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3 445k 0023134 5 ORNL-3293

l UC-4 — Chemistry
TID-4500 (17th ed., Rev.)

e -

 

THERMODYNAMIC PROPERTIES OF

MOLTEN-SALT SOLUTIONS

Milton Blander

 

 

 

OAK RIDGE NATIONAL LABORATORY
operated by

UNION CARBIDE CORPORATION
for the

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nor the Commission, nor ony person acting on behalf of the Commission:

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ORNL-3293
UC-4 — Chemistry
TID-4500 (17th ed., Rev.)

Contract No. W-7405-eng-26

REACTOR CHEMISTRY DIVISION

THERMODYNAMIC PROPERTIES OF MOLTEN-SALT SOLUTIONS*

Milton Blander

DATE ISSUED

0CT 12 1962

 

*This paper is to be presented as a chapter in Selected Topics in Molten-Salt Chemistry, Interscience
Publishers, New York.

OAK RIDGE NATIONAL LABORATORY
Qak Ridge, Tennessee - .. ~ e -

i

UNION CARBIDE CORPORATION OAK RIDGE NATION
’ 3 4456 0023134 5

M

 

 

 

 

 

 

 

for the
U.S. ATOMIC ENERGY COMMISSION

 
v

CONTENTS

INTRODUCT ION eiitiiitieeettireeteeee et esteeetetestessesasesseeseassseessansasssessssssssaasssasasnssessesseensasessasenbesnbennseatsaeseassasnsatnsasesuen 1
ol GENETAL ettt v e st ettt ea et e e Rt e e s e et na e ene e e e n e e nenenee 1

1.2 The Limiting Laws coiieicieiiiccieescceesesesreesiaesssessste e sanesesee e srnenesrnaessressnessaresssasssnnesssnsssasesssnnsssssesessses 1

1.3 The Temkin [deal Solution et n s st s e nn e 3

|.4 Salts Containing lons of Different Charge.......coconveiiieiniiiiiiicc et 7

[.5 Standard States and Units of Concentration ...ceecieciicnirencee et 9
SOLUTIONS WITH COMMON ANIONS OR COMMON CATIONS ..ottt v 11
1.1 Cryoscopic Methods of [nve stigation ...t 11

[1.2 Electromotive Force Measurements ......ccoiiieiiioniiniiiiiiiiiniine sttt enesressiesenne o 12

[1.3 Strongly lonic Salts Containing Monovalent Cations and a Common Anion .......cccocoiiiiiiniiincnnne 16

I1.4 Mixtures Containing Polarizable Cations and a Common Anion.......ccccceeiinieriecriincceneenrcnenee s 29

11.5 Binary Mixtures Containing Polyvalent 1ons. ..t 33

1.6 Discussion of Binary Systems with @ Common Anion ... s 52

[1.7  OBher SYSTEMS .ocviceiiciiiceicireiresie e et esns s ae s eeseasaen e sr e et smeeoeceneesaeeseen e e e bnasaenss s sansss st s srnesassrananns 57
RECIPROCAL SYSTEMS oottt ettt csr s e sseetesses s e essees e sass s se e be s sas e e s ebe s b e se et s b e e sassas s bennasaessbentesansanes 61
LT G@NEIAL ettt ettt ettt et er s e e s tae e mes e e st es e £ et e st e s e e e b e eRe s aene e se e e e e e e R Aba R s e b et e e e s ere s 61
111.2 The Random Mixing Nearest-Neighbor Approximation ..........ccocoiviiiiiiiniiiiinninninnnn e 63
I11.3 Corrections for Nonrandom Mixing: The Symmetric Approximation .......cccoeererieenienreeicncieinnas 67
I11.4 Comparison of the Symmetric Approximation with the Random Mixing Approximation ................ 69
[11.5 The Asymmetric ApProximation ..o eirirniiieiiecieetisecteseesteeea s eressts e s et e e sbessseassassesbesaresenssnnnes 73
[11.6 Conventional Association ConstaNTs ...cciiriiiiiieciee e sas s s saees 75
[11.7 Comparison of Theory with Experiments in Dilute Solutions .o, 77
11,8 Generalized Quasi-Lattice Calculations ... 81
111.9 Association Constants in Dilute Solutions .. 84
MISCELLANEODUS cooeeieteeieeieitecere e e eeeeetese s aeseseessesuessaassaessasasasesssasaessssansessessssasenssesnsissstesnsssssesesssessssssessnsssesnsnsnes 89

REFERENCES <ottt me s et s e e e e s e e b e s et e e s erbe st n e e e srne e e e s e e s rneasebassanseabas 93

 
THERMODYNAMIC PROPERTIES OF MOLTEN-SALT SOLUTIONS

Milton Blander

INTRODUCTION

1.1 Generadl

In this chapter the physical description of molten-salt solution thermodynamics will be dis-
cussed. Because of the large volume of work in this field this chapter cannot be comprehensive.
As the field of molten-sait solution chemistry is still in a rudimentary state, this must be con-
sidered as an interim report on some of its aspects,

The Gibbs free energy G and the Helmholtz free energy A are related to the chemical poten-

tial of the component i, i, of a solution by the relation

aG dA
<8_ T,Pny +m; = a—)T,V,nk+ni=,ui. (1.1.1)

Tli ni

For pure liquid and solid i the symbols ,u? and p?, respectively, will be used to represent the
chemical potential. Rational forms can be deduced for expressing the chemical potentials of
components of solutions by considering a hypothetical ideal solution. In choosing such a hy-
pothetical ideal solution, one must be careful to have it bear some resemblance to real mixtures,
and the equations derived should conform to the limiting laws which are valid for dilute solutions,
Since the equations derived should conform to the limiting laws, we will discuss these before dis-

cussing ideal solutions.

1.2 The Limiting Laws

Limiting laws can be derived for any solution that is dilute enough so that the enthalpy of
solution per mole of solute is essentially independent of the concentration of solute and the
equation

0 -
holds, where H is the total enthalpy of the solution, n, and Hc]’ are the number of moles and en-
thalpy of pure solvent, and n, and H; are the number of moles and partial molar enthalpy of sol-
ute.

If the solute has no ions in common with the solvent, then the ideal limiting laws may be

107,123

derived from statistical considerations. If a solute molecule dissolves to form & dif-

ferent species with v, particles (or ions) of kind %, then the number of ways of arranging the

ions of the solute in solution, or the number of configurations €1 is given by

(B,)
Q.,l-[ £ (1.2.2)

L1 7!

 

 
where 7, are the number of particles of type &, where B, is the number of ways of placing one
particle k& in the solvent, and, if the solution is dilute enough, (Bk)'Tirt is the number of ways of
placing 7, distinguishable particles k. The 7, ! in the denominator corrects for the indistinguish-
ability of all the particles of a given type. The B, may be all different but are all proportional to
7 4, the number of molecules of solvent. The entropy of mixing may be calculated from the relation

AS
~—=hQ. (1.2.3)
k

By using Stirling’s approximation one obtains

AS
- =2y Inny + 27,00 B, — 27, In 7y, + 2

=ka?z'2 In 7 +Evk772 In B, -~ Evk?z'z In (Vkr'1'2) +Evk7'z'2 , (1.2.4)

where B, =, 3, and 7, = v, 7,, and where 7, is the number of molecules of solute. From Egs.
(1) and (4) the ideal limiting laws for the solvent are
dAS - N4

] - —RTfl—] 2y, ZRT In (1 - N2, ), (1.2.5)

 

0 < 0
pr=my= =TSy =59 =-T

where N, is the mole fraction of the solute. For the solute the ideal limiting law is

dAS - n,
* ~

Po =pg==-T ——=2v, RT In— =2y, RT InN, , (1.2.6)

dn,y 7

where the term ,u; [= H; +2v, In (Bk/vk)] is the partial molar free energy of a standard state
chosen so that a solution of component 2 will behave ideally at extremely high dilutions. The
term .u; is a function of the concentration scale used. Equations (5) and (6) express the fact
that in a dilute solution the solvent obeys Raoult’s law and the solute obeys Henry’s law. The
fimiting laws given by Eqs. (5) and (6) are independent of the specific properties of the solvent
{except for the value of p;) unless the solvent has an ion or particle in common with one of the
species. |f the ions or particles formed from the solute upon dissolution which are already pres-

ent in the solvent are designated as I, then

0 I}(Bk)zk(Kflfl)! 127
LSRRI KRS ' -

 

where K; is the number of [ particles per molecule of solvent. If the K, are not very small, then
it may be shown that
o -—yc;:RT In (1« N,v) (1.2.8)

* Ao

,u2 - #2 = VvRT In N2 ’ (|°2°9)

 
where v = Z v, and is equal to number of independent particles which differ from those already
k*l
present in the solvent which are introduced upon the dissolution of one molecule of solute. To

illustrate this the solute KCl in the solvent AgNO, leads to a value of v = 2,37 but KNO, and
Ag,50, in AgNO, lead to a value of v = 1.37 Partially ionized solvents such as water can be
described by using more than one value of v. The dissolution of HCl in H,0 at concentrations

of HC| much lower than the concentration of H' from the self-ionization of water leads to a value
of v = 1. At concentrations of HC| high enough so that the self-ionization of water is suppressed,
v =2, Thus by choosing an ionic solute with @ common ion, a distinction can be made between
an ionizing and a non-ionizing solvent by testing the limiting laws. Care must be taken before
using this as a criterion of the ionic nature of the solvent to apply these considerations to solu-

tions that are dilute enough so that the limiting laws are valid.

1.3 The Temkin ldeal Solution'?”

Liquid salts are similar to solids in some of their aspects and differ considerably from solids
in important ways. A molten salt must be considered as an assembly of ions with the expected
alternation of charge as in solids, with the cations having anions as nearest neighbors and the
anions having cations as nearest neighbors. The enthalpies and energies of formation of solids
and liquids from the gaseous ions do not differ greatly, since the enthalpy and energy of fusion
is very small relative to the total lattice energy of the solid. The sharp increase of conductance
upon melting indicates that the melting process leads to ions of greater mobility than in the solid.

In the Temkin model, salts are considered as completely ionized. The strong Coulombic forces
in @ molten salt lead to a strong tendency for the alternation of charges such that cations are sur-
rounded by anions and the anions are surrounded by cations. [f a mixture of the two monovalent
cations A* and B* and the two monovalent anions X~ and Y~ is considered, then the anions re-
side in a region adjacent to the cations and the cations reside in a region adjacent to the anions
and the molten salt might be considered as a quasi-lattice, If the two cations and the two anions
respectively have the same physical properties, then the cations can mix randomly in the cation
region of positions which is adjacent to the anions, and the anions can mix randomly in the anion
region of positions which is adjacent to the cations. The total enthalpy and energy of the solu-
tion is the same as that of the pure components, and the heat of mixing and energy of mixing are
zero. The total entropy of mixing, AS, can be calculated from the total number of possible equiv-

alent and distinguishable configurations, ..,

AS . (7, +7Ag) ! (my + 7y} !
cnop=hnd | — ) T}, (1.3.1)

nA!nB'

 

7y ! nYl

where the 7, are the number of cations of kind i " and E']. are the number of anions of kind j~. By

using Stirling’s approximation

7ln%-7; (1.3.2)

 
then

~AS.,

 

= 1, InNA+nA InNB+nx lan+1:zY lnNY, (1.3.3)

where n_. and n; are the number of gram moles of ions i* and j=,and N, and N]. are the ion frac-

. L4 . e .
tions of cation i or anion j~ respectively.

 

 

 

A x
N, = , Ny = ,
n, +ng ny +ny
(1.3.4)
v ng . ny
B nA"'"B’ Y~nx+nY'
For any number of monovalent species
i
N, = —
A I
7
x
Ny =, (1.3.5)
n,
]
=AS
=2n.InN,+2n. InN. .,
7 1 7 1
The partial molar entropy of solution is then
< 0
(Sij - S:‘j) dAS . dAS . 0AS ..
———::lnNNn-—— = am - — ’ (ln3-6)
) on,. on, on .
if i j
and the chemical potential can be expressed by
Hi; = by = RT In NN (1.3.7)

Equation (7) is compatible with (1.2.8), when 7j is the solvent; if ij is the solute, Bij differs from
#o in (1.2.9) by a constant. By defining the activity of the component ij, @i by the equation

bi; = b =RT lna, (1.3.8)

then for a Temkin ideal solution

a; = NN (1.3.9)

If the solution contains only one anion as X~ and a number of cations, then for any component

such as AX for example, Ny = 1 and

Ay =Ny =Ny, (1.3.10)

 
where N,  is the mole fraction of the component AX. A similar relation holds if the cation A is
the only cation. Thus, if in a mixture of several simple* salts containing two ions each, and if
all of the components of the mixture contain one ion in common, the Temkin ideal activity of a

component is equal to its mole fraction. In an ideal mixture of one mole of AX with one mole of

BX, for example, the activity of AX and of BX are both ]/2

On the other hand, in an ideal mixture of one mole of AX with one mole of BY, the activities
of AX and BY are both ]/4 Thus the activity of a given mole fraction of an ionizing salt in a
mixture depends strongly on whether it has an ion in common with other salts in the mixture.
Even though the salts AY and BX have not been used, the activities of AY and BX are also ]/4
There are four different ions in this solution, and the restriction imposed by the condition of
electroneutrality reduces the number of independent thermodynamic components to three. [f,
as is unlikely, in all equilibria and phases n, = n, and ny = n,, then another restriction is

imposed on the solution and it is a two-component system. If, in some equilibria this condi-

tion is true, the solution may be termed a quasi-binary system for that equilibrium,

The condition of electroneutrality makes it necessary to choose electrically neutral com-
ponents. In the three-component system A*, B*, X=, Y=, for example, there are four possible

ways of choosing components

AX-BX-BY
AY-BX-BY
AX-AY-BX
AX-AY-BY

all of which are correct. For some compositions and choices of components a negative con-
centration of one of the components would have to be used. For example a mixture of 1 mole of
AX, 1 mole of AY, and 1 mole of BY, if described in terms of the components AX, BX, and BY,
would be composed of 2 moles of AX, 2 moles of BY and —1 mole of BX. Although this is a
thermodynamically valid method of description, it is usually more convenient to avoid negative
concentrations of components. Any partial molar value of the thermodynamic function T for the

component ij containing monovalent ions can be calculated in two ways by
= aT oT ar (1.3.11)
.. == -_— = _—— + _— ' - .

*Simple salts contain only two atomic ions.

 

 
where ni is the number of moles of the component ij. The use of the sum (GT/ani) + (aT/dnf) per-
mits one to avoid stating a choice of components. In general, the partial derivative of any thermo-

dynamic function T for a component A X_ will be given by

JoT oT > ( aT >
= r + s .
c?nArxs <c9nA anx

 

 

 

An ideal mixture of two different salts of the same charge type as a mixture of A X_and B Y

would give an expression for the total entropy of mixing of

 

~AS .
=n, In Ny +7ng InNg +ny, InNy +n, InN, (1.3.12)
and
~(5.. - 59)
ij ij
=rInN.,+sinN., (1.3.13)
R ! ]
so that
0 7 AIS
Hij = by = RT In N’ N (1.3.14)
and
NP AIS
a;; _NI.N]. . (1.3.15)

Another interesting definition of an ideal solution is that which is derived under the assump-
tion that all cations and anions are randomly mixed despite the differences in the sign of the
charge. Although this is undoubtedly a poor picture of any molten salt, it can give an idea of
the effect of the interchange of cations and anions on the cation and anion positions; since a
molten salt is not a rigid lattice, some ions of the same charge must occasionally be near

neighbors. For the pure salts ij containing only monovalent cations the entropy of mixing is

AS), Ay As?
=7II- |n2+n]. |n2=-—T-—-?, (].3.]6)

 

and for a random mixture of the four ions A*, BY, X~, and Y~

 

-AST 7, ng
=ny, In —8——— +ng h —a—m— —
nA+nB+nx+nY nA+nB+nx+nY
+ ny In ————— +n, h—, (1.3.17)
np+ngtny +ny mp +ng + Ry + Ry

and, since n, +np = ne + np, it can be shown that

AS AS? - Sn.ASY = 3. ASO
1 i ]

]
- S =3n. InN.+2n. InN.
R R P T

 

 

(1.3.18)

’

 
which is the same as Eq. (5). Thus, the assumption of random mixing of all the ions leads to the
same definition of an ideal solution for mixtures of monovalent ions as does the Temkin model in
this case,

This conclusion may be generalized since the configurational integral for = molecules of uni-

-BU; _
z]=f..fi_ (d7)27 (1.3.19)
(7 1)?

where d7is a volume element in configurational space B = (1/kT) and Uij is the total potential

univalent salt is

energy of asalt, ij, ina given configuration and the integration is overall configurations. For a

mixture of anions, j, and cations, i,

_BUmix (—-|)2 '.BUrnix
e — - e —

z oo~ o @0, (1.3.20)
it | | it | | (mH?

where 7 = 27, = EE].. The total free energy of mixing per mole is

AA_=A . ~SIN.N.A..==kT InZ_. +EZN1.N]. kT In Z (1.3.21)

m mixture 1] mixture

For the case in which the quam‘ify

|nf f Umix (d7)?" —EN N. Inf f -BU d’rz"}

 

is zero,* then
~AA=TAS. = -RT(Zn, In N, + En]. In N].) , (1.3.22)

which is equivalent to Eqs. (5) and (18) but has been derived without a model.

1.4 Salts Containing lons of Different Charge

Although the laws of ideal solution are unambiguous for ionizing salts of the same charge
type, expressions for salts of different charge types present a problem. Férland”® has given an
extensive discussion of this. For a system A*, B2* X~ for example one can consider that a
quasi-lattice exists with the anions occupying the anion region of the lattice and the cations
mixing on the cation portion of the lattice, For every B2* ion added from BX to a solvent A X a

. . . . . ”
‘“*vacancy’ is also added. lf, as is reasonable, there is a very large ‘‘concentration of vacancies

 

*One obvious condition for which this is true is when the two cations and the two anions respechvely
have the same physical properties. In this case, for any given ?eometrlc configuration of the ions, the po-
tential energy of the mixture (Umix) is the same as the potential energy of any one of the salts (U, ])

 
or “’holes’’ in the solvent liquid, then the added hole at very low concentrations will have no ef-
fect on the properties of the solution just as the presence of a common ion in the solvent sup-
presses the effect of a solute ion on the limiting laws. The total ideal entropy of mixing would

then be

~AS .
=n, In Nj+mg In Ng (1.4.1)
and
5, - 58 , S, -89
r == |n NA —-R——'—' = —|n NB ; (1.4.2)

where salt 1is A, X and salt 2 is BX. These equotions will be valid as long as the "*concentra-
tion of vacancies’’ in the solvent is large enough to buffer the added ‘’vacancies.”” Equations

(1) and (2) would hold for any valence types in such cases.

Fd'rland has also considered the cases, analogous to those found in solid solutions, in which
a divalent cation sait BX will dissolve in a monovalent cation salt A X by occupying one site and
creating a vacant site. |f the vacant site associates with the B2* cation, then the cation lattice
behaves as a mixture of monomers and dimers and an approximate expression stated by Fatland
and based on the calculations of the ideal entropy of mixing of molecules of different sizes33+62

is

 

~AS..
=n, INN, +ng InNZ, (1.4,3)
where N is an ion equivalent fraction of the ith ion.
" 7, \ 2ng
A—rzA+2nB' B nA+21zB'
~(5, = 59
=2InN, +Ng , (1.4.4)
T 0
—(52 - 52) ’ ’
= |n Ng = N4 (1.4.5)

The assumption in Eqs. (3), (4), and (5) is that the divalent ion B2* and the associated vacancy

are twice as large as the A* cation so that the entropy of mixing of cations is that of the ‘‘dimer”’
+ : ,

(B2*.vacancy) and the “monomer’ A*. Fatland has discussed a small correction term to these

expressions to account for the fact that at high B2* concentrations, where more than one vacancy

 
may be near a given B2*, one cannot distinguish which one should be part of the **dimer.”” If

the cation vacancy dissociates from the B2* ion, then

-AS ..
T=nA In N; +ng In NI; , (1.4.6)
5, - 59
2 = -2 [n N; . (|'4-7)
< 0
52 --S2 )
R = --2 In NB . (I.4.8)

Equations (7) and (8) have been derived for solid solutions and are probably not reasonablie pic-
tures of liquids where ‘‘vacancies' must exist even in the pure salts.
The very careful study of the CaCO,-M,CO, systems, where M= Na or K, by Fefland and co-

57.58 4nd were consistent with Eqgs. (1)

workers appeared to be inconsistent only with Eq. (6)
and (3).

Equations (1) through (8) are useful largely to obtain convenient forms for the expression of
chemical potential and may be generalized for mixtures of ions with different valences. The large
differences in the Coulombic interactions of ions of different valence make it improbable, ex-

cept for very special cases, that the entropy expressions (1) through (8) will be valid over a

large range of concentrations for real systems.

1.5 Standard States and Units of Concentration

As seen by the preceding paragraphs, reasonable concentration scales are the mole fraction,
equivalent fraction, ion fraction, and ion equivalent fraction although this chapter will, generally,
use mole and ion fractions. The mole ratio defined by R, = nz/n], where nyand 7, are the num-
ber of moles of solute and solvent, is sometimes convenient in dilute solutions when it differs
little from a mole fraction. The molarity scale {moles/liter) is sometimes convenient in a case,
for example, where experiments are compared with theoretical calculations made for a constant
volume process. The expression of concentrations on a molality scale (moles/1000 g solvent),
because of the large number of different solvents of different molecular weights, does not seem

to be well-chosen if one wishes ultimately to compare phenomena in different solvents.

Some definitions of the activity and activity coefficients of, for example, the salt B_X_ are

y2=y2+RT |na2==,u;+RT Ina2=p§'+RT |nA2U, (1.5.1)
2y
y2 flN’—_NS:}’Br Yx© (1.5.2)
10

2
Yq = — T (yg) yy)* . (1.5.3)
. %2 .
}/2 n---—--——r S: ‘yBr ‘yxs ’ (|.5.4)
NB NX
ay
vy = ————= gV (D)°, (1.5.5)
Ng" Ny

where the standard chemical potential ug is the chemical potential of the pure liquid salt, [,L; is
the chemical potential of a standard state chosen so that y; approaches unity as the concentra-
tion of all the solutes approaches zero, and [,L,? is the chemical potential of pure solid. It should
be noted that the value of ,u; depends on the concentration scale used and unless otherwise stated,
the definition of p; derived from the use of the ion fraction scale expressed in (4) will be used
here.* For the comparison of the solution properties of different mixtures containing salts of dif-
ferent melting points, the most convenient standard state is the pure liquid (supercooled if nec-
essary) since there will be no break in the temperature dependence of some of the derived activ-
ities at temperatures at which there are transitions in the solids. It is probably more meaningful
to compare liquid solution properties of a component with those of the pure liquid component. The
standard chemical potential ,u; is often conveniently used in dilute solutions. The usefulness of
any chosen standard state should be measured by the ultimate ability to measure the value of p
in that state,

It should be noted that the single ion activity coefficients, yg, v, y;, y; , etc., do not have
a strict thermodynamic significance except as a product for the ions in a neutral species or as a
quotient for ions with the same total charge. The use of single ion activity coefficients may often
be confusing and should be avoided if possible.

Excess chemical potentials may be defined by
E

By considering the equality

Y2

Y

#; - #g = RT |n ; (|.5.7)

 

: * , + ~ 9
then since y, approaches 1 as the concentration of B"* and X~" ions both approach zero, p, ~ o
is the excess chemical potential of the salt B X _ at infinite dilution and may be termed an excess

chemical potential of pure liquid B_X _ at infinite dilution,

 

*
*To convert from one scale to the other, the relations fy (mole fraction) = y; (molarity} = RT In V, = ]1;
{molality) + RT In (]OOO/M]) may be used, where V, is the volume of one mole of solvent and M, is the gram
molecular weight of the solvent.

 
11

SOLUTIONS WITH COMMON ANIONS OR COMMON CATIONS
I1.1 Cryoscopic Methods of Investigation

The limiting laws have been investigated mainly by cryoscopy and with emf measurements. A
description of the theory and experimental applications of these methods is given in sections |1
and 11.2.

Cryoscopic measurements have been made from thermal halts, visual observations, and by fil-
tration and analysis of solutions at equilibrium with a solid. For an equilibrium between a pure
solid A X_ (component 1) and a liquid mixture

dlnay, dlnN,"N,* /
- + =~ , (1.1.1)
d(1/T) d(1/T) d(1/T) R

where AHf is the enthalpy of fusion of A_X_and a, is the activity of the component A X _ in a

diny, AH

 

solution at equilibrium with the pure solid at the temperature T. This relation may be re-expressed

for the solubility of a slightly soluble salt A X .

* %
dlna, dlnN NS dlny, (- HD)

 

 

(1.1,2)

= + BT - ’
d(1/T) d(1/T) d(1/T) R
where (E; - lej) is the heat of the solution of solid A X _ to infinite dilution. In general, y; and
y; are not constant except in solutions dilute enough for the limiting laws to apply, and they must
be known in order to evaluate AH/ and (H" = HD) from cryoscopic or solubility measurements. The

term AH/ is a function of temperature:

0
AanAH;’ - fTT AC, dt (11.1.3)

where AH/ and AH? are the heats of fusion at the temperatures T and the melting temperature T,

respectively and ACP = Cp(liq) — C, (solid). If the heat capacities of the pure solid and the pure

p
liquid A_X_ can be expressed by

Cp=a+bT+cT"2;
then

Ac

AC, =Aa+ TAb + —. (11.1.4)
p 2
T
By introducing Eq. (1) and integrating one obtains

 
12

The considerable deviations from ideality of most mixtures of molten salts make it essential that
AH/ be obtained from calorimetric measurements except for a limited number of cases. The use of
phase diagrams to obtain a "‘cryoscopic’’ heat of fusion under the assumption of ideal solution be-
havior has been shown to be often in error.#3773 The terms containing the correction for ACP must
be included in a calculation of @, from measurements of the liquidus temperature. For example, if
ACP = 2 cal/deg mole at all temperatures and T,/T = 1.2, the error in a, would be about 2% if the
ACP correction were excluded. For T,/T = 1.5 the error is about 10%, and when To/T = 2 the error
in a, is about 31%. Since the values for the heat capacity for pure liquid have to be extrapolated
below the melting point, any errors in the extrapolation can be appreciable at large values of
T,/T. Table 1 gives a summary of selected values of AH? and the parameters for CP for solid
and liquid.45+73
Cryoscopic measurements have been used to test the limiting law expressed by Eq. (1.2.8).
Combining Egs. (1.2.8) and (5) and expanding the logarithms in the relation obtained, one obtains
the van't Hoff relation,
RT}
AT =
A

 

- vN, = (T, = T), (1.1.6)
/
for small values of N, and for values of AT small relative to T,. Equation (6) has been used to
investigate the limiting laws in many systems. The freezing point lowering of NaNO, by NaCl
obeys Eq. (6) to about 7 mole % of NaCl for v = 1,129,130 The compounds Na,CO,, Na,S0,,
NaBrQ,, Na,W0,, Na,MO,, Pb(NO,),, and LiNO, also gave apparent values of v = 1 in NaNO;;
K|04, LiCl, and CsCl led to values of v = 2, CaCl,, SrCl,, and BaCl, led to apparent values of
v=3,and LaCl, to v =4 in Nc:N03.!"]29 In molten AgNO, the solutes Ag,SO,, KNO,, and
Pb(NO,), led to values of v = 1, and PbCl,, K,Cr,0,, HgCl,, HgBr,, and Hgl, led to values of
v=3.80 |n molten KNO, the limiting law has been demonstrated for a number of cases, mostly
at concentrations of solute less than 1 mole %.8% And Na,30, in a solution with NaCl and

NaBr obeys the limiting law®2 and Eq. (1.2.8) at all concentrations.

I1.2 Electromotive Force Measurements

Measurements have been made in concentration cells with liquid junctions such as

AX_(N3)
BX

AX_(N,)

BY A, (11.2.A)

 

 

 

 

 

*It should be noted that in most other cases of systems consisting of a solvent containing a foreign cation

and a foreign anion, deviations from ideality are large ot the lowest concentrations of the studies cited so that
the limiting laws cannot be tested.

 
13

Table 1. Melting Points, Heats of Fusion, and Heat Capacities of Some Salts45:73

(C,=a+bT+ e/T?)

 

 

 

 

N T H ot H,_ Cplsolid) c,(iia)
Composition {°K) {keal/mole) a b 103 c % 10=5 a
LiF 1121 6.47 10.41 3.90 - 1.38 15.50
LiCl 883 4.76 (11.00)* (3.40)
LiBr 823 4.22 (11.50) (3.02)
Lil 742 3.50 (12.30) (2.44)
LiNO3 525 6.12 14,98 21.20 26.60
NaF 1268 8.03 10.40 3.88 -0.33 16.40
NaCl 1073 6.69 10.98 3.90 16.00
NaBr 1020 6.24 11.87 2.10
Nal 933 5.64 (12.50) (1.62)
N0N03(a.) 549(Tr) 0.81(Tr) 6.34 53,32
NaNO,(5) 579 3.49 35.70 37.00
KF 1131 6.75 11.88 2,22 - 0.72 16.00
KC1 1043 6.34 9.89 5.20 0.77 16.00
KBr 1007 ' 6.10 10.65 4,52 0.49
KI 954 5.74 11.36 4,00
KN03(CL) 401(Tr) 1.40(Tr) 14,55 28.40
KNO,(3) 611 2.80 28.80 29.50
RbF 1068 6.15 (11.33) (2.55)
RbCI 995 5.67 {11.50) (2.49)
RbBr 965 5.57 (11.89) (2.22)
Rbl 920 5.27 {11.93) (2.27)
CsF 976 5.19 (11.30) (2.71)
CsCl 218 4,84 (11.90) (2.28)
CsBr 909 5.64 (11.60) (2.59)
Csl 899 5.64 (11.60) (2.68)
AgCl 728 3.08 14.88 1.00 -2.70 16.00
AgBr 703 2.19 7.93 15.40 14.90
AgNO3(a) 433(Tr) 0.6 {Tr) 8.76 45,20
AQNO:;(B) 484 2.76 25.50 30.60
INumbers in parentheses are estimated values (K1),
The emf of the cell can be given by
A RT | 2 Ad (1.2, 1)
E = — In — . s
nF az’ diff 7

where Ad ... is the diffusion potential and the prime (”) denotes the left-hand electrode. In a
binary system, all that need be known in order to evaluate a2/a£ from the emf of cell (A) are the

Hittorf transference numbers of the components. For a system containing more than two com-

 
14

ponents, the gradients of concentration for each component across the liquid junction between
the left- and right-hand compartments need also be known.
To give an idea of the magnitude of A¢ (¢ for salts containing only monovalent ions for

example, let us consider the approximate equation for mixtures of AX and BX dilute in AX

A= T ax ~ fox (N, = N2) RT_bAB(N N2 (1.2.2)
diff_?T 2 2_-}';-—-be 2 2/ 1 s

where bi;‘ is the mobility of the ion of species i relative to j. We may cite three pertinent ex-
amples:
1. The ion A interacts so strongly with X= that it has a low mobility relative to

X=(b,x = 0). In this case A 4igs is small only when (N, ~ N7} is small.

2. The relative mobilities of A* and B* are the same. In this case Ad i is zero.?3
3. The A% ion is relatively more mobile than the B* jon. If b,y =mbyy, then
RT )

|f m is large, one must be especially careful to either correct for A 44 or to work in extremely
dilute solutions.

For solutions dilute enough so that Ag ;. is small, then the emf of the Daniell cell

AX CX
n ™ (1.2.B)
BX BX
is given by
1/
. RT (“cxm) i
AE = AF 4+ — |n —8 —, (11.2.4)
F (a )1/71
\ Axn
where
. RT RT
~+ (e, TR,
For cells of the type
BX (solid)|| BX (solid)
A[AX (Ny) AX (Ny) | A (11.2.C)
AY AY

 

in which BX is very insoluble and for concentrations of solutes low enough so that A 444 can be

neglected, the emf can be expressed by

RT %ax
AE = = — In

 

—, (11.2.5)
2AX

where a;x denotes the activity of AX in the left-hand electrode.

 
15

At concentrations where A¢ ... is negligible and low enough for the limiting laws to apply,

the emf of concentration cells (A) and (C) obey the Nernst equation

RT N

 

AX
AE=t— |n —, (11.2.6)
nF NA'X
and cell (B) will obey the equation
AE = AE +?|n ——i;/—'—. (”.2.7)
(N gy )"

The validity of Eqs. (6) and (7) are proof of the validity of Eq. (1.2.9) for the solute. Many ex-
amples of concentration cells and Daniell cells exist in the literature which illustrate the lim-
iting Nernst laws up to concentrations at least as high as 0.5 mole % and often for solutes of

the same valence type to more than 1 mole %. Some examples are AgNO, in Nc1N03-KNO3,46'60
in NaNO, (Fig. 1)7° and in KNO,;'" AgCl, CoCl,, PbCl,, ZnCl,, NiCl,, CdCl,, TICI, CuCl,

UNCLASSIFIED
ORNL—LR-DWG 40353

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4x4073
\o
b
2 0
\ SOLID LINE REPRESENTS NERNST SLOPE
I.\
3
10 N

g N\
S N,
=z 4
o N
< ©
o

\‘.
4 N
Q\
\\
°
N\
2 e
\\
1o~ %
0 40 80 120 160 200
EMF {mv)

Fig. 1. Demonstration of the Nernst Equation in AgNO3-NuNO3 Mixtures
from the Cell

AgNO,(N])
NaNO3

AgNO3(N1)

Ag
NuNO3

Ag

 

 

 

 

 
16

CrCl,, MnCl,, FeCl,, and SnCl,, in NaCI-KCl mixtures;*7+48 and PtCl,, PdCl,, BiCl,, AgCI,
NiCl,, CuCl, PbCl,, FeCl,, CdCl,, and TICl in LiCI-KCI mixtures.82 From cells of type (C)
the Nernst law, with silver solid-silver-halide electrodes, has been demonstrated for KCI in
LiNO,-KNO,,?? KBr and K1 in KNO, and in NaNO,-KNO, mixtures.?*

These illustrations indicate that for mixtures with a common anion the solvents obey Raoult's
law and the solute obeys Henry’s law in dilute solutions,* even for solutes with polyvalent cat-
ions. The high concentrations of charges in a molten salt, composed of monovalent ions, appar-
ently swamps out or partially cancels the high local-charge density of a given polyvalent cation
and, in a sense, the solvent must behave like a medium of very high dielectric constant in cases
where the solution contains only one anion. These cases in dilute solutions also indicate that
the effect of any “*holes’’ introduced into the solvent by the addition of polyvalent cations is
suppressed by the presence of ‘'holes’’ in the solvent, Although a molten salt seems to be a
highly concentrated ionic solution if the solvent ions are included, the effects of the solvent
on the ionic solutes having a common anion seem to be such as to make the properties of these
solutions simpler in less dilute solutions, than is the case with water or other non-electrolytes
as a solvent for salts. Similar checks of the limiting laws for ionic solvents containing poly-
valent ions are unavailable. Although measurements do not appear to lead to results of interest

in fairly dilute solutions, measurements in concentrated solutions are of more interest.

11.3 Strongly lonic Salts Containing Monovalent Cations and a Common Anion

The most revealing experimental work on mixtures of salts with monovalent ions are the
calorimetric measurements of the molar enthalpies of mixing of the alkali nitrates by Kleppa,”®
and Kleppa and Hersh.”® Although the alkali nitrates cannot be considered as good a prototype
of an ionic salt as the alkali halides, they are analogous to the alkali halides.

The molar enthalpy of mixing of two salts 1 and 2 is given by
AH =N, (H, = H) + Ny(H, = H), (11.3.1)

where H | and H, are the partial molar enthalpies of components 1 and 2. Enthalpies of mixing of
all of the ten possible mixtures of alkali nitrates were reported with measurements for seven of
the mixtures being reported in detail. In Figs. 2-4 are plotted some typical data for AH_ and
AH_/N,N,. The data may be represented by the expression

H =N ;Nyla+bN;+cN;N,). (11.3.2)
In Table 2 a summary of the values of @, b, and ¢ representing the data is given.

All of the observed enthalpies of mixing in mixtures of alkali nitrates are negative and are

more negative the greater the separation of the two alkali metals in the periodic system (and

 

*Molality is about an order of magnitude larger than mole or ion fraction in these cases. By standards used
for aqueous solutions, these are concentrated solutions.

 
17

UNCLASSIFIED

 

 

 

 

 

 

%3 c00 ORNL~LR-DWG 66347
S _
~.
o
o
N -475 5 o o on-3—
~ e 5
= $o —— 0= *
3 - '
2]\‘ -450
.0/
—100 o

 

 

~
yd

 

AHpy, (cal/mote)
@
O
%
N
\-
‘

 

>
o
\\
/
—v

 

|
N
O

~

 

 

 

 

 

 

 

 

 

 

 

 

0 Q.2 0.4 0.6 0.8 10
M,

Fig. 2 Total Molar Heats of Mixing (AHm) in NuN03LiNO3 Mixtures
(LiNO3 Is Component 1}.

also the greater the difference in size of the two cations). In all the systems an energetic

asymmetry in the enthalpies of mixing is present so that for a given pair of nitrates, the value
of AH_ is more negative in a mixture dilute in the large-cation nitrate than in a mixture dilute
in the small-cation nitrate. The parameter b is a measure of the energetic asymmetry. Assum-

ing that the form of Eq.(2)is correct, then the partial molar enthalpies are given by

Hy = HS=(a+2b = N2 + (4c = 26)N3 ~ 3eN$, (11.3.3)

Hy = H}={a~b=cN}+(2b+ 40N} = 3cNY , (11.3.4)

at N, =1, El - H? =a,and N, = 1, Ez - Hg = (a + b), where component 1 has a smaller cation
than component 2. Since both the a and the 4 are negative, the partial molar enthalpy of solu-
tion can be seen to be asymmetric. Only for systems in which the absolute value of AH s
small does it appear that the parameter c is negligible and that the term containing the concen-

trations to the fourth power are not necessary to represent the data.
18

UNCLASSIFIED

 

 

 

 

 

 

 

1500 ORNL-LR-DWG 66345
o
=9
@ —4400 "/)’
@ - -,
2 P
~
© 7 //
8 -1300 &
T /4"/
= 1200 >
X ~ 3
T /;/
< 4100 /y}’
—1000
- 320 -

 

 

—280 //

—240 fi \
o

-200 \

-160 / \

-120

 

 

 

Hy, (cal/mole)
™
.\
"

 

 

Ll
L

-80

—-40 / ~\

0 0.2 0.4 0.6 0.8 1.0
N,

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 3. Total Molar Heats of Mixing (AHm) in CsNO,.NaNO, Mixtures
(NuNO3 Is Component 1).

Kleppa, by using the enthalpy of mixing of 50-50 mixtures of the nitrates as a measure of the

magnitude of the effect, demonstrated the empirical relation

di"‘dz

2
4AH° S Ly ( > = Us? = ~14057 , (11.3.5)

dl+d2

where & = (d.' ~ d,)/{dy + d,), and d; is the sum of the radii of the cation and anion indicated,

and U is about =140 kcal, The value of U is about the same magnitude as the lattice energy of

 
19

UNCLASSIFIED

 

 

 

 

"o ORNL-LR—DWG 66346

% —2000 L-LR—DWG

£

~ L o

8 —1920 —

~— L —

QN '/. \GP\\

= -1840 o * k>
., ___-_-—--__-

X oo’ T

5 1760 Ko e =t~

 

—-480 —~,

7 <

 

 

|
W
n
O

~N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

< —160 / .
/ \
—80 ° s
0
0 0.2 0.4 0.6 0.8 1.0
N1

Fig. 4. Total Molar Heats of Mixing (AHm) in KN03-LiN03 Mixi'ures(LiNO3

Is Component 1).

the alkali nitrates. The results of Kleppa may be rationalized in terms of simple concepts. Since
the simplest binary mixtures are those containing monovalent cations and anions, simple solution
theories are more likely to apply to these mixtures than to mixtures containing polyvalent ions.

Although some of the relations discussed below will be naive, they will serve the main ob-
jective of this discussion, which is to relate the solution behavior of molten salts to fundamental
physical laws.

As discussed in a previous section, a molten salt may be compared to a quasi-lattice. Be-
cause of the alternation of charge, the quasi-lattice consists of two sublattices, one of cations,
and the other of anions which interlock so that the anions have cations as nearest neighbors and
the cations have anions as nearest neighbors. For a mixture of salts with a common anion, the
cation sublattice may be considered as being imbedded in a sea of anions. The anions are not
excluded from consideration, since the cation environment of a given anion will greatly affect
its relative position and energy. Since the solute and solvent in a mixture both have the same
anions as nearest neighbors as they do in the pure state, any solution effects are caused by

ions further away although these ions further away may, indirectly, affect the nearest-neighbor

anions.

 
20

Table 2. A Summary of the Parameters a, b, and ¢ Derived from the Heat of Mixing Data

for Binary Nitrate Systems

 

 

System T (°C) a (cal/mole) b (cal/mole) ¢ (cal/mole)
(Li-Na)NO, 345 - 464 -11.5 ~0
(Li-K)NO, 345 - 1759 - 87 - 463
(Li-Rb)NO, 345 — 2471 - 178 - 945
(Li-Cs)NO, 450 (- 3000)

(Na-K)NO, 345-450 ~408.5 -68 ~0
(Na-Rb)NO, 345 ~744.5 - 268 -36
(Na-Cs)NO, 450 -~ 1041 — 435 - 93
(K-RBINO, 345 (- 60)

(K-CsINO, 450 ~89.c ~87.5 ~0
(Rb-CsINO, 450 (- 14)

(Li-Ag)NO, 350 702 - 108 0
(Na-Ag)NO, 350 677 ~ 156 0
(K-Ag)NO, 350 -303 - 294 0
(Rb-AgINO, 350 ~ 944 - 337 ~297
(Li-TINO, 350 - 901 178 - 294
(Na-TINO, 350 131 241 ~0
(K-TINO, 350 447 - 17 ~Q
(Rb-TINO, 350 240 ~15 ~Q

 

%Pparentheses indicate uncertain data,

Molten-salts solutions differ from solid-salt solutions in an important respect, |n order to

place a large cation in solution in a solid salt having a small cation the structure near the for-

eign cation must be distorted. |n a solid, such a distortion is difficult as evidenced by the

rigidity of the lattice. Although there is some ability of the ions in a solid to adjust their

positions to minimize the energy,*4 the net effect is that the enthalpy of mixing of ionic solids

is positive, and there is a strong tendency for icnic solids having a common anion to be mutu-

ally insoluble if the cations are very different in size. The structures of molten salts are much

less rigid, and the salt can easily accommodate cations of different size.

The theory which can most easily be applied to mixtures of molten salts with monovalent ions

is the quasi-chemical theory of Guggenheim®? which is based on a quasi-lattice model. Since it

may safely be assumed that cations almost exclusively have anions as nearest neighbors in a solu-

tion containing only one kind of anion, all the nearest neighbors of the cations will be the same as

in the pure salts, and solution effects will be caused by ions further away than nearest nieghbors.

The nearest cation neighbors which are next nearest neighbors in the salt quasi-lattice might be

considered as a first approximation.

 
2]

If salt 1is AX and salt 2 is BX, then the potential energy of the ion triplet A* X~ A* may be
designated by V,,, of B*X~B" by V,, and of A*X-B* by Vu.78 To validly apply the quasi-
chemical theory to the model, V,,, V,,, and V,, must be assumed independent of the local
environment of the ionic triplets. Although this assumption is not correct, it may serve as an
initial working hypothesis. The molar excess free energy of solution and molar heat of mixing

of solution as calculated from the quasi-chemical theory will be given by®8

 

-——AAE NN, AT~ NN —A (11.3.6)
= - + .00 |, e
AH_ 22

T-N1N2A1—N1N2m + ... ‘ (”.3.7)

where A= (L Z°/2) (2V12 -V =Vy= N ZAe /2,1l is Avogadro’s number, and Z” is the

number of cation nextenearest neighbors of a cation.

Fériand37+38 has discussed the quantity (2V12 -V - V22) = A€’ in terms of the change of
the repulsions of next-nearest-neighbor cations. Fdrland represents the configuration of next-
nearest-neighbor cations and a nearest-neighbor anion as in Fig. 5 and calculates the Coulombic

energy change, Ae_, for mixing the cations in these two arrays of three hard spherical ions.

11N /dy =dy\?
Ae_= —e? <_ + —_> (_‘_—i> , (11.3.8)
dy d,)\d,+ 4,

where e is the electronic charge. The term —ez[(l/dT) + (1/d,)1, for a real ionic salt, can be re-

lated to the average lattice energy of the two salts composing the mixture and is analogous to the

UNCLASSIFIED
ORNL-LR-DWG 40446R

oo

 

‘ J

Fig. 5. Configuration of lons for the Calculation of Fgrland on the Change
of Repulsions of Next-Nearest Neighbors.

 
22

empirical parameter U in Eq. (5). A€_ is always negative and tends to be more negative the
greater the differences in the cationic radii. Except for small factors the form of Eq. (8) is ob-
viously related to the empirical relation (5).

Blander !5 has extended Férland's calculations to a hypothetical salt mixture which is ex-
tremely dilute in one component and which is represented by an infinite linear array of hard-
charged spheres. Although this model is unrealistic for a real three-dimensional salt, it does
serve to assess the effect of Coulombic interactions of longer range than the next-nearest neigh-
bors. The inset of Fig. 6 is a picture of a portion of the solution of one mole of the solute with
an interionic distance 4, in an infinite amount of solute. In Fig. 6 are plotted calcuiated values
of —AEC a?]/e2 vs A, where d, = d,(1 + A), and where A€_ is the energy of mixing per molecule
of solute. The values of A€_ are always negative and become more negative the greater the value
of A and are only about 0.4 times the magnitude of the values calculated from Férland’s simplie
model, |f the mutual dissolution of two salts 1 and 2 where salt 1 has the smaller cation is con-
sidered, then Blander's calculation indicates that a dilute solution of 2 in 1 will lead to a more
negative value of Ae_ than for a dilute solution of 1 in 2, Since Férland’s calculation predicts
a symmetry in the energy of mixing, the effect of the long-range interactions is to decrease the
total calculated value of | A€_| and to lead to a small asymmetry in the energy of solution. The

asymmetry effect means that the parameter A cannot be independent of composition.

UNCLASSIFIED
ORNL-LR-DWG 52098R

 

 

 

 

 

 

 

 

   

0.07
0.06 ——\
d d(lh’l)r—
008 OOOEOEE )OO —
- dum)l--

 

0.04 \

\
0.03
/
0.02 /

e

\
o \/

-06 -04 02 0 0.2 0.4 06 08 1.0
A

 

-AE
e2/d

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 6. Values of the Energy of Solution at Infinite Dilution (AE) in Units
of(ez/d) Calculated from an Infinite One-Dimensional Solidlike Model.

 
23

Care must be taken not to ascribe the observed negative deviations from ideal solution be-
havior or an asymmetry to only Coulombic effects. For example, polarization may also con-
tribute to the energy of solution. The ions in the solid-like linear model for the pure salt have
no field on them, but in the mixture represented in the inset of Fig, 6 there is an appreciable
field intensity on some of the ions which can polarize the ions.

Lumsden’! has calculated the effect of polarization of the anions by the cations in terms of
a one-dimensional model essentially including only nearest-neighbor and next-nearest-neighbor

ions. He obtained the relation

2 4 2
aF T 1 d, - d
Ae, = — x—ae? | — + — | (L—2) , (1.3.9)
P2 d, d, d,d,

where F is the field intensity on an anion between two cations of different size, and a is the po-
larizability of the anion. Polarization of cations, which may not be small, has been neglected.
Equation (9) is the same form as (5) and (8), and AEP is negative so that it should be difficult

to separate the purely Coulomb interactions from polarization interactions without a valid calcu-
lation of the relative magnitude of these two interactions. However, any simple extension such as

91 to three dimensions of a one-dimensional model for either the Coulomb

was made by Lumsden
or polarization interactions may lead to misleading values for their relative magnitudes.

If the solute in Fig. 6 is salt 2 in the solvent 1 where cation 2 is larger than 1, then the field
intensity on the anions adjacent to the solute cation is greater than if the solute is salt 1 and the
solvent salt 2, For polarizable anions, this would make the energy of mixing more negative and
contribute to the asymmetry effect. [f thermal motions are considered, then the tendency of ions
to reside longer in regions of high field intensity will also contribute to the asymmetry being in
a sense a ' 'positional’’ polarization. |f these simple considerations are valid for a real three-
dimensional salt, then at least part of the asymmetry effect is related not only to Coulombic
but also to polarization interactions by ions more distant than next-nearest neighbors. In any
theory of molten-salt mixtures it appears to be necessary, then, to include long-range interac-

tions, except under very special conditions.

The comparison of the measurements with the concepts discussed is straightforward. As
discussed, the parameter A for a molten salt in (7) is not independent of composition and

Kleppa’® has approximated the effective value of A as a linear function of composition,

A=a’+ b'N,, (11.3.10)

so that for values of =\ small relative to Z'RT, Eq. (7) becomes

AH =N Ny(a’+b°N)), (1.3.11)

which is the form of the experimental results in the three systems studied by Kleppa which ex-

hibit the smallest deviations from ideal solution behavior.
24

For values of =X not too small, Eqs. (7) and (10) lead to

~ , (a”+b'N,)?

AH = N N,|la"+b'Ny = 2N Ny —— (11.3.12)
Z'RT

Comparison with Eq. (11.3.2) shows that ¢ can be identified with —2\2/Z'RT. The precision of

the measurements is not high enough to detect a change of ¢ with composition. By using an

average value of A,

)\=a+b/2,

as a measure of A, Kleppa showed that a plot of ¢ vs 7\2/RT for the systems (Nq-Rb)Nos, (Na-
Cs)NOs, (Li-K)NO,, and {Li-Rb)NO, is consistent with a reasonable range of values of Z* of
10 to 12, This is the number of next-nearest neighbors in an NaCl type lattice and is only a
small variance with the number of next-nearest neighbors in some molten alkali halides.®”

Equation (11.3.11) corresponds to the random mixing of the cations on the cation sublattice.
The presence of the c term, if A varies linearly with composition, implied an appreciable non-
random mixing of the cations, and ¢ was termed a short-range order parameter by Kleppa.”®

It should be made clear that although the results of Kleppa have been rationalized in terms
of the modified quasi-chemical theory, a fundamental premise of the quasi-chemical theory is
that A is independent of composition. Consequently, the form of the theoretical equations de-
rived, based upon the quasi-chemical theory, although in correspondence with the empirical
Eq. (2), requires a sounder theoretical justification.

A justification of the form of the empirical Eq. (2) has been made by the methods in the
elegant work of Reiss, Katz, and Kleppa.'!® They used a method, which is essentially an
adaptation of the theory of conformal solutions,®® in which no model is used. The derivation
was made for ions behaving as hard-charged spheres with a sum of radii equal to 4 so that the

pair potential

 

ulr)=e0, r<d, (11.3.13q)
+7e2
u(r) = , r>d, (11.3.13b)
Kr

where r is the distance between the two ions in any given pair, and « is a dielectric constant
which is assumed constant for a set of salts with a common anion. The potential function can

be generalized to the form for a monovalent salt
1
u(r) = i—;f(r/d) , r>d. (11.3.14)

This is a less-stringent condition than (13b). Because of the relative rarity of anion-anion con-
tacts (except in salts as Lil), or cation-cation contacts, the contribution to the configurational
integral of configurations in which ions of the same charge are touching (or almost touching) is

very small and is neglected. As a consequence, except in these rare configurations, the total

 
25

contribution to the potential energy of a given configuration due to cation-cation or anion-anion
interactions is independent of small differences in the cation size, and only one parameter of
length d for the sum of the radii of a cation-anion pair is necessary for the description of a
pure salt. For a mixture of salts, two parameters are necessary, dy and 4,.

In the derivation of the theory, a single-component reference salt with the single parameter
of length d is transformed into either component 1 or 2 by varying 4. f g; = d/dz. where i = 1 or

2, then the configuration integral for the pure salt i is

-BUi
Z;=Z(g,) =f--f“g(;|)"3-(d7)2’7, (11.3.15)

where U, is the potential energy of the 2% ions (7 cations + 72 anions). Since the cation-anion

pair potential is

u {r) =if(—r)= g;ulg;7), (11.3.16)

then the total potential

~

I
>t
8] & EL

giuacle;) + 2 2, ucc (1.3.17)

where A represents anions, C cations, and the symbols A < A”and C < C” signify that the pair
potentials are added in @ manner so that no pair is counted more than once. The molar Helm-

holtz free energy,* A_ for pure salt i can be expressed as a series

A, dln Z
TRT InZ;=InZ(g)=(In 2) _; +{g; = 1) og;

g;= 1

(g, =12 /0%2InZ
Lle=1D " Y uee, (11.3.18)
2 agf

Ei=1

 

*Only the configurational part of the partition function is treated here. In calculations concerning changes
upon mixing, the ‘‘translational’’ part drops out and may be neglected. Although the equations that follow
were derived for hard sphere ions which interact with a genera?ized form of the Coulomb potential, the same
equations may be derived for more general potential functions. If the core repulsions of a cation-anion pair
are of the form /(gir) {(a special case of this form is the hard-sphere repulsions) and if the other interionic in-

teractions in the system are such that for any given geometric configuration of the ions, the permutations of
the two types of cations over all the cation positions do not lead to a change of the contribution of these
other interactions to the total potential of the system, then the equations derived will be the same as the
equations to be derived [Egs. (23—27;1 with different values of qSAC in the integrals which are contained in

the coefficients. Types of interaction which would be included in this category are not only Coulomb inter-
actions but also cation charge-anion multipole interactions and,for the cases in which the two cations have
relatively small or equal polarizabilities, all other charge-multipole and multipole-multipole interactions.
Salt mixtures which conform to this might be termed conformal ionic mixtures.
26

where Z(g.) has been expanded about g, = 1. Similarly, for a mixture of 1 and 2 the potential
energy is given by

 

 

 

 

U,=% = T (630 +3 3, up,’
= u ) + u r) + u
12 = & & E1M“AacC\8 X < 824Ac 82 & & tan
(N7} (N %) (N,7) (N7 (N 7) (N ,7)
+ Z 2z u r+ X 2z u v X T u , (11.3.19)
¢y < & ©1% & < ¢y %% & & &
and Z is given by
-BU
, 7! e 12 - 7l
Z.|2 = P —-——-——(d’T) n . 2(81’82) ’ (”.3.20)
Al (71)2 7y ln)!
and the Helmholtz free energy for one mole is
A dlnZ dlnZ
-—12 - (In 2) + ( —1)< ) + ( —U( )
- 81:8 =1 gI 82
kT 1482 dg 61.8y=1 g, ——

,lea - 1? <a2 In z)
2
81089~1 2 %2 g1:89=1

3% 1nZ
+(g; = Digy, = 1) -2 +..+ ZN NI N, . (11.3.20)
ag]agz _ t 1

The appropriate derivatives of Egs. {(15) and (20) were used to evaluate the first and second de-
rivatives of In Z contained in Eqs. (18) and (21). The values of A, A,, and A, thus obtained

were used to calculate the total excess Helmholtz free energy of mixing of N, moles of component
1 with N, moles of component 2 to form one mole of mixture

E
AAE — A, =N A = NyA, = RTEN, ln N, .

The first order terms cancel and the second order terms lead to

2—4 d. - d 2
73 4737 = Vo 0| (D17 %2 (11.3.22)
z d,d,

where Z, €, w, and a are related to the integrals characteristic of the “‘test’’ salt

(712

fff__L e
(71)2 '

42

AAE = NN
m

 

1

 

297kT

 
27

ff_¢(¢) (@,

where ¢, = u, - + r(du, ~/Jr) and a prime on A or C means that the integration is for two dif-

ferent anions or cations. From (22) it was shown that

dy—-d, 2
AAE = N N, T (T,V) <————> Y., (11.3.23)
™ dldz
dy—d, 2
E
AG” = NN, 0(T,p) <——-—> + ., (11.3.24)
" dld2
and
dy —d, 2
AH =N N, Q(T,p) (———) +..., (11.3.25)
7 d1d2

where [, 8, and Q are functions characteristic of a single *"test’’ salt. The influence of the factor
(a’l 152’2)2 on the thermodynamic excess functions is much weaker than the influence of the factor
(d, ~ d2)2. As a consequence, the form of Eqs. (23), (24), and (25) is similar to that implied by
Eqs. (8) and (9) and is consistent with the empirical relation (5).

The higher order terms in the theory of Reiss, Katz, and Kleppa were complicated. The higher
order terms are simplified by the choice of particular relative values of the perturbation param-
eters g, and g, so that (g, — 1) = (g, ~ 1).'7 This condition implies that for each particular

mixture a ‘‘test’’ salt is chosen. The calculations lead to the result

AAE SN N, POT 4 N NNy = N)O8% + [N NR + N NN, — N1+ L, (11.3.26)

where

P 1(32 zn)
kT 2 \z?2 1z

0 (_71'2F 72BD B® )
—=4 +— |,
kT 6z 2z% 3z3

 

 

R and § are complicated functions,

dl_'d2
0= |————1,
dl+d2

B=-B7’a,

D =B%% e+ (7 - Nl ,
28

B37
F=-

 

2
f fquc bac*Pacre” (d7)27

(71)?
+3(= -1 fquAC ¢AC'¢A’c”_.3-BU (d,r)i’r'z-
+ (7= N)(7 - 2)f...fquc bprcrbprrcrreBY (an?T

P, O, R, and S are characteristic properties of the test salt. In a similar fashion, the heat of

mixing may be shown to be
’ ’ 1ol
AH =N N PB4+ N NNy = NJQ B3+ [N NR 4 N NNy = N)2SI84 + ., (11.3.27)

where the prime signifies the proper temperature derivative of the primed quantity. The form of
Eq. (27) is seen to be consistent with (2) if a=P 82 -08% + RS* + 5%, b =203, and
¢ =-45’8%. This constitutes proof that the form of Eq. (2) is consistent with rigorous theory.
The methods of Reiss, Katz, and Kleppa may thus be used to support in a rigorous manner the
form of the empirical Eqs. (2) and (5), as well as the approximate form of Eqs. (8) and (?) which
had been derived on an intuitive basis.

Powers, Katz, and Kleppa’41%6 have measured volume changes of mixing of several com-
positions of each of the binary alkali-nitrate mixtures (No»K)NO3, (No-Li)NO3, (Nc-Rb)NOa, and
(Na-Cs)NO,. The average values of the quantity AVE/N]N2 are listed below:

Mixture Temperature (°C) AvE/N 1N, (cm3/mo|e)
(Na-K)NO3 350 0.26 +£0.08
425 0.28 +0.08
(Na—Li)NO3 310 0.26 £0.02
(Na-RB)NO, 340 0.82 £0.10
(NCI—CS)NO3 425 1.37 £0.12

All of these volume changes are positive and obey the approximate equation

AVE =N N, VB4,
where V”'= 22,000 cm?/mole. These positive deviations from the additivity of the molar vol-
umes, significantly, are found in mixtures in which the heats of mixing are negative. No satis-
factory theory has been proposed for this.

The only data on activities in mixtures of alkali halides with a common anion has been ob-
tained from cryoscopy. Unfortunately such data is not isothermal and uncertainties in the phase
diagram and in the heats of fusion as well as the necessity for precise measurement of liquidus
temperatures to obtain reasonable values of the excess free energies reduce the value of this
source of information. The component LiF in mixtures of LiF-KF, LiF-RbF, and LiF-CsF3.3¢
exhibit negative deviations from ideal solution behavior, which are more negative (the activity

coefficients are smaller) the larger the difference between the sizes of the two cations. The

 
29

same is true for the component LiCl in mixtures with KCI, RbC!, and CsCIl.''? This appears
to be in accord with the ideas presented in section ({1.3). However, the work of Cantor?! on
cryoscopy of NaF in mixtures with KF, RbF, and CsF have indicated that there is a small
positive excess free energy which becomes more positive in the order KF < RbF < CsF. Since
purely Coulomb or polarization interactions would be expected to always lead to negative de-
viations from ideal solution behavior, it is clear then that even in mixtures of the highly ionic
alkali halides other types of interactions are important. In the next section we will show that
these interactions may be, at least in part, related to the dispersion interactions of the solute

cations. Some discussion of this for alkali halides has been made.?!

1.4 Mixtures Containing Polarizable Cations and a Common Anion

In order to separate the various physical interactions which are significant in determining the
solution behavior of molten salts, it is advantageous to compare two different mixtures of salts in
which the major difference in the solution properties can be related to the differences in the prop-
erties of one ion. As an example, mixtures of alkali nitrates with silver or thallous nitrates would
be suitable for such a comparison with mixtures containing only alkali nitrates, since the differ-
ence in the properties of Ag* and T1* jons from those of Na* and Rb* is largely related to the rel-
atively high polarizabilities of Ag* and T1*,*+105

Kleppa has measured the heats of mixing of AgNO, and TINO, with all of the alkali nitrates
except CsNO3.77'79 By fitting his data to equation (11.3.2), where N, is the mole fraction of
either AgNO, or TINO,, Kleppa obtained the values of the parameters a, b, and c which are listed
in Table 2. The observed deviations from ideal solution behavior differ from those of the corre-
sponding mixtures of alkali nitrates with NaNO, or RbNO,. In addition to the interactions present
in mixtures of alkali nitrates, an additional interaction needs to be postulated to rationalize the ob-
served results. This difference has been shown to be in reasonable agreement with a calculation
of the London dispersion energy of interaction of next-nearest neighbors. '® The predominant term

of the London dispersion interaction energy between two ions is the dipole-dipole term,
k! rr —kl 6
Uty ==, CL/d°, (11.4.7)

where 5:”is a constant probably in the range of 1 to 2 and depends on the structure of the melt,
d for a pure salt is the cation-anion distance with the cation-cation distance assumed propor-

tional to 4, and 4 for a mixture is an average cation-anion distance. The paramater Cii is given

by101

3a ol
k1 kRCL

chlae ————, (11.4.2)
2 I +1,

 

*1t should be noted that although the Pauling radius of Ag+ ion is 1.26 A, the interionic distances in
AgCl and AgBr and the relative molar volumes of liquid AgNO, and NaNO, are more consistent with a

radius of about 0.95 A which is close to that of Na+.

 
30

where & and [ are the two cations, ais the polarizability of an ion, and I has been estimated !0
for the alkali cations, TI* and Ag*. Values of c and I are listed in Table 3. A calculation of

the contributions of this interaction, AUfP, was approximated from the dispersion energy change

represented crudely by the process AXA + BXB - 2AXB,
A
AULRE = 2048 —uhh - BB, (1.4.3)

where the solutes are AX and BX, where Sg’ 2 1.8, and where 2d, o =d, s +dgg.* Equation
(11.4.3) is an approximation to the contribution to AH?H'S/NTN2 so that the relation for molten

nitrates (11.3.5) is modified to become
XZAARDS = Us? + AUAB (11.4.4)

The value of U = =140 for alkali nitrates includes a small positive contribution from van der Waals'
interactions so that a correction is needed which will make AUff less positive.'® A cruder but

simpler approximation to AULB may be made in a manner similar to an approximation useful in
68

2
e x (forr - fuBE), (11.4.5)

where the values of C__ in Table 3 in conjunction with a value of S;’: 1.8 may be used with Eq.

nonelectrolyte solution theory.

(5) and ionic radii for roughly estimating AU4AB. From Table 3 it can be seen that C,, will be
quite large for Cs*, Rb* and K* ions and the positive term, AUfP, in (4) may be large enough
to cancel the negative values of U8? for mixtures of, for example, NaF with KF, RbF or CsF.

The calculations of Lumsden?®’ are in accord with this and this may be used to rationalize the

 

* A better approximation for dpg is [(a’i + d; )/2] ]/2, which differs little from (dA + a’B)/2 when d, is

not very different from dg. The factor for Sg' contains a small correction for interactions of longer range

than next-nearest neighhbors,

Table 3. Polarizability and Potential Parameter Used for Estimating

Cation-Cation van der Waals’ Interaction

 

 

lon a x 1324 Ix 1012
(em™) (ergs/molecule)
Lit 0.030 90.9
Na® 0.182 56.8
k* 0.844 38.2
Rb* 1.42 33.0
cst 2,45 39.0
Agt 1.72 30.0
+

Tl 3.50 30.0

 

 
31

! mentioned at the end of the previous section. However all of these methods

results of Cantor®
are approximate and are useful largely for semiquantitative estimates of solution behavior.

Laity®3 has shown that A 4.¢¢ is negligible in the cell

 

 

 

 

AgNO,4(N ) AgNO (N )
g Ag
NaNO, NaNO,
and the emf of this cell is given by
AE =RT In {a /aj) .
The measurements are consistent with the expression
uf =840 N2, (11.4.6)

where 1is AgNO, and 2 is NaNO,. The results did not exhibit the asymmetry in the heats of
mixing found by Kleppa for the same system. Although the total excess entropy is small rela-

tive to the total entropy of mixing, it is negative and is not small relative to AH_ or AGE;
TASE = AH - AG" = (=156 N; = 163)N N, ,
m

so that although Eq. (6) has the form for regular solutions the excess entropy does not appear
to be negligibly small.

There have been many studies of mixtures of silver halides and alkali halides using the
formation cell
AgX

WX X, (p = 1 atm) graphite,

 

y

where M is an alkali metal ion and X is a halide. The emf of this cell can be related to the

activity of AgX by
p]—-,ucl)z—F(E—-Eo)=RT|n ay, (11.4.7)

where 1 is AgX.

The most extensive work on these systems has been the work of Hildebrand and Salstrom
who studied mixtures of AgBr with LiBr, NaBr, KBr, and RbBr. 86,116,117 | Fig. 7 are plotted
values of #I]E for AgBr (component 1) vs Ng. Within the experimental precision, ,ufl‘: is independent

of temperature and can be represented by the equation
2
,ufl? = AN . (11.4.8)

Values of A are given in the table below and may be rationalized in terms of Eq. (4) using the
data in Table 3. Many studies of mixtures of AgCl with alkali chlorides have been made. Unfor-
tunately, there are significant differences between different measurements on the same systems.
The most reliable and consistent studies appear to be those of Salstrom '8 and of Panish %4 on

the (Li-Ag)C! and the {(Na-Ag)Cl systems. Although there is scatter in the high-temperature data

 
32

Volume Change

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mixture A (cal/mole) of 50-50 Mixture
(cm3/mo|e)
{Li-Ag)Br 1880 -0.13
(Na-Ag)Br 1050 +0.17
(K-Ag)Br ~ 1480 +0.27
(Rb-Ag)Br - 2580 +0.42
(Li-Ag)ClI 2100
{Na-Ag)ClI ~ 800
UNCLASSIFIED
ORNL-LR-DWG 66348
1600
/7
1200 /e
a
@
_ L~
LU-— /A/'a
i
o ®
0 fi
'
o
\.
\\Kar
-400 b~ -
o ®
N
\..
\ .
RN
-800
RbBr N\
-~
o
\
-41200
0.2 0.4 0.6 0.8 1.0
2
N2

Fig. 7. Values of the Excess Chemical Potential of AgBr (Component 1) in
Mixtures with Alkali Bromides.

 
33

of Panish, the results of both Salstrom and of Panish on the AgCl-LiCl system lead to positive
deviations from ideality, which follow, approximately, Eq. (8) with A = 2100 (cal/mole) from 500

d104

to 900°C. Small positive deviations from ideality have been foun in the AgCl-NaCl system.

There is too much scatter in the results to be able to represent ,u']‘: precisely, but crudely ,u’]‘? =
800 Ng. The work of Stern is consistent with these results. 24

In all of these chloride systems there is considerable scatter and uncertainty, and it cannot
be clearly shown that the data can be best expressed by an expression as Eq. (8) and that X is
truly independent of temperature.

Measurements of AgCI-KCI mixtures by Stern are doubtful. 2% The measurements of Mur-

gulescu and Sternberg indicate that for AgCl-KCI mixtures 102

ui = =1555 N2,

and that the excess entropy of mixing was nearly zero. However, the values of E observed by
Murgulescu and Sternberg differed from those given by Salstrom, Panish, and Stern by about 9 mv
at 500°C.

An interesting comparison with solid solutions is exhibited by Panish.'%4 Although the
molten salt system AgCl-NaCl exhibited only small positive deviations from ideality, the
measured deviations from ideality in the solid solutions were more positive. This illustrates
the fact that, aside from other effects, the accommodation of ions of different sizes in a given
material leads to a greater positive {or less negative) free-energy change in a crystal than in
a liquid.

124 4hat the volume change upon mixing of

It was pointed out by Hildebrand and Salstrom
50-50 mixtures of the four systems containing AgBr, which are listed on page 32, could not
be related to a weakening or strengthening of the interactions of the ions or with the devia-
tions from ideality. As with the results on alkali nitrates for AH_, the values of A;f:; vary

in a direction opposite in sign to that of AVE with variations in the cation.

I1.5 Binary Mixtures Containing Polyvalent lons

Although there has been much experimental work on mixtures containing polyvalent ions, very
little theoretical discussion based on fundamental physical principles has been published. This
section will be devoted to the presentation of thermodynamic data to give the reader an idea of
the magnitudes involved and, where enough data exist, to pointing out the correlation of proper-
ties of mixtures with the physical properties of the ions. Where it is considered necessary, a
discussion of the principles of measurements will be included. In the next section, a discussion
of these data and a critique of the description of these data in terms of '‘complexes’ will be made.

Kleppa and Hersh?7 measured the molar heats of mixing of Ca(NO,), with LiNO,, NaNO,,
KNO,, and RbNO, at 350°C. By using a heat of fusion of Ca(NO,), of 5.7 kcal/mole obtained by

 
34

* 0
-Hl

extrapolation of their measurements, the limiting heat of solution of liquid Ca(NO,), (E]

in Table 4) at 350°C obeyed the empirical relation
HY = H =0.3 = 225((r,,/2) = (r,/ N 2/1d, + 4,12,

which relates the radii of the divalent and monovalent cations (r++ and r+) to the observed heats of
mixing. The heats of solution decrease with increasing radius of the alkali cation. It should be
noted that the heat of solution in LiNO, is positive. No simple representation of the concentration
dependence of the molar heat of mixing was made. It was noted, however, that the slope of plots
of AHm/N][Ca(Noa)z is component 1] vs N, for mixtures with KNO, and RbNO3 had maxima at

Ny =0.25-0.33 (or at equivalent fractions N{ = 0.4-0.5). The results in these two systems prob-

ably can not be represented by an equation with as few terms as (11.3.2).

Table 4. Extrapolated Values of the Limiting Heats

 

 

of Solution of Ca(N03)2
— 0
Solvent HI ~ HY {kcal/mole)
LiNO3 +0.25
N(:lNO3 -0.9
KN03 ~3.0

 

The most extensive comparative studies of binary mixtures containing polyvalent ions have

30,3134 who measured the freezing point lowering

been the cryoscopic measurements of Cantor,
of NaF by polyvalent salts. NaF can be considered as a prototype of an ionic salt. In Fig. 8
are plotted the liquidus temperatures of NaF {component 1) in mixtures with the alkaline earth
fluorides. The upper line is the calculated liquidus temperature for an ideal solution with the

data contained in Table 1. For an ideal solution at the liquidus
_ nideal
ay =N, ,

and the activity coefficient in a real solution is given by
ideal

N

¥y =—

Ny

at the liquidus, where Ni]d“' and N, are the compositions of NaF in the ideal and real solu-
tions respectively at the same temperature. A freezing point fower than the ideal value means
that y, <1 so that the solutions all exhibit negative deviations from ideality. The smaller the
radius of the alkaline earth the greater the deviations from ideality.

The Ca?* ion has about the same radius as Na*, but the NaF-CaF, mixture exhibits negative

deviations from ideality. This illustrates the effect of charge. Deviations from ideality in the

 
35

UNCLASSIFIED
ORNL-LR-DWG 67567

 

1000

 

980

 

960

 

 

 

940

 

 

920

 

900

TEMPERATURE (°C)

 

880

 

 

860 \ CaFp
3 Mng

 

 

 

 

 

 

 

 

 

820

 

O 5 10 15 20 25 30
SOLUTE (mole %)

Fig. 8. Liquidus Temperatures of NaF in Mixtures with Alkaline Earth

Fluorides.

NaF-BaF, system are small. Since the Ba2* ion is larger than the Na* ion, the large size of
the divalent ion appears to, at least partially, compensate for the greater charge. The excess
free energies of NaF, #1]5’ at the liquidus temperatures in mixtures with the alkaline earth flu-
orides are plotted in Fig. 9 vs Ng (where 2 is the solute). (Note that these values of pfi? are
not isothermal.) For comparison with monovalent cation salts, data with LiF and KF as solutes
are also plotted. The Li* ion is about the same size as the Mg2* ion and both are smaller than
the Na* ion. If the LiF and the MgF, mixtures are compared with NaF, the deviations from

ideality in both appear to be negative, being much more negative in NaF-MgF, mixtures. On
36

UNCLASSIFIED
ORNL-LR-DWG 67571

50 KF

 

 

 

 

 

 

0 g ——

-50 \\ \.\ i,

 

 

 

 

 

 

\>\ CQFé

 

 

 

 

 

/////

 

 

 

 

 

 

 

 

 

 

 

 

-350
-400 \ MgF,
-450 \
) BEFz
-500
O 0.02 0.04 0.06 0.08
2
(1 _NNaF)

Fig. 9. Excess Chemical Potential of NaF in Mixtures with the Alkaline
Earth Fluorides LiF and KF as Calculated from Liquidus Temperatures.

the other hand, the K* ion is about the same size as the Ba2* ion, both being larger than the
Na* ion. The deviations from ideality of the solvent NaF in mixtures with KF and BaF, are
both small. A further illustration of the influence of charge is shown in Fig. 10, which gives
,ulls for NaF in mixtures with Can, YF3, and Thf:4 in which salts the interionic distances are
about the same. Al| of these illustrations show that the deviations from ideal solution behavior
are related by a function which appears to be monotonic in the charge of the solute cation, Z,
and in 'I/dz, where d, is the sum of the cation and anion radii of the solute. However, other
effects such as van der Waals' interactions, ligand-field effects on transition metal ions, etc.,
will be superimposed on the effects of charge and radius of the ions. Figure 11 gives a parallel
plot of yll‘: at 20 mole % of solute and the lattice energies of the solid solutes MnF,, FeF,, CoF,,
NiF,, and ZnF,. The measured cation-anion distance in solid MgF, is about the same or smaller

 
UNCLASSIFIED
ORNL-LR-DWG 67569

0 m — ]
-100 N

T
-200 \
N
N
N\

 

¥ C0F2

 

 

 

 

-300 \\\\
3 \
2 -400 O Y Fy —————-
w
o
Iy £
3

-500 \

 

 

 

 

 

 

 

 

 

 

 

 

~600 \
—700 ) % ThF,
-800
0 0.010 0.020 0.030
2
(1- NNGF)

Fig. 10. Excess Chemical Potentials of NaF in Mixtures with CaF,, YF3,
and ThF, as Calculated from Liquidus Temperatures,

than those of these transition metal ions. The greater negative deviations from ideality found for
mixtures with the transition metal fluorides are therefore not related solely to the radii of the ions.
Since the pattern of the lattice energies with a maximum at NiF , or CoF, is explained by ligand-
field theory for octahedral or to tetrahedral symmetry respectively, 103 then the pattern of ,ufi; and
the differences from the NaF-MgF, system suggest that the change of the ligand-field effect upon
dissolution is related to the deviations from ideality of NaF. Having monovalent ions as next-
nearest neighbors in the mixture, as compared to divalent ions as next-nearest neighbors in the
pure transition metal fluorides, probably leads to a greater ligand field and a great ligand-field
stabilization of the solute component in the mixture than in the pure salt. Whatever the specific
structure of the melt and of the ligands about the transition metal ion, it is apparent that the
effect of the ligand-field stabilization on the solvent is in the same order as might be expected

from ligand-field theory for the solute.

 
38

UNCLASSIFIED
ORNL-LR-DWG 67576

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

740
- T20 \
o
o
0
5 -1
a / P
léJ 700 -
w / //
w //
Q —~
C 2
< -
- 680 _T

P
-~
-
~
-
-~
-
-~

660 &<

500
—~ p—
o
L \\'
w450 -7
> -
- -~
C -~
n -
B -

-
w -
Y 400 7 >
Q P
= -~
-
Q _-
- ~
~
< 350 =
o &
W =2 //
& -7
L
300
Man Fer CoF2 NiF2 Cqu ZnF2
SOLUTE

Fig. 11. Parallel Plot of the Lattice Energies of Some Transition Metal
Fluorides and the Excess Chemical Potential of NaF in Mixtures with Transi-

tion Metal Fluorides.

Cantor34 has also made cryoscopic measurements on NaF with ZrF4, HfF4, ThF4, and UF,

as solutes. In Fig. 12 are plotted values of yfi? Vs Ng at the liquidus for these four mixtures. The
deviations from ideality are all more negative than those for the alkaline earth fluorides which fure
ther illustrates the effect of charge. The effect of radius appears to be reversed for these tetra-
valent salts, since Zr** which has the smallest radius also has the smallest negative deviation
from ideality. The cause of this is not clear, although steric hindrance related to anion-anion
contacts in the coordination shell adjacent to the tetravalent ion has been suggested as a limit-

ing factor.®4 Thus any tendency by a tetravalent ion to have a high coordination number might

 
39

UNCLASSIFIED
ORNL-LR-DWG 67568

 

 

-100 \\

—200 \
-300 k

 

 

{cal)

 

) -400 \ \
i A\
-500 \ M HfFg —

ZrFa

-600 —— - \\

NoF

 

 

 

-700 e S

 

 

 

 

 

 

 

 

 

-800

 

0 0.010 0.020 0.030

2
(1 _NNOF)

Fig. 12. Excess Chemical Potentials of NaF in Mixtures with ZrF,, HfF4,
UF4, and ThF4 as Calculated from Liquidus Temperatures,

be sterically limited for ions as small as Zr** and this might limit the magnitude of the devia-
tions from ideal solution behavior.
Another comparative study including polyvalent ions covering a relatively broad range of con-

centrations was the emf measurements by Yang and Hudson ' 33 by use of cells of the type

MCI
7 C]2 .
(LiCl-KCI eutectic)

For M = Pb2* Cd?*, Zn?* Mg?*, Be2*, activities were calculated from the relation
0 0y _ 1

In the five systems measured, the deviations from ideality were always negative (y] <N, In

Table 5 are given values of (;LI]E/Ng) at 800%K calculated from the measurements in the most dilute
40

solutions of the solute (component 1) with values of E? contained in Table 6 for the calculations.
Except for ZnCl,, the values of (,u‘?/Ng) are more negative the smaller the radius of the divalent
ion. These quantities in addition to the influence of radius are influenced by other factors which
are probably significant for polarizable cations such as Zn?*, Cd?*, and Pb2*. It should be noted
that the measurements of Takahashi !28 differ from those of Yang and Hudson on mixtures of PbCl,
with the LiCl-KCl| eutectic mixtures.

Another series of comparative studies of molten-salt mixtures was made by Lantratov and Ala-
byshev®3 by using the cell

M,Cl, B
: M”C[n 2!

where M, = Pb2* Cd?*, and Zn?* and M,, are alkali and alkaline earth metals. They were able to
observe the effect of a change in the diluting chloride M;,Cl_ on the activity of M,Cl,. In Fig. 13
values of lu’f, at 500°C for PbCl, in mixtures with LiCl, NaCl, KCI, and BaCl,; at 600°C for
CdCl2 in mixtures with NaCl, KCI, and BcCIz; and at 500°C for ZnCI2 in mixtures with NaCl,

KCI, and BaCl,, are plotted vs the square of the mole fraction of the other component.

Table 5. Values of (,u?/Ng) in Dilute Solutions of Divalent Chilorides
in the LiCl-KC| Eutectic Mixture

 

 

Solute (1} /N3) N,

PbCI, ~1,410 0.984
cdcl, ~ 4,620 0.993
ZnCl, ~ 8,650 0.984
MgCl, - 8,150 0.986
BeCl, 14,680 0.998

 

Table 6. Values of the Parameters in the Equation Eo =a + bT(°C),
Where E®is the EMF of the Cell M/MC!_/Cl, ( P = 1 atm)

 

 

Salt a b x 104 References
AgCl 1,046 1 ~2.92 116
PbCl, 1.5855 -6.25 131
cdcl, 1.7188 -6.29 89
ZnCl, 1.9200 -6.95 131
MgCl, 2.9823 ~6.73 89
BeCl, 2.6205 ~8.60 133

PbBr, 1.424 - 7.4 86, 115

 

 
4]

UNCLASSIFIED
ORNL-LR-DWG 66349

 

 

 

 

 

 

6000 l 1
600°C / 500°C 500°C i
5000 =
/@ -
/ QO
v X
5 v
4000 O O
o
1} N
O
E
>~ 3000
o
o
W_
i
|

 

2000 ~ o /
() ~
/ N /fl © S
/
(J\"b O’b

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/
1000 [ & L © &
__Noc,‘\ ’\/
poCl2
\
0 ~ ,/%°(J =
S PbCl,-BaCl,
PbCl, - LiC|
-1000 :
0 02 04 0 02 04 0 02 04
2 2 2
Ny N Ny

Fig. 13. Excess Chemical Potentials of CdCl,, PdCl,, and ZnCl, in Mix-
tures with Other Chlorides as Obtained from EMF Measurements.

The values of E? at 600°C used by Lantratov and Alabyshev were 1.4987 v for ZnCl,, 1.3382
for CdCIz, and 1.2215 for PbC|2, which are —4.3, =3.2, and 11.0 mv different from the values cal-
culated by using the data in Table 6. Because of these differences in EY, the values in ,u’.lg must
be considered uncertain by about 200 cal. Discrepancies in values of E? reported by different
workers are common and are probably related to the solubility of metals in their own pure salts.
In Table 7 are given values of y, at 600°C and values of the total volume change per mole of
mixture, AV, at N, = 0.5. With an increase in the size of the alkali cation in mixtures with
alkali chlorides values of v, decrease and values of AV increase. These relative variations
in the deviations from ideal solution behavior and volume changes are in the same direction as
was observed in alkali nitrates and in mixtures of AgBr with alkali bromides. As in the meas-
urements of Yang and Hudson, mixtures of alkali halides with ZnCl,, CdCl,, and PbCl, exhibit

less negative deviations from ideal solution behavior in that order
E E E
(Yznc1, <¥cact, <¥puci, M Fzact, <Hcact, < HpbC,) -

Values of ,u’f at 589°C from measurements in the F’bBrz-KBrB“j system are more negative than
those in the PbCl,-KCl system (E? valves for PbBr, given in Table 6) but differ appreciably

from the measurements of Reid '8 in the same system.

 
42

Table 7. Values of ¥ and Motar Volumes of Mixing in Equimolar Binary Mixtures of Chlorides at 600°C

 

 

 

 

 

 

PbCI, cdcl, ZnCl,

Diluent

2 AV (cm®/mole) ¥4 AV (em®/mole) ¥4 AV (em®/mole)
LiCl 1.064 +0.39° ~0.053¢
NaCl 0.86 +1.19 0.68 +0.20 0.41 ~1.38
KCl 0.46 +1.66 0.36 +0.97 0.19 +0.35
CaCl, 1.25
BaCl, 0.95 - 0.43% 0.84 - 1.85 0.58 ~2.63
“Ni ;o = 0.466.
PN iy = 0.428.
N o = 0.428,
“’NBC,c|2 = 0.4, T = 650°C.

Senderoff, Mellors, and Bretz ! 19 measured the activities of CeCl in cells
I 7! 3

CeCl, (N )
MCI (N )

e 21

 

 

where M = Na or K. In Fig. 14 are plotted values of y'f vs Ng at 800°C for CeCl, in mixtures
with NaCl and with KCIl. At low concentrations the excess entropy of solution appeared to be
negative. However, the large scatter in the experimental results indicated in Fig. 14 makes any
quantitative conclusions uncertain. It is apparent that the CeCl3 is stabilized considerably by
dissolution in both NaCl and KC| with KCI having a larger effect.

Measurements have been made of the activities of the divalent chlorides in PbCI,-KCl mix-
tures,%” PbCl, mixtures with LiCl, NaCl, KCI, and RbCI,?8 MgCl, mixtures with LiCl, NaCl,
KCi, RbCl, 100 ZnClz-RbCI,w and BeC!z-NoCI("B — and have been reviewed.?%:%? Since these
measurements were made by generating chlorine by electrolysis within the cell itself, the state
of the chlorine gas is undefined and the E® usually differed from those which are given in Table
6. The activity coefficients did follow the expected relative order (that is, for a given divalent
chloride they decreased with an increase in size of the alkali cation), but the absolute values
are probably unreliable.

The measurements of Laitinen and Liu®2 and of Flengas and Ingraham?7+#8 provide another

comparative study of the effect of alkali halide solvents on solutes. They used a cell

\ MCI (N,)
(LiCI-KCI eutectic)

PtCI2 (Nz)

Pt I1.5.A
(L.iCl-KC! eutectic) ( )

 

 

 
43

UNCLASSIFIED -
ORNL-LR-DWG 67577

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14,000 '
12,000 A
800°C * /
— 10,000 e /
[
9 CeCly-KCl /
S 8000
2 %
Ww. 6000 P ; }/
| / CeCl3-NaCl
4000 //
» ,,//
2000 A
%/
0 —y
0 0.2 0.4 0.6 0.8 1.0
2
NZ

Fig. 14. Excess Chemical Potential of CeCl, in Mixtures with NaCl and
KCl as Obtained from EMF Measurements.

in which the solutes PfCI2 and MCIfl were shown to be dilute enough to obey the Nernst equation,

 

 

(N2)1/2
By adding AE; to AE;, obtained from the cell
PtCl, (N.)
t 2?2 (LiCI-KCI eutectic)|Cl, , (1.5.B)
(LiCl-KC!| eutectic)
where
1

AE, = AEY + 20 log ——
= + og
B B !
F 1/2
v,)
the standard formation potential is obtained from which the chemical potential of formation of

MCl_ may be calculated by using the relation

* * * *
Hyci =-nFE" = -nF(AE, + AE) .

There are not enough good values of E® for the pure liquid solutes at 450°C to make a mean-
ingful comparison of E" and E°. Flengas and Ingraham, from the cells

AgCl
NaCl-KCl (50-50 mole %)

MCl»

Ag
NaCl-KC! (50-50 mole %)

 

 

 
44

and

Ag

 

AgCl
NaCl-KC!

 

NoCl-KCI|CI2

dilute enough in the solute to obey the Nernst relation, obtained values of the standard-formation
potential of MCI_, E*, in the NaCl-KCl solvent at 700, 800, and 900°C. Values at 700 and 900°C
are listed in Table 8.

For a comparison, values of E in NaCI-KCl (50-50 mole %) extrapolated to 450°C are given
in column 3 of Table 9. Column 4 gives the differences between the values of y* in the two sol-
vents. |t can be seen that the values of p* are always more negative in the NaCI-KC| mixture
as expected due to the fact that the effective cation radius in the NaCl-KCI] mixtures is greater
than in the LiCI-KC] mixtures. Large differences in the two solvents are apparent for the tran-
sition metal halides.

In column 4 of Table 8 are listed values of E? for the pure liquid at 900°C, Where E® was
unavailable the standard formation potential for the pure solid is given in parentheses. The
values of E® are taken from Hamer, Malmberg, and Rubin.® It should be noted that the value of
EQ for ZnCl, is very different from that calculated from Table 6. In view of the discrepancy,

both (1 — %) in Table 8 and values of uZ cited earlier must be considered questionable for

Table 8. Values of E , E%, and ;i = 1% in NaCI-KC! (50-50 Mole %) Mixtures

 

 

 

£ E°, T

700°C 900°C 900°C 900°C
MnCI, +2.051 +1,967 1.766 - 9260
znCl, +1.705 +1.605 1.438 -7200
CrCl, +1.603 +1.523 ~1.307 - 9970
TICI +1.510
CdCl, + 1,465 1.279 (700°C) ~ 8580 (700°C)
FeCI2 + 1.365 +1.293 1.084 - 9640
CrCl, +1.270 +1.140 (1.060)%
PbCI, +1.235 +1.150 1.076 ~ 3420
SnCl, +1.215 +1.135 ~1.255 +5530
CoCl, +1.169 +1.070 0.939 - 6050
CuCl +1.105 +1.055 0.903 - 3510
NiCI2 + 0,985
AgCl +0.845 +0.795 0.805 230
CuCl, +0.675 +0.603

 

%The standard formation potential for the pure solid.

 
45

Table 9. Values of E* (volts) at 450°C in LiCl-KC! and NaCIl-KCI| Mixtures and Calculated Differences
in the Standard Chemical Potentials

 

 

met E"(LiClKCl) E*(NaCl-KCI)? i (NaCl-KCl) = p(LiCI-KCI)
MgCl, +2.796

MnCl, +2.065 2.135 ~3230
AlCl, +2.013

ZnC|2 +1.782 1.835 - 2450
TICH +1.587

crcl, +1.641 1.715 -3420
cdcl, +1.532 1.535 ~69
FeCl, +1.387 1.465 ~ 3600
PbCI, +1.317 1.352 ~ 1615
SnCl, +1.298 1.315 ~ 785
coCl, +1.207 1.277 —~ 3230
CuCl +1.067 1.145 - 1800
GaCl, +1.10

InCly +1.051

NiCL, +1.011

AgCl +0.853 0.905 ~ 1200
sbCl, +0.886

BiCl, +0.804

HgCl2 +0.7

PdCl, +0.430

PfCI2 +0.216

AuCl ~0.095

 

*
“Values of E in NaCl-KC| mixtures are extrapolated from higher temperatures.

ZnCl,. The values of E? have, in general, an uncertainty large enough so as to make com-
2 g9 Y 9 9

*
parisons of u — u® semiquantitative. It is clear, however, that the transition metal halides
are greatly stabilized by dissolution in alkali halides, the stabilization being greater the

larger the radius of the alkali cation.

1M1 2

Reznikov, improving on the method of Treadwe!l and Cohen, 128 made measurements

of the activities of MgCl, in mixtures with KCl and with NaCl at 750, 850, and 950°C by using

the heterogeneous equilibrium

1
MgCl, 4-502 — MgO + Cl, .

 
46

Reznikov cites evidence that pure MgO is the solid phase in contact with the melt and that the
known solid solution of MgO with MgCl, is unstable at high temperatures. The equilibrium partial
pressures of O, and Cl,, for the above reaction were measured for pure liquid MgCl, and for mix-

tures of MgCl, with NaCl and KCI.

PC|2 be,

172
0 1/2
<P02> Po., “mgcCl,

where pure solid MgO and pure MgCl, are taken as standard states having activities equal to unity,

K=

 

where pgl and pg are the equilibrium partial pressures of Cl, and O, measured at equilibrium
2 2
with pure liquid MgCl, and pure solid MgO, and p, and p are the partial pressures at equi-
2 2

librium with a mixture. Reznikov approached the equilibrium from two sides and his results do
not differ greatly from, but are probably more reliable than those of Treadwell and Cohen. His

values of the activities of MgCl, are listed below:

 

 

MgCl,-kcl MgCl,-NaCl
T (°C) Mole % MgCl, {50 mole % MgC|2)
100 75 50 33.3
750 1.0 0.47 0.10 0.010 0.15
850 1.0 0.50 0.11 0.011 0.19
950 1.0 0.54 0.12 0.013

The activity coefficients in the mixture with NaCl are higher than in the mixture with KCl as
expected.

Other measurements using heterogeneous equilibria of the melt with a gas phase include the
work of Blood'? and co-workers on the standard free energy of formation of NiF, in NaF-ZrF

and LiF-BeF, mixtures using the equilibrium
Ni + 2HF = NiF, +H, .

The equilibrium quotient K, given by

2
PHF

was constant in dilute solutions indicating that NiF, obeyed Henry's law. (HF and H, at the
temperatures and pressures involved are essentially ideal gases.) The standard free energy
. - . - » . . - *
(chemical potential) of formation of NiF, in its standard state in solution (1y;g ) could be
2

calculated from the equation

* 0
#Nin :ZGHF - RT In Ky -

 
47

Vapor pressure measurements afford a method of measuring activities in molten-salt mix-
tures. Unfortunately, the large number of complex compounds found in the vapor often make
it difficult to analyze vapor-pressure data. As there is no general discussion of this in
standard texts, some of the principles involved in deriving activities from vapor-pressure
measurements will be discussed.®?

The chemical potential of a component in a mixture is related to the fugacity, /, of the

component
'u"|=RT ]n f-l ’ (”.5.])
and for the pure liquid #(1) =RT In f?,
/i
o _p?_—_RTln —=RT In a, . (11.5.2)
i

The fugacity is defined in such a way that /,/p, + 1 as P+ 0, where p, is the partial pressure of
the component in the vapor and P is the total pressure.

In investigations of salt vapors it is generally assumed, and will be assumed here, that, except
for the formation of associated species or compounds in the vapor, the vapor behaves ideally so
that the fugacity of a species in the vapor is equal to the partial pressure of the species.* if only

a monomer is present,
0 Py
py=p;=RTIn—=RTIna,. (11.5.3)
0
Py

If a vapor with a monomer vapor molecule represented by M, at total pressure, P, associates into

several species

(M])2 (M])3 (M])i
M —_ —_— ——
] < < < I

2 3 i

 

 

 

where (M!)2 is a dimer, (M])3 a trimer, etc., then the total pressure P is (if there is only one com-

ponent in the vapor)

P=( )+ @)+ (pdy+...=2(p,);, (11.5.4)

where (p,), is the partial pressure of the associated species (M,).. When the vapor is at equilib-

rium with a mixture (or pure substance)

ity(mixture) = p,(vapor) .

 

* Although this assumption may be valid, it has never been investigated. It is probable that at pres-
sures approaching one atmosphere in alkali halides some of the interactions of the dipoles in alkali halide
vapors are large enough to have an appreciable effect on the fugacity of the vapor even when the molecules
are too far apart to be defined as an essociated species.

 
48

One total mole of component 1 in the vapor would have the chemical potential {per mole)

 

(6y) (p4) (p4) (29); (),
L L g b =Yt (11.5.5)

,u](vapor) = (p)] +7 (u - 3 (,u)3 + =

where (1), the chemical potential of the associated species, is given by {); = RT In {p,); if the
non ideality of the vapor is due to the association only, and (#,)./7P is the number of moles of

species i in a portion of gas containing a total of one mole of M,. Because of the equilibrium,

(Ju)] =— =, {(11.5.6)

Combining Eqgs. (5) and (6) we get

,u](vopor) ={p); =—, (11.5.7)

so that

Py, RT (p),

=——In

P, iKY,

I

=RT|I‘161

 

 

py =g =RT In - (11.5.8)

In order to measure the activity of the component 1, one need only know the partial pressures at
a given temperature of one species containing 1 only which is in equilibrium with a mixture which
is in equilibrium with a mixture or with the pure liquid component. At low pressures Eq. (8) is
valid for component 1 independently of all other species in the vapor.

The heat of vaporization AH]. of species j is given by

d(‘u]./T) Rd In b

v @ M =

The variation in total pressure with temperature for any mixture or any number of species is
P

_ ]
P ==L M= - XA, 11.5.10
d(1/T) k P d(1/T) Zp i, X AH,, (11.5.10)

Rd In P p]-d]n b;

 

where X], is the mole fraction of species j in the vapor, and j can be any species.
Vapor pressures of mixtures have been measured by several methods. The Rodebush and boil-

ing point methods#:21.33 120

make a measurement of the total pressure P. In the transport methods
the vapor at equilibrium above a liquid is swept away with a known volume of inert gas and ana-
lyzed. If only one component of the liquid is vaporized, then the apparent vapor “‘pressure,’”

PT' is

 

=PIT=p,+20p)), +3(p )y + ... =Zilpy),, (11.5.11a)

 
49

where PT* is an apparent ‘‘pressure’’ calculated assuming that the only species is the monomer.

For more than one component

Tr
P,"= 2, (11.5.115)

where P:' is the apparent transport pressure for the nth component and v is the number of mole-

cules of 7 in species ;.

The vapor composition at equilibrium with a given liquid to obtain association constants may
be analyzed by using more than one experimental method under a variety of conditions and partial
pressures. A complete analysis of the vapor in equilibrium with a mixture requires knowledge of
the association constants for all of the species in the vapor. This analysis may be extended to
the case where more than one component is vaporized. The precision of partial-pressure meas-

vrements decreases very markedly the greater the number of species in the vapor.

The vapor pressure of ZrF  in equilibrium with mixtures of ZrF , with LiF'21 NaF 120 gnd
with RbF33:121 have been studied by the transport method '29:121 and by the Rodebush tech-

nique.3® Values of ,u';‘: for ZrF, at 912°C are given in Fig. 15 with the value of png {calculated
4

from the equation given by Cantor for the vapor pressure) at the measured melting point of 912°C;

log pngd (mm) = 12.542 — 11,360/T (X) . (11.5.12)

At high ZrF , concentrations the major species in the vapor is ZrF ,. Deviations from ideality are
large, and are larger the larger the alkali cation. Some of the values of the vapor pressure used
in these calculations were not directly measured but were extrapolated from other temperatures.
Although Sense and co-workers report vapor pressures of the alkali fluorides in these mixtures,
they do not in any case properly correct for the presence of associated species in the vapors.
Cantor and co-workers33 have reported that in RbF-ZrF , mixtures the excess entropies are
positive and that the excess enthalpies as obtained from temperature coefficients of vapor pres-
sure data exhibit both positive and negative values. Although these conclusions are more reli-

able than those obtained from emf data, the temperature coefficients are subject to large errors,

The most thorough study of the vapor pressures of a molten-salt mixture is that of Beusman,®
who partially studied LiCl-FeCl, mixtures and studied more completely KCI-FeCl, mixtures at
temperatures from about 850 to 1000°C by using the Rodebush technique for measuring the total

pressure and the transport method for measuring the vapor composition.

In the vapor above mixtures of KCI and FeCl, the presence of the species FeCl,, Fe,Cl,,
KFeC|3, KCI, and K2C|2 was consistent with his measurements. Calling these species 1, 2,
3, 4, and 5 he could solve for the number of moles of each of these species in a unit volume of
vapor and, hence, for the partial pressures at equilibrium with the melt from measurements of the

total pressure of salt (P == _RT/V), and by a chemical analysis of the chemical compounds swept

 
50

UNCLASSIFIED
ORNL-LR-DWG 67572

 

16,000

ROF -ZrF,

14,000 4
912°¢ /

12,000 /

 

 

 

 

 

— 10,000

@

o

£

> NaF - ZrFy
O

: ]
L V.

1

|

 

// o
4

A
/.

S

2000 M//)/

N

 

 

 

 

 

 

 

 

 

Fig. 15. Excess Chemical Potentials of ZrF, in Mixtures with LiF, NaF,
and RbF as Obtained from Vapor-Pressure Measurements.

out by a unit volume of gas from above the pure components and from above the mixtures. He solved
the simultaneous equations
n,=ny+tnyg+ngtn, +n,,

ne =ng+n,+ 2,

2 "y

”1=KFeC|2”2 5/
nU

2

nyg=Kecr s <_P .

 
51

where _ is the total number of moles of all species in a unit volume, 7 _and n. are the number of

Fe and K ions in a unit volume, and K and K are the dissociation constants in pressure
FeC|2 KClI

units for the dimer of the subscript component and were evaluated from data obtained with the pure
materials KCl and FeCl,. The presence of a trimer in LiCl vapors made this procedure very im-
precise since the calculations, which essentially involve subtracting large numbers, are much more
sensitive to errors in the measurements when more species are involved. In Fig. 16 are plotted
Beusman's values of uf for the two components FeCl, and KCI at 900°C derived from the values

of the partial pressures. It is apparent that values of uF are all negative. Calculation of #II:;CI

from ,uléeaz by integrating the Gibbs-Duhem relation
E E
N,duy + Nyduy =0

leads to about the same values as were measured.
The deviations from ideality of KCl are somewhat greater than those for FeCl,, and the appar-

ent values of both the excess entropies of mixing and partial molar heat of solution are positive.

Barton and Bloom? have measured the vapor pressures of PbCl,-KCl, CdCl,-KCl, and CdCl -

113

NaCl mixtures at 900°C by using boiling point and transport methods. At concentrations of

UNCLASSIFIED
ORNL-LR-DWG 67574

 

 

 

 

 

 

 

 

 

 

 

 

10,000
,'LFeCIa
8000 »
900°C
Hxel
3 6000 /o 4
£ @
g /
S ////
°
w 4000 L/
1
:
2000 //
0
0 0.1 0.2 0.3 0.4 0.5
e

Fig. 16. Excess Chemical Potentials of KCl and FeCl, in KCl-FeCl,

Mixtures as Obtained from Vapor-Pressure Measurements.

 
52

less than 60 mole % alkali halide they could neglect the volatilization of the alkali halides and
vapor compounds containing alkali halides. Their results are in fair agreement with the emf meas-
urements of Lantratov and Alabyshev®® on the PbCl,-KCl| system. They found that the apparent
deviations from ideal behavior in the CdCl,-KCl| system were smaller than in the CdCl,-NaCl sys-

tem. The measurements in the systems containing CdCl, are open to question.

1.6 Discussion of Binary Systems with a Common Anion

The results in the previous section exhibit certain very general features for mixtures of salts
of a monovalent alkali cation with salts of a polyvalent cation. The most obvious feature is the
variation of the thermodynamic properties with cation radius and charge. The deviations from ideal
solution behavior of both the alkali ion salt and the salts with polyvalent cations usually become
more negative (or less positive) with an increase in the radius of the alkali cation and with an in-
crease of the charge or decrease of the radius of the polyvalent cation. This type of behavior has
often been ascribed to *‘complex ion formation'’ or to ‘‘complexing.’'’*3+62:85:97 Thig terminology
has been used so freely and in so many different senses that some of the “‘explanations’’ of solu-
tion behavior in terms of "‘complexes’’ are merely redundancies of the observed facts and add noth-
ing to the understanding of solution behavior in terms of physical concepts. As a consequence, in
this section, a discussion and critique will be given of this concept,

Among the most reasonable and careful considerations of the concept of complex ions are those
of Flood and Urnes>> and Grjotheim.®? Flood and Urnes, for example, discuss the liquidus curves
of RbCl, KCI, and NaCl in mixtures with MgCl,. They reason that a mixture of an alkali halide
with an alkali salt of a large divalent anion will exhibit only small deviations from ideality. Evi-
dence for this comes from the apparently negligible deviations from ideality found in the liquidus
curves of Na,50, in mixtures with NaCl and with NaBr.?2 (Note the work of Cantor on the parallel
effect of cation radius.) Flood and Urnes propose that the component M,MgCi , containing the
MgCl42' grouping would exhibit small deviations from ideality based on the Temkin definition.

Thus at low concentrations of MgCl,,

n n — 2n
ci™ MCI MgCI2

Ao = = (11.6.1)
MC . e
n + 7 7 -~ n
cI- Mc142‘ MCI ™ TMgCl,

 

The procedure of Flood and Urnes is essentially a redefinition of components. They show that the
liquidus temperatures’® (and activities) for KCI and RbCl are in reasonable agreement with Eq.
{1). The liquidus temperatures (activities) of NaCl in NaCl-MgCl, mixtures exhibit positive devia-
tions from the calculations based on Eq. (1). This was ascribed to a partial dissociation of the
MgC|42" ion. Thus, by the redefinition of components, and by the careful choice of systems, rea-
sonable correlations with the data were obtained.

Although such a procedure has the advantage of being simple, there are many criticisms which

can be made, The major criticism, perhaps, is that this method can be applied to very few systems

 
53

and does not lead to quantitative predictions which can be made a priori. For example, the lig-
uidus temperatures of NaF in NaF-BeF , mixtures are too low to be described by any redefinition
of components which is consistent with possible structural concepts. Since the Be?” ions are so
small, a coordination of Be2" cannot be expected larger than four and yet a mixture of, for ex-
ample, NaF and Na_BeF, would have to be described as exhibiting negative deviations from ideal
solution behavior. On the other hand, no reasonable choice of a ‘‘complex ion'' grouping or com-
plex component can be invoked to explain the small deviations from ideality of NaF in NaF-BeF,
mixtures, Further, although the thermodynamic data may be described by choosing a particular
complex component, this does not necessarily imply the existence of the ions of this component in
the melt. Except for the very stable (relative to the separate ions) complex ions as NO,~, PO43",
and 5042', a simple comprehensive description of the solution behavior of mixtures with a common

anion cannot generally be made with only one *“complex ion'' and little can be learned about solu-

tion behavior a priori from such an approach.
The absence of a simple explanation of the solution behavior of molten-salt mixtures is evident

in NOF-ZI‘F4 mixtures shown in

from the analogy between a in HCI-H O mixtures and a

H,O ZrF
2
Fig. 17. In water the O-H interaction is very strong so that it is only slightly ionized. At low

UNCLASSIFIED
ORNL-LR-DWG 66350

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0
//,
4
7
0.8
7/
0.6 Y
7/
02 //
0.4 /
/
/
//
/ /
/ /
// NaF - ZrF, A
//
/
0
0 0.2 04 0.6 08 10

Na

Fig. 17. Plot of the Activity of H,0 in H,0-HCl and H,0-HBr Mixtures at
25°C and the Activity of ZrF, in ZrF -LiF, ZrF -NaF, and ZrF ;-RbF Mixtures
at 912°C.
54

, , : + - .
concentrations of HCI the solution may be understood in terms of solvated H and Cl~ ions inter-
acting in a dielectric medium, In mixtures with HBr the activity of water is even lower than with

o : , +
HCI. The limiting law at concentrations of HCI greater than the concentration of H™ from the self

ionization of water is

0 [aY]
,quo—,quo = RT In (]—QNHCI), (11.6.2)
since HC| behaves as two particles. lfy,,  is defined by
2
0
- =RT In N , (11.6.3)
FH,0 THH 0 H,0YH,0
then at low concentrations
1 - 2NHC!
- <. (11.6.4)

YH.0

Interionic interactions of H® and Cl™ will cause ¥y _o to differ at higher concentrations from the
value given by (11.6.4). Beyond the range of validity of the Debye-Hiickel theory, this is unpre-
dictable although there is a persistence of the negative deviations from ‘‘ideal’’ solution behavior.

In NaF-ZrF , mixtures, the solvent ZrF , may be considered to be more highly ionized than
water. Consequently, the self ionization of ZrF , and the one particle limiting law will probably
hold to higher concentrations than in water so that the deviations from ideality in dilute solutions
based on an equation such as (I1.6.3) will be smaller in the Nc:F-ZrF4 mixture than in HCI-HQO
mixtures. Apparently the smaller (less negative) deviations from ideal solution behavior in the
NaF-ZrF , mixture as compared to HCI-H,0 mixtures persist at high concentrations. Just as with
water, the farger the size of the *‘foreign” ion (CI~ and Br~ in H,0 and Li*, Na*, and Rb*in
ZrF4) the greater the deviations from ideality. This does not explain the observed solution be-
havior in NaF-ZrF , mixtures but merely suggests that any fundamental explanation in concen-
trated solutions is at least as difficult as in concentrated solutions in water where it is clear that
the H* and CI~ ions are solvated but no valid quantitative predictions can be made in terms of
structural concepts. Because of these apparent inadequacies of the concept of ‘‘complex ions'’ in
describing solution behavior in mixtures containing one type of anion it is in order to discuss and
attempt to classify some of the effects and interactions which have been included in the terms
““‘complex ion'’ or '‘complexing’’ in the hope that such a procedure would be more instructive and
useful in future attempts at deriving quantitative theories. Most definitions of ‘‘complex ions'’ or
of “‘complexing'’ fall into two categories.

In the first category a '‘complex ion’’ is usually conceived as a microscopic structural entity.
A complex ion can be most clearly defined as a grouping of at least one central cation and near-
neighbor anions having a particular configuration. |f each grouping is isolated from others and
shares no anions, then the grouping is a finite complex. NO, ™, P043_, and 5042_ ions are finite

complexes, If the groupings are all interconnected by shared anions, then infinite complexes are

 
 

55

present. By these definitions all pure salts are infinite three-dimensional complexes and very di-
lute solutions of one salt in another always contain finite complex ions although the configura-
tions of all the finite complexes are not necessarily uniform. X-ray, uitra-viclet, infra-red, Raman
spectra, and other methods of investigating structure are means for investigating these complex
ions.

“‘complex ions’’ or ““complexing’’ are used to describe a tend-

In the second category the terms
ency to stabilization. This is the least satisfactory use of this terminoclogy, since so many differ-
ent interactions and concepts are included in this usage that less information is conveyed than by

the use of the word stabilization.

For pure materials as for example AgCl, NiF, or HgCl,, specific interactions (van der Waals’
ligand field and covalent binding) give rise to more negative values of the energy of formation from

the isofated ions than might be expected for alkali or alkaline earth halides where Coulomb inter-

actions are relatively more important. |n solutions the tendency to '‘complexing’’ or toward the
stabilization of a component in solution is characterized by negative values of uE. Some of the

solution effects which influence the values of p& are:

(a) Coulomb effect. The discussion in section Il shows that Coulomb interactions in mixtures
of salts containing monovalent ions lead to negative values of u. This effect appears to be pres-
ent in mixtures containing polyvalent cations, Long-range interactions are very significant in this
effect and as a consequence a quantitative description of this effect in terms of finite complexes

can only be fortuitously correct.

(b) Polarization effect. The field intensity at an ion position will, in general, not be zero be-
cause of ionic motions and because of the difterent sizes and charges of cations. For example, an
anion having two cations the same size but of different charge as near neighbors will tend to have
a Coulomb field intensity on it. As a consequence, the electrons on the anion and the thermal mo-
tions of the anion will be "polarized’’ so that the negative charges reside a greater fraction of the
time near the cation with the higher charge. In a pure molten salt this effect will be expected to
be smaller than in a mixture, and the net contribution will lead to a relative stabilization of the

mixture (negative contribution to the deviations from ideal solution behavior).

(c) van der Waals' interactions. As in mixtures containing monovalent cations these interac-
tions usually will lead to a positive contribution to the deviations from ideal solution behavior for
systems containing polarizable cations. To illustrate with a clear-cut example, the systems NaCl-
PbCl, and AgCI-PbCl, might be compared. In the former the measured deviations from ideal solu-
tion behavior of PbCl, are negative (#Ebcl < 0), and from the Gibbs-Duhem equation it can be
shown that P’EoCl is also negative, Measurzements in the latter system] 16 indicate that p'fgCl (and
“EbCI ) is essentially zero at all concentrations. The major differences between these two sys-
tems are probably related to the high polarizability of the Ag’ ion as compared to Na* and hence
to the contribution to van der Waals’ interactions. Quantitative estimates of the magnitude of this

effect in such systems are tenuous at present.
 

56

(d) Ligand field effects.'®3 These interactions will tend to stabilize pure salts of transition
metal ions and particular configurations of near-neighbor anions will tend to be more probable.
Such stabilization, regardless of the specific symmetry of the near-neighbor anions, will tend to be
monotonic with the strength of the negative ligand field. For given anions as near neighbors to a
particular transition metal ion in a mixture, the negative ligand field will be attenuated by more
distant cations with the attenuation tending to be smaller the smaller the charge and the larger the
radius of these other cations. The dissolution of a transition metal salt, NiF, for example, in an
alkali fluoride would lead to a replacement of next-nearest neighbor Ni2* by monovalent alkali cat-
ions. This will lead to a stabilization of NiF, (,ufiiF < 0), which would be more pronounced the
larger the alkali cation. The influence of ligand-field interactions will be limited by steric require-
ments and in mixtures with alkali metal salts will probably lead to negative contributions to the
deviations from ideal solution behavior of both components,

(e) Packing and steric effect. To satisfy the tendency toward local electroneutrality it is
probable that small highly charged cations will tend to have a larger number of anions as near
neighbors than cations of low charge. Any energy changes (stabilization) related to this effect

will be sterically limited in accordance with the values of the anion-cation radius ratios.

All of the factors mentioned are included in the concept of ‘‘complex ion"' of or “‘complexing’’
when it is applied to stabilization, Some of these effects may be concomitant with a foreshorten-
ing of cation-anion distances (e.g., coulomb, polarization, and/or ligand field) or with a tendency
toward specific configurations of anions about cations (e.g., ligand field and/or packing). In all
cases, these factors influence the free energy differences between pure salts and salts in solution,
It may be preferable to refer to the observation of negative values of uF as a stabilization, since
such a stabilization is not necessarily related to the observation of a ‘‘complex ion'’ as a struc-
tural entity. By this usage, no unwarranted implications about the structure of the melt need to be
made,

The existence of solid or gaseous compounds which are made from the two salts in a solution
cannot be used as evidence that particular ““complex ions'’ are formed in solution. Although many
of the factors and interactions which lead to relatively greater stability of gaseous and solid com-
pounds may also give rise to negative deviations from ideal solution behavior, many of the factors
influencing the structure of solids or gases have no counterpart in liquids. For example, in solids
steric repulsions of the ions are more important than in liquids and have a strong influence on
structure; and in gases the entropies of association are generally negative and give rise to a
strong influence in favor of forming the simplest compounds. Kinetic definitions of ‘‘complex
ions’’ in terms of the lifetime of a grouping or of the relative mobility of ions2? cannot be clearly

related to equilibrium thermodynamic properties or to ‘‘complex ions’’

as a structural entity unless
these lifetimes are very long.
. . , : E : ,
Since there is no adequate theory for most binary mixtures, u~, for any component in a given
mixture containing polyvalent cations, must be estimated empirically by comparison with known

systems containing mixtures of the same charge type and the same anion. Keeping in mind the
57

types of interactions which influence the values of ,uE, reasonable estimates may be made by anal-
ogy with known systems or by interpolation., The development of a theory, as, for example, by the
extension of the perturbation theory of conformal ionic mixtures to mixtures containing cations of
different valence, would be an aid in such estimations and might be used to confirm empirical re-

lations such as was proposed by Kleppa.

I1.7 Other Systems

Measurements of the activities of lead halides in the mixtures F’bC|2-ZnC|2 (ref 131) and
PbBr-ZnBr,, (ref 117} indicated small negative deviations from ideal solution behavior in the for-
mer and small positive deviations from ideality in the latter. (Calculations of the activities of
ZnCl, in the first system by use of the Gibbs-Duhem relation were in reasonable agreement with

]26) I

activities calculated from measurements of the partial pressures of Zn(.:l2 in this system, n

these two systems there is no difference between the mole fraction of a component (NPbX ) and
the product of the ion fractions (NPbNi =Np, =Np,x_) and there is little ambiguity in defining
2
activity coefficients, On the other hand, in a system as F’bCIz-F’bBr2 there is some c:mbigui‘ty,28
: 2 y2 N2
since Np /Ny =Ny = prxz.
cients depends on the type of compounds, |f the lead halides were very stable molecular com-

In such systems, consequently, the definition of activity coeffi-

pounds and did not react with each other (were not molten salts), then the activity coefficient

PbX, ™ NPbx2}’Pbx2°
lecular, and where one might consider the exchange PbCl, + PbBr, == 2PbCIBr), in order to be

would be defined by « For ionizing salts {or where the compounds are mo-
consistent with the limiting laws, the activity coefficient is better defined by a, , =
NowNX¥Yppx_ * On this latter (and more realistic) basis, the activity coefficients, Yppge,. N
F’bC|2-F’|:>Br2 mixtures, are larger than unity. V17 The choice is not always clear-cut as many com-
pounds cannot be strictly classified as either molecular or ionic salts,

Very few other measurements on binary systems that have a common cation have been made.

Precise measurements by Toguri, Flood, and Fgrland®® on the exchange equilibria

Cl, + 2MBr = 2MCl + Br, (1.7.A)

in LiCl-LiBr, NaCl-NaBr, and KCI-KBr mixtures were used to investigate the activity coefficients

of the alkali halides in these mixtures. The equilibrium constant for (A) is

 

2 2 2
NMCE}’MCllblzs.r2 YMcl
K = _ K’ , (11.7.1)
M N2 2 M 2
MBrYMBr cl, YMBr

where K;A is the measured equilibrium quotient, Taking the logarithm of K|, and using as a first

approximation for the activity coefficients in any one binary system

2 a2
RT Iny,c =ANyg, 9nd RT Iny,g, =AN{cy,

 
58

then
RT In K’ = RT In K + 2ANZ - N2 ). (1.7.2)
Plots of RT In K* vs (N2 —~ N2 ) led to the values of A in Table 10, which indicate small posi-

MBr MCl
tive deviations from ideality. For these relatively large and polarizable anions, packing or van der

Waals' effects have been proposed as possible contributing factors. To contrast this, an analysis
prop p g

4 with the data in Table 1 indicates small nega-

of the liquidus temperatures of LiF-LiCl mixtures
tive deviations from ideal solution behavior for both components. Since the F~ ion is smaller and
less polarizable than Br~™, it would seem that at least one of these two properties of the ions is

significant,

Table 10. Values of A (cal) from Equilibrium Measurements

in Binary Systems with a Common Cation (M)

 

 

 

Li Na K
MBr-MClI 150 350 530
M2Cr 207.-M 2Cr04 0 ~300 ~ 500
Similar measurements of the equilibrium
M,Cr,0,=M,Cr0, + %Cr,0(solid) + %02 (11.7.B)

. . . 4+ o+ +
in molten mixtures of chromates and dichromates have been made?? for M = Li ,Na, K, or TI".

The equilibrium constant, if CrO42_ = X%~ and Cr2072' = Y2-, is given by

3/4
Nx(?’oz) Y, X Y, X
Ky = ———— - K}, ; (11.7.3)
N Y Y
Y M,Y M,Y

 

 

by using the approximation that RT In YMox = ANAZA yand RT Iny, = ’\N:\ x the values of A
M, X M2Y
10 and are seen to be small, When M* was an alkali ion the stability of M,Cr,0O, relative to

could be obtained from the slope of a plot of In K!:A vs (N ). These are given in Table
M,CrO, increased as the size of the M?* cation increased and consequently the equilibrium con-
stant K, (and the equilibrium quotient K‘;) for reaction (B) decreased with an increase of the size
of M*. This is also true for the equilibrium in reaction (A). These facts are useful for anticipat-
ing some of the properties of molten reciprocal salt systems discussed in section lll. For example,
consider the equilibrium (B) in a mixture of N02Cr207 and N02Cr04. The equilibrium constant is
given by

~RTInK., =F®° + 1 Fd - F° (11.7.4)
Na N02Cr04 2 Cr2C):3 Nu2Cr207

 
39

and is, of course, dependent only on the properties of the pure reactants. If the composition of the
mixture is altered so that the Na* ion is gradually repiaced by K * ion, the equilibrium (B) will
gradually go more to the left and K will decrease. When very little Na®ion is left and the melt
is essentially a mixture of K,Cr,0, and K,CrO,, the value of K will be equal to the value of

Kl‘( in the mixture containing only the K cation. For this case one obtains from Eq. (3)

In K =InKg=InK  ~In (yNuzx/yNa2Y)

=In K, —In (szx/szY), (1.7.5)

where the activity coefficients are all in a solution containing mainly K* ions and very little Na*

ion. Introducing Eq. (4), one obtains

In (yNazx/yNazY) = In (K /K) + In (szx/)’sz)
Ayo
- + In (szx/szY) , (11.7.6)

 

where A#O is the free energy change for the reaction of the liquids in (C)
N<:2Cr207 + K2Cr04\‘: K2Cr207 + Na,CrO, (11.7.C)

and the last term in Eq. (6), In (yK x/yK Y), can be seen to be small in this case from the data in
Table 10. The value of Au® is negative and the ratio of the activity coefficients of the components
Na,CrO, and Na,Cr, O, is much greater than unity, and in simple cases such as this, Na,CrO, ex-
hibits positive deviations from ideal solution behavior and Na,Cr,0, exhibits negative deviations
from ideal solution behavior. Thus N02Cr04, which is a member of the stable pair in reaction (C),
exhibits positive deviations from ideality and Na,Cr,O,, which is not a member of the stable pair,
exhibits negative deviations from ideal solution behavior. This tendency is present in all recip-
rocal systems. Flood and Maun4? have measured In K”as a function of the ion fraction of Na* in
mixtures of Na*, K, Cr042', and Cr2072' ions. A plot of In K’ vs Ny, given in Fig. 18 can be

seen to be nearly linear in the cation fraction. The data fitted the equation

In KNa,K=NNaIn Kiat Ny In KK+bNNaNK’ (11.7.7)

where b is —0.2 at 662°C. Similar measurements in the TI7, K+, Cr042", Cr2072' system are
plotted in Fig. 18. The quantity b is discussed by Flood and Maun, is related to the proper-
ties of binary mixtures made up from the four ions in the system,??and is probably small when all
the binary systems have small deviations from ideal solution behavior. These properties of recip-
rocal systems have been used in an ingenious derivation of a zeroth order theory of these sys-
tems.®3+%4 A more complete description of reciprocal systems is given in section |ll.

Since linear relations are often useful from a practical point of view, two linear relations which
apply to ternary systems having a common anion will be stated.>® These apply to ternary systems
in which the solution properties of two of the components (components 1 and 2) do not differ greatly,

mixtures of these two components do not exhibit large deviations from ideal solution behavior,
60

 

 

 

 

 

 

 

 

 

 

 

 

UNCLASSIFIED
ORNL-LR-DWG 67570
0.5
0
<X
< -05
o
o
-1.0
C
-1.5
0 0.20 ©.40 C.60 0.80 1.00
Nna OR V14
Fig. 18. Measured Values of the Equilibrium Quotient for the Reaction
(11.7.B).

and the properties of these two components differ significantly from the third component (component
3).

In these cases, some of the properties of the ternary mixtures may be estimated from the prop-
erties of binary mixtures composed of the three components.

To illustrate, the measurements of Férland37+98 on the partial pressures of CO, in equilibrium

with CaCO, which is component 3 in mixtures of Na,CO,, K,CO,, and CaCO 4 will be used. The
CO, is in equilibrium with CaCO,.

CoCOJsqution)# CaO(solid) + COz(gas)_

The components Na,CO, and K,CO, have negligible partial pressures of CO,, and the partial pres-

sure is proportional to the activity of CaCOy, in solution. Férland derived the relation having a

form similar to the equation below which at a constant mole fraction of component 3 is

N
1
Iny, (in ternary) =——_In (in binary 1-3 mixture)
V3 N+ N, & Y
N2

— —JlIn {in binary 2-3 mixture) ~ N, N_»", (I1.7.8)
N, +N, V3 Y 172
61

where »” is related to properties of binary mixtures of 1 and 2 and the last term in (8) is negligible
when the properties of 1 and 2 differ little. Although Fgrland derived this equation (in somewhat
different form) under the restrictive assumption of regular solutions, the modified result {Eq. 8) is
probably much more general. The form used here has been changed to avoid the ambiguity associ-
ated with the concentration scale to be used.

The measurements of p ., in mixtures of CaCO, with Na,CO,, with K,CO, and with an equi-

molar mixture (Nc:,K)CO:3 at a constant mole fraction of C(JCO3 was consistent with the equclfion58
NNu
In pcoz[(Na,K,Ca)CO3] =—-———NNO T In pcoz[(Na,Ca)CO3]
NK
+ _NNO — In pcoz[(K,Co)CO3] , (11.7.9)

which is consistent with Eq. {8). A similar relation for component 1 at constant mole fraction of

component 3 is
In Yy (in ternary) = In Y1 (in binary 1-3 mixture) + b"'N2 + b'"N% , (11.7.10)

where »°”’ is probably small when components 1 and 2 do not differ greatly in their properties. To
illustrate this Christian,3* in unpublished work on the partial pressure and activities of ZrF, at

912°C in mixtures of ZrF ,, UF ,, and NaF containing 54 mole % of NaF, demonstrated that the

measurements fitted the equation

In erF4=|n 0.049+4.12NUF4. (11.7.11)

Further tests in other systems of such linear relations would be of interest,

RECIPROCAL SYSTEMS
111.1 General

Reciprocal systems are mixtures of salts containing at least two cations and two anions. The
nature of this class of systems lends itself to theoretical treatment. Perhaps the most obvious,
and most naive treatment of this class of systems considers the reciprocal chemical reactions. For

, . : oy oy , .
example, in the simplest type of reciprocal system A*, B' X7, Y7, the reciprocal reaction

Ay{lia) . gx(lia) == px(lia) _ gy(lia) (IN.1.A)

is considered, [t is easily shown that for the system to conform to Temkin's definition of ideality
it is necessary that the free energy change (AG% or Ay?\) and the enthalpy change (AHOA) for the
equilibrium (A) be zero and that all of the binary systems AX-BX, AX-AY, AY-BY, and BX-BY
form ideal solutions. Except for isotopic mixtures all of these conditions are probably not

realized in any real systems.
62

|f A,u% for reaction (A) is negative, then there will be a tendency for the components AX and
BY to exhibit pasitive deviations from ideal solution behavior, and if A,ug is positive there will be
a tendency for AX and BY to exhibit negative deviations from ideal solution behavior., That these
are tendencies and not necessary consequences can be illustrated in a simple manner. The excess
free energy (or enthalpy) of dissolution of a small amount of AY to form an ‘‘infinitely’’ dilute so-
lution in BX can be calculated from the sum of A,u2 (or AH%) with the sum of the excess chemical

potentials (or enthalpies) for the processes

AX(“q) —_ Ax(dilute solution) A-"LB , (111.1.B)

BY(qu) s BY(dilufe solution) A-"LC , (|||]C)

Thus, in dilute enough solutions, the excess chemical potential of solution of AY is A,uOA + Apg +
Ap and hence the sign and magnitude do not depend on Apg alone. There is a rough correla-
tion between A,ug (or AH%)* and the deviations from ideal solution behavior and the types of be-
havior which are exhibited in solid-liquid phase equilibria.>

In the previous section Ay and Ap have been discussed. The term Ap% (or AH%) is related
to a variety of types of interactions. When values of A,qu cannot be obtained from tables, it is
sometimes useful to be cognizant of one of the major influences on Apg, that of coulomb interac-
tions.

For the alkali halides, for example, the largest contribution to Ap% is the Madelung term

. 2( 11 1 )
- Ae + - - ,
dAX dBY dAY dBX

 

where dil' =TT where 7. is a cation radius and r is an anion radius. |t can be shown that if

T <rgandr, > Tyr OF T, >7p and r, < ry then the Madelung term is positive. This tendency
leads to the general reciprocal Coulomb effect which is valid for all the alkali halides. This ef-
fect is such that in a reciprocal system with two cations and two anions the two stable components
(stable pair) as evidenced by A,ui are the small cation-small anion component and the large cation-
large anion component. These two components** would tend to exhibit positive deviations from
ideal solution behavior and the other two negative deviations. From a consideration of the Made-
lung term one would expect positive deviations from ideal solution behavior for the stable pair to
increase in the order [NaF-KCI] < [LiF-KCI] < [LiF-CsCH < [LiF-CsBr]. The last two systems ex-
hibit such large deviations from ideal solution behavior as to have liquid-liquid miscibility gaps
which have been observed.® The reciprocal Coulomb effect probably applies for salts of different

valence containing nonpolarizable ions and is in such a direction that in a given system the salts

 

*The criteria which are discussed and used by Bergman and associates are the values of AG(A'u) of the

solids at room temperature which in view of their crude correlations are equivalent to Ap_o or AHO.

**All four components are not indepsendent of each other and only three of the four are true components
in the Gibbs sense,

 
63

with the smallest or most highly charged cations and smallest or most highly charged anions will
tend to be a member of the stable pair.

Obviously, the Coulomb effect is not the only important one and many deviations from the gen-
eralization are to be found, especially for systems containing polarizable ions. For example the

reaction
AgNO ; + NaCl == AgCl + NaNO,

has a large negative value of Au® or AH? (about —15 to =17 keal/mole) which is considerably more
than the Coulomb effect and which is probably the result of the large stabilization of AgCl by van
der Waals' (London dispersion) interactions '®! of Ag” and CI~,

By contrast with the binary systems discussed in section |l, the interactions are, in general,
much larger in reciprocal systems as they are mostly between nearest-neighbor cations and anions
rather than next-nearest neighbors and consequently one would expect to find many reciprocal sys-
tems with very large deviations from ideal behavior. In the following chapter some of the theories
will be discussed which have been advanced for these systems beginning with the simplest approx-

imation and continuing with approximations of increasing complexity.

{I.2 The Random Mixing Nearest-Neighbor Approximation®*

This derivation is based on the Temkin quasi-lattice model. For the simplest member of this
class of systems, that containing the two cations A* and B* and the two anions X~ and Y™, the
model is an assembly of charges in vacuo and consists of two interlocking sublattices, one a lat-
tice of the cations A* and B* and the other of the anions X~ and Y™, The nearest neighbors of the
cations are anions and of the anions are cations.

The total entropy of mixing is AST/R =-2n InN, - En]. In N, and for any component is Eij -
S:.)]. =~R In NN, where 7 and j are cations and anions respectively, All of the ions have the same
coordination number Z. The model is restricted so that all of the ions of the same charge are the
same size. This restriction eliminates any difference in the long-range Coulombic interactions be-
tween either A” or B* ions or X or Y ions and their respective environments, and limits the
model to short range extra-Coulombic effects which are assumed to be nearest-neighbor interac-
tions.* The form of the equations derived will probably apply even to systems with different-size
ions.

In Fig. 19 is a two-dimensional representation of the quasi-lattice. If the pair interaction
energy of AT.Y = is €, of B*.X~, €,, of AtX-, €5, and of B*.Y-, €4 then

A€=€4+€3-€2-El-—--fi-—, (111.2.7)

 

*|f random mixing is assumed, or for a dilute solution only one pair need be the same size to eliminate
differences in the Coulombic interactions.
64

UNGLASSIFIED
ORNL-LR-DWG 31157A

 

 

 

 

 

 

 

 

 

 

gt ry— Bt r- gt r- gt r-
y= |4t @ B* + y- gt @ Bt
gt y- Bt y- Bt y- Bt r-
. . )
[ *A* —
gt ry- Bt y- Bt y- gt r-
y~ |4t @ gt + y~ 8t @ g*
Bt r=— BY r- gt y- Bt r-

Fig. 19. Two Dimensional Quasi-Lattice Representation of the Process
A++ X~ &= AX in the Solvent BY,

where A¢€ is the energy change for the interchange of the circled X~ and Y~ ions and is the energy
of formation of the ion pair A*-X=. If there is random mixing of the cations and of the anions on
their respective sublattices, then the fraction of positions adjacent to any given cation occupied
by a given anion will be equal to the ion fraction of that anion. The assumption is made that the
relative energy of each A*-X~ pair is Ae. This is equivalent to the assumption of the noninter-
ference of pair bonds or to the assumption of the additivity of bond energies. Since the total num-
ber of positions adjacent to any ion is equal to n.7 or E].Z, then the total energy or enthalpy of the

solution is

E :ZAZNY(E + K) +ZB ZNX(€2 + K) +5A2Nx(€3 + K) +EBZNY(64 +K)=H

; ] (111.2.2)

T f

where ZK is the value of the energy of interactions of the A* or the B* ions with ions beyond the

nearest-neighbor anions. The partial molar enthalpy or energy of solution is

0 _ & 0 /
HI.]. - Hz.]. = Ez.]. - Ez.]. =2 (1=~} - N].) ZAE (111.2.3)

where the — sign is pertinent when ij is AX or BY, and the + sign is pertinent if it is AY or BX.
Remembering that u.. = H.. - TS, then
17 ty 1y

i =gy =t (1= N)1=N) ZAE + RT In NN, (1.2.4)
and

RT In Yii = i(]—Nl.)(]—N].) ZAE . (111.2.5)

 
65

The derivation of Eq. (5) is implied by the work of Flood, Fgrland, and Grjotheim who have, how-

ever, emphasized a somewhat more general relation. Equation (5) is strictly valid only for cases

in which AE is small relative to RT so that one might reasonably be close to random mixing of the

ions. The form of Eq. (5) is probably valid in some cases where there is only a small deviation

from random mixing and is instructive and important for the qualitative understanding of solution

behavior, Flood, Fgrland, and Griotheim propose a method for making a crude estimate for ZAE

from the heat contents of the pure components. Figure 20 is a two-dimensional quasi-lattice repre-

sentation of the metathetical reaction (l11.1.A) for which the heat change is AH® per mole.* Since

all ions of the same charge are the same size, only extra-Coulombic nearest-neighbor interactions

are changed in this reaction, Since the number of nearest neighbors for each of the salts is 7l per

mole of salt, then for the reaction AH® = ZAE, if each pair interaction energy were the same. In

real systems the pair interaction energies are probably a function of the number and kinds of anions

which are nearest neighbors to a given cation so that AE will not be truly constant and will only

be roughly approximated by (AH%/7).

A relation analogous to (5), but somewhat more general, has been derived by Flood, Férland,

and Grjotheim>*

where Ayo is the change of chemical potentials for the metathetical reaction (I11.1,A).

RT Iny, = +(1=N)(1-N) Aul,

 

o ~
*For many reciprocal salt pairs probably ASO = ( so that AHO = A‘u .

ORNL-LR-DWG 31158A

0

UNCLASSIFIED

 

 

Aty a4t y- Bt x- Bt x~
y™ a4t r- a4t + x~ g% x gt
Aty oAt yo gt x- gt x~
r *’j \

At x- AY x° gt vy~ Bt r-
x~ At x4t + x~ BY v~ BT
At xT At XxT gt vy~ B8t y-

Fig. 20. Two Dimensional Quasi-Lattice Representation of the Metatheti-

cal Reaction AY(liq) + BX(lig) &= AX(lig) + BY(liq).

(111.2.6)
66

If the deviations from ideality are large enough, then the solution will tend to separate into two
liquid layers. Since the theory is symmetrical in composition, the upper consolute temperature,
T, below which temperature two liquid phases will form, will be at a composition such that N, =
Ny=Ng=N,= ]/2 It may be calculated from Eq. (5) or (6) by setting the derivative da, /dN ,
equal to zero in mixtures of AY and BX, where Na=Ny=N,y=Ng=N, and Npy is the mole
fraction of AY in a mixture made up from the salts AY and BX. The expression for the upper con-

solute temperature derived from Eq. (5) is

ZAE ., AHO
- = (1.2.7)
€ 4R 4R
and from Eq. (6)
A 0
_2r (111.2.8)
€ 4R

To illustrate Eqs. (5), (6), (7), and (8) let us consider the dissolution of a mole of liquid AgCl
in NaNO,, where the ions Ag+, No+, NOa—, and Cl~ correspond to AT, B X", and Y~ respectively.
From published data®?:78:114 51 the pure salts, Au® = +17 keal and AH® = +15 keal at 455°C. It
can be seen from Egs. (I11.1.5) and (l11,1.6) that the components AgCl and NaNO .,
bers of the stable pair, should exhibit positive deviations and AgNO ; and NaCl should exhibit

which are mem-

negative deviations from the Temkin ideal-solution behavior. The results are similar for the sys-
tem Ag ' I(+, NO3-, Cl™. In both these systems the calculated upper consolute temperature is well
above the melting point of all the possible components that can make up the system, and two im-
miscible layers are present in this system. However the measured upper consolute temperature is
much lower than that calculated from Eqs. (7) and (8). Similarly in the system Li*, K, CI~, F~,
where the stable pair is LiF-KCI, the values of A,uo and A% at 1000°K are about +17 keal;3159.78
yet two liquid layers have not been detected in the quasi-binary system LiF-KCl,>! although the
calculated consolute temperature is very much higher than the measured liquidus temperatures.
Clearly Au® and Al are not the sole measure of the deviations from ideality in reciprocal molten-
salt systems, In mixtures for a given class of salts, such as alkali halides, they probably serve
as a guide to the relative deviations from ideality. For example, the positive deviations from ide-
ality in LiF-KCI quasi-binary mixtures are greater than for the NaF-KCI mixtures. The values of

51 An analysis of the quasi-binary

Ap® for these two systems are +17 and +8 kcal respectively.
liquidus temperatures for LiF-KCl and NaF-KC! in which the stated components exhibit positive
deviation from ideal behavior and of the liquidus temperatures for LiCI-KF and NaCl|-KF mixtures
in which the stated components exhibit negative deviations from ideal behavior has shown that Eq.
(5) or (6) only describes the solid-liquid equilibria in a semiquantitative manner.3' The short-
comings of these two equations stem from a variety of possible reasons. Fgrland®® has discussed
the influence of those interactions which reciprocal systems have in common with binary systems

containing either two cations and one anion or two anions and one cation. As discussed in section

Il, these interactions are of longer range than nearest-neighbor interactions. Fgrland has discussed
67

this possibility for the hypothetical case in which this effect can be described in terms of the equa-
tions of regular solutions. From the derived relations it can be shown that if the binary systems ex-
hibit negative deviations from ideality, then the correction terms to Egs. (5) and (6) are in a direc-
tion which makes the activity coefficients smaller and which lowers the calculated upper consolute
temperature. Although this correction is in the right direction, it is not large enough to lead to a
good correspondence of calculations with experiment. As discussed in the following sections it
will be shown that two other important effects which have been experimentally demonstrated are
present. One effect is related to the nonrandom mixing of the ions which, except for extremely
small deviations from ideality, leads to magnitudes and a concentration dependence of the devia-
tions from ideality which are very different from Eqgs. (5) and (6). The second effect is the non-

additivity of pair bond interactions.

I11.3 Corrections for Nonrandom Mixing: The Symmetric Approximation

For the case in which AE is not very small relative to RT, corrections for nonrandom mixing
of the ions must be included. Flood, Fefland, and Grjotheim have given a preliminary discussion
of nonrandom mixing.>* Explicit calculations based on the nearest-neighbor quasi-lattice model

14,18 1 1d Blander and Braunstein. ' 2

have been made by Blander,
In the following sections approximations based on the quasi-lattice model will be used to cal-

culate the effect of nonrandom mixing (or associations) on the calculated thermodynamic proper-

ties of the model system., These calculations will also be related to conventional association con-

stants for associations of the A" and X ions to form ““complex ions”’
+ - Hm~-
mAY & aX :_AAm Xn {m=n)

and will be used to illustrate some of the properties of these constants. It should be noted that
some of the relations derived may also be derived without the use of a quasi-lattice model.'? The
model is useful in defining the parameter Z and in the statistical counting in the theoretical calcu-
lations.

In dilute solutions of A* and X ions in BY most of the associated species {or ‘‘complex
ions'’) Aanflm"") are isolated from one another by solvent B* and Y~ ions and are easily defin-
able. This is in sharp contrast with solutions having only one kind of anion where complex com-
pounds are not easily defined since all cations will have the same anions as near neighbors re-

gardless of the properties of the solution.

12 63

The symmetric approximation'“ is essentially the quasi-chemical theory of Guggenheim.®® |n
this approximation as in the others in section ||| only nearest-neighbor interactions are taken into
account, The assumption is made that the interaction of any given adjacent pair of ions is the
same independent of the local environment. A given A* ion may interact as many as Z X™ ions and
a given X~ ion may interact with as many as Z A* jons with the relative energy of each interaction

being Ae. The total number of the pairs A*-X", B+-X-, A*Y", and B*-Y is Z(n, + nB)n. If v’

 
68

is defined as the fraction of positions adjacent to the A* ions that are occupied by X~ ions, then

the number of pairs of each kind and the total energy of such pairs are given below:

Type of Pair Totol Number Total Energy
Aty Zn (1 =Y M =R ~5’ 0
B x~ Zlne~n, ¥ WM =5] 0
Atx” Zn, YN =5’ Zn, Y'AE
By~ Zng—nc+n, ¥ M =R, -5/ 0

For simplicity the relative energies of the pairs other than A*-X~ are arbitrarily set at zero. This
makes no difference in the final results. R’ and R; are the number of positions adjacent to all the
A" and B” ions respectively; S. is the number of positions adjacent to the A% ions occupied by X~

ions; and Sl: is the number of positions adjacent to the B* jons occupied by X ions. The number

 

of ways of distributing these pairs, w_, is
(R; + R))!
@ = TR AT (11.3.1)
(R, — SNSI(R, = S)LS]
6

As in the quasi-chemical approximation,®3 when w, is summed over all possible values of Y7 the

value for the total number of configurations is incorrect. A normalizing factor can be calculated to

correct this so that the combinatory formula is

o sTURE = sDIUST IR - ST, + 11 [y + m )AL 3
T SIUR =SOSR, = SN T (g WL (2, )Y (2 )1 (13-

”
a a a b

 

where the superscript dagger (T) on a symbol signifies the value of that quantity for a random dis-
tribution of ions so that Y1 = Ny-
The most probable distribution is obtained by maximizing QS' under the condition of constant
total energy and constant number of ions involved and is given by
Y NX -~ NA Y

_ , 111.3.3
1-Y ]-—NA-NX+NAY B ( )

 

where 8 = exp (~AE/RT), and where the absence of a prime (*) on Y (or QS) signifies the value of
that quantity in the most probable distribution. The total energy is

—AE.=Zn, YAE ==AH_., (111.3.4)

and the total entropy of mixing is

ASp=kInQ_ . (111.3.5)

 
69

The total Helmholtz free energy can be calculated from Egs. (4) and (5). The following equation is

obtained for the partial molar free energy by differentiating the Helmholtz free energy

 

ST Rt inn, Ny () (111.3.6)
-— - n ! oty
Fay ~Hay ATY \7C Ny
where ,ui'l; is the chemical potential of AY in its standard state* and

zZ

. 1-Y
YAY {or yAY) = (W) . (111.3.7)

Because of the symmetry of the problem, Eq. (7) is valid for all of the components by merely rede-
fining Y and AE.

In this approximation (as well as the random mixing approximation) the assumption of the non-
interference or additivity of pair interactions has been made so that the energy of attachment of an
A* or an X~ ion to any X~ or A* ion respectively is always A€ independent of the number of other
ions attached to the A* or X~ ions taking part in the attachment. Thus the energy change for the

process

X Hm+1wn)

Am Xn+(m-—n) . A+.\.fiAm+] )

and for

AL X T X T = X )

are the same and are independent of the values of m or n. As will be discussed later this places
restrictions on the relative values of the successive association constants, The A* and X~ ions
associate if A6 <0 and Y > N, and they will be solvated by the B*and Y™ ions if Ae> 0 and

Y <N,. When Ac =0, Y = N, and the mixture obeys Temkin's definition of ideality.

l{1.4 Comparison of the Symmetric Approximation with the Random Mixing Approximation

Calculations from Eqgs. (111.3.3) and (111.3.7) probably lead to a more realistic description of
reciprocal systems than calculations from (I11.2.5) and (I11.2.6). For a mixture AY-BD, the upper

consolute temperature, T_, can be calculated from the condition

(da p y/dN yy) = (AN, Ny y 5 y/dN 5 y) =0

 

*
*Note that in the model ,ui'l; = #?\Y =y if the sol*vent has an ion in common with AY, The most con-
venient standard state to use if ;i is not the same as 1 in a real system depends on the concentration of
¥ . :
solutes. For example in a solution dilute in A "or Y , Hay is convenient and in the solvent AY, ,uiY is

convenient,

 
70

The problem is simplified because of the symmetry of the model so that the upper consolute tem-

perature falls at N, =Ny =N, =N, = ]/2 The solution is

 

Y
ZAE Z-2 4 16 8
—=-2ZIn =4 +=+ b=yt (11.4.1)
RT, 7 zZ 3z Z

For very large values of Z, Eq. (1) approximates Eq. (111.2.7).

z ZAE/RTC
4 5.5
5 5.1
6 4.9
o0 4.0

As can be seen from the above table, for a given value of ZAE, nonrandom mixing gives rise in
this case to a lower calculated consolute temperature than is calculated under the assumption of
random mixing. In Table 11 it can be seen that the consolute temperatures calculated from the
symmetric approximation are less unreasonable than those from the random mixing approximation
using the same parameters in the calculation. The parameters, A,uo, are those given by Flood,

Fykse, and Urnes.>!

It has been assumed that Auo = ZAE, and a reasonable value of Z = 4 has
been used in the calculations,
A calculation of Yay from the two approximations is also given in Table 11 along with values

measured at the liquidus temperature at 50 mole %. The symmetric approximation (again for Z = 4)

Table 11. Calculated and Measured Parameters at 50 Mole %

 

AY-BX LiF-kCI®! LiF-NaC1%4 NaF-KCI®!

 

Ai® (£ ZAE) (keal/mole) 17.1 9.1 8.0

Random mixing approximation
(TC (°K) 2150 1140 1310

Yay 7.8 3.2 2.7

Symmetric approximation (Z = 4)

T_(°K) 1560 830 730
Yay 4.7 2.8 2.4
Measured temperature (liquidus) (°K) 1045 973 1010
Y oy (from measurements)® 3.2 2.6 1.8

 

ayAY at liquidus temperature where AY is the alkali fluoride.

 
71

leads to values of the activity coefficients of LiF and NaF, which are much closer to those de-
rived from the measurements than those calculated from the random mixing approximation. The dif-
ference between the experimental results and the calculations from the symmetric approximations
is small enough so that the correction for long-range interactions proposed by Fgrland®® and men-
tioned in section [ll.2 may be large enough to account for the differences.

To illustrate this for a particularly favorable case, in Table 12 are given vaiues of YLiF

[yL iF(meas)] in LiF-KCI mixtures calculated from the liquidus temperatures®'+%4

using the heats
of fusion in Table 1. Also given are values of ¥L;r calculated from the random mixing and the

symmetric approximations [yLiF(symm)]' In the last column is given

yLiF(symm)
2 2

yLiF(meus)

where, in this case, N¢=Ng =Ngey The form of this quantity (A log yLiF) is consistent with
the form of the relation given by Férland®® for the correction factor, A log y , y, which is to be
added to log Yay in order to account for the influence of interactions of longer range than nearest

neighbors when these interactions obey the equations for regular solutions
2 2
RTAlogy, = Nedx,y * NyXa,g * NgNyIN (A, =20 + Ny (A, =201, (111.4.2)

The terms A, and A, are related to the deviations from ideality in AX-BX and AY-BY systems re-
spectively and Ay y accounts for the same type of long-range interactions as Ay and Ay but in mix-
tures containing both X™ and Y™ ions. Similarly A, and Ay are related to the deviations from ide-
ality in AX-AY and BX-BY systems respectively and in mixtures containing A" and B* AA,B refers

to the same type of interactions as Ay and A, The magnitude of )\A,B and AX,Y are probably

Table 12. Activity Coefficients of LiF in the LiF-KC| Quasi-Binary

 

 

 

YLiF yLiF(symm)
Liquidus Temperature N, .o From Symmetric Random ‘°9m Nyct
Measurements  Approximation  Mixing LiF

1078 0.90 1.10 1.13 1.08

1068 0.80 1.35 1.45 1.38 0.72
1056 0.65 1.98 2.40 2.71 0.68
1053 0.60 2.30 2.93 3.70 0.67
1045 0.50 3.24 4.72 7.84 0.66
1040 0.46 3.77 5.88 11.17 0.67
1028 0.35 6.28 12.02 34.36 0.67
1020 0.30 8.33 17.97 62.44 0.68

1005 0.25 11.44 29.05 123.6 0.71

 
72

closely related (perhaps a weighted average) to A, and A, and A, and Ay respectively. These
parameters are discussed in section II. In the system discussed here (LiF-KCI) the last term in
Eq. (2) is probably small. From the last column of Table 12 a value of (’\XY + )\AB) of - 3200
cal/mole is calculated if the last term is neglected. This is reasonable for the interactions of the
ions involved (see section II}. This unexpectedly good agreement is probably fortuitous in view of
the approximate nature of the equations for regular solutions as applied to molten salts and the
agreement may not be as good in other systems. However, further detailed investigations of such
systems, especially in reciprocal alkali halide mixtures, would be interesting for comparison with
these considerations where the symmetric approximation is used for nearest-neighbor interactions
and Eq. {I11.4.2) is used as a correction factor. This correction factor when included in the calcu-

lations of the consolute temperatures, T will lead to much more realistic values than are calcu-

lated from Eq. (1).

It should be borne in mind that neither the symmetric approximation nor any other approxima-
tion which contains the implicit or explicit assumption of the additivity of pair interactions can be
generally valid for all molten salts and that neither can give better than semiquantitative results,

This will be discussed in a later section.

If Eq. (111.3.3) is solved for (1 = Y) in terms of (3~ 1), N,, and N, then

 

AI
1-v) -b+ \/b2—4ac
- - 2a
and for small values of ac/bz,
c ac 2a%c?
(1=Y)=-— {14+ —+ + o], (111.4.3)
b b2 b*

where a = N, (B~ 1), b=1 + (NgNy =N, N B =1, e ==N
and taking the logarithm of y, . one obtains

v Substituting Eq. (3} in (111.3.7)

ac 202(.“2

Iny,y=Z1In <]+b—2+ v

+...]=ZInb. (111.4.4)

 

The meaning of the symmetric approximation is made ciear by Eq. (4). Since N, and N appear in
exactly the same way in b and in the product ac, the interchange of particular numerical values of

N, and N will lead to the same value of y, . If N, and N, are variables, then the function YAy
is symmetric about the line N, = Ny By expanding the logarithms in Eq. (4), one obtains

Z
In YAy =—ZNBNX(‘8_ ])+§-[(NBNX)2+2NANBNXNY](B__'|)2

Z[ ) 6N, N )2 3 (B -1)°3 111.4.5
“g(NB“fl +6Ng NN NG + 3NN (NN THB=-1)  + o (1IL4.5)

 
73

The remaining terms are sums of products of (8 -~ 1)? and (NB NX)!’_”(NA NY)", where p > 3 and
p>nZ0. If (B~1)is small, then only the first term is important and

~ AE
(B~1)=(e~QB/RT _n T _Z 4 ..,
RT

so that Eq. (5) reduces to Eq. (|11.2.5), which was calculated from the random mixing approxima-
tion, For small enough values of N, N, or Ng Ny the higher terms in Eq. (5) are small relative to
the first, so that

Iny,y = —NgNyZ(B=1), (111.4.6)

which has the same form of the concentration dependence as Eq. (111.2.5) but does not contain the

implication that there is random mixing of the ions,

I11.5 The Asymmetric Approximation

One of the weaknesses of the symmetric approximation is the assumption of the additivity of
pair interactions which means that in dilute solutions, for example, the energy for forming the pair
AX from AT and X~ in the solvent BY is the same as forming AX2- from AX and X~ and A2X+ from
AX and A*. Measured association constants in dilute aqueous solutions indicate that this is not
valid, especially if the central cation is polyvalent.® Thus, any generalization of the theory which
includes a description of polyvalent cations and other special interactions must include a correc-
tion for the fact observed in aqueous and molten-sait solutions and discussed in a later section

that species such as, for example, Cd2C|3+

are not stable in dilute solutions whereas CdCl, is
stable in solution. In the theory which follows only monovalent ions are considered for simplicity.
However, most of the relations derived for the association constants in dilute solutions apply to
systems containing polyvalent ions as well.

The approximation given in this section is the asymmetric approximation which accounts for
species as AX (CdCl,, CdC|3_, AgCl,~, AgC|32') and neglects ionic groupings as A X
(Cd2C|3+, A92C|+). The applicability of this approximation to real systems will depend on the
specific nature of the system. The purpose of the approximation is to derive relations which relate
the influence of asymmetry of the ionic interactions to the thermodynamic properties of the solu-

tion,

In the asymmetric approximation'® the anion portion of the lattice is divided into two regions,
a and b. Region a contains all anion positions adjacent to one Ation and (Z = 1) B* ions, and re-
gion b contains all other anion positions. In a solution dilute enough in A* to neglect positions
adjacent to two A" ions, the number of positions in region a is ZnAYL = L, and in region & is
n(nx +n,)~L,=Lg. The X~ ions in region a are more stable by the energy Ae. If A€ is nega-

tive, the concentration of X~ ions in region a will be greater than in region & or, in other words,

there will be an association of AT and X ions. |f X’ is the concentration in ion fraction units of

 
74

-, . , . . .. : + . ,
the X ions in region a, then it is also the fraction of positions adjacent to A" ions occupied by
X~ ions. The A and X ions associate when X” > N, and are solvated when X“ <N, . If M} and

’
MB
nxn - M;. The total number of ways of mixing the anions in the anion region of the lattice, and

are the total number of X" ions in region a and & respectively, then M} = Zn XN and Mg =

the cations in the cation region of the lattice, (0°, is

L;!Lé![(nA + flB)n]!
Q= . 111.5.1
a (L;—M;)!M;!(Lé-Mé)!Mé!(nAm!(an)! ( )

 

By using Stirling’s approximation for the factorials and maximizing 17 under the condition of con-

stant total energy and constant total number of particles, the most probable distribution is calcu-

 

lated:
T | Tk 8. (11.5.2)
1-Xx 1-ZN, (1 -X) =Ny
The total energy of dilution in the solvent BY is
—AEg., = Zn, XAE , (111.5.3)
and the total entropy of mixing is given by
ASp=klInQ . (111.5.4)

By combining Eqgs. (111.5.3) and (111.5.4), the total Helmholtz free energy of dilution can be calcu-

lated.
Total Number Total Energy
L, Zn N 0
L‘B (nx+nY—ZnA)n 0
M, Zn , X1 Zn, XAE
MB (nX—ZnAX)n 0

Differentiating the total Helmholtz free energy to calculate the chemical potentials of the four

salts AX, AY, BX, and BY when the solvent is BY, one obtains

 

PFax ~Hax Ny =ZN, X X Z=
R N, (1~ X)Z 1+ ——— , (111.5.5)
RT 1-7ZN,(1-X)-N, B(1 - X)
. Z1
Py Ay v - x)? [1 . 111.5.6
—_—=n -_— —_— et
RT A "B -X) ' ( )
“Bx*”;x Ny =ZN, X

_ln N , (111.5.7)

B ’
RT ]—ZNA

 
75

0
Fgy “HBy 1-7ZN, (- X) =N,

=InN . 111.5.8
" 1-2N, ( )

 

RT

. . , + . - .
In the asymmetric approximation a given A" ion can have as many as Z X~ ions as nearest
. , - + . : i
neighbors, but a given X ion can only have one A" ion as a nearest neighbor. Thus only associa-

tions to form the groupings AXn+(]_”)

are taken into account and the groupings containing more
+. . -
than one A~ ion are completely neglected. The energy of attachment of each successive X ion to

a given AY ion is the same so that the energy for the association
AX M=n) X" =2 AX 7", 0523 (z-1),

is A€ independent of the value of n. This places restrictions on the relative values of the succes-

sive association constants as will be shown in the next section,

I11.6 Conventiona! Association Constants

The meaning and interpretation of the symmetric and asymmetric approximations can be made
more evident in terms of conventional association constants in dilute solution. |f the ions A™ and

X~ of the two solutes AY and BX in dilute solutions in the solvent BY associate as

AT+ X~ = AX
AX + X7 == AX,"

AX,”+ XT &= AX, 7, ete.

AX + AT =4, X", etc.

with the association constants being respectively K, Ky Kg, etc,, and K4 etc., then the associ-
ation constants may be related directly to parameters contained in the two approximations. It has

been shown?> that the thermodynamic association constants may be evaluated from the derivatives
of In y;Y or In y;X by the relations which have been derived under the reasonable assumption that

in very dilute solutions all species obey Henry's law.

dlny, 9 Iny.
AY B X
= | -k, (11.6.1)
R
B X AY
R Av=0 R av=0
Rpx=¢ Rgx=0
2 2 *
“lny,y 0% Inygy ,
(8% - K2_2K, K, (111.6.2)
IR2 dR. . OR
B X Bx 7% Ay
Rpvy=p Ravy=0
R

 
76

2 * 2 *
a ]n }/AY 8 In }/BX 2
2T = K2-2K K, (111.6.3)

2
AY/ . IRy
AY=0 Ray=0

Rpx=0 Rpgx=0
where Ry = flz‘j/nBY and where n is the number of moles of the solute component ij (AY or BX in
this example). The association constants are in mole ratio or mole fraction units which are the
most rational units in molten-salt solutions. These relations are not unique for calculating the as-
sociation constants and many other derivatives of functions of the activity coefficients may be
used. |t should be noted that there is a single limit of the derivatives of the single-valued func-
tions In y:j at infinite dilution of all solutes. Therefore these equations define true thermodynamic
association constants under conditions where the calculation procedure includes solutions dilute
enough so that all species may be reasonably expected to obey Henry’s law. By using Egs. (1),
(2), and (3), expressions for association constants have been calculated from the asymmetric and

symmetric approximations [Eqs. (111.5.2), (i11.5.6), (111.3.3), and ([11.3.7)] and are given below:

 

 

Association Constant Asymmetric Approximation Symmetric Approximation
K, Z(B-1) Z(B-1)
zZ-1 zZ -1
E (e S
Z -2 zZ-2
Ky ( ; %B—l) ( ; %B~U

K. ()= (222 -

n
- —(B-1
12 2 2
This table makes the differences between the symmetric or quasi-chemical and the asymmetric ap-
proximations clear. In both approximations
K, 2K, 3K nK

3 n
Z-1 Z-2 Z-n+1"

A
[[7aN

 

1 z (111.6.4)

1
— n
Z
which are the statistical ratios of Adams and Bjerrum. 149 Thus these approximations are shown to
be equivalent to the Adams-Bjerrum ratios® in dilute solutions. In the symmetric approximation
K, =K, but in the asymmetric approximation the effective value of K., is - ]/2, which is essen-
tially equivalent to zero. Although a negative value of an association constant is meaningless
thermodynamically, it can be understood in terms of the model. |f all the ions are randomly mixed
and the solution is ideal, all the K's are zero. Since in the asymmetric approximation the condition
has been introduced that no more than one A* ion be a nearest neighbor to any one X~ ion, then

: : , - + oy .
there is less than a random number of A" ions in positions near an A*-Xion pair. Thus the effec-

tive value of K, must be less than zero. This will occur if the A¥ ions repel each other.

 
77

It is clear that these two approximations can correspond exactly to real systems only for
special cases or in very dilute solutions, where only the first association to form AX is important.
However approximate these models are, they are still useful for semiquantitative descriptions of
solution behavior. Moreover, as will be shown in the next section, both models lead to a predic-
tion of the temperature coefficient of the first association constant, K., for the association of mon-
atomic ions which is correct within the experimental precision of measurements which have been

made.

1.7 Comparison of Theory with Experiments in Dilute Solutions

Measurements of y:\gNO , the activity coefficients of AgNO, in the mixture Ag *' B*, CI7, and

NO, ™ dilute in Ag+ and Cl~, have been made using the concentration cell

AgNO,
ag|2MNOsil Be A, (111.7.A)
BNO, B0,

where B is an alkali metal ion {or a mixture of alkali metal ions). In Fig. 21 are plotted measured
values of —log yA aNO, vs Ry at 385°C at two values of RAgNO3‘ The solvent BNO, in this
case is a 50-50 mole % mixture of NaNO ,-KNO,.  The activity coefficients decrease with increas-
ing concentration of KCl, the decrease being sma[ler the larger the initial concentration of AgNO,.
Obviously the concentration dependence of —log VA NO, is very badly approximated by (l11.2.5) or
(111.2.6) and the magnitudes of —log VA aNO, would requn'e very improbable values of ZAE or Ay
(about —300 kcal/mole). This large discrepancy is undoubtedly related to nonrandom mixing of the
ions. A comparison of these measurements with calculations based on the asymmetric and symmet-
ric approximations is made in Fig. 21 and shows that the measured concentration dependence of
-log y;\gNO corresponds only roughly to these approximations. At low RAgN03 both approxima-
tions are essentially the same and at the higher chloride concentrations indicate a lower activity
coefficient than is measured. This, probably, stems from the fact that in this system

~1
K2<< 5 )(B—l).

N

 

The activity coefficients at the higher concentrations of AgNO, lie between the two approximations

Z-1

 

indicating that - 4 < K, << >(B — 1). The same is true if the solvent is pure NaNO, or

KNO, with measurements in KNO, being closer to the asymmetric and in NaNO, to the symmetric
approximation, These comparisons indicate that these two approximations, although much more re-

alistic than the random mixing approximation, can be, at best, semiquantitative. One reason for

 

*|n these dilute solutions RAgNO3: NAg and RKCI = NCl'

 
78

UNCLASSIFIED
ORNL-LR-DW7S 57004

 

0.3 ’

7 =385° Ragno, = 0.2 X107 3.

SYMMETRIC OR /

ASYMMETRIC APPROXIMATION /
0.2

7

 

 

AN

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

> _ -3
2 / Fagno,= 22 *10
> .
o 7 /"_
o o
' K
0A //.‘/< ///
' -
/ ’ SYMME TRIC APPROXIMATION
S .///\\ r | i
////.,// ASYMMETRIC APPROXIMATION
p
§
7
4
0
0 10 2.0 3.0

Rey X 103

Fig. 21. Comparison of the Concentration Dependence of Measured Values
of —log ngNO in NaNO4.KNO, (50-50 Meole %) Mixtures with Theoretical

Calculations Based on the Symmetric and Asymmetric Approximations.

this is the nonadditivity of pair bond interaction energies in dilute solutions. This means that the
relative values of successive association constants do not, in general, correspond to the values
given on page 76. In the next section a generalization of theory will be made which will include

the possibility of the nonadditivity of pair bond interactions.

The theoretical evaluation of K, however, is meaningful for certain systems and in solutions
dilute enough in A" and X~ so that the most important species is AX, the temperature dependence
of the activity coefficients (and of K]) is predicted by these two approximations. Measurements
of —log V;«gNOB in dilute solutions of Ag* and CI™ inthe three solvents Nc1N03,70 KNO3,23'94 and
50-50 mole % N0N03-KNO369 mixtures were compared to theory, By comparing the approximation

- . . *
which was closest in concentration dependence to the measured vaive of ~log Yagno, ot low con-
3

centrations of Ag' and Cl~ values of K, could be evaluated and are given in Table 13. This pro-

cedure for evaluating K| has been shown to be equivalent to more conventional extrapolation pro-

 
79

Table 13. Values of AE.l Obtained from the Comparison of Theory with Experimental Dato

 

—AE.I {kcal)

T (OK) K] =Z(B_'|)(a)

 

 

Asymmetric Approximation, Ag+, K+, Cl-, N03-

623 6.12 5.85 5.62 553
643 6.17 5.89 5.66 498
658 6.21 5.93 5.69 460
675 6.17 5.87 5.64 396
696 6.18 5.88 5.63 348
709 6.17 5.86 5.62 315

Symmetric Approximation, Ag+, No+, cl, NO3-

604 5.10 4.83 4.62 277
637 5.12 4.84 4,62 226
658 5.17 4.88 4.65 205
675 5.10 4.81 4.57 176
696 5.13 4.83 4.59 160
711 5.12 4.81 4.56 146
773 5.14 4.82 4.55 110

Asymmetric Approximation, Ag+, (Nu+, K+), Cl-, N03-

 

506 5.6 5.4 5.2 1050

551 5.57 5.33 5.13 644

658 5.67 5.38 5.15 302

752 5.72 5.40 5.13 180

801 5.6, 5.24 5.04 133
(“)K] in mole fraction units.

cedures?® if used correctly. To evaluate the parameter AE(AE ) contained in the theoretical ex-

pression for K, a value of Z must be assumed where

K,=Z(B~-1)=Zlexp (-AE,/RT) - 1], (11.7.1)

In molten salts a range of values of Z which covers all reasonable possibilities is 4 to 6. In Table
13 are given values of AE, calculated for values of Z = 4, 5, and 6. |n any one system and for any
one value of Z the values of AE, thus calculated, within the estimated experimental error, do not

vary with temperature. This means that Eq. (1) correctly predicts the temperature coefficient of K

in these systems. In the NaNO ,-KNO, system this prediction is correct over a range of 295°C and

 
80

for a variation of K, by a factor of about 8. At low enough concentrations of Ag*and CI” so that
the only important species is AgCl the variation of —log y‘AgN03 with temperature, within the ex-
perimental precision, is also correctly predicted. This is illustrated in Fig. 22 which gives a plot
of —log y:\gNoa in a dilute solution of Ag” and Cl ™ in NaNO, at several temperatures. The dashed
lines were calculated from the symmetric approximation using the parameters given in Table 13,

Using these essentially constant values of AE, leads to an excellent correspondence of the calcu-

lated and measured valuves of —log y:\gNO at low concentrations of Ag* and CI~,
3

UNCLASSIFIED
ORNL-LR-DWG 44359

 

 

 

 

   
   

 

 

 

 

 

0.4
7=331°C
4 364°C
R, =0.30x10"°
AgNO3z ™ 7
------ THEORETICAL CALCULATIONS j
0.3 ‘
402°c
O
2
202
™
2 -
g ‘ { 500°C

 

 

 

7,

04

 

 

 

 

 

 

 

 

0
2 3 4 (x10™3)

o 1
RNaCI

Fig. 22. Comparison of the Temperature Dependence of —log y;\gNO in
3

Nr.:NO3 with Theoretical Calculations,

 
81

The theory, in essence, leads to a prediction of the ‘‘configurational’’ contribution to the en-
tropy of association so that from measurements at one temperature one may also calculate the heat

of association, AH ]':

dinK, -AH! dlnZ(B-1)

= = ] . (11.7.2)
) R d( %

 

d( ‘/T

Since AE, is independent of temperature,

AH? = AE, (B_B_.]_) (111.7.3)

where it is to be remembered that AE, can be calculated from measurements at a single tempera-
ture. Equation (3) for AH ] may be confirmed {within the experimental precision and within the
range of values of AE, for the three values of Z) from the values of K, given in Table 13. Other
reported values of AH T which differ from Eq. (3)38 were calculated from too few points and over

too short a range of temperatures to be significant.

I11.8 Generalized Quasi-Lattice Calculations'?

The comparison of both the symmetric and asymmetric approximations with experiments make it
evident that less stringent restrictions on the relative energies of association are necessary for a
comprehensive theory. In this section a generalized calculation based on the quasi-lattice model
will be discussed. The purpose is to calculate more general expressions for some of the higher
association constants.

For simplicity, the assumption is made, as in the asymmetric model, that the solution in the sol-
vent BY is so dilute in A jons that one can neglect all groupings of A" and X ions containing
more than one A jon. From a calculation of the partition function for the assembly of AY, B X,
and Y ions calculations were made of the Helmholtz free energy, the chemical potential for the
component AY, and, hence, the activity coefficients of AY, y:\Y, in terms of the ion fractions of the
ions, Z and 3., where 3. = exp (~AA /RT) and AA, is the “*specific bond strength’’ or the ‘‘spe-

cific Helmholtz free energy change’’ for the association

([FaN

AX{Z=D | X —a AX{1 ), 15iS7.  (I1.8.A)

i
In this approximation AA, # Ad, £ AA, # AA;in general. |t must be kept in mind that the symbol
AXS.]"i) represents an A" ion having i X ions and (Z — i) Y~ ions as nearest neighbors. Thus
(AAl./n) is the free energy change for exchanging one X~ ion in the body of solution with a par-
ticular Y ion adjacent to the A” ion in the grouping AXSET.). The term AA . is related to partition
functions for the individual ions involved in the association (A) (which is really an exchange of

ions) so that

AA;=~RT In| —r], (111.8.1)

 

 
82

where 774/ and Wq;; represent the product of the partition functions of the individual ions, &, in-
volved in the association process (A) evaluated before and after the association process respec-

tively. If the partition functions are separable so that

- =E,/RT
Thi = 9ri® ’

where g, represents a partition function for the internal degrees of freedom of the ion of type %,

then

 

AA; = AE, - TAS, = (RE/ = XE/.) = RT In , {111.8.2)
z 1 i z ki N

T4
and the “*specific'’ entropy term, AS., contains only contributions from the internal degrees of free-
dom and excludes statistical or combinatory factors for the groupings of ions. For negligible

~o

changes in the internal degrees of freedom of the ions involved in the association process AS. =0
and (dAA./dT) = 0. This is the case for the values of AA | in the systems cited in Table 13,
The statistical mechanical calculation'* leads to the equations for some of the successive as-

sociation constants {in mole fraction units)

(111.8.3)

Ky=2Z(B;~1, (a)
K1K2=-Z—(%:-]—)(Blfiz—2fil+1), (b)
K1K12=Z_(Z;!;])‘(181512-251+]), (c)
K1K2K3=Eg_—;.i(z-—;a(818283—38182+3,8]—1), (d)

Z(Z - 1NZ-2)(Z - 3)
4!

 

Ky KoKgKy = (B1B2B3By =481 By B3+ 68,8, =48, +1). (e
Equation (111.8.3a) is the same as the expression for K, given in the table on page 76 if AA | =
AE |. The terms in Z are spatial and statistical factors and the terms in B; are related to the bond
energies, For the case in which AAy=AA,=AA;=AA and B, =3, = B3 = B, the statistical
ratios of Adams and Bjerrum apply.

Some of the relations derived from (111.8.3) exhibit surprising properties. For example by di-

viding (111.8.3b) by (111.8.3a) one obtains the expression

B,-B
K, =[(z~1)/2] {(32-1“(%) . (111.8.4)

It can be seen from (111.8.4) that K, depends not only on Z and B, butalsoon B,. If B, is small,
this dependence may be relatively significant. If, for example (2 - ]/B]) > B, > 1, then there ex-

 
83

ists a tendency for the association* of AX with X to form AX2_ and yet the values of K, may ap-
pear to be negative.** This unusual and apparently contradictory result arises because of the re-
quirement that the conventional association constants, K., be almost zero in an almost ideal solu-
tion. The standard states for some of the associated species under this requirement which is
inherent in the commonly accepted methods of describing associations in solution cannot be under-
stood in g simple way and |lead to unusual properties for weak associations. An analogous situa-
tion occurs when gas virial coefficients are interpreted in terms of clusters,” /74

The assumptions made in the calculation of Eq. (3) are that the ith X ion attaching itself to
an A% ion can do so in (Z — i + 1) equivalent positions. Different relations would be obtained
under different assumptions. If, for example, only a linear AXz- ion triplet can form, the second

X~ ion has a nonzero value of A4, in only one of the (Z — 1) sites near an AX ion pair which is not

already occupied by an X™ ion, For linear AX2— then

Ky Ky = [z(z = 0/20{18, B/(Z = W] + 1 = [2/(Z - VIR, } (111.8.5)

and

BQ"1
B]"']

 

i 1
Ky=%|By—-Z +

The stepwise association constant for formation of linear AX2" is Ké and would be smaller than
K, for a nonlinear grouping even with the same values of Z, 8,, and B, Thus the comparison of
the Eqs. (3b) and (5) demonstrates in this simple case the general principle that the greater the
tendency toward ''directionality’’ in a ''bond'’ the lower will be the association constant, if all
other factors are equal,

Equations (5) and (3a) lead to conclusions differing from those of Bierrum7 on the ratios of
successive association constants for linear AX,". For values of 8, = 8, >> 1 for example,
KI/Klz 2 27, where Z is a maximum coordination number, In Bjerrum's derivation this number is a
characteristic coordination number N, For a common case in which 8, << 8,, N is two and much
smaller than Z. The error in the calculation of Bjerrum arises from the fact that when the total pos-
sible number of X ligands is restricted to N in his derivation, the total number of positions adja-
cent to a spherical A¥ ion which are available to the first ligand is simultaneously limited to N al-
though the first ligand is actually able to attach itself in any one of Z positions.

The equations discussed in this section can be derived for nearest-neighbor interactions inde-
pendently of the lattice model. The coordinator number Z in such a derivation would be the ratio
of the volume of the first coordination shells adjacent to a mole of A* ions to the volume of a mole

of solvent anions. Such a derivation would apply to polyvalent cations,

 

*f AA:' is negative and Bi > 1, there will be a tendency toward the association of AXS._z_-l'i) and X~ to
form AXf.]-'i).

**Negative values of K, are meaningless thermodynamically, and apparently negative values usually

mean a repulsion of the ions involved rather than the assumed association,

 
84

II1.9 Association Constants in Dilute Solutions

In this section a compilation is given of association constants (in mole fraction units) which
have been measured in reciprocal molten-salt systems, Measurements of associations invelving the
Ag* ion have been largely made with cells of the type (I11.7.A) and the most reliable measurements

for associations involving T1¥, Pb2*, and Cd?* with halides from cells of the type

A(NO )
AgX(solid) n
Aq |BX AgX(solid)| | (111.9.A)
g g . .7,
BX
BNO,
BNO,

using silver-solid-silver halide (AgX) electrodes where At s T|+, F’b2+, or Cd2*. The emf of
cells (I11.7.A) and (111.9.A) may be related to the activity and activity coefficients (y*) of AgNO,
or BX respectively. To avoid confusion, it should be emphasized that these activity coefficients
are defined so as to encompass all solution effects including ionic associations (‘‘complex ion'’
formation). At concentrations where Henry's law is obeyed by all species (probably true at ¢on-
centrations below 0.5 mole %) it represents only those deviations from ideal solution behavior
which are caused by association in solution, This usage is simpler than the usage most often em-
ployed in aqueous solutions where deviations from ideal solution behavior are subdivided into “*ac-
tivity coefficient’’ effects (related to the ionic strength) and an effect due to associations. Ther-
modynamic association constants may be computed from these measured activity coefficients by an
extrapolation method.?% Some of the association constants cited here have been recalculated from
the data in the literature.2* In cases where errors in calculating association constants may be sig-
nificantly larger than the errors stated by the original workers and not enocugh data were available
to correct the calculations, the association constants are given in parentheses or omitted, From
the tabulated association constants (Tables 14 and 15), values of A4, were calculated from Egs.
(111.8.3) for Z = 6 and are given in Table 16 for monatomic ions. For other values of Z, AA. would
be somewhat different {for Z = 4 the AAz. would be more negative by about 0.4 to 0.6 kcal) but the
differences between the different values would be about the same. The differences in AAin Table
16 are related to the association constants (for K;>> 1) by AAI.'—- AAiH Z_RT In Kz.'/KI.".

In every case where measurements were made at more than one temperature for associations in-
volving monatomic ions only, values of AA, for a given association in a given solvent and for Z = 4,
5, or 6 were independent of temperature within the experimental uncerfuinties.* Thus it appears
that, for monatomic ions, the temperature variations of K, and K, as well as of K, may be pre-
dicted from Eqgs. (111.8.3) by using constant values of AA, and it appears that the entropy of as-

sociation is largely the *‘configurational’’ entropy calculated from the quasi-lattice model. For

 

*There did appear to be trends in the variation of AAI. with temperature in some cases. The total varia-
. . . . +
tions were smaller than the experimental errors in all coses except for AAI for the formation of CdBr in 50-
50 mole % NaNOa-KNoa, where the variation of AA'I was slightly larger than the estimated experimental

errors,

 
85

Table 14. Campilation of Associatian Constants from EMF Measurements (see also Table 13)

 

Associating

 

T (°K) Solvent lons K] K2 Kl2 References
675 NaNO, Agt 4B 633 246 280 95
711 500 180 200
733 430 151 167
773 325 103 120
606 NaNO ,-KNO , (53-47 mole %) Agt+cl” 381 145 38
647 302 97
649 Agt+Be” 1,008 (360) 38
687 781 (199)

528 Pb2t s Br” 199 39
576 153

579 67

529 cd?t i ee” 1,170 550 39
547 1,030 510

571 810

513 NaNO ,-KNO 4 (50-50 mole %) TiteBe” 31 15 27
519 Agt+CN™ 230,000 140,000 80,000 10
559 220,000 105,000 60,000 93
599 190,000 50,000 36,000

513 cd?t 1B 1,520 680 ~0 25
573 990 450 ~Q

513 cd?t 4 ” 5,330 2,200 ~Q 25
563 3,130 1,300 ~Q

513 Pb2* 4 Br” 250 125 ~Q 27
573 170 85 ~Q 92
623 KNO, Agt+cI” 553 215 <40 94
658 460 169 20

709 315 17 <40

676 Agt 4B 932 370 293 2
711 768 285 230

725 728 273 208

747 617 228 174

773 540 195 145

675 Agt+1” 5,420 2,700 3,555 2
636 Ag+4-5042_ 11.6 132
681 12.1

706 12.7

722 13.3

513 LiNO 4-KNO 5 (80-20 mole %) cd?t 4B 4,300 1,700 26
513 (6535 mole %) 3,600 1,600

444 (50-50 mole %) 7,500 3,300

513 3,000 1,300

513 (40-60 mole %) 2,500 1,100

513 (26-74 mole %) 2,300 1,000

553 (40-60 mole %) T1 4 Be” 56 30 27

 

 
86

Table 15. Association Constants from Other Measurements

 

 

Solvent T (°K) Species K] Method
129 + +
Nc:xNO3 580 CdCl 190 =50 Cryoscopy
PbCIt 60 +20 Cryoscopy
LiN03-KN0335 (50-50 mole %) 453 cdcit (900)% Polarography
Pocl? 270 +80 Polarography
NaNo3-|<No3“ (53-47 mole %) 523 cdcl’t (250) Solubility
573 cdci? (300) Solubility
523 poctt (200) Solubility
573 PbCtt (85) Solubility

 

2P arentheses indicate uncertain data.

associations of Ag* with 5042- or CN~ values of AA, decrease with temperature indicating an en-
tropy of association larger than the ““configurational’’ entropy of association. Values of the
negative of the '"specific bond free energy'’ (where comparisons can be made) are in the order
5042' <CI=<Br= < I~ <CN-and T!I* <Pb?* < Cd?* < Ag*, which is comparable to the order
found in water and, in general, the values of RT In K, (K, in mole fraction units) are roughly com-
parable to those found in water. Values of AA, (or AEI.) do not correspond precisely to the pre-
diction of Flood, Férland, and Grjotheim (section I11.2), but the relative magnitudes can be corre-
lated with the heat (or free energy change) for the reciprocal reaction {I11.1.A). For the silver
halides for example, the heat changes for a reaction as (l11.1.A) are in the same order as AA

101 Superimposed

and may be correlated largely with non-Coulombic (van der Waals) interactions.
upon the non-Coulomb interactions is a reciprocal Coulomb effect illustrated in the two-dimensional
representation in Fig. 23.

The major change in the association of A" and X is the interchange of nearest-neighbor A¥Y”
and B X~ pairs to form A*X ™ and B*Y " pairs as illustrated in the lower part of the figure. A cal-
culation of the nearest-neighbor Coulomb energy change (for ions which touch each other) indicates

that this contribution to AA | is e2(]/dAY + 1/de -~ VdBY - 'l/dAx) and has the sign given be-

low:

Contribution to AA i

- rA<rB rx<rY
- rA>rB rx>rY
+ rA<rB rx>rY
+ r,>r r, <r

 
87

Table 16. Average Values of **Specific Bond Free Energies,"”’ AAz. (kcal/mole), for Z = 6

for the Association of Monatomic lons in Molten Nitrates

 

Solvent Compesition (mole %)

 

 

Association - ~A4, ~A4, ~AA,,  References
lons LiNO,, NaNO,, KNO,
Agtici” 0 100 0 4.59 70
0 53 47 5.04 4.8 38
0 50 50 5.12 69
0 0 100 5.64 5.5 94
Ag*+Br” 0 100 0 6.2, 6.0 6.2 95
0 53 47 6.6, (6.2) 38
0 0 100 6.8, 6.7 6.4 2
Agt+1” 0 0 100 9.1, 9.4 9.8 2
Tteee” 40 0 60 2.1 27
0 50 50 1.8 27
cd?t i cl” 0 100 0 4.0 24, 129
cd?t 4 Be” 0 53 47 5.5, 5.6 39
0 50 50 5.75 5.8 25
80 0 20 6.64 6.7 26
65 0 35 6.58 6.6 26
50 0 50 6.33 6.4 26
40 0 60 6.16 6.2 26
26 0 74 6.06 6.1 26
cd?t 41" 0 50 50 6.99 7.0 25
Pb2t L c1” 0 100 0 2.8 24, 129
Pb2ty Be” 0 53 47 3.64 3.8 39
0 50 50 3.85 27, 92

 

with the magnitude being dependent on the relative differences in size. For example, for the asso-
ciation of Ag+ and Cl” in KNO, this contribution to A4, is about 2.6 kcal/mole more negative than
in NaNO, and for the association of Ag "' and Br™ about 1.4 kcal/mole more negative in KNO, than
in NaNO . This nearest-neighbor Coulomb contribution to the differences between solvents is in
the right direction but is over twice as large as the measured differences in A4, in these systems
given in Table 16. This is probably related largely to the influence of long-range interactions
which cannot be assessed for a realistic three-dimensional model but for a one-dimensional model

the long-range interactions can be shown to attenuate the effect.??

 
88

UNCLASSIFIED
ORNL—-LR—DWG 604584

 

dO+A,+4,)

Fig. 23. Two Dimensional Representation to lllustrate the Reciprocal
Coulomb Effect.

Within the experimental errors the measured constants in mixtures of two nitrates (a and &) obey

the linear relations
In K](in mixture) = N, in K](in pure a) + N, In K](in pure b) (111.9.1)
and
AAl(in mixture) = N, AA](in pure a) + N, AA 1(in pure b} . (111.9.2)

Surprisingly, values of ~AA, for the formation of CdBr* and CdBr, are larger in LiNO,-KNO; mix-
tures the larger the mole fraction of LiNO, and are larger than in the corresponding NaNO,-KNO,

 
89

mixtures, These particular association constants are, therefore, not related to the radii of the sol-

vent cations by a simple monotonic relationship.

MISCELLANEQUS

Solutions of gases in molten salts are of interest to theoreticians because of their innate sim-
plicity, The rare gases, mainly because they interact relatively weakly with most substances, form
the simplest of such solutions. Measurements of the solubility of helium, neon, argon, and xenon

13,61

have been made in molten fluoride solvents, All of the solubilities obeyed Henry's law

C,=K (TP _, (1)

where C,is the concentration of gas in the salt in moles per em? of melt, Kp isa Henry's law con-
stant, and Pg is the gas pressure in atmospheres. A simpler method of expressing solubilities for

theoretical treatment is
C,= KC(T)Cg , (2)

where Cg is the concentration of gas in the gas phase in moles per cm®, The use of Eq. (2) and of
K_, which is unitless, eliminates those trivial additive (and usually relatively large) contributions
to the entropy of solution which are related to the arbitrary choice of concentration units, Henry's
law constants for rare gases are given in Table 17; those for HF in NaF-ZrF , mixtures (discussed

in the following paragraph) are given in Table 18. The solubility of the rare gases increases with

an increase of temperature and with a decrease of the size of the gas atom. A calculation of the

enthalpy of solution and the standard entropy of solution was made by using the equations

dln Kp AH
a1/1T) R
and
d(RT In K_)
— _AsY,
dT

and is given in Table 19. In all cases, the entropy of solution is @ small negative number for the
rare gases. |f the gas phase concentrations were expressed in pressure units (atmospheres), then
values of the standard entropy would be obtained by adding, to the entropies in Table 19, —=R(1 +
In R’T), where R’ is the gas constant (em? atm/deg mole), and R is the gas constant in entropy
units. The free energy of solution (AF_ = —RT In K_) may be estimated roughly by the free energy
of formation of holes the size of the rare gas atom

C

_RT In — - 18.084% = ~RT In K_,

Cg

where d is the gas atom radius in Angstroms, and o is the surface tension. This approximation

neglects curvature of the holes and interactions of the gas and liquid. 1°7

 
90

Table 17. Henry's Law Constants for Noble Gases

 

 

 

Measured
o moles
Solvent Gas T{("C) Kp( )X ]08 K x ]03
3 c
cm atm
LiF-NaF-KF {46.5-11.5-42.0 mole %) He 600 11.3 0.7 8.09
650 13.7
700 17.5 0.2 14.0
800 23.0 £ 0.7 20.3
Ne 600 4,36 £0.20 3.12
700 7.51 £0.22 6.00
800 11.18 £0.26 9.84
Ar 600 0.90 +0.04 0.645
700 1.80 £0.04 1.43
800 3.40 1 0.03 2.99
NGF-ZI’F4 (53-47 mole %) He 600 21.6 1.0 15.5
700 29.2 0.7 23.3
800 42,0 +1.3 37.0
Ne 600 11.3 +0.3 8.09
700 18.4 £ 0.5 14,7
800 24.7 0.7 21.7
Ar 600 5.06 £0,15 3.62
700 8.07 +0.08 6.44
800 12.0 £0.6 10.6
Xe 600 1.94 1.39
700 3.56 2.84
800 6.32 5.56

 

By contrast, gases which interact strongly with components of the solvent have much higher
solubilities than the rare gases. The solubility of HF in NaF-ZrF, mixtures for example is much
higher than of the rare gases and increases with an increase in the concentration of NaF indicating
that the strong interactions (negative) of HF in solution are with NaF. Measurements of the solu-
bility of water in molten LiCl-KCl mixtures however indicated that the water solubility did not in-
crease significantly (except at 390°C) with increasing concentrations of LiCl with which com-
ponent water has a relatively strong interaction. The data are given in Table 20. Note that the
units are in micromole of H,O per mole of solution per millimeter pressure. The heats of solution

appeared to be —5 and — 11 keal/mole in the 50 and 60% mixtures respectively.

 
92

Kinetic measurements have been used to measure equilibrium constants3%+4° for the acid-base

reactions,

Cr2072— +NO," = N02+(so|) + 2Cr042_ , (3)

$,0,27 + NO,~ == N0, Y(sol) + 2507, (4)

in NaNO ;-KNO ; mixtures (53-47 mole %). (Note that N02+ was assumed, The data fit N205
equally well.) The limiting step in the reaction was the removal of N,O, by sweeping out its de-
composition products in a stream of gas bubbles. Since the evolution of the gas was dependent on
rates of diffusion into the gas phase and the rate of bubbling, the kinetics of the reaction were not
related to properties of the solution, By an extrapolation procedure the equilibrium constants for
(3) and (4) could be deduced and are given in Table 21 and indicate that 52072'" is a stronger
“acid" than Cr2072". The values of K , were so small that heavy metal ions had to be added to

the solution to increase the rate of gas evolution by removing Cr042_ from solution,

Table 21. Equilibrium Constants® for Reactions (3) and (4)

 

 

O
T (°C) K Kg
-14
235 8.5 x 10 0.026
275 3.8 x 10~ 12 0.038

 

a . .
In molality units,

 
93

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Q7

ORNL-3293
UC-4 — Chemistry
TID-4500 (17th ed., Rev.)

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