter

aL

for polar-polar mixtures Appendix 9.4

-

Brelvi and O'Connell Correlation for Partial Molar

Volumes

Table 1 ^: Characteristic volumes

Appendix 9.5

-

Gypsum Solubility Study Parameters at 25OC Table 1: Binary solution partimeters for the Pitzer equations Table 2 : Mixed electrolyte solution parameters for the Pitzer

equations

Table 3: Gypsum solubility product at ^25OC

560 560 563 564 564 566 567 568 568 575 577 584 586 588 589 606 595 6 4 1 650 663 675 683 683 683 684 685 685 69 1 691 6 9 1 692 692 693 699 700 7 0 1

7 0 1 702 703

704 705 706 706 706 706

X. AP P EN

D1

CES ?ll APPENDIX A

-

COMPUTER PROGRAMS FOR SOLVING EQUILIBRIA 713

PROBLEMS

APPENDIX B

-

SELECTED THERMODYNAMIC DATA 721

APPENDIX C

-

COMPILED THERMODYNAMIC DATA SOURCES FOR 737 AQUEOUS AND BIOCHEMICAL SYSTEMS: A n

Annotated Bibliography (1930

-

¹⁹⁸³⁾

INDEX 843

xvi

NOMENCLATURE

A

-

Debye-Huckel constant, log base 10, equation ( 4 . 3 1 )

A+

-

Debye-huckel constant

for

osmotic coefficients, log base e, A'

-

Debye-Huckel constant for activity coefficients, log base e,

equation ( 4 . 6 4 ) equation

a i

-

^activity

of

species i. equation ( 2 . 2 1 )

at

-

parameter used by Criss and Cobble's correspondence principle, equation ( 3 . 3 2 )

aW

-

water activity

a

-

distance of closest approach

or

core size. equation ( 4 . 3 3 )

n ^-

^{parameter for} Guggenheim's activity coefficient equation, log

B

-

interaction parameter for Bromley's activity coefficient B Y

-

parameter in Pi tzer's activity coefficient equation,

1 0 basis. equation ( 4 . 3 9 ) equation, equation ( 4 . 4 4 ) equations ( 4 . 6 3 ) . ( 4 . 6 6 )

bt

-

parameter used by Criss and Cobble's correspondence C 9

,Cp

-

heat capacity

Cpo

-

partial molal heat capacity

C

-

molarity

C

-

salt concentration in the Setschgnow equation (7.1) D

-

dielectric constant. equations ( 4 . 1 1 , ( 4 . 9 8 )

d

-

solution density, equation ( 2 . 3 4 1 , Chapter V l l l dL

-

solvent density

dW

-

pure water density, equation ( 4 . 9 8 ) E

-

intrinsic energy

e

-

electronic charge. equation ( 4 . 2 ) principle, equation (3.32)

-

Pitzer parameter, equation ( 4 . 6 5 )

-

xvii

F f

f Y

f V

gi

- G Gi gi

j H

- H Hi

I

K Ka q

KS

P

- KT

ko k k L

M

M S

m

NA

"i P

-

NRTL

parameter of equations

( 4 . 7 0 1 ,

- rational activity

( 2 . 3 3 ~ )

- Pitzer electrosta - fugacity of vapor

Chen's activity

( 4 . 7 1 )

coefficient (mo

coefficient equation,

e

fraction based), equation

ic function, eq ation

( 4 . 5 2 )

species

V .

equation

( 3 . 2 0 )

- number of grams of

i ,

i=O for solvent - Gibbs free energy

- partial molar Gibbs free energy, equation

( 2 . 1 4 )

- radial distribution function, equation

( 4 . 5 1 )

- ^enthalpy

- Henry'

s

constant

- partial molar enthalpy, equation

( 2 . 1 5 )

- ionic strength, equation

( 4 . 2 7 )

- equilibrium constant, dissociation constant

- equilibrium constant. equation

( 3 . 1 8 )

- solubility product, equation

(3.13)

- thermodynamic equilibrium constant, equation

( 3 . 6 )

- partial molal compressibility, equation

( 3 . 4 0 )

- Boltzmann's constant, equation

( 4 . 2 )

- Setschhnow salt coefficient, equation

(7.1)

- Ostwald coefficient

- solute molecular weight. equation

( 2 . 3 4 )

- solvent molecular weight, equation

( 2 . 3 4 )

- molality

- Avagadro'

s

number, equation

( 4 . 2 )

- number of moles of

i

- pressure, total vapor pressure

-

p a r t i a l p r e s s u r e

= -log K

-

M e i s s n e r ' s i n t e r a c t i o n p a r a m e t e r , e q u a t i o n (4.46)

-

g a s law c o n s t a n t

-

r a d i u s of t h e f i e l d a r o u n d a n i o n , e q u a t i o n ( 4 . l a )

-

e n t r o p y

-

g a s s o l u b i l i t y i n p u r e w a t e r , S e t s c h e n o w e q u a t i o n (7.1)

-

g a s s o l u b i l i t y i n s a l t s o l u t i o n . S e t s c h e n o w e q u a t i o n ( 7 . 1 )

-

t e m p e r a t u r e , K e l v i n s

-

t e m p e r a t u r e , C e l s i u s

-

volume

-

p a r t i a l molar volume, e q u a t i o n (2.16)

-

l i q u i d mole f r a c t i o n

-

v a p o r mole f r a c t i o n

-

a n y e x t e n s i v e t h e r m o d y n a m i c p r o p e r t y , e q u a t i o n ( 2 . 8 )

-

molar a c t i v i t y c o e f f i c i e n t , e q u a t i o n (2.3313)

-

n u m b e r of c h a r g e s on ion i

-

D e b y e - H u c k e l c o n s t a n t , log b a s e e

-

n o n r a n d o m n e s s f a c t o r in Chen's e q u a t i o n f o r act c i e n t s , e q u a t i o n (4.70)

-

p a r a m e t e r i n c o r r e s p o n d e n c e p r i n c i p l e e q u a t i o n , (3.36a)

v i t y c o e f f i -

e q u a t i o n

-

p a r a m e t e r for Gugyenheim's a c t i v i t y c o e f f i c i e n t e q u a t i o n . e q u a t i o n ( 4 . 3 8 )

-

p a r a m e t e r i n c o r r e s p o n d e n c e p r i n c i p l e e q u a t i o n , e q u a t i o n

-

P i t z e r i n t e r a c t i o n c o e f f i c i e n t ( 3 . 3 6 b )

-

Y i t z e r i n t e r a c t i o n c o e f f i c i e n t

-

P i t z e r i n t e r a c t i o n c o e f f i c i e n t

r -

reduced activity coefficient, equation ( 4 . 4 5 ) Y

-

molal activity coefficient, equation ( 2 . 2 2 )

6

-

Bromley parameter

for

the additive quality of ion interac- tion, equation ( 4 . 4 5 )

€gas

-

Lennard-Jones energy interaction, equation ( 7 . 1 8 )

O i j

-

Pitzer's coefficient for like charged ion interactions,

v i

-

partial molar Gibbs free energy

or

chemical potential, equations ( 5 . 3 2 1 , ( 5 . 3 3 )

equation ( 2 . 1 7 )

u i O

-

reference state chemical potential. equation ( 2 . 2 1 )

V

-

sum of cation and anion stoichiometric numbers

V

-

harmonic mean of v + and v-, equation ( 4 . 3 8 )

-

v i

-

stoichiometric number of ion i

n -

osmotic pressure, equation ( 4 . 5 1 ) P

-

charge density, equation ( 4 . 1 )

T

-

NWTL binary interaction parameter

of

Chen's activity coefficient equation. equations ( 4 . 7 0 ) . ( 4 . 7 1 )

,4

-

osmotic coefficient, equation ( 2 . 3 0 )

$a

-

ionic atmosphere potential, equation ( 4 . 1 4 )

J'ijk

-

Pitzer triple ion interaction parameter, equation ( 5 . 3 2 ) J l r

-

ionic potential

$t

-

total electric field potential, equation ( 4 . 1 ) Q

-

apparent molal volume

Subscripts : A

-

^anion

a

-

anion

C

-

cation

C

-

^cation

g

-

gas

1

-

any species j

-

any species L - liquid

m - molecular species S

-

sal t

U

-

molecular (undissociated species V - vapor

W

-

water

1,3,5...

. ^-

^cations

2 . 4 , 6 , . .

. ^-

^anions

* ^-

molecular species

Superscripts :

el - electrostatic ex

-

excess

lc

-

local composition

P - P i t z e r long range contribution of Chen's equation, equation ( 4 . 8 7 )

tr - trace amount

0

-

indicates reference state of pure (single solute) solutions

u

-

infinite dilution

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I nt rod uction

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INT

RODU CT

ION

In the past several years, interest in electrolyte phase equilibria has grown significantly. This growth in interest can be attributed to a number of evolving application areas and factors among which are:

o Recognition of the necessity to reduce pollutant levels

in

process waste The removal of sulfur by formation of gypsum is an example water streams.

of such an application.

o Development of new flue gas scruhbing systems using regenerative processes.

Scrubbing of C1, from incinerator streams and SO2 from flue gases are specific application examples.

o Recent escalation of the prices of oil and gas leading to the study and development of synthetic fuel processes i n which ammonia. carbon dioxide, and hydrogen sulfide are produced as by-products which usually condense to form aqueous solutions. Sour water strippers and amine scrubbers are specific processes developed in this area.

Most of the application areas mentioned above concern the vapor-liquid phase equilibria of weak electrolytes. However, in the past several years, consider- able interest has also developed in the liquid-solid equilibria of both weak and strong electrolytes. Application areas and factors that have affected this growth in interest include:

o Hydrometallurgical processes. which involve the treating of a raw ore or concentrate with an aqueous solution of a chemical reagent.

o The need of corrosion engineers to predict the scale formation capabilities of various brines associated with oil production or geothermal energy production.

o The need of petroleum engineers to predict the freezing or crystallization point of clear brines containing sodium, calcium, and zinc chlorides and bromides to high concentrations.

o The need for waste water clean up customarily done by precipitation of heavy metals.

o Sea water desalination.

o Crystallization from solution in the manufacture of inorganic chemicals.

3

AQUEOUS

ELECTROLYTE THERMODYNAMICS

o

Specific ion electrolytes o Ion exchange

Specific processes which typify these application areas are:

o Treatment of gypsum which is formed in waste water cleanup.

o Several processes involving formation of Cr(OH)3. These processes include:

-

cooling tower blowdown

-

plating processes

-

manufacture of chrome pigment

U s e of a simple solubility product (e.g. Lange's Handbook) for Cr(OH), is invalid since precipitation involves intermediate complexes which f a r m to a significant degree.

These are just

a

few of the application areas of electrolyte phase equilibria which have generated an interest in developing a better understanding of aqueous chemistry. In contrast to other systems, in particular hydrocarbon systems, design-oriented calculation methods are not generally available for electrolyte systems. In the undergraduate education of chemical engineers, little if any mention of electrolyte thermodynamics is made and most chemical engineering thermodynamic texts ignore the subject completely. If an engineer is exposed to electrolyte thermodynamics at all during undergraduate education, the subject is taught

on

a rudimentary level so that many misconceptions may arise. For example, in manufacturing chrome pigment, noted above, use of the solubility product in order to determine solubility leads to very large errors since the solubility product approach totally ignores formation of complexes and their attendant effect on system state. Ry contrast. for hydrocarbon systems, most engineers are presented with a basic groundwork that includes design-oriented guidelines for the calculation of vapor-liquid equilibria of simple a s well as complex mixtures of hydrocarbons. In addition, during the last decade, the education of a chemical engineer has usually included the introduction to various computer techniques and software packages for the calculation of phase equilibria of hydrocarbon systems. This has not been the case for electrolyte systems and even now it is generally not possible for the engineer to predict phase equilibria of aqueous systems using available design tools.

I. .Introduction

For

processes involving electrolytes, the techniques used in the past and even some still in use today at times rely heavily on correlations of limited data which a r e imbedded into design calculation methods oriented towards hydrocarbon systems. Worse yet, until recently, limited use of the limited data available has occasionally led to serious oversights. For example, in the C1, scrubbing system noted earlier, the basic data published in Perry's Chemical Engineers' Handbook which has been used for years is actually in error. Hampering the improvement of the design tools or calculative techniques have been several restrictions including :

o A lack of understanding of electrolyte thermodynamics and aqueous chemistry.

Without this understanding, the basic equations which describe such systems cannot be written.

o

The lack of a suitable thermodynamic framework for electrolytes over a wide range of concentrations and conditions.

o The lack of good data for simple mixtures of strong andlor weak electro- lytes with which to test or develop new frameworks.

o The diversity in data gathering because of the lack of a suitable thermody- namic framework. A s a result of different thermodynamic approaches, experimental measurements to develop fundamental parameters have often led to the results being specific to the system studied and not generally useful for applications where the species studied are present with other species.

Fortunately, in the last decade, because of the renewed interest in electrolyte thermodynamics for the reasons described earlier, a considerable amount of work in the field of electrolyte thermodynamics has been undertaken. Techniques for the calculation of the vapor-liquid equilibria of aqueous solutions of weak electrolytes are being published. New, improved thermodynamic frameworks for strong electrolyte systems are being developed on a systematic basis. With these developments. DIPPR established, in 1980, Research Project No. 811. The objective of this project i s to produce a "data book" containing recommended calculation procedures and serving as a source of thermodynamic data either through recommended tabulated values or through annotated bibliographies which point t o suitable sources.

5

AQUEOUS ELECTROLYTE THERMODYNAMICS

In order to meet the objectives of this project, several phases were established.

The specific phases involved:

1) Definition of the project scope.

2) Gathering available data and literature references for the preparation of test data sets and data tables contained in the report.

3) Review of thermodynamics, techniques and recent developments in order to select those techniques to be evaluated in the final report.

4) Testing and comparison of the various techniques against selected test data sets.

5) Development of the handbook in order to present the results of the project in a useful and readable form.

This book is a result of DIPPR Research Project No. 811. In it, the reader and user will find a systematic presentation of electrolyte thermodynamics, from the basic definitions of equilibrium constants of ionic reactions. to the prediction of activity coefficients of various species in rnulticornponent aqueous solutions of strong andlor weak electrolytes and the resulting phase equilibria calculative techniques. For several systems, data are presented and calculative techniques are illustrated. The goal of this book is for the engineer, faced with the need for solving industrially important problems involving aqueous solutions of electrolytes, to be able to understand t h e possible alternatives available and apply them to the problem at hand, either with available data or through available data prediction and analysis techniques. Several examples

will

be used to illustrate the calculative techniques necessary for different types of problems. The examples chosen are of a size that can be solved with limited computer facilities. The techniques can be expanded for more complex problems.

In order to better understand the basis for the chapters which follow. let us consider the formulation of a predictive model for a particular aqueous based electrolyte system. The example chosen involves water-chlorine. The reactions to be considered are:

I. Introduction

H,O(vap)

=

HzO(aq) Cl,(vap) = Cl,(aq)

Cl,(aq) + H,O = H(ion) + Cl(ion) + HClO(aq1 HClO(aq) = H(ion) + ClO(ion)

H,O(aq) = H(ion) + OH(ion)

The problem is to predict the resulting phase distribution and phase compositions.

Or, in other words:

Given: Temperature (T), Pressure (PI and inflow quantities H,O(in) and Cl,(in) Determine:

1) Total vapor rate.

V

2) Rate of HzO(aq)

3) Vapor phase partial pressures, pHzO and pClz

Liquid phase concentrations, usually expressed in molality (gm moles solute per 1000 gms solvent-Hz O(aq)), mHCIO(aq), mC12 (aq)

,

"H(ion), mOH(ion), mCl(ion), mClO(ion)

The problem, stated above for water-chlorine, is typical of all calculations involving electrolytes.

For the water-chlorine system above, a set of ten equations is required in order to solve for the ten unknowns just described. These equations are:

Equilibrium Equations: Equations 1-5

Equilibrium K equations are written, one for each reaction.

equations are of the form:

As we shall see. these 'ip

n

^"iR (mi,) 'i R (miP)

n

^{( y )v'ip}

i P i P i R (

K =

where.

K = The thermodynamic equilibriun constant; a function of T and P.

AQUEOUS ELECTROLYTE THERMODYNAMICS

yip, yiR = Activity coefficient or, for vapors, fugacity coefficient of the ith product and reactant respectively; a function of T, P and canposi t ion

vip, viR = Stochianetric coefficient of the ith product and reactant

mip, miR = Molality

or,

for vapors, partial pressure of ith product and respectively

reactant respectively.

For

our H20-Cl2 system (using y for activity coefficient, a for activity, f for fugacity coefficient and p for vapor partial pressure) w e thus have five such equations :

- -

‘H(ion) % ( i o n ) ‘Cl(ion) mCl(ion) YHCIO(aq) %ClO<aq) K ~ 1 2 ( a s ) yC12 ( a q ) m C12 (aq) ‘H20(aq)

m m

KHCIO(aq) = YHCIO(aq) %ClO(aq)

%,O(aq) aH 2 0 ( aq)

ion) H( i o n ) ‘CIO( io n ) C ~ O ( i o n )

-

-

YH( ion) “‘HC ion) ‘OH( ion) “‘OH( ion)

Electroneutrality Equation: Equation 6

The electroneutrality equation states that the solution is. a t equilibrium, electrically neutral. Generally stated, the equation is:

total molality of cations = total molality of anions or, for our H20-C12 system:

m m ^m

% ( i o n ) = OH(ion) + C l ( i o n ) + CIO(ion)

1. introduction

Material Balances: Equations 7-10

The requisite material balances. In this case, four such balances are needed to make the number of equations equal to the number of unknowns. The equations are:

Vapor Phase

= %,avap) ⁺

P c ~

(vap)

These ten equations can, with a reasonable computer, be solved for the ten unknowns in question. Alternatively, by carefully organizing the calculations and making some simplifying assumptions, for a simple system such a s CL,-H,O trial and error using a calculator is also feasible. What has been understated thus far is that, embedded in equations ^1-5, the K equations, is the essential complexity of the electrolyte calculations. The variables, K(T,P) and y(T,P.m) are often highly nonlinear functions of the state variables shown. The purpose ^of this book is thus to describe:

1) The underlying physical chemistry theory which governs determination of 2) Practical methods for calculation, estimation or extrapolation

of

these values.

K and ^y.

9

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I I:

Thermodvnamics

II

of

Solutions

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THERMODYNAMICS

OF

SOLUTIONS

In order to calculate the equilibrium composition of a system consisting

of

one

or

more phases in equilibrium with an aqueous solution of electrolytes, a review of the basic thermodynamic functions and the conditions of equilibrium is impor- tant. This is particularly true inasmuch as the study of aqueous solutions requires consideration of chemical and/or ionic reactions in the aqueous phase as well as a thermodynamic framework which is, for the most part, quite different from those definitions associated with nonelectrolytes. Therefore, in this section we will review the definition of the basic thermodynamic functions. the partial molar quantities, chemical potentials, conditions of equilibrium, activities, activity coefficients, standard states, and composition scales encountered in describing aqueous solutions.

Basic Thermodynamic Functions

The thermodynamic properties of a system at equilibrium consists of two types of properties, intensive and extensive properties. The most common intensive properties encountered are the temperature, T. and pressure,

P,

which are independent of the size of a measurement sample and are constant throughout the system. In fact, our definition of true equilibrium, to be described later requires T and P to be uniform throughout the system and the constituent phases.

The most common extensive properties are volume, V and mass. As one would suspect, these extensive properties are proportional to the size of a measurement sample.

The thermodynamic properties most often encountered in describing phase equilibria of a system are functions of the state of the system. This is important since the calculation of these thermodynamic properties depends only on the existing state of the system and not the route by which this state has been reached. The following energy and energy related properties are extensive properties if they refer to the system as a whole:

The intrinsic energy E

The enthalpy H = E + P V

AQUEOUS ELECTROLYTE THERMODYNAMICS

The entropy S

The Gibbs free energy

The heat capacity at constant pressure G =

H - TS

Cp =

(g)p

The above thermodynamic properties are intensive properties when their values are expressed on a per mole basis. Other useful relationships which we will encounter are obtained through differentiation of these basic thermodynamic functions and include :

dE = TdS

-

PdV (2.1)

riH = TdS

+

VdP (2.2)

dG = -SdT + VdP (2.3)

These relationships express the first and second laws of thermodynamics for a closed system. Furthermore, from the last expression above we can also obtain

Rewriting the

T h i s equation which will be constants and

(%)*

⁼

(

% )

P =

-s

definition of the Gibbs free energy,

can be arranged to give the Gibbs-Helmholtz equation, an expression very useful in calculating the effect of temperature on equilibrium is given by:

-H

-

Solutions

-

Basic Definitions and Concepts

The pure substances from which a solution can be made are called the components, or constituents of a solution. The extensive properties of a solution are deter- mined by the pressure, temperature, and the amount of each constituent. The intensive properties of a solution are determined by the pressure, temperature and the relative amounts of each constituent. or in other words b y the pressure, temperature and composition of the solution. For aqueous solutions. the most commonly used measurement of composition of the solution is the molality, m.

Molality is defined as the number of moles of a solute in one kilogram of the solvent, and for aqueous solutions the solvent is water. One of the advantages of using the molality scale for concentration is that it is independent of temperature

11. Thermodynamics of Solutions

and thus, the density of the solution does not need to be known in order to determine the composition on a mole basis as would be required with the unit of concentration, molarity. The molality of a solute i in water is given by 1000gi/(M go) or 1000

ni/go

where gi and go are the number

of

grams of solute and solvent,

M

is the solute mo!ecule weight and ni is the number of gm-moles of solute.

The thermodynamic analysis of solutions is facilitated by the introduction of quantities that measure how the extensive thermodynamic quantities

(V,

E,

HI)

G ,

...I

of the system depend on the state variab!es T , P, and ni. This leads to the definition of partial molar quantities where, i f , w e let Y be any extensive thermodynamic property, we can define the partial molar value of Y for the ith component as:

‘i = ( e ) T , P , n j + j ^(2.8)

where nj stands for all the mole quantities except ni. It is important to note that the partial molar

(or

partial mold, which has the same meaning) quantities pertain to the individual components of the system and are also properties of the system ^as a whole. Furthermore

dY = (%)T.P,nj+l + ( e ) T , P , n jS2 dn2 +

...

-

- -

dY = Y1 dnl

+

Y2 dn2 + Y3 dn3 +

...

or dY =

c - Yi

dni (2.9)

i

Partial molar quantities are intensive properties of the solution since they depend only on the composition of the solution, not upon the total amount of each component. If w e add the several components simultaneously, keeping their ratios constant, the partial molal quantities remain the same. W e can thus integrate the above expression keeping “1, “2,

...

ⁱⁿ constant proportions and find, while holding temperature and pressure constant, that

-

-

Y = Yl”l + Y2%

+ ...

or Y

= c

Yini i

Furthermore, w e can differentiate this expression

to

obtain d Y = Y1 dnl

+

r q d Y +

Y2

dn2 + npY2 +

...

(2.10)

15

AQUEOUS ELECTROLYTE THERMODYNAMICS

Since

-

dY =

c

Yi dni i

a s has previously been shown, our total differential, equation

O = C n i i Since Y represents any G , for

Y we

obtain the O = C n i

i

C ni dYi

-

i ^(2.11)

( 2 - 9 ) we

can

then substitute this expression

for

dY into (2.111, and obtain

extensive property, if we substitute the G i b b s free energy, Gibbs-Duhem equation

&i

-

which is very useful in the thermodynamics of aqueous solutions.

The partial molar quantities of interest in aqueous solutions molar Gibbs free energy, enthalpy and volume which are defined

The partial molar G i b b s free energy is also known or defined

(2.13)

a r e the partial respectively as:

(2.14)

(2.16) a s the chemical

<2.17)

Equilibrium

-

Necessary Conditions

With aqueous solutions of electrolytes

w e

have two types ^of equilibrium to consider: phase equilibrium and chemical or ionic reaction equilibrium. Phase equilibrium of interest are primarily vapor-liquid and liquid-soEd

,

though vapor- liquid-solid is often of great importance as. for example, in carbonate systems.

The necessary condition of phase equilibrium is that the chemical potential of any species i

in

phase a is equal to the chemical potential of that same species i in phase b or

Visa = p i , b (2.18)

16

11. Thermodvnamics

of

Solutions

For chemical or ionic equilibria in a particular phase, the condition of equilibrium is of the same form as the chemical equation. Thus if the reaction at equilibrium is represented by

a A + h B = c c + d D (2.19a)

the condition of chemical equilibrium in a particular phase would be denoted by aPA ⁺ =

clJc

⁺

db

which can be represented in a generalized form as c v i Ui = ₀

(2.19b)

(2.20) where v i is the stoichiometric coefficient of species i in the reaction of interest; it is positive if the species is a product and negative if a reactant.

The two conditions of equilibrium for phase and chemical equilibrium can be combined to represent the heterogeneous liquid-solid equilibrium of an aqueous solution of a salt B in equilibrium with the solid of salt B

P B , s = lJBB,aq

The chemical potential of the solid crystal salt B is in phase equilibrium with the dissolved salt B in the liquid or aqueous phase. In aqueous systems w e are primarily dealing with salts of strong electrolytes, which i n water dissociate completely to the constituent cations and anions of the salt. The chemical potential of the dissolved salt is then given by

pB.aq = 'c pc ⁺ va Fa

where vc and va represent t h e stoichiornetric number of cations and anions while p and pa are the chemical potential of the cation and anion respectively. Thus t h e condition of equilibrium for a strong electrolyte dissolved in water and in equilibrium with its crystalline phase becomes

P B , s = v c P c ⁺ v a pa

Activities, Activity Coefficients and Standard States

It is generally more convenient in aqueous solution thermodynamics to describe the chemical potential of a species i. in terms of its activity. ai. The basic relationship between activity and chemical potential was developed by G. N. Lewis who first established a relationship for the chemical potential for a pure ideal gas, and then generalized his results to all Systems to define the chemical potential of species i in terms of its activity ai as

pi(T) =v;(T)+ RT In (ai) (2.21)

17

AQUEOUS ELECTROLYTE THERMODYNAMICS

H e r e poi is a reference chemical potential or the standard chemical potential at an arbitrarily chosen standard state. The activity is a measure of the difference between the component's chemical potential at the state of

interest

and

at

its standard state. Thus as the chemical potential of component i approaches the chemical potential of component i at its arbitarily chosen standard state, the component's activity approaches unity.

For aqueous solutions in which the composition of the solution is expressed in terms of molality, the arbitrarily chosen standard state is the hypothetical ideal solution of unit molality at the system temperature and pressure. It is chosen ^so that a s the molality approaches zero. the ratio of ailmi tends to unity. This ratio ailmi is called the molal activity coefficient, yi.

and

Yi = ailmi Yi-1 as m i 4 0

p i =

u:

+ RT In

(vimi)

^(2.22)

The standard state chosen must be such that these equations hold a t all pressures and temperatures. I n these equations there appears to be an inconsistency of units, since the activity coefficient, Y i , is dimensionless, whereas molality has the units moleslkilogram. ^To avoid this inconsistency the activity coefficient should be defined as

Yi = ainP/mi ^(2.23)

where mo is the unit molality. For convenience, particularly in writing expres- sions for equilibrium constants, the normal convention in aqueous chemistry is to omit the writing of m0 and use the form ai = Yimi with t h e understanding that the activity and the activity coefficient are dimensionless.

W e have defined the activity and activity coefficient of a species i. Unfortu- nately, in solutions ^of electrolytes we find that we cannot make a solution containing only cations or only anions and need to introduce a mean or average activity coefficient. For example, when one mole of NaCl is dissolved in a kilogram ^of water, w e have created a one mold solution of NaCl which fully dissociates

to

form one mole of sodium ions and one mole of chloride ions. The chemical potential of the dissolved sodium chloride is given by

11. Thermodynamics of Solutions

or in the more general

a 'a

= vc P C + v 'salt aq

= ( V :p +

va

pi) +

v

KC ln(ycmc)

+

vaKl? ln(yama) (2.24)

C C

'salt aq

Rearranging and designating ( v c P: +

va

as

ugalt ,

we obtain aq

From this expression comes the definition of the mean activity coefficient.

Y,,

in terms of the ionic activity coefficients yc and ya. The mean activity coefficient is the property which is determined or calculated from experimental measurements. A similar expression results for the mean rnolality m+. which is not generally used in reporting experimental measurements :

-

v

_c

v

a ) l l v Y, - = 'Yc Ya

m, ^- = (mc v ^ma v

(2.26)

(2.27) where v = vc

+

va, the stoichiometric number of moles of ions in one mole of salt. The expression for the chemical potential of the dissolved salt could be

(2.28) Since we are concerned with solutions we need also to determine the activity of the solvent which, for our particular interest in aqueous solutions, is the activity of water, aw. The activity of water is related to the chemical potential of water by

pw = p; + RT In aw ^(2.29)

where the standard state is, by convention, pure water at the system temperature and pressure. Thus for the pure solvent, water, the activity aw = 1.

In real solutions, the activity of water is quite close to one as the dilution ratio of the dissolved salts increases. In order to accurately represent the activity of water for dilute solutions. several significant digits would be

AQUEOUS

ELECTROLYTE THERMODYNAMICS

required. To avoid this problem, in many compilations of data it is common to tabulate data in terms of the osmotic coefficient,

9 .

The osmotic coefficient i s defined for any aqueous solution as:

-1000 In aw

'

^{= 18.0153 C} _i ^{vi mi (2.30)}

,The osmotic coefficient tends to approach unity a s the solution becomes infinitely dilute. The Gibbs-Duhem equation, equation ^(2.13). which was presented earlier at a constant temperature and pressure, can be expressed as:

C ni p i = 0 (2.31)

i

This relation can be applied to an aqueous solution of a single solute salt in water and, after suitable manipulation, the following expression representing the interrelationship, between the water activity. aw, the mean activity coefficient of the dissolved salt, y*, and the osmotic coefficient.

4,

can be obtained:

rn

d(ln(m y, _- 1) = -55.50837 d(ln

aJ

= vd(m

4)

(2.32)

This expression can be used in different ways. I n the treatment of experimental data one of the most common methods of determining activity coefficient data involves the measurement of the water vapor pressure of the aqueous solution a s a function of concentration through various experimental techniques. Through proper manipulation of this equation and the acceptance of ^a theory of ionic solution behavior. the above expression can be used to calculate the mean activity coeffi- cient from the experimentally determined osmotic coefficient. This expression is also useful for calculating the water activity a s a function of concentration if one has developed a model of the behavior of the activity coefficients with concentration and the parameters of the mean activity coefficient model have been experimentally determined.

The relationships expressed in this chapter, while discussed on a rnolality scale, hold true for calculations based on other scales. The basic definition of the activity coefficient is:

a

(rn)

yi

- -m

_i

on a m l a l scale:

yi ^- m l a l activity coefficient

(2.33a)

20

11. Thermodynamics

of

Solutions

m

-

m l a l i t y

a . ( m )

-

a c t i v i t y on m l a l basis i

ai(c) yi =

-

_'i

on a molar scale:

- m 'i

i - m

C

on a mle fraction scale:

-

ra f i

a r activity coefficient a r i ^ty

a . ^(x) f i

- - +

i

ional activity coefficient

(2.3313)

(2.353

-

rmle fraction

X i

a.(x) ₁

-

a c t i v i t y on a m l e fraction basis

When necessary; activity coefficients may be converted from one scale to another via the following relationships presented by Robinson and Stokes (1) ^:

f , _- = (1. + .001 Ms v m) y,

-

(2.34a)

d + .001 c ^(V Ms - MI

f,

^{= do}

d

-

-001 c M y? =

(

^{d o}

(2.34b)

(2.3412) (2.34d)

where: v ^,

-

s t o i c h i m e t r i c mmber = v+ + v- d

-

solution density

do

-

solvent density

M

-

mlecular weight of the solute kls

-

m l e c u l a r weight of the solvent

Throughout this book the molecular weights used a r e from the NBS Tech Note ²⁷⁰ Series (11-16).

AQUEOUS

ELECTROLYTE THERMODYNAMICS

Occasionally, reference is made to the "reduced activity coefficient", as in Meissner's work discussed beginning in Chapter IV. It is defined as:

r

^E yl/z+z-

where ^z+ and ^z, are the absolute value of the charges on the cation and anion.

2 2