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NSRDS

Electrolytic Conductance

And the Conductances of the

Halogen Acids in Water

u.s. DEPARTMENT

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NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. Today,

in addition to serving as the Nation’s central measurement laboratory, the Bureau is a principal

focal point in the Federal Government for assuring maximum application of the physical and

engineering sciences to the advancement of technology in industry and commerce. To this end

the Bureau conducts research and provides central national services in four broad program

areas. These are: (1) basic measurements and standards, (2) materials measurements and

standards, (3) technological measurements and standards, and (4) transfer of technology.

The Bureau comprises the Institute for Basic Standards, the Institute for Materials Research, the

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THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

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1 Headquarters and Laboratories at Gaithersburg:, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

- Located at Boulder. Colorado 80302. Located at 5285 Port Royal Road, Springfield, Virginia 22151.
UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary,

NATIONAL BUREAU OF STANDARDS# Lewis M. Branscomb, Director

I « I

Electrolytic Conductance and the Conductances of the

Halogen Acids in Water

Walter J. Hamer and Harold J. DeWane

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

NSRDS

NSRDS-NBS 33

Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. (U.S.), 33, 37 pages (May 1970) CODEN: NSRDA

Issued May 1970

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C 13.48:33), Price 5U cents
NATIONAL BUREAU OF STANDARDS

JUL 2 3 1S70

QC/O n

US 73

DO,

Foreword

j The National Standard Reference Data System provides effective access to the quantitative

data of physical science, critically evaluated and compiled for convenience, and readily accessible

through a variety of distribution channels. The System was established in 1963 by action of the

President’s Office of Science and Technology and the Federal Council for Science and Technology, with responsibility to administer it assigned to the National Bureau of Standards.

The System now comprises a complex of data centers and other activities, carried on in aca- demic institutions and other laboratories both in and out of government. The independent operational status of existing critical data projects is maintained and encouraged. Data centers that are components of the NSRDS produce compilations of critically evaluated data, critical reviews

of the state of quantitative knowledge in specialized areas, and computations of useful functions

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experimental techniques. They are normally closely associated with active research in the relevant

field.The technical scope of the NSRDS is indicated by the principal categories of data compilation

projects now active or being planned: nuclear properties, atomic and molecular properties, solid state properties, thermodynamic and transport properties, chemical kinetics, and colloid and

surface properties.

The NSRDS receives advice and planning assistance from the National Research Council

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activities. This assembly of advisory services contributes greatly to the guidance of NSRDS

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The NSRDS-NBS series of publications is intended primarily to include evaluated reference

data and critical reviews of long-term interest to the scientific and technical community.

Lewis M. Branscomb, Director

© 1970 by the Secretary of Commerce on behalf of the United States Government.

Library of Congress Catalog Card Number: 79-604209

II
Preface

This document represents the initial effort in the compilation and critical evaluation of available data on the electrolytic conductance of aqueous solutions of electrolytes. A readily available source of such data is presently needed since the last such compilation and evaluation was made

in the early 1930’s. Although, for the most part, these data are available in the literature, they

have not been brought together in a single volume nor have they been treated in a consistent manner. For example, some data in the literature are given in international ohms, others in absolute

ohms. Methods used to obtain limiting equivalent conductances differ widely. Also most of the

literature data are expressed on the old atomic weight scale of 16 for naturally occurring oxygen

rather than on the 12 C scale. All data reported here have been converted to absolute ohms, and

the 12 C scale of atomic weights; data presented in subsequent reports will be treated in a similar manner. Modern theories of conductance, especially those developed by Peter J. W. Debye, Lars

Onsager, and Raymond M. Fuoss are or will be used in the critical evaluation of the data, in particular for the determination of limiting equivalent conductances. Also when the available data

are of sufficient accuracy, modern theories will be used to determine apparent “ion-size” parameters; such determinations will not be made in the present document, however. In general, the

data on equivalent conductances will be presented in tables. Equations, based in part on theory,

which give equivalent conductances as a function of concentration will also be included. In some

cases, empirical equations will be used in the entirety, but these will reproduce the experimental

results which are, in essence, the essential product desired by the user. For example, empirical

equations are used to express the conductances of HC1 at —10 and —20 °C; in the limit of zero concentration the solutions are solid, i.e., in the frozen state. In the present document, definitions relating to the conductance of electrolytic solutions are first presented. These are followed by some general considerations of the migration of ions

and general laws governing the movement of ions under applied potential gradients as are in- volved in electrolytic conductance. Conductance relations (equations) are then given, and these

are followed by a condensed treatment of the Debye-Hiickel-Onsager-Fuoss theories of electrolytic

conductance, given in general terms (a more detailed treatment is given in Appendix A, but even

there the minor details and mathematical complexities are omitted). Values of the parameters,

namely B i, B2 , E\, and E2 in the theoretical equations are given for temperatures from 0 to 100 °C

for the convenience of those engaged in conductance measurements. They are not needed, however, by those who are interested only in data on the equivalent conductance of an electrolytic

solution. Finally, tables of data on the electrolytic conductance of HF, HC1, HBr, and HI are given

for various concentrations and temperatures in tables 10 through 19. The constants governing the

dissociation of HF from 0 to 25 °C are given in table 9.

Ill
For the convenience of the reader a glossary of symbols is given at the end of the document.

In 1968, the Commission on Symbols, Terminology, and Units of the Division of Physical

Chemistry of the International Union of Pure and Applied Chemistry in a tentative proposal

(IUPAC Bulletin No. 32) did not include “equivalent conductance” among its list of definitions,

but replaced it by “molar conductance” and assigned the symbol A to the latter. Accordingly, if this proposal is eventually adopted, the term “equivalent” in the present document should be

replaced by “molar” and the definitions of “molar conductance” and “limiting molar conductance”

given on page 1 eliminated (historically, molar conductivity, as defined on page 1, had use in determining the mode of ionization of salts when it was not possible to determine the equivalent weight

of the salt). Under this proposal of IUPAC, the meaning of a mole of salt would need to be clearly

specified. Thus, the molar conductance of MgCU, for example, could be given as

A(£ MgCl2)= 129 a-1 cm2 mol-1 or as A(MgCl2 ) = 258 D_1 cm2 mol_1

(see IUPAC Bulletin No. 32) depending on how a mole of MgCU is specified (of course, Avogadro’s

constant would apply only to the latter definition).

Acknowledgment

The authors wish to express their appreciation to Professor Raymond M. Fuoss of Yale University for his advice during the course of this work.

IV
Contents

Page Foreword II Preface Ill

1. Definitions and symbols 1

2. General considerations 3

3. General laws . 3

4. Conductance relations 4

5. Theoretical expressions for equivalent conductance... 5

6. Determination of Ao 8

7. Equivalent conductances of HF, HC1, HBr, HI 8 HF 8 HC1 10 HBr 12 HI 12

8. Conclusions 13

9. References 13

10. Appendix A. The Debye-Hiickel-Onsager-Fuoss

theory of electrolytic conductance 14

A. Bibliography for Appendix A 18

11. Appendix B. Densities of aqueous solutions of HF 18

12. Appendix C. Use of the Systeme International d'Unites 19

13. Tables •. 19 Table 1. Specific conductances of standard aqueous

solutions of KC1 19 Table 2. Physical properties of water 20

Table 3. Values of the Debye-Hiickel-Onsager constants B i and B2 for equivalent conductances for 1-1 aqueous solutions from 0 to 100 °C 20 Table 4. Values of the Fuoss-Onsager E\ and E2 constants for equivalent conductances for 1-1 aqueous solutions from 0 to 100 °C 21 Table 5. Differences in the values of Bt and B2 from those given in table 3 if the dielectric constants of water determined by Owen et al.

Page

are used instead of those of Malmberg and

Maryott 21 Table 6. Differences in the values of E\ and E2 from those given in table 4 if the dielectric constants of water determined by Owen et al. are used instead of those of Malmberg and

Maryott 22 Table 7. Values of the Debye-Hiickel constants Ac and Bc for activity coefficients of aqueous

solutions from 0 to 100 °C 22 Table 8. Values of A0 for HF 23 Table 9. Values of the constants governing the

dissociation of HF 23 Table 10. Equivalent conductances of aqueous

solutions of HF at 0, 16, 18, 20, and 25 °C 23 Table 11. Equivalent conductances of aqueous

solutions of HC1 from —20 to 65 °C 24 Table 12. Interpolation equations, constants, and coefficients for A for HC1 25 Table 13. Equivalent conductances of aqueous

solutions of HBr at 25 °C 26 Table 14. Equivalent conductances of aqueous

solutions of HBr from —20 to 50 °C 27 Table 15. Interpolation equations, constants,

coefficients, and standard deviations of fits for A for HBr 28 Table 16. Equivalent conductances of aqueous

solutions of HI at 25 °C 29 Table 17. Equivalent conductances of aqueous

solutions of HI from — 20 to 50 Tl 30

Table 18. Interpolation formulas for A for HI 31 Table 19. Limiting equivalent conductances of HF, HC1, HBr, and HI in water at 25 °C 31

14.

Glossary of symbols 32

V
— , I

ss
Electrolytic Conductance and the Conductances of the

Halogen Acids in Water

Walter J. Hamer and Harold J. DeWane

Definitions, symbols, general principles, and general laws related to the electrolytic conductance

of aqueous solutions are presented. The general laws considered are Coulomb’s law for charged

bodies, Poisson’s equation relating the electrostatic potential to charge distribution, and the Stokes and

Oseen laws for the velocity of a sphere in a fluid medium. The relations between electrical resistance,

electrical conductance, specific resistance, specific conductance, and equivalent conductance are

set forth. Theoretical expressions for the equivalent conductance as derived by Debye, Onsager, and Fuoss are given in general form and in a somewhat more detailed fashion in an appendix. The general

methods of treating the equivalent conductances of ionophores and ionogens, especially in regard

to the determination of the limiting equivalent conductance, the degree of ionic association, and the degree of ionic dissociation are discussed. Data on the equivalent conductances of the halogen acids,

hydrofluoric, hydrochloric, hydrobromic, and hydriodic acids in water are given for a wide range of concentration and temperature.

Key words: Conductances of HF, HC1, HBr, and HI; electrolytic conductance; theories ot electrolytic

conductance.

1. Definitions and Symbols

Conductance, cr — The conductance of a conductor

of electricity is the reciprocal of its electrical

resistance ( R) and its unit is the reciprocal “absolute” ohm, ohm-1

, or mho.

Specific conductance, crsp — The specific con- ductance, or conductivity, of a conductor of electricity is the conductance of the material between

opposite sides of a cube, one centimeter in each

direction. The unit of specific conductance is ohm-1 cm-1 or mho cm-1

. Electrolytic cell constant, Jc — The cell constant

of an electrolytic cell is the resistance in ohms

of that cell when filled with a liquid of unit re- sistance.

Equivalent conductance. A — The equivalent

conductance of an electrolytic solution is the

conductance of the amount of solution that contains one gram-equivalent of a solute (or electrolyte) when measured between parallel electrodes

which are one centimeter apart and large enough

in area to include the necessary volume of solution.

Equivalent conductance is numerically equal

to the conductivity multiplied by the volume in

cubic centimeters containing one gram-equivalent

of the electrolyte. The unit of equivalent conduct- ance is ohm-1 cm2 equiv-1 (frequently, in the

literature the unit is given simply, although in- correctly, as ohm-1

, so that it may be comparable

to the unit for conductance, in general).

Limiting equivalent conductance, Ao — The

limiting equivalent conductance of an electrolytic

solution, Ao, is expressed by Ao=lim ( a-C0rJc

)

where (Ton. is solution conductance corrected for

solvent conductance and c is the equivalent concentration. Ao is the value which A approaches

as the solution is diluted so far that the effects

of interionic forces become negligible (and dissociation, in the case of ionogens, is essentially

complete).

Molar conductance, Aw — The molar conductance

of an electrolytic solution is the conductance

of a solution containing one gram mole of the solute

(or electrolyte) when measured in a like manner to equivalent conductance. Seldomly used.

Limiting molar conductance, A^— The limiting

molar conductance of an electrolytic solution,

A(,”, is expressed by Ag'^lim (o-C0rr/m) where

<xcorr is solution conductance corrected for solvent

conductance and m is the molar concentration.

A^ is the value which Am approaches as the solution

is diluted so far that the effects of interionic forces

become negligible. Seldomly used.

Degree of dissociation (or ionization) in general,

a — The degree of dissociation (or ionization) of an electrolytic solution is the percentage of solute

(or electrolyte) in the dissociated (or ionized)

state in solution. Classically this degree is obtained

from conductance measurements from the ratio,

A/Ai where A; is the equivalent conductance an

electrolytic solution would have at some finite

concentration if it were completely dissociated

into ions at that concentration. (See ionogens).

This symbol is also used to denote the fraction

of free ions in a solution when simple ions, ion

pairs, and clusters higher than ion pairs are present.

(See ionophores).

Degree of association, (1— a) — The degree of

association of an electrolytic solution is the percentage of ions associated into nonconducting

species, such as ion-pairs. (See ionophores).

Ionic equivalent conductance, The ionic

equivalent conductance is the equivalent conduct- ance of an individual ion constituent of the solute

(or electrolyte) of an electrolytic solution. This

1
symbol is also used to designate the equivalent

conductance of complex ions, ion pairs, ion clusters,

etc., in combination with simple ions.

Limiting ionic equivalent conductance, \o— The

limiting ionic equivalent conductance of an individual ion constituent of the solute (or electrolyte) of an

electrolytic solution is given by A.0 = lim (A/c). This

c —>0 symbol is also used to designate the limiting

equivalent conductances of complex ions, ion

pairs, ion clusters, etc., in combination with

simple ions.

Ionic mobility, a — The mobility of an ion at any

finite equivalent concentration is the velocity

with which the ion moves under unit potential

gradient. Its unit is cm2 sec

-1 volt -1 equiv-1 or cm2 ohm-1 F

-1 where F is the Faraday expressed

in coulombs (or ampere seconds) equiv -1

. Limiting ionic mobility, a

0 — The limiting mobility

of an individual ion of a solute (or electrolyte) is given by u° =lim u.

c—»0 Kohlrausch law of independent migration of

ions — The value of the equivalent conductance,

as the concentration approaches zero, is equal to the sum of the limiting ionic equivalent conduct- ances of the ions constituting the solute of the

electrolytic solution.

Transference (or transport) number, t — The

transference number of each ion of a solute (or

electrolyte) in an electrolytic solution is the fraction

of the total current carried by that ion, and is given

by the ratio of the mobility of the ion to the sum of

the mobilities of the ions of the solute constituting

the electrolytic solution.

Interionic attraction — The electrostatic attrac- tion between ions of unlike charge (sign).

Interionic repulsion — The electrostatic repulsion

between ions of like charge (sign).

Ion atmosphere (or continuous charge distribution)— In the electrostatic effects between ions the

term ion atmosphere denotes a continuous charge

distribution, or charge density, p (r), which is a continuous function of r, the distance from the

reference ion, rather than a discrete or discontinuous charge distribution. The ion atmosphere

extends from r—a to r— 0(Vlls

) ~ °°, where V is the

volume of the system, and acts electrostatically

somewhat like a sphere of charge — e at a distance,

k

- 1

, from the reference ion of charge -fie (see below

for definition of k

-1

). Thickness or average radius of ion atmosphere,

k

- 1 — The average distance of the ion atmosphere

from the reference ion in angstrom units. This

average distance decreases in magnitude with the

square root of the ionic concentration. Mathematically, /c -1

is the distance at which the average

charge, dq, in a spherical shell of volume 4>nr2dr

reaches a maximum using the continuous density,

p (r), approximation.

Ion size or “ion-size ” parameter, a (or a,) — The

ion size is formally considered to be the sum of

the ionic radii of the oppositely charged ions

in contact. The ion size is also called the “distance

of closest approach” of the ions, or the “ion-size”

parameter. Generally the ion size is greater than the

sum of the crystal radii, and the “ion-size” parameter may include several factors which contribute

to its numerical value.

Electrophoretic effect — The slowing down, owing

to interionic attraction and repulsion, of the move- ment of an ion with its solvent molecules in the

forward direction by ions of opposite charge with

their solvent molecules moving in the reverse

direction under an applied electrical field (potential gradient).

Relaxation-field effect— The delay in the ion

atmosphere in maintaining its symmetry around

a central ion as the central ion moves in the for- ward direction under an applied electrical field

(potential gradient).

Osmotic-pressure effect— An enhancement in

the velocity of the central ion, in the direction

of the applied external field, as a result of more

collisions on the central ion from ions behind the

central ion than from ions in front of it. Viscosity effect — An alteration in the velocity

of a given ion as a result of the contribution to the

bulk viscosity owing to the ions of opposite charge.

This effect applies to ions of large size.

Walden’s rule, A0170— Walden’s rule states that

the product of the limiting equivalent conductance

of an electrolytic solution, A0 , and the viscosity of

the solvent, 170, in which the solute (or electrolyte)

is dissolved is a constant at a particular temperature. Walden’s rule is an approximation which

would be valid only for ions which behave hydrodynamically like Stokes spheres in a continiuum

(see later for Stokes Law).

Debye-Falkenhagen effect — The increase in the

conductance of an electrolytic solution produced

by alternating currents of sufficiently high fre- quencies over that observed with low frequencies

or with direct current.

Wien effect— The increase in the conductance of an electrolytic solution produced by high electrical

fields (potential gradients).

Dissociation-field effect — The increased dissociation (or ionization) of the molecules of weak electrolytes under the influence of high electrical fields

(potential gradients).

Ionophores— Substances, like sodium chloride,

which exist only as ionic lattices in the pure crystalline form, and which when dissolved in an appropriate solvent give conductances which change

according to some fractional power of the concentration. Such solutions possess no neutral molecules

which can dissociate, but may contain associated

ions.

Ionogens — Substances, like acetic acid (HAc),

which, although in the pure state are nonelectrolytic neutral molecules, can react with certain

solvents to form products which rearrange to ion

pairs which then dissociate to give conducting

2
solutions. As an example:

HAc + H2 0^HAc'H2 0

*±H;i 0+ • Ac-^H30+ + Ac2. General Considerations

Three factors cause ions to move in a medium in which they may exist. These are (1) thermal motion

of a random nature, (2) flow of the medium as a whole, and (3) forces acting on the ions. The last may be internal or external or both. Internal forces

may arise from concentration gradients, velocity

gradients (which produce tangential stresses in

viscous flow), temperature gradients (Soret effect),

and electrostatic forces due to the ions themselves.

External forces may be produced by pressure

changes, gravitational fields, or the application of an

electric field. Ions in an electrolytic solution are neither created nor destroyed during their motion

under a dc field, i.e., they follow the equation of

continuity, analogous to the equation of continuity

in hydrodynamics which states that matter is con- served in liquid flow. In moving through a medium,

ions must overcome friction whatever may be its cause. If an ion has a mobility of u\, its velocity i>, under a force 3F will be SF uu Since the coefficient

of friction of the ion, p\, equals l/ui, the ionic velocity

under the force 3F is given by tFIpu The “absolute”

mobility of a body in the Systeme International d’ Unites (SI) and MKSA systems of units is the

velocity in m per s attained under a force of 1 N;

in the cgs system of units the “absolute” mobility

of a body is the velocity in cm per s attained under a force of 1 dyne. In practice, however, when dealing

with ions, the unit of force is taken as a unit potential gradient acting on a unit charge. Letting Ui denote “absolute” mobility of an ion and u{ “electrical mobility”, the velocity attained by an ion

under unit potential gradient is:

Ui = u'J \zi\e = Nu[l \zi\F (2.1)

where N is Avogadro’s constant and F the Faraday.

Since the ionic equivalent conductance, denoted

by A,-, is equal to Fu\ eq (2.1) may be written:

Ui = u'J \zi\e = Nkil \zi\F2 (2.2)

The “electric mobility” is directly proportional,

therefore, to the “absolute” mobility and the applied potential gradient. Since 1 V = [1/299.7925(1)] esu of potential (cgs system) and the elementary

charge e = 4. 80298(7) X 10 -10 esu of charge, a

field of 1 V/cm exerts on an ion a force of 1.60210(3)

X 10 _11 |zj|dyn. (See appendix C for SI units.) The

figures in parentheses give the uncertainty in the

last decimal arising from the uncertainties in the

physical constants.

In the treatment of electrolytic conductance at any selected constant temperature presented here,

it is assumed that internal and external forces

acting on the ions are restricted to electrostatic

forces between ions, the virtual forces due to local

concentration gradients produced by interionic

forces, and an applied electrical field.

3. General Laws

In dealing with electrolytic conductance the

following must be considered:

(1) Coulomb's law for charged bodies: This law

states that the force, 3F

between two charges,

ei and e2 , directed along a line between the charges,

is directly proportional to the magnitude of the

charges and inversely proportional to the square

of the distance, r, of their separation. If a material

medium is present the force is weakened. This law

may be represented by

&= (eie 2 /e r

2 )ri (3.1)

where n is the unit vector in the direction from

e\ to e2 and e denotes the dielectric constant of

the material medium.

(2) Poisson s equation relating the electrostatic

potential to charge distribution: This law states that for any point (ion) in a medium located by

three space coordinates, the divergence of the

gradient of the potential, TV is proportional to the charge density at that point (ion). This law,

in cartesian coordinates, may be represented by

div grad ^° = V * Vf°=V2 lf0 ==:^

(3.2)

where e again is sthe dielectric constant of the

medium, and pc is the average charge density.

(3) Stokes law: This law states that the velocity,

v, of a sphere of radius, r, moving under a force cF

in a medium will be inversely proportional to the

viscosity, 17, of the medium and the radius of the

sphere. This law may be represented by

v = SFj()TTr)r (3.3)

For ions moving under an electrical force, JV, this

equation becomes

Vi — ^el6,trv ri~ ^Cziel^irqri (3.4)

where X is the vector electric field in the x direction,

Zi the ionic valence, and e the elementary charge.

(4) Oseen law: This law is for a volume force rather

than the directional force of Stokes law. This law

may be represented by

d\= [d&F+ (d& * r)r/r2 ]/87n7r (3.5)

which gives the velocity dx produced by a volume

force d&T (acting at the origin), at a point located

by the vector r in a medium whose viscosity is 77.

372-265 0 - 70 -2 3
For ions moving under an electric force, the

volume force is replaced by the electrical force,

and the velocity of the ion is taken in the x or

field direction.

4. Conductance Relations

Since electrical conductance, cr, is the reciprocal

of electrical resistance, R, it may be expressed as

o-tn- 1 )

1 R(n) (4.1)

The conductivity data given in this monograph

are based on (or referred to) the standards of

Jones and Bradshaw [l] 1

, i.e., their data were used in determining the cell constant of conduct- ance cells. Jones and Bradshaw with great care measured the specific conductance of aqueous

solutions of potassium chloride at three different

concentrations and three different temperatures.

Their specific conductances (reported in international electrical units) are given in table l

2

, cor- rected to absolute electrical units. The conversion

from international to absolute units was made using

the relation [2]: The resistance of a homogeneous substance of

uniform cross-sectional area, A, and length, /, is given by

R — R S p (4.2)

1 international ohm (USA)= 1.000495

absolute ohms.

For data obtained outside of the United States the

relation:

where Rsp is known as the specific resistance (or

resistivity). Rsp is a characteristic of a substance

under given physical conditions and is numerically

equal to the resistance between opposite sides of a cube of the substance, one centimeter in each direction. The unit of Rsp is H cm. It follows that

a=

\

=

W~l=(Jsv T ^ Cm 1 (4- 3)

where cr sp is the specific conductance (or conductivity) in units of U

_1 cm -1 or mho cm -1

. In terms of Ohm’s law ( E=iR ) the specific con- ductance is given by

o- S p= HEa O- 1 cm-' (4.4)

where i is the electric current and Ea , the potential

applied to a centimeter-cube sample of the conductor, the conductance being measured between

a pair of opposite faces of the cube.

Since conductance cells are not normally con- structed with uniform cross section and length,

eq (4.3) cannot be used to compute cr sp from meas- urements of cr. Instead the cells are calibrated with

a conducting solution whose crsp has been measured

when A and / are uniform and accurately known;

this crsp may be represented by (crsp)g where the

subscript s denotes standard. Then eq (4.3) for

the calibrating solution becomes:

and

{or Sp)s=crs ^j- = crs Jc (4.6) A S where Jc is the cell constant. Therefore (cr sp) e, the specific conductance of an experimental solution is given by

(cr S p) e = 0\sp = (TeJc = CT

c (4.7)

1 international ohm (mean) = 1.00049

absolute ohms

has been used. Experimental data obtained since

January 1, 1949, are presumed to be in absolute

units; data obtained prior to January 1, 1949,

have been converted from international to absolute

units using the above relations. (The conversion

from international to absolute units officially took

place - on January 1, 1948, but it is assumed that

approximately one year, from January 1, 1948, to January 1, 1949, was required to put the transition

into effect if the authors were not specific about the

units they employed.)

Jones and Bradshaw gave their results in terms

of a “demal” solution. A “denial” solution is defined as a solution containing a gram mole of

salt dissolved in a cubic decimeter of solution at zero degrees Celsius. In calculating the value of

“demal” they used the atomic weights of 1933.

Fortunately they also gave the grams of KC1 added

to 1000 g of solution and no change is necessary

in their values of the specific conductance as a result of changes in atomic weights to the unified 12 C

scale of atomic weight (the specific value of their

“demal” solutions differs but not the number of

grams added to 1000 g of solution).

Although specific conductance is useful in comparing metallic conductors, it has little direct importance in dealing with solutions. As the

concentration of solutions may be varied at will,

comparisons of the conductances of solutions

containing equivalents or fractions therefore are more significant. Equivalent conductance is defined

by:

A = o- sp ( 1000/c) fl~ 1 cm 2 equiv -1 (4.8)

where c is in equivalents per liter. The factor 1000/c

replaces A/l in eq (4.3), and, therefore, the equiva1 Figures in brackets indicate the literature references on p. 13.

1 All tables are presented in a group beginning on p. 19.

4
lent conductance is the conductance of that amount

of an electrolytic solution which contains one equivalent of solute (or electrolyte) when placed between

parallel planes 1 cm apart and of sufficient area to retain the volume expressed in liters of solution;

the conductance is measured normal to the planes.

The equivalent conductance may also be ex- pressed in terms of Ohm’s law by

the electrophoretic and relaxation-field effects in

particular in a hydrodynamic and electrostatic con- tinuum concluded that A should vary linearly with

the square root of c in dilute solutions. Somewhat

later Onsager [4] improved on their model by in- cluding the thermal movement of the reference ion,

and considering ions as point charges obtained the

relation:

A = ——

-1 cm 2 equiv 1 (4.9)

ri <|C

Accordingly, the equivalent conductance is numerically equal to the current in amperes that would

pass through such a solution if a potential gradient

of 1 V were applied across the electrodes, all disturbing effects being absent.

If the manner of dissociation (or ionization) of the

solute constituting an electrolytic solution is not

known, the molar conductance rather than the

equivalent conductance is used. It is defined by

Am = crsp (1000/ra)n -1 cm2 mol -1 (4.10)

where m is the gram moles of solute (or electrolyte)

dissolved in 1000 cm3 of solution.

The equivalent conductance of an individual ion

\i is given by (+ ion taken for example):

A.+ — Fu+Cl~ l cm2 equiv -1 (4.11)

where F is the Faraday and uthe ionic mobility. The

equivalent conductance of a binary electrolyte is equal to the sum of the individual equivalent con- ductances of the ions, or, A= A.+ + X- — F(u+ + u_)fl -1 cm2 equiv -1 (4.12)

Since the transference number of an ion is given by

t+ = u+(u+ + U-) or t- = u-l(u+ + U-) (4.13)

it follows that

|z+z-|e 2 Ao<? / 47rNe 2 \

1/2 3ekT(l + V^) \ 1000ekTj

F2 67nqN (k+l+k-l)

47rNe2 lOOOeAT

fl 1 cm2 equiv 1 (5.1)

where _ |z+z-| ( A.q + X 0 ) (|z+|+|z-|)(|z+|\J + 1-2-1 A(7

and z+ is the valence of the positively charged ion, z- the valence of the negatively charged ion, \o the

limiting equivalent conductance of the individual

+ and — ions, e the relative dielectric constant of

the solvent, r) the viscosity of the solvent, and

T= absolute temperature in kelvins, defined in

the thermodynamic scale by assigning 273.16 K to the triple point of water (freezing point of water = 273.15 K). The 13th General Conference on Weights and Measures in 1967 changed the unit

of temperature and temperature interval from

“degrees Kelvin” to simply “kelvin” (symbol K).

The constants have the values:

k = Boltzmann constant = 1.38054(18) X 10-16 erg K

-1 A^= Avogadro constant = 6.02252(28) X 1023 mol -1

e = elementary charge = 4.80298(20) X 10 -10 esu F = 96,487.0(1.6) coulomb (g-equivalent) -1

,

k+ —+ A and A._ = t - A (4.14)

5. Theoretical Expressions for

Equivalent Conductance

The equivalent conductance, A, of an electrolytic

solution varies with the concentration, c; for very

dilute solutions A varies almost linearly with the

square root of the concentration but for concentrated

solutions the relation between A and c is quite complex. Kohlrausch, late in the nineteenth century,

found that the equivalent conductance of a large

number of electrolytic solutions, in the dilute range,

varied with a fractional power of the concentration;

he was inclined to favor the square root as the fractional power.

In 1923 Debye and Hiickel [31 by considering

interionic attraction and repulsion in general and

where the numbers in parenthesis in each case

represent established limits of error, namely, three

standard errors based on the standard deviations

of the data and applied to the last digits in the

listed value of the physical constant. These values

of the physical constants are those recommended

in 1963 by the committee on fundamental constants

of the National Academy of Sciences — National

Research Council [5]. (See appendix C for SI units.)

The so-called thickness of the ion atmosphere

around a central ion, k

- 1

, appears as its reciprocal

in equation 5.1 as

(5.2)

5
where /, the ionic strength, is equal to (c/2)

(v+z^ + v-zl.) and v+ and V- are the number of

positive and negative ions in one molecule of

electrolyte, respectively.

For uni-univalent (or 1—1) electrolytes, eq (5.1)

may be simplified to: A= A0 — (B 1A0 + B2 ) V7 n-1 cm2 equiv- 1 (5.3)

where

B,=

6ekT( l + Vl}2 ) (KlVJ) l

' Ke^-'12

<5- 4>

co this theory were calculated from J(a), discussed

later. Values of Ei and E2 for temperatures from 0 to 100 °C are given in table 4 for 1—1 electrolytes.

Differences in the values of B\ and Bo and of

E\ and E2 from those given if the values of the

dielectric constant of water determined by Owen,

Miller, Milner, and Cogan [12] are used instead of

those of Malmberg and Maryott are given in tables

5 and 6, respectively.

Later, Fuoss [lb, 17] extended eq (5.6) to asso- ciated electrolytes (associated ionophores) and gave:

A = A0 — {B 1 A0 T- B2 ) Vac 4- Eac log ac

and + J (a)ac — K.AcaylA H

-1 cm2 equiv -1 (5.11)

B2 = -r^-—r (k/ V7) n

_1 cm2

/

1/2 equiv 1/2 (5.5)

Snrjly

where B2 and B2 are the coefficients of the relaxation and electrophoretic terms, respectively.

Physical properties of water, especially the

dielectric constant and viscosity needed to calculate B 1 and B2 for aqueous solutions are given in

table 2. Values of B 1 and B2 for aqueous solutions, on the volume basis, from 0 to 100 °C, are given in

table 3 for 1-1 electrolytes. (Conductivity data

are usually reported on the volume basis).

The equation of Onsager is generally referred to as the limiting law for equivalent conductance.

It gives the tangent to the conductance curve at zero concentration.

In 1932, Onsager and Fuoss [13] recalculated the

electrophoresis term, using charged rigid spheres

to represent ions and later Fuoss and Onsager

[14] (see also Fuoss and Accascina [15]) by also

considering ions as charged spheres rather than

poipt charges and with the retention of higherorder terms in treating the relaxation-field effect

obtained for 1-1 electrolytes:

A = A<> — (5iA 0 4-Z? 2 ) Vc + Fclnc

+ J(a)c O

-1 cm2 equiv -1 (5.6)

where B 1 and B2 have the same significance as given above, and

II >0 1 ro (5.7)

where = 2.302585 /c 2a

2 62 /24c (5.8)

and

£2 = 2.302585 ko6R2 /16cm (5.9)

and a function J(a), discussed later, where

b = e

2laekT (5.10)

where (1 — a) is the fraction of an ionophore asso- ciated in ion pairs and related to the association

constant by the mass action equation:

(1 — a) = KA o.

2cy2

l mol-1 (5.12)

where yc is the activity coefficient, generally

obtained by some theoretical equation [18], such as:

log yc =—

c V7/(l +Bc a V7) (5.13)

where / is the ionic strength, a, as before, is the

“distance of closest approach of the ions” and

A c and Bc are theoretical constants, the values

of which are given from 0 to 100 °C in table 7 (the

subscript c means that y,A, and B are on the volume

basis). The Arrhenius hypothesis that

a — A/A,- (5.14)

where A, is the equivalent conductance of the free

ions, is used to calculate a. Equation 5.11 is used

to determine approximate values of A,- by first taking a = A/A0 and then successive arithmetical

approximations are carried out until self-consistent

values of A, and a are obtained for each concentration.

Fuoss and Kraus [19] using the Bjerrum [20]

theory for ion pairs gave

4 nN / e

2 \3

for the association constant where K 1 is the dissociation constant and Q(b ) is the definite integral

i:

exp (x)x~4dx where x = e

2 lrekT, and the other

symbols have the significance given above. Fuoss

and Kraus tabulated values of Q as a function of b. Denison and Ramsey [21] used a Born cycle to show

that Ka should be a continuous function of eT

and for a 1-1 electrolyte obtained:

and a is the “ion-size” parameter (actually a „ _ t u\ 1 cancels in (5.8) and (5.9)). Values of a, according A — A^expf ) mo (5.16)

6

t
where K'A was given unit value by neglecting dimensions or taking K'A as 1 liter per mole. Later, Gilker- son [22] using Kirkwood’s [231 partition function

obtained:

Ka = K°A exp ( b ) / mol-1 (3.17)

where K°A included the effect of solute-solvent

interaction and the free volume of the solute. In 1958, applying Boltzmann’s distribution to the

problem and considering the solvent as a continuum,

Fuoss [24] obtained:

^=S) exp(6,/moH ,5 ' 18)

For the halogen acids in water, considered in

this document, it was not necessary to include a consideration of Ka ; accordingly, numerical values

of Ka as a function of a are not given here.

The method of determining a for ionogens is discussed later. At a later date Fuoss, Onsager, and Skinner

[25] retained the Boltzmann factor explicitly in

its exponential form and introduced the dimensionless variable, r, which is the ratio of the Bjerrum

distance, near which most ion-pair distribution

functions have a minimum, and the Debye-Hiickel

distance, 1 /k. This variable is a rational reduced

variable for the description of electrolytic solutions; two solutions at different dielectric constants

would be equivalent electrostatically; for example,

their activity coefficients and relaxation fields would

be identical. The variable r is equal to [6(0.4342945)

£V] 1/2

. With these changes, Fuoss and Onsager

obtained:

A = A0 — (B iA0 + B2 ) c

1 ' 2 + Ec log T

2 + L(a)c — K^Aocy2 fl _1 cm2 equiv-1 (5.19)

for the equivalent conductance of highly dissociated 1-1 electrolytes and

A= Ao~ (B 1 A0 + B2 ) (ac) ll2 + Eac log ar2 + L(a) ac — K^Aoctcy2 cm2 equiv -1 (5.20)

for associated electrolytes or associated ionophores.

Fuoss, Onsager, and Skinner [25] also pointed out

that there are other sources of linear terms in the

conductance equation. The volume of the ions, as suggested by Steel, Stokes and Stokes [26] and the

c

1/2 term in the dielectric-constant expression of

Debye and Falkenhagen [27] lead to a term linear

in c. Accordingly, the coefficient of the c term may

be complex consisting of a number of factors. In

1969 Chen [28] considered the interaction between

the relaxation field and the electrophoretic flow

which Fuoss and Onsager had omitted. Chen found

that this consideration lead to an additional term of

the order of c log c with the result that the coefficient of c log c term in eqs (5.19) and (5.20) becomes

F1 A0 — 2E2 and the coefficient L(a

in eqs (5.19)

and (5.20) (as well as the J(a) coefficient in eqs

(5.6) and (5.11)) acquire a different functional form

from that published by Fuoss, Onsager, and Skinner

[25] in 1965. Since the L(a

and J(a) functions now

have only historical interest, numerical tables for

them are not included here.

The final conductance equation then has the form:

A = A0 — (B\Ao + B2 )c ll2 -\- (E 1 Ao — 2E2 )c log c + ke c — K^Aocy2 O- 1 cm2 equiv -1 (5.21)

for highly dissociated electrolytes and

A = A0 — (B iA 0 + B2 ) (ac) ll>2 + (Ei A0 — 2E2 ) ac log ac + ke ac — K^Aoacy2 li -1 cm2 equiv -1 (5.22)

for associated electrolytes or associated ionophores.

As of this date, ke is considered an empirical constant, although its value will be related to the

“ion-size” parameter.

A method [29] of obtaining A, and thus a for iono

gens may be illustrated as follows: A, is obtained

using the Kohlrausch principle of independent ion

migration and the assumption that solutions of

alkali salts and inorganic hydrogen acids are totally

unassociated into ion aggregates at all concentrations. For acetic acid (HAc), for example, the

procedure would be as follows:

(a) Determine the equivalent conductance of

acetic acid (HAc) at a series of concentrations,

(b) Determine the equivalent conductance of

sodium acetate (NaAc), HC1, and NaCl also at a

series of concentrations,

(c) Calculate A,( HAc) from the relationship:

^i(HAc) = ^H+ ^Cl+ ^Na+ k Ac — ^Na ~~^-Cl (5.23) = \H "f XAc (5.24)

which follows from the Kohlrausch principle and in

which X represents the ionic conductances of the

ions denoted by subscripts,

(d) Calculate values of AHAc /A;( HA c )= a for

various values of the stoichiometric concentration

which would give values of the degree of dissociation of the acid.

This procedure of obtaining A, entails a short

series of approximations since the ionic concentration for which A/A, must be calculated cannot be

known a priori.

Values of a and A,- of ionogens may also be

obtained by a series of successive approximations

using the procedure discussed for ionophores.

For dilute solutions the limiting law of Onsager is

7
used. Values of a at higher concentrations may be

obtained if the E and higher coefficients are known.

As a start these may be estimated and then iteration

is made until values of a, A,-, E, and the higher

terms are self consistent. In many cases, complex ionic equilibria exist in aqueous solutions and it is not possible to categorically cover all of these in a general fashion.

Instead each case will be considered individually

where necessary, as, for example, for HF considered

later in this document.

A somewhat more detailed treatment of the

Debye-Hiickel-Onsager-Fuoss theory of electrolytic

conductance is given in appendix A.

The temperature range has been limited since wax

or wax lined cells had to be used owing to the

corrosiveness of HF. This situation could be

remedied, today, by using polyethylene containers

or other containers of plastic. The data in the

literature on HF have been reported on various

concentration scales, namely, mole percent, weight

percent, volume dilution, etc. These data were all converted to the molarity basis using available

data on the density of aqueous solutions of HF.

The density data were fitted to polynominals. the

constants for which are given in appendix B. The dissociation of HF is controlled by the two

equilibria:

6. Determination of A0

The method outlined by Fuoss and Accascina [15J was followed, where possible, in determining A0 . As a start the Shedlovsky [30] function A„ given by

\'

0 = (A + B2 c

112

) I {l— BiC 11

'2

) O

-1 cm'2 equiv -1 (6.1)

is calculated from observed values of A and the

theoretical constants B\ and B2 . Values of Aq are then plotted against c and a preliminary value of A0 obtained by extrapolation to c = 0. This preliminary

value of Ao is then used to compute B\Ao + B2 and E (or FhAo — 2E2 ). Then values of A' given by

A' = A+ (B,Ao + B2 )c'i*-Ec = Ao

+ ke c H-1 cm2 equiv -1 (6.2)

are plotted against c and the intercept at c = 0 gives

A0 with the slope giving the empirical constant, ke . In some cases the procedure must be iterated until

consistent values of A0 (between (6.1) and (6.2)) are obtained. The empirical constant corresponds to the older functions J (a) or L(a) where E was equal

to FhAo — E2. Note here that E — E\A0 — 2E2.

7. Equivalent Conductances of HF,

HC1, HBr, HI

Of the four halogen acids HF, HC1, HBr, HI all are unassociated ionogens except HF which is not

only incompletely dissociated at finite concentrations, but exhibits association of the fluoride ion

and the undissociated molecule. These four acids

are considered in order. All data, where necessary, were converted to the Jones-Bradshaw [1] con- ductance standard, the 12 C scale of atomic weights,

and the absolute electrical units [2]. All data were programmed for an IBM 7090 computer.

HF

(a ) Equilibria. Data on the equivalent conductance

of HF are available only at 0, 16. 18. 20, and 25 °C.

HF H+ + F- (7.1)

and

HF2-^HF+F- (7.2)

with the first one more significant for dilute solutions below 0.001 molar. The equilibrium constants

for these equilibria are given, respectively, by:

a H c H

+ c F -yH

+ 7F

_ mol /' aHF chf7hf

(7.3)

and

a HFaF ahf:

cHFCF~Thf7f

c hf7 7hf1

mol / -1 (7.4)

where a, c, and y denote, respectively, the activity,

concentration, and activity coefficient of the species

denoted by the subscripts. If we let y and y2 be

the ratios, respectively, of the concentrations of

F

_ and HFy to the stoichiometric concentration.

C, of HF, and assuming, as a start, that all activity

coefficients are unity, we have:

K

%Qyir±isl^ C(+)moU-,

1 y — 2y3 (7.5)

A = —- - - y 2y:i) ~ — mot M (7.6)

ya 73

The approximations given in eqs (7.5) and (7.6) are obtained by setting 1 —7—273= 1. Now the observed conductance of HF is given by:

A — 7A.0 +73A0 O -1 cm2 equiv -1 (7.7)

8
where Ao is the sum of the limiting equivalent

conductances of H + and F

- and A0 is the sum of

the limiting equivalent conductances of HF.7 and H +

. Solving the approximate versions of eqs (7.5)

and (7.6) for y and y3 and substituting in (7.7)

gives:

A ( 1 -h c/A-) 1/2 = ( A„ VK/Vf

+ (\o y/~K I k) Vc n-J cm2 equiv -1 (7.8)

This equation may be converted to a linear form [35]

by multiplying by Vc, adding and subtracting

cAoV^/A to the right side, dividing by (l + c/A) 1/2

, squaring both sides, and simplifying; these steps

lead to: cA2 = A§K + [2Xo/A 0 — 1 + ( 1 — Xo/A 0 ) 2 /(l

+ k/c)]AlKc/kQ,~2 cm4 equiv -2 (7.9)

At low concentrations the term ( 1 — Ao/Ao) 2

/ ( 1 + k/c)

vanishes while at high concentrations it approaches

asymptotically the limit (1 — A 0 /Ao) 2

. Therefore,

this may be neglected when Xo/A 0 is sufficiently

close to unity to render (1 — X 0 /A 0 ) 2 negligible with

respect to (2X 0/A 0 — 1). Accordingly, equation 7.9

reduces to: cA2 = A2 K + c(2A 0 A 0 — \l)Klkil~2 cm4 equiv -2

(7.10)

If we now introduce the ionic activity coefficient, y, and the ionic mobility coefficient, m'

we have:

1-A/Ao

A%K

(2AqAq — A ) K

(7.11)

Values of y and m

are given, respectively, by:

log y = — A c VcA/Ao (7.12)

and

m

’ = 1 — (Z?iAo + B>) A7

1 VcA/Ao (7.13)

where A c is the Debye-Hiickel constant, given by:

2vN \

1/2 e

3 / 1 1000/ 2.302585A 3/2 \ T3l2 e

312

(7.14)

and B\ and B> are the Debye-Hiickel-Onsager constants given, respectively, by eqs (5.4) and (5.5) and

A c is the Debye-Hiickel constant in eqs (5.13), values

of which are listed from 0 to 100 °C in table 7. A plot of values of the left side of eq (7.11) against

c (

— A/Ao) gives a straight line for molarities

^ /O 2 mol-1 / 2

between 0.004 and 1.0. The intercept when

c(l — A/Ao )=0 gives AIK and the slope of the line

gives (2A 0 A 0 — Al) K/k. Thus, since A0 and A 0 are known K and k can be evaluated (see below under

(b).

(b) Values for Ao. Values reported for A0 for HF

are given in table 8. The limiting equivalent con- ductances of the ions at 25 °C as compiled by

Robinson and Stokes [41] yield 405.0 for Ao for HF

(their values were converted to absolute units here).

Erdey-Griiz, Majthenyi, and Kugler [42] using

Shedlovsky’s [43] A0 values for HC1 and NaCl and

their value for NaF obtained 405.04 on the old

Parker conductivity standard which becomes 405.09 on the Jones-Bradshaw standard. Using 126.39 for

A0 for NaCl [41], 105.43 for Ao for NaF [42], and

426.06 for Ao for HC1 (see later in this document)

gives 405.10 for Ao for HF; this value was selected

here as the most reliable value for A0 for HF at 25 °C.

Wooster [35] gave 255 and 404 for Ao at 0 and

25 °C, respectively. Using the ratio 405.10/404,

Ao becomes 255.69 at 0 °C. From a linear plot of A0 against 1 IT one obtains 354.29, 365.85, 377.26 for

Ao at 16, 18, and 20 °C, respectively. Wooster [35]

gave 437 and 275.4 for Ao at 25 and 0 °C, respectively.

On converting to the above basis and using a

(A 0 -l/r) plot, Ao values of 276.15, 383.08, 395.62,

407.99, and 438.19 are obtained, respectively, at 0, 16, 18, 20, and 25 °C.

(c) Equivalent conductances of HF. The available

data on HF at 25 °C appear in papers by Deussen

[37], Fredenhagen and Wellmann [34], Thomas and

Maass [44], Ellis [45], and Erdey-Griiz, Majthenyi.

and Kugler [42]. Thomas and Maass reported their

results to only one decimal place and Erdey-Griiz

et al. only for very dilute solutions; the latter’s data also appear low on a plot of eq (7.11). The data

of Deussen, Fredenhagen, and Wellmann, and of

Ellis agree within their experimental uncertainty

and were fitted to eq (7.11). Data at 16 and 20 °C

were obtained only by Roth [36] while data at 18 °C

were obtained both by Roth [36] and Hill and

Sirkar [31]. The data of Hill and Sirkar, however, were very much lower than those of Roth and were

inconsistent with the data obtained by other experimenters at other temperatures. For 0 °C, data were

obtained by Deussen and Hill and Sirkar; the

latter data showed erratic changes with concentration and Deussen’s data at 0 °C were, therefore,

selected as the more reliable.

All of these data were fitted to eq (7.11). From a

plot of the values of the left side of eq (7.11) against

c(l— A/A0 ), shown for 25 °C for example in figure 1, the values given in table 9 were obtained

for K and A. The s x values, given in table 9. are the standard deviation with which eq (7.11) was

fitted over the concentration range of 0.004 to 1.0 N for 0 °C; of 0.006 to 0.2 N for 16. 18. and

20 °C; and 0.004 to 1.0 N for 25 °C.

9
Eq 7.11 may be rearranged to give for A

A = AJK(y)

2 (1-A/A„)C

+

(2AoA0 -A2K) ^^ (1 — A/Ao)2

O-1 cm2 equiv-1

It was also found that

(m'/y) 2 (1 — A/A0 ) and (m'ly) 2 (1 — A/A0

Values of ji, j[ , y2 , and j'

2 follow:

t j i

• t h J2

•t h Sx

°c n-1 cm 2 equiv~ l

0 1.37 0.307 1.28 0.363 0.7

16 1.28 .230 1.26 .320 1.2

18 1.28 .230 1.26 .320 1.1 20 1.28 .230 1.26 .320 1.1 25 1.34 .268 1.29 .341 0.6

may be expressed, respectively, by:

(1 — A/A0 ) =ji+j[ log C (7.16)

and

(y)

2 (l-A/A0 ) 2 =72 +y; log C (7.17)

The sx values are the standard deviations with

which eq (7.18) was fitted over the concentration

range of 0.004 to 1.0 M for 0 and 25 °C and for

0.006 to 0.2 M for the other temperatures.

Values of the equivalent conductances of HF

for rounded concentrations are given in table 10. Values at other concentrations, within the ranges

given, may be calculated from eq (7.18).

Therefore, eq (7.15) may be written:

(2A0X0 A~K)

A = ^lK{ji+j[ log C)C+

1/2

O'

+y; iogC) n-1 cm2 equiv-1 (7.18)

HC1

(a) Data at 25 °C. The equivalent conductances of

HC1 at 25 °C have been reported by Ruby and Kwai

[46], Hlasko [47], Howell [48], Shedlovsky [43], Saxton and Langer [49], Owen and Sweeton [50],

Klochko and Kurbanov [51], Stokes [52], and Murr

and Shiner [53]. Of the earlier data given in the

Figure 1. Plots used to obtain values for K and k governing the dissociation o/HF.

A. No corrections made for activity coefficients or changes in ionic mobilities with concentration.

B. Corrections made for activity coefficients. C. Corrections made for activity coefficients and changes in ionic mobilities with concentration.

10
International Critical Tables only the data of

Hlasko and Howell are considered here; their data were the most recent and were for higher concentrations where there is a sparsity of data.

It was necessary to divide the data into four con- centration ranges to obtain reasonable least square

fits to the interpolation equation:

A = A0 — (B\ Ao + #2 )

1 / 2 + Ec log c + Ac

+ Be312 + Cc2 + Dc^l2 fl -1 cm2 equiv-1 (7.19)

(This equation is an extension of eq (5.21) which is required for higher concentrations. When higher

concentrations are included the coefficient of the

c term differs from that obtained when lower concentrations are used). At low concentrations the

last two terms on the right were insignificant. The

four concentration ranges were as follows:

Concentration range

Sx References

(molarity) O-OMl

a-' cm2 eqiiiv~ x 0.05 [43], [49], [50], [52], [53]

0 . 01 -0.1 .14 [43], [46], [47], [49], [50],

0. 1-3.0 .10

[52]

[50], [51], [52]

3.0-11.6 .15 [47], [48], [50], [51]

The sx values are the standard deviation with which

the following equations fit the experimental data:

0-0.01 M

A = 426.06-158.63 c

J / 2 + 185.76 clog c + 747.385 c — 2095.71 c

3/2 O-1 cm2 equiv-1 (7.20)

0.01-0. 1M

A = 426.06- 158.63c 1 / 2 + 173. llc-f 43.515c3 / 2 — 345.46c2 O

-1 cm2 equiv -1 (7.21)

0.1-3.0;4f

A = 426.06 - 158.63c 1 / 2 + 221 .501c - 252.771c3 / 2 + 115.606c2 — 19. 5824c5/2 fl -1 cm2 equiv-1 (7.22)

3.0-11.6M

A = 426.06 - 158.63c 1 / 2 + 143.554c - 116.628c3/ 2 + 35.2535c2 — 3. 56231c5/2 fl -1 cm2 equiv -1 (7.23)

The coefficient of the c

1/2 term was obtained from

Ao and the B\ and B2 coefficients of the DebyeHiickel-Onsager theory. For the dilute range the

coefficient of the c log c term was obtained from A0 and the E 1 and 2E2 coefficients given in table 4

(actually 185.767 which was rounded to 185.76).

For concentrations above 0.01 M it was found that

the c log c term was not needed. It was also found

that the value of A0 depended somewhat on the source of the data used in the dilute range. The

data of Shedlovsky, Saxton and Langer, Owen and

Sweeton, Stokes, and Murr and Shiner below 0.01

M were used to obtain A0 , according to the procedure outlined by Fuoss and Accascina [15], with

the following results:

Experimenters

Number

of measure- ments

Ao

Shedlovsky 11

O -1 cm2 equiv~ x 426.00

Q

-1 cm 2 equiv~ x 0.07

Saxton and

Langer 10 426.35 .08

Owen and

Sweeton 5 426.55 .07

Stokes 9 426.40 .04

Murr and

Shiner. 20 426.06 .02

Since s x for the A0 value obtained from the data

of Murr and Shiner was the lowest their A0 value was selected (evident on reference to eqs (7.20) to

(7.23)).

The equivalent conductances of HC1 at rounded

concentrations at 25 °C, and the experimental

range covered, calculated by eqs (7.20) to (7.23), are given in table 11.

(b) Data at other temperatures. Values of the

equivalent conductance of HC1 at —20, —10, 0, 10,

20, 30, 40, and 50 °C, given here, are based on the

data of Haase, Sauermann, and Diicker [54], except

that at 50 °C the values in the dilute range (0 to 0.01 M) were derived from the results of Cook and

Stokes [55]. Values at 5, 15, 35, 45, 55, and 65 °C are based on the results of Owen and Sweeton [50].

These results were consistent with those obtained

at 25 °C and discussed above.

The equivalent conductances of HC1 at rounded

concentrations for these temperatures and for the

experimental range covered are included in table 11. The data in each case, except below 0 °C, were

fitted to the equation:

A = A0 — Sc 1/2 +Ec log c + Ac + Bc'i,2Jr Cc'2 + Z>c 5/ 2 0 -1 cm2 equiv-1 (7.24)

372-265 0 - 70 -3 11
where S = ZCAo + B> and B i, B% and E are the

theoretical Debye-Hiickel-Onsager-Fuoss coefficients. Below 0 °C the empirical equation

A = A + Be + Cc2 + Dc3 + Ec4 + Ec5 + Gc6 El~ 1 cm2 equiv -1 (7.25)

was used. In this case A obviously doesn't represent

Ao as the infinitely dilute solution would be in the

solid or frozen state. Nevertheless eq (7.25) may be

used for interpolation purposes.

Values of the coefficients of eqs (7.24) and (7.25)

and the concentration value over which they apply

are given in table 12. The 5 values given in column

9 are the standard deviations of the fit. HBr

(a) Data at 25 °C. Data on the equivalent con- ductance of HBr at 25 °C are based on the measure- ments of Dawson and Crann [56]. Hlasko [47], and

Haase, Sauermann, and Diicker [54]. Only the last mentioned are recent data. As with HC1 the data were divided into concentration ranges (in this case three) to obtain reasonable least square fit of the data with the equation:

A = A0 — Sc ll2 + Ec log c + Ac + Be312 + Cc2 + Z)c5/2 0

_1 cm2 equiv-1 (7.26)

The c In c term was not required above 0.01 M, and

for the dilute range the c

5/2 term was negligible.

The concentration ranges with the conductance

equations follow (the s values are the standard

deviations of the fit):

0.0000 — 0.014/ (5 = 0.09; references [47, 56])

A = 427.74- 159.02c1 / 2 + 186.65c log c + 899.72c - 4289.3c3/ 2 + 8912.1c2 H-1 cm2 equiv" 1 (7.27)

0.01 — 0.104/ (5 = 0.09; references [47, 56])

A = 427.74- 159.02c 1 / 2 + 210.8272c — 209.547c3/ 2 H_1 cm2 equiv -1 (7.28)

0.10 — 3.04/ (5 = 0.08; references [47, 56])

A = 427.74- 159.02c 1 / 2 + 192.8124c- 177.2474c3/ 2 + 53.93903c2 -3.746204c5/ 2 O -1 cm2 equiv -1 (7.29)

3.0— 7.54/ (5 = 0.09; references [47, 54])

A = 427.74 - 159.02c 1 / 2 + 130.5906c - 101 ,6916c3/ 2 + 29.49103c2 — 2.843402c5/ 2 fl -1 cm2 equiv -1 (7.30)

As for HC1 the coefficient of the c

1/2 term was

obtained from Ao and the B i and Bo coefficients

of the Debye-Hiickel-Onsager theory.

The equivalent conductances of HBr at rounded

concentrations at 25 °C, calculated by eqs (7.27) to (7.29) for the experimental range covered are given in table 13.

(b) Data at other temperatures. Values of the

equivalent conductance of HBr at —20. — 10. 0. 10.

20, 30, 40, and 50 °C are based on the recent

measurements of Haase, Sauermann. and Diicker

[54]. The equivalent conductances of HBr at rounded

concentrations for these temperatures and for the

experimental range covered are given in table 14. The data in each case, except below 0 °C, were

fitted to the equation:

A = A0 ~~ Sc 112 + Ac + Be312 + Cc2 + Dc5/ 2 H -1 cm2 equiv -1 (7.31)

with a zero standard deviation. The coefficient S

again equals /?iA0 -t-/?2 . Below 0 °C the empirical

equation used for HC1. namely, eq (7.25) was again

used. Values of the coefficients of eqs (7.31) and

(7.25) (as applied to HBr) are given in table 15

together with the concentration range over which

they apply. The coefficients are empirical and the

coefficient of the c term is low in terms of modern

theory.

HI

Only a few measurements have been made on the

electrolytic conductance of HI and. with the exception of those of Haase, Sauermann. and Diicker in [54], these were made over 30 years ago. The Inter- national Critical Tables (1929) cited two papers

for 18 °C, one by Loomis in 1897 [57] and the other

by Heydweiller in 1909 [58]. For 25 °C, the work of

Ostwald in 1903 [59], of Washburn and Strachan

in 1913 [60], and of Strachan and Chu in 1914 [61] were cited. Ostwald's results were thrown in doubt

when Bray and Hunt [62] showed that his data for

HC1 were in error by about 3 percent and presumably his data on HBr were also questionable.

Washburn and Strachan's data were nullified by

analytical errors made in establishing the con- centration of the solutions; Strachan and Chu

corrected their data. Only three references to work

on HI were found that had not been listed in ICT.

Two of these contained the same data. Hlasko and

Wazewski [63] and Hlasko [47]. The third one was

by Haase et al. mentioned above [54]. Solutions of HI may also contain traces of iodine although

the experimenters made attempts to prevent HI

decomposition.

(a) Data at 25 °C. The data of Strachan and Chu,

Hlasko, and Haase et al. at 25 °C were combined

and fitted to a polynomial. The Ao for HI was

calculated from the ionic values listed by Robinson

and Stokes and converted to absolute units. The

equations are from 0.0000 — 0.014/ and 0.01 — 7.04/

,

12
respectively, for the equivalent conductance as a function of c at 25 °C

A = 426.45 - 158.72c 1 / 2 + 185.96c log c + 1085.7c!— 1 1678c3/ 2 + 107071c2 — 394591c5/ 2 fl -1 cm2 equiv -1

, (7.32)

A = 426.45 - 158.72c 1 / 2 + 248.885c - 256.566

3' 2 + 96.2195c2 — 12.3887c5/ 2 fl -1 cm2 equiv -1 (7.33)

with a standard error of 0.2. The coefficient of the

c

1 / 2 term is based on the Ao value and the B\ and

B2 coefficients of the Debye-Hiickel-Onsager theory.

The coefficient of the c log c term is based on the

A0 value and E\ and 2Ez coefficients of the DebyeHiickel-Onsager-Fuoss theory.

The equivalent conductances of HI at rounded

concentrations for 25 °C and for the experimental

range covered, as calculated by eq (7.33), are given

in table 16.

(b) Data at other temperatures. Values of the

equivalent conductance of HI at —20, —10, 0,

10, 20, 30, 40, and 50 °C are based on the recent

9. Re

[1] Jones, G., and Bradshaw, B. C., J. Am. Chem. Soc. 55,

1780 (1933).

[2] NBS Circ. 459, 7 (1947).

[3] Debye, P., and Hiickel, E., Physik. Z. 24, 185, 305 (1923).

[4] Onsager, L„ Physik. Z. 27, 388 (1926); 28, 277 (1928).

[5] Chem. & Eng. News, p. 39, Nov. 11, 1963.

[6] Thiesen, M., Wiss. Abh. der Physikalisch-Technischen

Reichsanstalt 4, No. 1, 1904.

[7] International Critical Tables 3, 25 (1928).

[8] Swindells, J. F., Coe, J. R., and Godfrey, T. B., J. Research

NBS 48, 1 (1952) RP2279.

[9] Hardy, R. C., and Cottington, R. L., J. Research NBS 42,

573 (1949) RP1994.

[10] Coe, J. R., and Godfrey, T. B., J. App. Phys. 15, 625 (1944).

[11] Malmberg, C. G., and Maryott, A. A., J. Research NBS 56,

1 (1956) RP2641.

[12] Owen, B. B., Miller, R. C., Milner, C. E., and Cogan, H. L.,

J. Phys. Chem. 65, 2065 (1961).

[13] Onsager, L., and Fuoss, R. M., J. Phys. Chem. 36, 2689

(1932).

[14] Fuoss, R. M., and Onsager, L., Proe. Nat. Acad. Sci. 41,

283 (1955); Fuoss, R. M., and Onsager, L., J. Phys. Chem.

61, 668 (1957); Fuoss, R. M., J. Am. Chem. Soc. 79,

3301 (1957); Fuoss, R. M., and Kraus, C. A., J. Am. Chem.

Soc. 79, 3304 (1957).

[15] Fuoss, R. M., and Accascina, F., Electrolytic Conductance,

Ch. XIV and XV (Interscience Publishers, Inc., New York,

1959).

[16] Fuoss, R. M., J. Am. Chem. Soc. 79, 3301 (1957).

[17] Fuoss, R. M., and Kraus, C. A., J. Am. Chem. Soc. 79,

3304 (1957).

[18] Hamer, W. J., Nat. Stand. Ref. Data Ser., Nat. Bur. Stand.

(U.S.), 24, 276 pages (Dec. 1968).

[19] Fuoss, R. M., and Kraus, C. A., J. Am. Chem. Soc. 55,

1019 (1933).

[20] Bjerrum, N., Kgl. Danske Vidensk. Selskab. Math.-fys.

Medd. 7, No. 9 (1926).

[21] Denison, J. T., and Ramsey, J. B., J. Am. Chem. Soc. 77,

2615 (1955).

[22] Gilkerson, W. R., J. Chem. Phys. 25, 1199 (1956).

measurements of Haase, Sauermann, and Diicker

[54], The equivalent conductances of HI at rounded

concentrations for these temperatures and for

the experimental range covered are given in table

17. The data in each case, except below 0 °C, were

fitted to eq (7.31). Below 0 °C the empirical equation used for HC1 and HBr, namely, eq (7.25) was

used except that the c

4

, c

5

, and c

6 terms were not

needed at —20 °C and the c

5 and c6 terms were not

needed at — 10 °C. The equations derived for each

temperature are listed in table 18; the s values given

in the footnote refer to the standard deviation of the

fit of the equations.

8. Conclusions

In table 19 the limiting equivalent conductances

of HF, HC1, HBr, and HI obtained herein at 25 °C

are compared with the best previous data. The

agreement is quite good. It is interesting that A

0 does not show a consistent trend with the atomic

number of the halide.

Except for HF, for very dilute solutions the con- ductances are in accord with the Debye-HiickelOnsager limiting law.

[23] Kirkwood, J. G., J. Chem. Phys. 18, 380 (1950).

[24] Fuoss, R. M., J. Am. Chem. Soc. 80, 5059 (1958).

[25] Fuoss, R. M., Onsager, L., and Skinner, J. F., J. Phys.

Chem. 69, 2581 (1965).

[26] Steel, B. J., Stokes, J. M., and Stokes, R. H., J. Phys. Chem.

62, 1514 (1958).

[27] Debye, P., and Falkenhagen, H., Physik. Z. 29, 121, 401

(1928).

[28] Chen, M. S., Dissertation, Yale University, June 1969.

[29] Maclnnes, D. A., The Principles of Electrochemistry,

Ch. 18 (Reinhold Publishing Corp., New York, 1939).

[30] Shedlovsky, T., J. Am. Chem. Soc. 54, 1405 (1932).

[31] Hill, E. G., and Sirkar, A. P., Proc. Roy. Soc. A83, 140

(1910).

[32] Domange, L., Compt. rend. 198, 469 (1934).

[33] Winteler, F., Z. angew. Chem. 15, 33 (1902).

[34] Fredenhagen, K., and Wellmann, M., Z. Phys. Chem.

A 162, 454 (1932).

[35] Wooster, C. B., J. Am. Chem. Soc. 59, 377 (1937).

[36] Roth, W. A., Ann. Chem. 542, 35 (1939).

[37] Deussen, E., Z. anorg. Chem. 44, 310 (1905).

[38] Wegscheider, R., Z. Phys. Chem. 69, 620 (1909).

[39] Pick, H., Z. Phys. Chem. Nernst-Festschrift, 360 (1912).

[40] Davies, C. W., and Hudleston, L. J., J. Chem. Soc. 125,

260 (1924).

[41] Robinson, R. A., and Stokes, R. H., Electrolyte Solutions, 2nd. ed., p. 465 (Butterworths Scientific Publications,

London, 1959).

[42] Erdey-Gruz, T., Majthenyi, L., and Kugler, E., Acta Chim.

Acad. Sci. Hung. 37,393 (1963).

[43] Shedlovsky, T., J. Am. Chem. Soc. 54, 1411 (1932).

[44] Thomas, D. K., and Maass, O., Can. J. Chem. 36, 744 (1958).

[45] Ellis, A. J., J. Chem. Soc. 4300 (1963).

[46] Ruby, C. E., and Kwai, J., J. Am. Chem. Soc. 48, 1119

(1926).

[47] Hlasko, M., J. Chim. Phys. 26, 125 (1929).

[48] Howell, O. R., J. Chem. Soc. (1929) 162.

[49] Saxton, B., and Langer, T. W., J. Am. Chem. Soc. 55,

3638 (1933).

[50] Owen, B. B., and Sweeton, F. H.. J. Am. Chem. Soc. 63,

2811 (1941).

13
[51] Kloehko, M. A., and Kurbanov, M. Sh., Akad. Nauk SSSR,

Izvest. sek. Fiz-Khim. analiza 24 , 237 (1954).

[52] Stokes, R. H., J. Phys. Chem. 65 , 1242 (1961).

[53] Murr, B. L., Jr., and Shiner, V. J., Jr., J. Am. Chem. Soc. 84, 4672 (1962).

[54] Haase, R., Sauermann, P. F., and Diicker, K. H., Z. Phys.

Chem. Neue Folge 47, 224 (1965).

[55] Cook, B. M.. and Stokes, R. H., J. Phys. Chem. 67, 511

(1963).

[56] Dawson, H. M., and Crann, T. W., J. Chem. Soc. 109, 1262

(1916).

[57] Loomis, E. H., Ann. Physik. und Chemie 296 [3], 60, 547

(1897).

[58] Heydweiller, A., Ann. Physik. 30 , 873 (1909).

[59] Ostwald, W., Lehrbach der allgemeinen Chemie, (Engel- mann, Leipzig, 1903).

[60] Washburn, E. W., and Straehan. E. K., J. Am. Chem.

Soc. 35 , 690 (1913).

[61] Straehan, E. K., and Chu, V. G., J. Am. Chem. Soc. 36, 810 (1914).

[62] Bray, W. C., and Hunt, F. L., J. Am. Chem. Soc. 33, 781 (19H).

[63] Hlasko, M., and Wazewski. D., Bull, intern, acad. Polonaise, No. 4-5A, 181 (1928).

[64] Ref. 29, page 342.

[65] Harned, H. S., and Owen, B. B., The Physical Chemistry of Electrolytic Solutions, 3rd ed., p. 231 (Reinhold Publishing Corp., New York, 1958).

[66] Ref. 41, page 465.

10. Appendix A. The Debye-Hiickellytic Conductance

According to the Debye-Hiickel theory of electrolytic solutions, in the electrostatic effect between

ions, the discrete charge distribution is replaced by

a continuous charge distribution, or charge density,

p(r), or “ion atmosphere” which is a continuous

function of r, the distance from a reference ion. The

“ion atmosphere” extends from r—a to r=0(V1/3

) ~

where V is the volume of the system, and acts electrostatically somewhat like a sphere of charge — e at a distance k

_i from the reference ion of charge

+ e. The distance k

-1

is considered the thickness

of the “ion atmosphere”.

Thermal or Brownian motion of the ions will

tend to disturb this distribution but not entirely so, and Debye and Hiickel used the Boltzmann principle to express the ionic distribution as a function

of the ratio of the electrical and thermal energies.

The localized concentration of a given species of

ions is equal to the average or bulk concentration

of that species multiplied by the exponential function exp (— U/kT) where U gives the potential

energy at the point under consideration. If we have

a solution containing Ni ions of species i(i — 1 , 2

for simple ions and 1,2 . . .5 for electrolyte mixtures) in a volume V, the localized (average) con- centration riji of ionic species i in vicinity of ionic

species j is given by

riji = niexp (— Uji/kT) (A.l)

where ni = Ni/V. The potential energy t/p of an

i-ion with charge z,e near a 7-ion is given by z/eT]’

where e is the elementary charge, z; the valence of an i-ion, and TT is the electrostatic potential at the

location of the i-ion, stated in terms of a spherical

coordinate system with the origin at the location of

the 7-ion (or reference ion). The thermal energy is given by kT. Since f/ = z,exBj) eq (A.l) may be written:

(A.2)

Onsager-Fuoss Theory of ElectroSince each i-ion has a charge z,e the net charge

density of i-ion in the vicinity of a 7-ion, pj, is given by

pj = ^

n/z/e exp (— neW^/kT) (A.3)

i Debye and Hiickel also used Poisson’s equation

to relate pj and Tj, namely,

V*F«==i%j (A.4)

where e is the dielectric constant of the medium in which the ions exist. This relation states that at any point in a medium located by three space

coordinates (x, y, z; r, d, $) the divergence of the

gradient of the potential is proportional to the

charge density at this point. Eliminating pj between

eqs (A.3) and (A.4) gives — 477 V2 xE? = V /i,-z/e exp (— z,e 'B9/&7’) (A.5)

J £

J

This relation implies a contradiction of the superposition principle of electrostatics, namely, that

the potential due to an assembly of charges is simply

the sum of the potentials due to each of the charges

acting alone. The left side of eq (A.5) is a linear

function of 'E whereas the right side is not; it is exponential. Debye and Hiickel resolved this

dilemma by restricting the application to dilute

solutions where the ionic potential energy is small

compared with the thermal energy (see eq (A. 7)

below).

For unperturbed electrolytic solutions the

“ionic atmosphere” is assumed to be spherical.

Therefore, it is most convenient to express V2 xE;

in spherical coordinates, thus,

f

1 dr\ dr j —^ mzie exp ( — Zietyj/kT)

riji = rii exp (— Zie^j/kT) (A. 6)

14
If we expand the exponential function as a series

and drop quadratic and higher terms (quadratic

terms may be retained for symmetrical electrolytes because ne2 vanishes), since the iproblem

is restricted to dilute solutions, we have

ld_

r

2 dr

mzfe2^

(A.7)

where k is defined by

K = (A.8)

eqs (A.4) and (A.7), k

2 ^° = — 47rp/e eq (A. 12) may

be written

ziej = I K

2y¥^er2dr (A. 13)

Jdj

Combining eqs (A. 11) and (A. 13) gives

Zjej= I K

2 erA exp (-Kr)dr (A. 14)

Jaj

which upon integration in parts gives

zjej = [—eA ( kt + 1 ) exp (—#cr)]g) = eA (Ka,j+ 1) exp (— Ka,j) (A.15)

k has the dimensions of reciprocal length and in- creases with the square root of the concentration;

1 Ik may be considered the average radius or thick- ness of the “ionic atmosphere.” Equation (A.7)

is consistent with the superposition principle since

both sides are now linear functions of the potential.

Since rn = CiN/1000 where A is Avogadro’s number

and c, is the concentration in moles per liter, eq (A.8)

may be written:

Accordingly,

. _ zjej exp (Kdj)

e(l + Kdj)

which upon substitution in eq (A. 11) /gives

it/q/ \ __ W exp [ic(aj-r)]

er(l + Kdj)

(A. 16)

(A.17)

K =

4<7re 2N\

1/2 / v

1000AT/ VS ZrC!' (A.9)

Since the ionic strength, /, is defined as 1/2 ^ zjCj

i eq (A.9) may also be written

k = (A. 10)

Equation (A.7) has the general solution:

/M (r)=J4 exP ( - Kr)

+ /l' eXpUr>

(A. 11) J r r

where A and A' are integration constants. They may

be evaluated upon the following considerations.

First A' must equal zero since —» 0 as r increases.

The evaluation of A rests on the principle of electroneutrality. The total charge in the space around a given charge ej must be exactly equal to this

charge but of opposite sign. The total charge is obtained by integrating the density over the entire

space around thus

~m= (A. 12)

where 47ri^dr is the total volume of the shells in the

“ionic atmosphere” around the central ion. The integration is from the “distance of closest approach”

of the ions given by the sum of the effective radii

of the ions in “contact” to infinity. Since from

At the distance of closest approach r~dj, so that

^)'(r)

z) ei =ZjCj ziejK

edj ( 1 + Kdj) edj e ( 1 + Kaj

(A.18)

where the first term on the right is the potential

at the surface of the ion due to the charge on the

ion itself and is independent of the concentration.

The second term on the right is that portion of 4^(r)

due to the “ionic atmosphere” and is dependent on

the concentration through the value of k. In the foregoing we have considered electrolytic

solutions in the absence of external fields. We

shall now consider such solutions when subjected

to potential gradients as encountered in electrolytic conductance. In the first place, ions are neither

created nor destroyed during their motion under a dc field in an electrolytic solution (this is known as the equation of continuity analogous to the equation

of continuity in hydrodynamics which states that

matter is conserved in liquid flow). In the second

place, we have two main effects as a consequence of

the interactions between ions: the reldxdtionfield effect and the electrophoretic effect. When an

electrolytic solution is subjected to a potential field the “ionic atmosphere” leads to these two effects.

When the “ionic atmosphere” is unperturbed,

i.e., not exposed to an applied electrical field or shearing force tending to cause ions to move

relative to the solvent it is assumed to have spherical

symmetry. However, when the ion is caused to move under an applied electrical field the spherical

symmetry of the “ionic atmosphere” is disturbed.

If a particular kind of ion moves to the right, for

example, each ion will constantly have to build up

15
its ionic atmosphere to the right while the charge

density to the left gradually decays. The rate at which the atmosphere to the right forms and the one on the left decays is expressed in terms of a quantity known as the time of relaxation of the

“ionic atmosphere.” The decay occurs exponentially and the return is asymptotic to the original

random distribution. The time required for the

“ionic atmosphere” to fall essentially to zero is given by 4qd where 6 is the time of relaxation and

q is defined by

z+z- / A.+ + X- \ ^ z+ + \z+X+ + Z-\~)

(A.19)

with z+, z- , k+, and having the same significance

as given previously.

The asymmetry of the ion atmosphere owing to the effect of the time of relaxation leads to an excess charge of opposite sign behind the moving ion.

This excess charge of opposite sign causes a retardation of the moving ion. Onsager by using the

Debye-Hiickel approach, taking care of the

Brownian movement of the ions, and dropping

higher terms in the equation of continuity obtained:

AT, =- grad ^(„m = —2-= k* (A. 20)

' 3ekTl + Vq

for the relaxation effect for a dilute electrolyte

solution dissociating into' two kinds of ions. Here

q is defined by eq (A.19), X is the electrical field

intensity acting in the x direction and X\ is the

relaxation field acting in the same direction but

in the opposite sense, as a result of the perturbation

of the “ionic atmosphere.” The resultant field is then:

X+Xr=X+—

—^kX (A. 21 ) 3ekT 1 + \fq

The total electrical force acting on the ion is then

,(* + *0^4+^^) (A.22)

which leads to a movement of the ion with a velocity,

vf of:

v) = ej(X +Xi)uf = ejX ^1 4

(A.23)

where uf is the “absolute” mobility of the ion.

Since the product eie2 is negative, the relaxation

field opposes the external field. The other factor, the electrophoretic effect,

arises from the tendency of the “ionic atmosphere”

shell to move with its associated solvent molecules

under an applied electrical field in a direction oppoe\e2 3ekT 1 4- V<7 K I U

site to the motion of the ion. The forces, kj, acting on the ions must be balanced by other forces acting on the solvent molecules, ks , hence:

X njkj = — nsks (A.24)

where bulk concentrations are denoted by nj

and ns. At a distance r from the central ion and

within a volume element, dV, the localized con- centration for the ion will differ from the bulk

concentration and the directed ion force near

a 7 -ion will be ^ njjkjdV which differs from the gross

volume average force mk\dV because of the

i 7 -ion concentrations within its vicinity (see later).

For the solvent molecules, the force remains un- changed and is given by ns k„dV for the volume ele- ment dV. The net force acting on the volume

element near a /-ion is, therefore,

(\ njiki + nsks\ dV= ^

( n,-,- — m)kidV (A.25)

' t ' i A spherical shell of radius r and thickness dr is, therefore, subject to the resultant force, d

,

given by:

dSP = ^ ( nji — rii ) ki ( \nr2dr) ( A.26)

i

Neglecting the asymmetry of the “ion atmosphere”,

Fuoss used the distribution function:

nji =ni [l-ei V?/kT+$ (e^f/kT) 2

] (A.27)

to express (% — ni

)

as a function of r and the

Debye-Hiickel expression (eq (A. 17)) for the potential T/j° (r). Thus, the resultant force on the shell is given by:

dfX = 47rr

2 ^

[— n &,ej ( 1 4- Ka ,

)

i exp (— Kr)/ekTr exp

(

ko,) + n,e,ej(l 4- Ka,) 2

exp (— 2/cr)/2(e/f7V) 2 exp (2/<«, )] X Zjedr (A.28)

where ki is replaced by the electrical force, Ae,.

Debye and Hiickel originally considered that this

force imparted a velocity to the reference ion which

they assumed followed Stokes law (eq (3.4)) for a

rigid sphere having a radius = b (hydrodynamic

radius). Later Onsager considered that the force

(eq (A.28)) imparted a velocity to the reference ion

and its atmosphere which he assumed followed

Stokes volume equation:

dvj' — d^ffjriqr (A. 29)

where p is the viscosity of the medium and the

average radius = r, thus eliminating the hydrodynamic parameter b (see also Fuoss and Accascina

[15], p. 164). The total velocity, At;/, produced by

16
all the ions in the atmosphere on the 7-ion is then

given by the integral of the combination of eqs (A.28)

and (A. 29) from the “distance of closest approach”

of the ions, a, to infinity, or:

2ziz2 ql{l + Vq) we have:

/ e

2 W , ezj \

At)j'= (| r)j

j

[-A, exp (— nr

+ Ao exp (— 2Kr)/r]dr (A. 30)

Inserting values of A\ and A2 from (A. 28) gives:

Av" = — 2Xej meySiqekTK ( 1 + Ka)

+ Xej( 1 + Ka) 1Ei(2Kd)

^ rc,ef/3i7(e/T) 2 exp (2Ka) (A. 31)

where Ei (2 Ka) is the exponential integral e~2K,'dr. For symmetrical electrolytes ei = e2 and the second

term of equation (A.31) vanishes. Thus,

Av" = — 2Xejei ^ nie']/3riekT(l + k a) (A.32)

Multiplication of the numerator and denominator

of (A.32) by k and substitution in the denominator

of the value for k

2 given by (A. 8) give for Av"

Av" = —XejKl6Trr)(l + Ka) (A. 33)

which is the original result presented by Onsager.

If a = 0 (ions are point charges)(A.33) becomes:

Av" = —XejKl6irr} (A. 34)

the limiting value derived by Onsager.

Combining (A.23) for the relaxation effect and

(A. 34) for the electrophoretic effect gives:

vj = vj + Av" = Aur+ £irrkv=g

u? - Kl67rn

)

,A- 35)

for the velocity of the ion. Since ejXu* = Vj 0 this

equation may be written, for an electrolyte containing two kinds of ions:

/ 4>7re2 N Y1

'2 r

-——

V1000A-77

Vz ' Cl + ^C2- (A. 37)

Onsager replaced Ci and c2 by equivalent concentrations (c is then the same for both types of ions),

here denoted by c; hence:

(e2 W

Vj = Vj°-{tekf Vj°

/ 477e2 /V\ 1/2 UOOOAT/ V(| Zi I + |z2 |)c. (A. 38)

For uni-univalent electrolytes zi = z2 =l and

r=2 - V2 ,

/e2 (2- V2)

Vj = Vj°~{- '6ekT

Vjo +

' 8 ire2N Y12 r- :ioooat/

Vc (A.39)

Now for a potential gradient of 1 V/cm,

V= 1/299.7923 esu

and letting Vj/299.7925 = Uj, the mobility, we have:

Uj = Ujo - e

2 (2 — V2)

6ekT

Ujo- ezi 6(299.7925)77r?)

(87

\1000A77

V^‘ (A.40)

Now, since UjF=kj, where kj is the equivalent

conductance of the 7-ion, eq (A.40) may be written:

/ e

2 (2-V2

k> kj °

' 6eAT

kj° ezjF \

6 (299. 7925) 77V

/ 87re

2NYl2 n

\ 1000AT/ VC> (A.41)

Since e = 299. 7925 F/N eq (A.41) may also be

written:

| ziz2 | q

3ekT i +

KVj0 ezj

67717 K (A. 36)

where the valence of the ions, z, is now introduced.

Upon introduction of k as defined by (A. 9) and substitution of z

2 Ci "b z|c2 for ^ zfc,- and W for

kj— k

/e-(2 — V2) zjP \

\ 6eAT j " 6Nnri)

/8ne2 NYl2 r

(lOOOAT/

^

(A. 42)

Now the equivalent conductance, A, of an electrolytic solution is the sum of the equivalent con17
ductances of the constituent ions, thus:

A = Ao

e

2 (2 — V2) | z iz 2 1 . (zi+ z 2 )F'- \

6ekT ° 6Nttti )

/ 87reW \

1/2

V 1000AT / Vc H-1 cm2 equiv 1

which may be written in the form:

A = Ao~ (R1A0 + R2 ) Vc H

- 1 cm 2 equiv

where B 1 and #2 denote, respectively,

6ekT \ 1000kT )

and

5 ,

(z,+z2 )F2 / 87reW V/2

()N7TTj \ 1000AT /

1

(A.43)

(A.44)

(A.45)

(A.46)

(See also eqs (5.4) and (5.5)). Equation (A.44) is the

limiting law of Onsager.

The Onsager equation, as stated above, gives the

tangent to the conductance curve at zero concentration; it is based on point charges as the model

for the ions. In 1932, Onsager and Fuoss recalculated the electrophoresis term, using charged

rigid spheres to represent the ions. For symmetrical

electrolytes the electrophoretic term BoC112 was

replaced by BoC 112

1 (1 + Ka)

Later in a series of

papers Fuoss and Onsager gave a more comprehensive treatment in v,nich they retained the

higher terms in the Boltzmann factor and the

equation of continuity and gave a list of numerical

values for certain transcendental functions related

to the exponential integral functions. In their comprehensive treatment they considered ions as charged spheres rather than point charges, included

the osmotic-pressure and viscosity effects, and

used Oseen’s equation for the volume force rather

than Stokes’ equation for directional force to obtain

the ionic velocity in the field direction. A rather

complicated expression for the electrophoretic

term resulted; it was shown however, by Fuoss and

Hsia that the function could be closely approximated up to about 0.1 Mfor 1-1 electrolytes in water

by the 1932 result of Onsager and Fuoss. In 1969

Chen found that Fuoss and Onsager had omitted an interaction between the relaxation field and the

electrophoretic flow, which leads to an additional

term of the order of c log c with the result that the

coefficient of the c log c term in eqs (5.6) and (5.11)

of the main text becomes FiA0 — 22? 2 and the L(a

)

function (see eqs (5.19) and (5.20)) acquires a dif

ferent functional form1 from that published in 1965

The replacement of E2 by 2Eo has little effect on

the values of the limiting conductance obtained

by extrapolation because E2 is small compared

to E 1 Ao. The final equations resulting from the

theory are eqs (5.21) and (5.22) in the main text. A. Bibliography for Appendix A

[1] Fuoss. R. M., and Accascina. F., Electrolytic Conductance

(Interscience Publishers, Inc., New York, 1959).

[2] Harned. H. S., and Owen, B. B.. The Physical Chemistry of

Electrolytic Solutions, 3rd ed., Ch. 4 (Reinhold Publishing

Corp., New York, 1958).

[3] Robinson, R. A., and Stokes, R. H., Electrolyte Solutions,

2nd ed., Ch. 7 (Butterworth Scientific Publications,

London, 1959).

[4] Fuoss, R. M., and Onsager, L., J. Phys. Chem. 66, 1722

(1962); 67, 621, 628 (1963); 68, 1 (1964).

[5] Fuoss, R. M., Onsager, L., and Skinner, J. F., J. Phys. Chem.

69, 25_81 (1965).

[6] Fuoss. R. M., and Onsager, L., Proc. Nat. Acad. Sci., U.S.

[7] Fuoss, R. M., and Onsager, L., J. Phys. Chem. 61, 668

(1957).

[8] Fuoss, R. M., J. Am. Chem. Soc. 79, 3301 (1957).

[9] Fuoss, R. M.. and Onsager, L., Proc. Nat. Acad. Sci., U.S. 47, 818 (1961).

[10] Onsager, L., and Fuoss, R. M.. J. Phys. Chem. 36, 2689

(1932).

[11] Onsager, L„ Physik. Z. 27, 388 (1926); 28, 277 (1927).

[12] Debye, P„ and Hiickel. E.. Physik. Z. 24, 185. 305 (1923).

[13] Fuoss, R. M., and Hsia, K. L., Proc. Nat. Acad. Sci., U.S. 57, 1550; 58, 1818 (1967).

[14] Chen, M. S., Dissertation, Yale University, June 1969.

41, 274, 1010 (1955). 11. Appendix B. Densities of aqueous solutions of HF

d = A + Bx + Cx2 + Dx3 + Ex4 + Fx5 x = weight percent

t Range A Bx\0:i CX 105 Dx 107 £X108 FX1010 s

a Xl04

°c

0

wt %

0-30 0.99987 3.92 *1.14 12.5

0 0-55 .99987 4.705 -4.451 5.675 29.2

0

15

0-100

0-36

.99987

.99913

4.6518

3.53

-3.5739

*5.15

-1.8571 2.3603 -2.3029 73.1

5.0

15 0-54 .99913 3.507 *6.143 5.0

18 0-30 .99862 3.97 -1.56 5.0

20 0-30 .99823 4.06 - 1.85 3.9

25 0-36 .99707 4.39 — 1.71 3.0

25 0-66 .99707 4.8587 -7.2689 *1.8184 -1.6497 9.8

a — standard deviation of 1 it to equation 18 °C — E. G. Hill and A. P. Sirkar 31]

* - X 106 20 °C - F. Winteler [33]

0 °C — E. G. Hill and A. P. Sirkar [31] 25 °C — K. Fredenhagen and M. Wellmann [34]

15 °C — L. Domange [32]

18
12. Appendix C. Use of the Systeme International d’Unites

In 1964 the National Bureau of Standards adopted

the International System of Units (abbreviated SI

for Systeme International). This International System of Units was defined and given official status

in a resolution of the 11th General Conference

on Weights and Measures which met in Paris

in October 1960. The SI is based on the meter

(m) as the unit of length; the kilogram (kg) for

mass; the second (s) for time; the ampere (A)

for electric current; the Kelvin (K) for temperature;

and the candela (cd) for luminous intensity. Of

these units, those for mass, length, time, and temperature are independent; that is, a definition of one does not depend on definitions of others. However, the ampere and the candela involve other units

in their definition. For example, the ampere involves

the units of length, mass, and time and is defined

as the magnitude of electricity that, when flowing

through each of two long parallel wires separated

by one meter in free space, results in a force between two wires (owing to their magnetic fields) of

2X10-7 newton (N) (kg m s -2

) for each meter of

length. The candela needs not concern us here.

As has been customary heretofore in dealing

with the electrolytic conductance of solutions, the

cgs (centimeter-gram-second) system of units was

used in this paper. In converting to the SI system

the following changes are required:

General— The unit of length is changed from the

centimeter to the meter, the unit of mass from the

gram to the kilogram, the unit of force from the

dyne to the newton, and the unit of energy from

the erg to the joule (J). Concentrations are then

expressed in kilomoles or kiloequivalent per cubic

meter. The ionic velocity (t>) is given in m s

_1 rather

than cm s

_1 and the “electrical mobility” of an ion see page 3) would be the velocity attained by an

on under a unit of potential gradient of 1 V per

meter rather than 1 V per centimeter. Accordingly,

equivalent conductances would retain the values

commonly used, but with the unit being m2 0

_1

kequiv -1

. The elementary charge is also given in

coulombs rather than in electrostatic units and ion

sizes are expressed in fractions of the meter rather

than in angstroms.

Theoretical constants in the expressions for

equivalent conductances and activity coefficients — In converting to SI, the following changes are required:

(1) In equations 4.8, 4.9, 4.10, 5.1, 5.2, 5.15,

5.18, 7.14, A.9, A.10, A.37, A.38, A.39, A.40, A.41,

A.42, A.43, A.45, and A.46 1000 (moles per liter) is replaced by 1 m3

. (2) Since the unit of charge is not defined through

Coulomb’s law on SI, e is replaced by e/ (47reo) 1/2 in

all equations where e appears. The constant, eo is known as the permittivity of free space and has the

value 8.8541 7(3)>fcO- 12 C

2J- 1 m-1

. (3) The constants in the various equations have

the following values:

k = 1.38054(18) X10~23J deg-'

N

6.02252(28) XlO26 kmoU1 e= 1.60210(2) X10“ 19 C

F = 9.64870(16) X1017C kequiv- 1 and the unit of viscosity is 10_1 N s m

-2

. Here as

in the text, the numbers in parentheses represent

established limits of error.

13. Tables

Table 1 . Specific conductances of standard aqueous solutions o/KCl

Demal* Grams KC1 per 1000

g solution in vacuum

Specific conductances, f1

1 cm at

0 °C 18 °C 25 °C

1.0 71.1352 0.065144 0.097790 0.111287

0.1 7.41913 .0071344 .0111612 .0128496

.01 0.745263 .00077326 .00121992 .00140807

*Based on the international atomic weights of 1933.

19
Table 2. Physical properties of water

Temperature Density " Viscosity 6 Dielectric constant c

t d 0

e

°C g/ml cP

0 0.99987 1.787 d 87.74 (87.90) 5 .99999 1.517 85.76 (85.90)

10 .99973 1.306 83.83 (83.95)

15 .99913 1.138 81.95 (82.04)

18 .99862 1.053 80.84 (80.93)

20 .99823 1.002 80.10 (80.18)

25 .99707 0.8903 78.30 (78.36)

30 .99567 .7974 76.55 (76.58)

35 .99406 .7194 74.83 (74.85)

38 .99299 .6783 73.82 (73.83)

40 .99224 .6531 73.15 (73.15)

45 .99025 .5963 71.51 (71.50)

50 .98807 .5471 69.91 (69.88)

55 .98573 .5044 68.34 (68.30)

60 .98324 .4669 66.81 (66.76)

65 .98059 .4338 65.32 (65.25)

70 .97781 .4044 63.86 (63.78)

75 .97489 .3782 62.43 (62.34)

80 .97183 .3547 61.03 (60.93)

85 .96865 .3340 59.66 (59.55)

90 .96534 .3149 58.32 (58.20)

95 .96192 .2976 57.01 (56.88)

100 .95838 .2822 55.72 (55.58)

a M. Thiesen [6]; International Critical Tables [7].

b J. F. Swindells, J. R. Coe, and T. B. Godfrey [8], R. C. Hardy and R. L. Cottington [9]; and J. R. Coe and

T. B. Godfrey [10].

c C. G. Malmberg and A. A. Maryott [11].

d Values in parentheses are those found by B. B. Owen,

R. C. Miller, C. E. Milner, and H. L. Cogan [12]; values

above 70 °C were calculated from their equation expressing the values obtained at lower temperatures as a function of temperature.

TABLE 3. Values of the Debye-Hiickel-Onsager constants Bi and B 2 for equivalent conductances for l—l aqueous

solutions from 0 to 100 °C

t B x B~,

°C l 112 equiv~ 112 H-1 cm2 1

112 equiv~3,2 0 « 0.2211 (1) a 29.817 (6)

5 .2227(1) 35.206(8)

10 .2244(1) 40.995(9)

15 .2261(1) 47.169(10)

18 .2272(1) 51.060(11)

20 .2280(1) 53.722(11)

25 .2300(1) 60.639(13)

30 .2321(1) 67.906(14)

35 .2343(1) 75.508(16)

38 .2357(1) 80.240(17)

40 .2366(1) 83.449(18)

45 .2391(1) 91.711(19)

50 .2416(1) 100.31 (2)

55 .2443(1) 109.20 (2)

60 .2471(1) 118.42 (3)

65 .2499(1) 127.94 (3)

70 .2529(1) 137.79 (3)

75 .2560(1) 147.94 (3)

80 .2593(1) 158.41 (3)

85 .2627(1) 169.95 (4)

90 .2662(1) 179.99 (4)

95 .2698(1) 191.32 (4)

100 .2737(1) 202.71 (4)

" The numbers in the parentheses gives the ± un- certainty in the last decimal arising from the uncertainties

in the physical constants.

20
TABLE 4. Values of the Fuoss-Onsager Ei and E2 constants for equivalent conductances for 1—1 aqueous

solutions from 0 to 100 °C

t Ei Ez a

°c 1 equiv~ x fl

_1 cm2 / equiv~2

0

b 0.4922(3) b 9.72(1)

5 .4992(3) 11.56(1)

10 .5066(3) 13.56(1)

15 .5146(4) 15.72(1)

18 .5196(4) 17.10(1)

20 .5233(4) 18.06(1)

25 .5325(4) 20.56(1)

30 .5422(4) 23.23(1)

35 .5526(4) 26.08(1)

38 .5591(4) 27.88(1)

40 .5637(4) 29.11(1)

45 .5754(4) 32.32(1)

50 .5876(4) 35.73(2)

55 .6008(4) 39.33(2)

60 .6145(4) 43.13(2)

65 .6287(4) 47.14(3)

70 .6439(4) 51.37(3)

75 .6599(5) 55.83(3)

80 .6768(5) 60.55(3)

85 .6945(5) 65.42(4)

90 .7132(5) 70.63(4)

95 .7328(5) 76.10(4)

100 .7538(5) 81.77(5)

3 Note that 2E2 is involved in the final conductance

equations (eqs (5.21) and (5.22)).

b The numbers in the parentheses give the ± uncertainty in the last decimal arising from the uncertainties

in the physical constants.

Table 5. Differences in the values of Bi and B 2 from

those given in table 3 if the dielectric constants of

water determined by Owen et al. are used instead of

those of Malmberg and Maryott*

t AB, ABi

°C D12 equiv ~ ij

fl

_1 cm2 Iff2 equiv~3/2

0 -0.0006 -0.027

5 - .0005 -.029

10 - .0005 -.029

15 - .0004 -.026

18 - .0004 -.028

20 - .0003 -.027

25 - .0003 -.024

30 - .0001 -.014

35 - .0001 -.010

38 0.0000 -.006

40 .0000 0.000

45 .0001 .006

50 .0002 .02 55 .0002 .04 60 .0003 .04

65 .0004 .07 70 .0005 .09 75 .0006 .11 80 .0006 .13 85 .0007 .16

90 .0008 .19 95 .0009 .22 100 .0010 .26

*Owen et al. value minus Malmberg-Maryott value.

21
Table 6. Differences in the values of Ei and E2 from

those given in table 4 if the dielectric constants of water

determined by Owen et al. are used instead of those

of Malmberg and Maryott*

t ae2 °

°c l equiv~l Cl- 1 cm2 1

112 equiv-312

0 -0.0027 -0.04

5 - .0025 -.04

10 - .0021 -.04

15 -.0017 -.03

18 -.0017 -.04

20 -.0015 -.04

25 -.0012 -.03

30 - .0007 -.02

35 - .0004 -.01

38 - .0002 -.01

40 0.0000 0.00

45 .0002 .01 50 .0008 .02 55 .0010 .04 60 .0013 .06

65 .0021 .10 70 .0024 .13 75 .0028 .17 80 .0033 .19 85 .0039 .24

90 .0045 .29 95 .0051 .34 100 .0057 .41

*Owen et al. value minus Malmberg-Maryott value.

a Note that the total difference is 2AE2 ; see eqs (5.21)

and (5.22).

Table 7. Values of the Debye-Hiickel constants Ac and Bc for activity coefficients of aqueous solutions from 0 to 100 °C

(On a volume basis)

t A c Bc

°C l 112 mol ~ 1/2

l

112 mol112

0 0.4918 0.3249

5 .4953 .3256

10 .4990 .3264

15 .5029 .3273

18 .5054 .3278

20 .5072 .3282

25 .5116 .3292

30 .5162 .3301

35 .5212 .3312

38 .5242 .3318

40 .5263 .3323

45 .5318 .3334

50 .5374 .3346

55 .5434 .3358

60 .5495 .3371

65 .5559 .3384

70 .5625 .3397

75 .5695 .3411

80 .5767 .3426

85 .5843 .3440

90 .5921 .3456

95 .6001 .3471

100 .6087 .3488

22
TABLE 8. Values of Ao for HF

t Values Experimenters

°c fl~ 1 cm2equiv~ 1 0 255 C. B. Wooster [35]

16 350.3 W. A. Roth [36]

18 361.7 W. A. Roth [36]

18 364 E. G. Hill and A. P. Sir- kar [31]

20 374.2 W. A. Roth [36]

25 393.2 E. Deussen [37]

25 399 R. Wegscheider [38]

25 397.3 H. Pick [39]

25 400.2 C. W. Davies and L. J. Hudleston [40]

25 400.2 K. Fredenhagen and M.

Wellmann [34]

25 404 W. A. Roth [36]

25 404 C. B. Wooster [35]

Table 10. Equivalent conductances (fl -1 cm2 equiv~ x ) of aqueous solutions o/HF at 0, 16, 18, 20, and 25 °C

c 0 °c 16 °C 18 °C 20 °C 25 °C

mol Z _1 0.004 106.7 140.5

.005 97.7 128.1

.006 90.9 112.9 114.6 116.3 118.8

.007 85.5 105.7 107.3 108.9 111.4

.008 81.1 99.9 101.3 102.9 105.4

.009 77.3 95.0 96.4 97.8 100.4

.01 74.2 90.8 92.1 93.5 96.1

.02 56.2 67.7 68.8 69.8 72.2

.05 39.3 46.8 47.6 48.3 50.1

.07 34.7 41.3 41.9 42.6 44.3

.10 30.7 36.4 37.0 37.7 39.1

.20 24.8 29.6 30.1 30.7 31.7

.50 20.4 26.3

.70 19.5 25.1

1.0 18.8 24.3

Table 9. Values of the constants governing the dissociation of HF“

t K k

°C mol l~ x mol H-1 cm2 equiv~ x 0 0.00109 0.413 0.7 16 .000782 .362 1.2 18 .000755 .355 1.1 20 .000731 .347 1.1 25 .000684 .381 0.6

a See eqs (7.3) and (7.4).

23
Table 11. Equivalent conductances (O 1 cm'2 equiv ') of aqueous solutions o/HCl from —20 to 65 °C

c -20 °C -10°C 0°C 5 °C 10 °C 15 °C 20 °C 25 °C 30 °C 35 °C 40 °C 45 °C 50 °C 55 °C 65 °C

mol l~ l

0.0001 296.4 360.8 424.5 . 487.0 547.9 577.7 606.6 662.9

.0005 295.2 359.2 422.6 , 484.7 545.2 575.1 603.5 660.0

.001 294.3 358.0 421.2 483.1 543.2 573.1 601.3 657.8

.005 291.0 353.5 415.7 . 476.7 535.5 564.4 592.6 647.3

.01 288.6 350.3 411.9 472.2 530.3 558.7 586.5 641.2

.05 280.3 339.9 398.9 456.7 512.4 565.6 616.9

.1 275.0 333.3 391.1 446.8 501.1 552.8 602.8

.5 228.7 254.8 283.0 308.1 336.4 360.7 386.8 411.9 436.9 461.1 482.4 508.0 552.3

1.0 211.7 235.2 261.6 283.9 312.2 332.2 359.0 379.4 402.9 424.8 445.3 468.1 509.3

1.5 196.2 216.9 241.5 261.5 287.5 305.8 331.1 349.4 371.6 391.5 410.8 431.7 469.9

2.0 182.0 199.9 222.7 240.7 262.9 281.4 303.3 321.6 342.4 360.6 378.2 398.0 433.6

2.5 131.7 168.5 184.3 205.1 221.4 239.8 258.9 277.0 295.8 315.2 332.0 347.6 366.7 399.9

3.0 120.8 154.6 169.5 188.5 203.4 219.3 237.6 253.3 271.5 289.3 304.8 319.0 336.9 368.0

3.5 85.5 111.3 139.6 155.6 172.2 186.5 201.6 218.3 232.9 248.6 263.9 279.4 292.1 308.6 337.2

4.0 79.3 102.7 129.2 143.4 158.1 171.5 185.6 200.0 214.2 228.4 242.2 256.6 268.2 283.6 310.1

4.5 73.7 94.9 119.5 132.0 145.4 157.4 170.6 183.1 196.6 209.5 222.5 235.2 246.7 260.2 284.7

5.0 68.5 87.8 110.3 121.3 133.5 144.4 156.6 167.4 180.2 191.9 204.1 215.4 226.5 238.4 261.0

5.5 63.6 81.1 101.7 111.4 122.5 132.3 143.6 152.9 165.0 175.6 187.1 197.1 207.7 218.3 239.1

6.0 58.9 74.9 93.7 102.2 112.3 121.2 131.5 139.7 151.0 160.6 171.3 180.2 190.3 199.7 218.9

6.5 54.4 69.1 86.2 93.8 103.0 111.0 120.4 127.7 138.2 146.8 156.9 164.8 174.3 182.7 200.4

7.0 50.2 63.7 79.3 86.0 94.4 101.7 110.2 116.9 126.4 134.3 143.3 150.8 159.7 167.2 183.5

7.5 46.3 58.6 73.0 78.9 86.5 93.3 100.9 107.0 1J5.7 122.9 131.6 138.1 146.2 153.1 168.1

8.0 42.7 54.0 67.1 72.4 79.4 85.6 92.4 98.2 106.1 112.6 120.6 126.7 134.0 140.4 154.1

8.5 39.4 49.8 61.7 66.5 72.9 78.7 84.7 90.3 97.3 103.2 110.7 116.4 123.0 128.8 141.5

9.0 36.4 45.9 56.8 61.2 67.1 72.5 77.8 83.1 89.4 94.8 101.7 107.1 112.9 118.3 130.1

9.5 33.6 42.3 52.3 56.4 61.8 66.8 71.5 76.6 82.3 87.3 93.6 98.7 103.9 108.8 119.7

10.0 31.2 39.1 48.2 52.0 57.0 61.6 65.8 70.7 75.9 80.5 86.3 91.1 95.7

10.5 28.9 36.1 44.5 48.0 52.7 56.9 60.7 65.3 70.1 74.3 79.6 84.1 88.4

11.0 26.8 33.4 41.1 44.4 48.8 52.6 56.1 60.2 64.9 68.7 73.6 77.7 81.7

11.5 24.9 31.0 38.0 41.1 45.3 48.5 51.9 55.3 60.1 63.6 68.0 71.7 75.6

12.0 23.1 28.7 35.3 38.0 42.0 48.0 55.6 62.8 70.0

12.5 21.4 26.7 32.7 39.0 44.4 51.4 57.9 64.8

13.0 20.0 24.9

24
TABLE 12. Interpolation equations, constants, and coefficients for A for HCT', b

, c

A = A + Be 4- Cc2 + Dc3 + Ec4 + Fc5 + Gc6

CJ o A B C D FX 10-' Fx 10-2 Cx 10-4 5 c

-20 188.486 -63.0162 16.8512 -2.88586 2.76616 -1.36131 2.68729 0.1 3.5-13.0

-10 225.132 -58.7706 12.3280 -1.88762 1.71785 -0.819972 1.58613 0.1 3.5-13.0

A = A0 — Sc ll2 -\- Ec log c + Ac-h Be31

'2 4- Cc2 + Dc312

t °C Ao S E A B C D s c

0 265.03 88.42 137.963 -177.413 92.7510 -18.2200 0.07 0.5-3.

0 265.03 88.42 57.5126 -43.3796 12.0843 -1.10359 0.10 3.5-12.5

5 297.42 101.45 125.32 558.938 - 1752.15 0.10 0 -0.01

5 297.42 101.45 103.189 414.041 -2474.81 3477.47 0.01 0.01-0.1

5 297.42 101.45 143.672 - 169.686 78.5958 - 13.3975 0.09 0. 1-3.3

5 297.42 101.45 76.6573 -60.7656 17.6613 -1.69413 0.08 3.3-12.0

10 329.78 115.00 167.354 - 194.674 89.2927 -15.1538 0.14 0.5-3.

10 329.78 115.00 87.9928 -69.0294 19.9793 -1.91012 0.12 3.6-12.5

15 362.05 129.03 154.83 576.443 -1512.88 0.12 0 -0.01

15 362.05 129.03 30.2129 1403.63 -5817.09 7203.38 0.06 0.01-0.1

15 362.05 129.03 178.321 - 204.494 92.4328 - 15.3605 0.08 0. 1-3.4

15 362.05 129.03 104.249 -83.4819 24.7903 -2.44755 0.06 3.4-11.5

20 394.16 143.59 175.522 - 150.370 36.5182 0.01 0.5-3.

20 394.16 143.59 107.020 -81.4337 23.2970 -2.21043 0.14 3.2-12.5

c 25 426.06 158.63 185.76 747.385 -2095.71 0.05 0 -0.01

c 25 426.06 158.63 173.105 43.5147 -345.464 0.14 0.01-0.1

c 25 426.06 158.63 221.501 -252.771 115.606 - 19.5824 0.10 0. 1-3.0

c 25 426.06 158.63 143.554 -116.628 35.2535 -3.56231 0.15 3.0-11.6

30 457.65 174.13 202.319 - 166.121 39.2792 0.01 0. 5-3.2

30 457.65 174.13 139.378 - 105.934 30.5564 -2.94264 0.15 3.2-12.5

35 488.91 190.06 218.01 937.656 -2717.23 0.07 0 -0.01

35 488.91 190.06 62.4686 2225.76 - 10030.0 13176.0 0.06 0.01-0.1

35 488.91 190.06 264.592 -295.287 133.716 -22.5186 0.10 0.1-3.

35 488.91 190.06 157.608 -119.583 34.5545 -3.34375 0.14 3.3-11.5

40 519.75 206.42 284.745 -310.940 139.117 -23.3201 0.01 0. 5-3.6

40 519.75 206.42 165.451 -122.051 34.7804 -3.33198 0.13 3.6-12.5

45 550 12 223 24 251.84 960 692 -2068.98 0.10 0 -0.01

45 550.12 223.24 143.352 1716.09 -8031.14 10523.2 0.10 0.01-0.1

45 550.12 223.24 311.829 -343.170 155.721 -26.4600 0.11 0. 1-3.3

45 550.12 223.24 196.780 - 148.978 43.3720 -4.25969 0.12 3.3-11.5

25
TABLE 12. Interpolation equations, constants, and coefficients for A. for HC1", b

, c — Continued

A = A0 —Sc 1/2 +Ec log c + Ac + Be31

'2 + Cc'2 + Dc512

t°C Ao S E A B C D s c

50 579.94 240.42 269.31 3501.14 -90352.7 1060726 -4254317 0.15 0 -0.01

50 579.94 240.42 301.922 -296.692 117.737 - 17.2082 0.08 0.5-3.

50 579.94 240.42 188.591 -133.516 37.1664 -3.49219 0.09 3.7-12.5

55 609.17 258.02 287.27 1258.14 -3704.21 0.10 0 -0.01

55 609.17 258.02 255.518 877.709 -4758.72 6156.27 0.06 0.01-0.1

55 609.17 258.02 361.044 -390.808 176.840 -30.1100 0.10 0.1-3.

55 609.17 258.02 229.922 - 170.083 49.1104 -4.82221 0.01 3. 3-9.

65 665.55 294.26 324.15 5257.15 -117982 1107235 - 3375389 0.09 0 -0.01

65 665.55 294.26 565.968 -1787.61 4760.06 -5501.35 0.13 0.01-0.1

65 665.55 294.26 413.887 -439.598 196.840 -33.1617 0.24 0. 1-3.4

65 665.55 294.26 267.714 - 193.878 55.3978 -5.41006 0.01 3. 4-9.

a The dimensions of the coefficients are such that their product with the appropriate power of c is cm 2 equiv -1

. 6 The c log c term was not used for temperatures of 0, 10, 20, 30, and 40 °C because data were not available below

0.5 N.

c Equations also given in text.

TABLE 13. Equivalent conductances (fl 'em2 equiv ~ x ) of aqueous solutions o/HBr at 25 °C

c A c A c A

mol /“' 0.0002 425.5mol /

' 0.04 402.7

mol l~ x 0.80 345.3

.0003 425.0 .05 400.4 .85 342.6

.0004 424.6 .06 398.4 .90 339.9

.0005 424.3 .07 396.5 .95 337.2

.0006 424.0 .08 394.9 1.0 334.5

.0007 423.7 .09 393.4 1.5 307.6

.0008 423.4 .10 391.9 2.0 281.7

.0009 423.2 .15 386.0 2.5 257.8

.001 422.9 .20 381.4 3.0 236.8

.002 421.1 .25 377.5 3.5 217.5

.003 419.7 .30 374.0 4.0 199.4

.004 418.5 .35 370.8 4.5 182.4

.005 417.6 .40 367.7 5.0 166.5

.006 416.7 .45 364.7 5.5 151.8

.007 415.8 .50 361.9 6.0 138.2

.008 415.1 .55 359.0 6.5 125.7

.009 414.4 .60 356.2 7.0 114.2

.01 413.7 .65 353.5 7.5 103.8

.02 408.9 .70 350.8 8.0 94.4

.03 405.4 .75 348.0 8.5 85.8

26
Table 14. Equivalent conductances (H-1 cm2 equiv~ l ) of aqueous solutions of HBr from — 20 to 50 °C

c -20 °C -10 °C 0 °C H—

1 o

o n 20 °C 30 °C 40 °C 50 °C

mol l~ x 0.50 240.9 295.9 347.0 398.9 453.6 496.8

.75 234.7 284.9 339.0 387.2 433.8 480.6

1.00 229.6 276.0 329.0 380.4 418.6 465.2

1.25 221.7 265.8 314.9 362.8 401.8 442.9

1.50 209.5 254.9 298.9 340.6 381.8 421.4

1.75 198.3 243.1 284.6 327.3 366.2 404.8

2.00 150.8 188.6 231.3 271.8 314.1 350.5 387.4

2.25 143.4 180.1 219.6 258.3 296.9 332.3 367.0

2.50 136.8 171.7 208.3 244.8 281.7 316.0 349.1

2.75 131.1 164.1 198.3 232.9 267.6 301.2 333.2

3.00 125.7 157.2 189.5 222.2 255.0 287.8 318.6

3.25 120.6 150.8 181.7 212.4 244.3 275.4 304.9

3.50 116.1 144.1 174.6 203.2 234.4 263.7 291.9

3.75 87.1 112.0 137.6 167.4 194.7 224.2 252.2 279.4

4.00 84.0 107.5 132.3 160.2 186.8 214.2 239.7 266.9

4.25 80.9 103.1 127.7 153.2 179.0 204.5 228.8 254.6

4.50 78.0 99.0 123.0 146.4 171.2 195.1 218.8 242.6

4.75 75.1 95.1 117.7 139.9 163.1 186.1 209.3 231.3

5.00 72.3 91.4 112.6 134.0 155.7 178.2 199.6 221.3

5.25 69.6 87.8 107.8 128.3 148.7 170.5 190.0 211.4

5.50 67.0 84.2 103.1 122.7 142.1 162.8 181.4 201.8

5.75 64.4 80.6 98.6 117.3 135.7 155.3 173.2 192.4

6.00 61.8 77.2 94.3 112.0 129.6 148.0 165.4 183.4

6.25 59.3 73.8 90.1 106.9 123.7 140.9 157.8 174.7

6.50 56.8 70.7 86.0 102.0 118.0 134.1 150.5 166.3

6.75 54.4 67.7 82.1 97.2 112.5 127.6 143.3 158.4

7.00 51.9 64.6 78.4 92.6 107.1 121.4 136.3 150.8

27
TABLE 15. Interpolation equations, constants, coefficients, and standard deviations offits for A for HBr a ’ b

A = A + Bc + Cc2 + Dc3 + Ec4 + Fc5 + Gc6

t °c A B C D E F G 5 c

-20

cr" 1 cm2 equiv " 1 153.2186 — 23.74458 1.984868 -0.09437269 0.0 3.85-7.12 - 10 269.7145 -96.49953 23.22557 -2.354479 0.0 2.12-3.43

-10 -2182.739 2809.984 -1391.051 359.8999 -51.69493 3.913505 -0.1220707 0.1 3.43-7.08

A - A0 - Sc l ' 2 + Ac + Be312 + Cc2 + Dc5' 2 (s = 0.0)

t °C Ao S A B C D c

0

cr~

1 cm2equiv~ 1 267.47 88.96 383.7746 -924.1801 909.7568 -318.2496 0.65-1.27

0 267.47 88.96 221.8995 -206.9480 21.53078 18.09947 1.27-2.09

0 267.47 88.96 303.0124 -422.2153 209.3597 -35.63960 2.09-3.42

0 267.47 88.96 1030.928 -1472.506 711.2473 -115.0276 3.42-4.78

0 267.47 88.96 53.84782 -33.86838 6.592146 -0.2034687 4.78-7.05

10 331.59 115.41 517.3115 -1217.696 1138.968 -378.8095 0.65-1.18

10 331.59 115.41 - 14.86989 305.0183 -326.0611 94.43531 1.18-2.11

10 331.59 115.41 418.1359 -573.9359 276.5141 -45.09855 2.11-3.29

10 331.59 115.41 -39.21039 125.1741 -77.38015 14.15616 3.29-4.81

10 331.59 115.41 66.16035 -43.75662 10.46823 - 76.65289 4.81-7.01

20 398.05 144.48 182.2736 - 162.3090 103.8241 -48.40040 0.65-1.23

20 398.05 144.48 565.2512 -928.6667 560.8808 -118.3103 1.23-2.12

20 398.05 144.48 581.8613 -844.4378 433.9378 -76.25953 2.12-3.25

20 398.05 144.48 479.9294 -643.1428 305.0230 -49.29015 3.25-4.76

20 398.05 144.48 174.1218 - 164.6092 57.37973 -6.869268 4.76-6.98

30 460.14 174.71 952.2020 -2645.858 2806.021 -1017.448 0.65-1.23

30 460.14 174.71 2796.582 -6065.851 4514.621 -1130.280 1.23-2.04

30 460.14 174.71 251.6347 - 189.8346 19.28790 9.967165 2.04-3.32

30 460.14 174.71 114.8066 -62.75681 6.837291 1.154146 3.32-4.74

30 460.14 174.71 -1.499032 63.95883 -37.66776 6.133478 4.74-6.91

40 523.06 207.21 771.4765 -1706.708 1518.772 -480.7958 0.65-1.21

40 523.06 207.21 1596.403 -3278.587 2389.530 -593.4768 1.21-2.00

40 523.06 207.21 622.8765 -851.3158 423.1395 - 72.47076 2.00-3.74

40 523.06 207.21 1231.781 - 1628.505 743.0089 -114.2550 3.74-5.08

40 523.06 207.21 280.8612 -268.7814 96.02265 -11.81506 5.08-6.89

50 584.43 241.50 647.1861 -1412.163 1392.202 -504.9734 0.64-1.25

50 584.43 241.50 1870.611 -3887.601 2867.502 -719.6425 1.25-2.01

50 584.43 241.50 850.1113 -1230.670 646.5982 -116.9136 2.01-3.32

50 584.43 241.50 28.24088 109.7581 -84.59918 16.49943 3.32-4.70

50 584.43 241.50 75.99004 3.194942 -18.72547 4.101969 4.70-6.85

a The dimensions of the coefficients are such that their product with the c quantity is fl _1 cm2 equiv -1

. b The c log c term was not used for these temperatures because data were not available below 0.5 N.

28
Table 16. Equivalent conductances (fl _1 cm2 equiv~ l ) of aqueous solutions of HI at 25 °C

c A c A c A

mol l~ l 0.00045 423.2

mol l~ x 0.0085 413.7

mol l~ x 0.35 377.4

.00050 423.0 .0090 413.4 .40 374.9

.00055 422.9 .0095 413.1 .45 372.3

.00060 422.7 .010 412.8 .50 369.8

.00065 422.6 .015 410.3 .55 367.3

.00070 422.4 .020 408.3 .60 364.8

.00075 422.3 .025 406.6 .65 362.2

.00080 422.2 .030 405.2 .70 359.7

.00085 422.0 .035 403.9 .75 357.1

.00090 421.9 .040 402.8 .80 354.5

.00095 421.8 .045 401.7 .85 351.9

.0010 421.7 .050 400.8 .90 349.2

.0015 420.7 .055 399.9 .95 346.6

.0020 419.8 .060 399.1 1.0 343.9

.0025 419.1 .065 398.3 1.5 316.4

.0030 418.5 .070 397.6 2.0 288.9

.0035 417.9 .075 396.9 2.5 262.5

.0040 417.3 .080 396.3 3.0 237.9

.0045 416.8 .085 395.6 3.5 215.4

.0050 416.4 .090 395.1 4.0 195.1 .0055 415.9 .095 394.5 4.5 176.8

.0060 415.5 .10 394.0 5.0 160.4

.0065 415.1 .15 389.5 5.5 145.5

.0070 414.8 .20 385.9 6.0 131.7

.0075 414.4 .25 382.9 6.5 118.6

.0080 414.1 .30 380.1 7.0 105.7

29
TABLE 17. Equivalent conductances (Vi 1 cm2 equiv *) of aqueous solutions of HI from —20 to 50°

c -20 °C -10 °C 0 °C 10 °C 20 °C 30 °C 40 °C 50 °C

mol l~ l 0.4 253.9 300.7 354.5 411.9 453.6 501.8

0.6 249.1 293.6 344.6 401.1 441.5 488.4

0.8 242.4 285.5 333.9 389.1 429.3 474.8

1.0 234.8 276.9 322.7 376.3 416.5 460.8

1.2 226.6 267.9 311.4 363.1 403.4 446.3

1.4 218.2 258.8 300.2 349.8 390.0 431.4

1.6 209.9 249.7 289.1 336.6 376.5 416.3

1.8 201.8 240.7 278.4 323.7 363.0 401.2

2.0 194.0 231.9 268.1 311.1 349.6 386.1

2.2 147.7 186.6 223.4 258.2 299.0 336.3 371.2

2.4 143.5 179.5 215.1 248.8 287.3 323.3 356.5

2.6 138.8 172.8 207.2 239.7 276.1 310.5 342.2

2.8 133.9 166.5 199.5 231.0 265.4 298.1 328.2

3.0 99.9 129.0 160.5 192.0 222.7 255.1 286.1 314.7

3.2 97.4 124.1 154.7 184.8 214.6 245.2 274.5 301.7

3.4 94.2 119.5 149.1 177.8 206.7 235.7 263.4 289.2

3.6 90.6 115.0 143.7 171.0 198.9 226.4 252.6 277.3

3.8 86.9 110.9 138.3 164.3 191.2 217.4 242.3 266.0

4.0 83.6 106.9 132.8 157.6 183.5 208.4 232.5 255.3

4.2 80.9 103.0 127.3 151.0 175.6 199.6 223.1 245.3

4.4 79.3 99.2 121.6 144.3 167.5 190.7 214.1 235.9

30
Table 18. Interpolation formulas for A for HI "• d - e

t°C

-20 A = — 188.5 + 268.0c - 79.17c2 + 7.294c 3 -10 A = 55.399 + 157.67c - 82.804c 2 + 16.276c 3 - 1 . 1560c4 0 A = 266.31 - 88.70c'/2 + 280.455c - 370.407c 3/ 2 + 176.385c 2 - 29.2746c 5/ 2 10 A = 330.58 - 1 15.18c'/2 + 248.768c - 298.965

3/2 + 132.508c 2 - 20.8142c5/ 2 20 A = 394.66 - 143.70c'/2 + 302.982c - 379.036c3/2 + 177.129c 2 - 29.2963c

5'2

» 25 A = 426.45 - 158.72c'/2 + 185.96c log c + 1085.7c - 11678c 3/ 2 + 107071c 2 - 394591c5/ 2

c 25 A = 426.45 - 158.72c'/2 + 248.885c - 256.566

3' 2 + 96.2195c 2 - 12.3887c 5/ 2 30 A = 457.93 - 174.20c'/2 4- 370.263c - 445.069c3/2 + 198.665c 2 - 31.2569c5/2 40 A = 519.72 - 206.42c'/2 + 317.597c - 312.945c 5/ 2 + 112.255c 2 - 13.6656c5/2 50 A = 579.38 - 240.28c'/2 + 354.327c - 331 . 163c5/2 + 1 10.209c 2 - 1 1.7132c5/ 2

a The c log c term was not used for temperatures other than 25 °C because data were not available below 0.4N.

» 0.0000- 0.01 N.

c 0.01N— 7. ON.

d The dimensions of the coefficients are such that their product with the c quantity is fl _1

cm 2 equiv -1

. e Statistical Information.

t °c s n Standard

error

-20 0.07 5 0.03 -10 .14 7 .05 0 1.05 15 .3 10 .75 15 .2 20 1.63 15 .4 25 1.03 30 .2 30 1.14 12 .3 40 1.40 13 .4 50 2.38 13 .6

Table 19. Limiting equivalent conductances

{n~'cm2>equiv~ 1 )of HF, HC1, HBr andWl in water at 25 °C

This

paper

Maclnnes

[64]*

Harned

and Owen

[65]*

Robinson

and Stokes

[66]*

HF 405.10

25 °C

405.01

HC1 426.06 425.95 425.94 425.9.5 HBr 427.74 428.0, 427. 7» 427.74 HI 426.45 426A 426.4» 426.44

*Values converted to absolute electrical units.

31
14. Glossary of Symbols

a, activity; also ion size, “ion-size” parameter, or “distance of closest approach” of ions.

b, Bjerrum ion parameter in conductance (see

eq. (5.10)).

c, concentration in equivalents per 1000 cm3

. d, density.

e, elementary charge.

i, electric current.

ji, j[, j-2 , j2 , constants in the expression relating A

for HF with the concentration of HF (see p. 10).

k, Boltzmann constant; also constant relating the

equilibrium between HF7 and HF and F~

(see table 9).

k,, general forces acting on ions.

A's , general forces acting on solvent molecules.

/, length in cm; liter. m, molar concentration.

m' , ionic mobility coefficient.

m, meter.

n, localized average concentration of ions.

rii, number of i ions per volume V.

riji , average concentration of i ions in the vicinity

of j ions.

q , function in theory of conductance (see p. 5).

r, distance of separation of charges,

ri, unit vector.

Sj-, standard deviations for equation fits.

t, transference or transport number; temperature.

u, ionic mobility.

u°, limiting ionic mobility.

uf, “absolute” mobility of an ion.

u\, “electrical ionic mobility.”

v, velocity.

y, ratio of the concentration of F~ to the stoichiometric concentration of HF.

73 , ratio of the concentration of HF2 to the stoichiometric concentration of HF.

z\ , ionic charge.

A , area in cm 2

. A', A", integration constants (see equation (A. 11 )). A c , coefficient in the limiting law of Debye and

Hiickel for activity coefficients, on molar basis

(see table 7). A, B, C, D, E, F, G, constants in polynomials.

B\, coefficient of the relaxation term in the theory

of conductance (see table 3). By, coefficient of the electrophoretic term in the

theory of conductance (see table 3). Br , coefficient of ion-size term in the Debye-Hiickel

theory for activity coefficients, on molar scale

(see table 7). C, stoichiometric concentration.

E, constant in Fuoss-Onsager theory of conductance.

Ei, constant needed to calculate E (see table 4). E2 , constant needed to calculate E (see table 4). E„, applied potential.

F, Faraday.

/, ionic strength.

J(a), coefficient of c term in the Fuoss-Onsager

theory of conductance.

Jr, cell constant.

K, dissociation constant.

K\, association constant.

K°a, association constant including effects of solutesolvent interactions and the free volume of the

solute.

L(a), coefficient of the c term in the Fuoss-OnsagerSkinner theory of conductance.

M, molarity.

N, Avogadro constant.

Nj, number of i ions.

r b Q{b), the definite integral exp {x)x Adx where

x = e

2 lrekT.

R, electrical resistance.

Rsp, specific electrical resistance or electrical

resistivity.

S, constant equal to B\Ao + B’.

T, Kelvin temperature.

U, potential energy.

V, volume.

W, function in the theory of conductance and

related to q (see page 17).

X, vector electric field in the x-direction.

force.

electrical force. a, degree of dissociation or ionization or complex

formation.

1 —a, degree of association.

7. activity coefficient,

e. dielectric constant,

to, permittivity of free space.

r\, viscosity.

Tjo, viscosity of solvent.

k, the reciprocal of the average radius of an ionic

atmosphere.

cr, electrical conductance.

cr sp, specific electrical conductance or electrical

conductivity.

(crsp)s, standard specific electrical conductivity.

cr p , electrical conductance of an experimental

solution.

X, ionic equivalent conductance.

A„, limiting ionic equivalent conductance.

v, number of ions in one molecule ot electrolyte.

pr , charge density.

Pi, coefficient of ionic friction.

r, relation of the Bjerrum parameter and 1 Ik. A, equivalent conductance.

Ao, limiting equivalent conductance.

A'", molar conductance.

A,"', limiting molar conductance.

O, function of volume (see p. 14).

VF\ electric potential.

32

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