1500
GEORGE E. BLOMGREN
Vol. 66
first approximation the effect of coulombic terms The free energy of hydrochloric acid (and precan be estimated by the Born treatment, whereas sumably also the “availability” of protons) therethe change of solvent basicity and solvating power fore decreases as the solvent is enriched with Nare the most important chemical factors. If methylpropionamide. I t is impossible to say the effective ionic radii are assumed to be un- whether this effect is entirely electrostatic or changed as the solvent composition is altered, tthe whether a part of it is to be attributed to an inBorn treatment predicts that a decreased ionic free creased average basicity of the solvent. Neverenergy will result from an increase in the dielectric theless, both N-methylpropionamide and N,N-diconstant. Increased basicity of the solvent will methylformamide produce a decrease in rate in have a similar effect. spite of their different, effects on the dielectric conUnfortunately, no measurements of primary stant of the mixtures. It therefore would seem medium effects in mixtures of water and E-methyl- that the change of basicity is the predominant facpropionamide have been made. However, the tor. At any rate, the result should be a decreased standard potential of the cell: Pt, Hz; HCl(m), amount of the protonated form of acetal and a AgCl; Ag in N-methylacetamide a t 40’ has been consequent decrease in the rate of the reaction. determined recently by Dawson and co-~vorkers.3~ This explanation is consistent with the qualitaOn the mole-fraction scale, their results give EON = tive success of the Y values as an index of the rate +0.064 v., as compared with EON = -0.005 v. for the same cell with pure water as the solvent. The behavior, while no simple relationship between primary medium effect is therefore negative and rate and dielectric constant appears to exist. Acknowledgments.-The authors are indebted logf”Hc1 = -0.557. to Dr. R. T. Leslie and Mr. E. C. Kuehner for the (33) L. R. Dawson. R. C. Sheridan, and 1% C. Eckstrom, J. Phys. distillation of the acetal. Chem., 6 6 , 1829 (1961).
THEORIES OF FUSED SALT SOLUTIONS BY GEORGE E. BLOMGREN Parma Research Laboratory, Union Carbide Corporation, Parma 50, Ohio Receiued March 17, 1969
The current status of theory and experiment in the field of fused salt solutions is discussed. An application of regular solution theory is shown to be inadequate to explain recent experimental results on silver salt solutions. ,4 derivation of a regular solution theory for ternary solutions, which is more general than earlier theories, is given and, although available experimental data are of solutions too dilute for the additional terms in the theory to have a sensible effect, some of the limitations of the application of regular solution theory are clearly shown by the derivation. Finally, a inolecular theory of dilute fused salt solutions is given which is found to be in semiquantitative agreement with measured heats of solution. No satisfactory extension of the theory to concentrated solutions was found.
I. Introduction In recent years a considerable amount of interest has been shown in the thermodynamic and mechanical properties of fused salt solutions.’ Advances in the techniques of handling materials at high temperatures have enabled reasonably precise measurements to be made on mixtures of fused salts which, even in the simplest cases, show some deviations from ideal behavior. Fused salts have an advantage over non-ionic liquids in that, for materials for which reversible electrodes can be constructed, the electromotive force of a solution can be measured. The e.m.f. is related to the partial molar free energy of the measured component, and the temperature dependence of the e.m.f. is related to the partial molar entropy of that component. The properties of the other component in a binary solution can be obtained from the same data by well known methods. This technique has been successfully applied by several workers*-* to various silver salts mixed ( 1 ) For reference to recent work see G. E. Blomgren and E. R. Van Artsdalen, “Annual Review of Physical Chemistry,” Vol. 11, Annual Reviews, Inc., Palo Alto, Calif., 1960 pp. 273-306. ( 2 ) J. H. Hildebrand and E. J. Salstrom, J . Am. Chem. SOC., 64, 4257 (1933). and references cited therein. (3) R. W. Laity, ibid., 79, 1849 (1957).
with group I and I1 salts. Alternatively, the excess partial molar free energy may be obtained from phase diagram studies.8 Thermal properties of fused salt solutions have not received much attention in the past, except in phase diagram studies, but recent developments in high temperature calorimetry should lead to increasing interest in this type of measurement. Most notable of recent work in this field are the measurements of the heat of solution of alkali metal nitrates by Kleppa and Hersh’ and the measurements of alkali halide solutions by Aukrust, BjGrge, Flood, and Forland.* Measurements of molar volumes of fused salt solutions have often been carried out in conjunction with electrical conductivity measurements in order to obtain equivalent conductivities. Since the excess volume has not been the main concern, many of these determinations are not very precise, (4) K. H. Stern, J . Phys. Chem., 60, 679 (1956). (5) M. B. Panish, F. P. Blankenship, W. R. Grimes, and R. F. h-ewton, ibzd., 62, 1325 (1958). (6) M. B. Panish, R. F. Newton, W. R. Grimes, and F. F. BlankenShip, ibid., 63, 668 (1959). (7) 0. J. Kleppa and L. S. Hersh, J . Chem. Phys., 84, 351 (1961). (8) E. Aukrust, B. Bjiirge, H. Flood, and T. Forland, Ann. N. Y. Acad. Sci., 79, 831 (1960), Art. 11.
August, 1962
THEORIES OF FUSED SALTSOLUTIONS
but some of the better measurements show interesting results. For example, Van Artsdalen and Yaffeg showed that, even for many simple alkali halide mixtures, there are small, but definite positive deviations from ideal volume behavior. Byrne, Fleming, and Wetmorelo measured the AgNO3NaN03 system and observed large positive excess volumes. The theoretical development of molten salt solutions has met with some success in correlating some of the excess properties. Hildebrand and Salstromz successfully applied the zeroth-order regular solution theory to explain the excess partial molar free energy of AgBr in va,rious metal bromide solutions. Ternkin11 developed an ideal solution theory tlo deal with more complicated mixtures. Flood, Forland, and Grjotheim12 extended the zeroth-order regular solution theory to systems involving two cations and two anions. This theory was further developed by Blander and Braunstein to the first-order or quasi-chemical approximati0n.13’~~All of the theories mentioned above have relied on a rigid lattice model for the liquid solution, as does the work in this paper. It may seem at first that the representation of a liquid salt solution by a crystalline lattice would be grossly inadequate, since the coulombic forces are long range and might be expected to change radically from the solid to the liquid phase and vary greatly from one liquid to another. It is observed, however, that heat,s of fusion for salts are not exceptionally large compared to the over-all iattice energy of the solid (albout 3-5% for many salts). This must mean that the coulombic energy does not change greatly on fusion, since this is by far the largest contribution to the lattice energy, even though long range ordering can no longer be present. Thus, in a calculation of excess properties, the lattice approximation may not be too severe. The restriction of rigidity, i e . , that, the ions are fixed to their lattice sites, may be more serious. Both entropy and volume effects, and to some extent the free energy, are difficult to explain on this basis. The discussion of specific theories which follows will bring out this point more clearly. 11. Regular Solution Theories
A. Binary Solutions.-Binary solutions are taken t o mean solutions with a common anion (cation) and twcl different cations (anions). The early studies of Hildebrand and Salstromz on (Ag-M)B r solutions showed the excess partial molar free energy to be independent of temperature to within the experimental uncertainty of the data. This implies that the excess entropy of the solutions is zero, and one might expect the zeroth approximation of regular solution theory15 to (9) E. R Van Artsdalen and I. 8. Yaffe, J . Phys. Chsm., 59, 118 (1955). (10) J. Byrne, H. Fleming, and F. E. W. Wetmore, Can J Chem., 30, 922 (1952) (11) M. Temkin, Acta Phvszcochzm. URSS. 20, 411 (1945). (12) H. Flood, T Forland, and K Gristhelm, 2. a n o ~ g allgem (“hem., 276, 289 (1954). (13) M. Blander, J . Phys Chem., 63, 1262 (1959). (14) hl. Blander and J. Braunstem. Ann. N. Y. h a d . Sca , 79, 838 (1960), Art. 11.
1501
apply. Thus, one considers an anion lattice of fixed dimension and allows the cations to be mixed on the cation lattice. The mixing of the cations will be determined by a single interaction energy parameter, w , which is defined as w :=
1
WAB
- 2- (WAA
+
WBB)
(1)
where WAB, WAA, and WBB are the interaction energies of an A-B pair, an A-A pair, and a B-B pair, respectively, and A and B stand for Ag+ and M+. An ideal solution results from w = 0 and deviations from ideality are described in terms of intermolecular pairwise interactions of cations, which, in salts, are next nearest neighbors. The excess partial molar free energy is given in this approximation by FAE
= ngzw or
PIE= n22w
(2)
where n B is the cation fraction of B and n2 is the mole fraction of the salt 2. By plotting the measured values of FIE us. n2, Hildebrand and Salstrom2found good linear plots of the data for LiBr, NaBr, RBr, and RbBr mixed with AgBr, and the slopes of the lines gave values for w. This approach is phenomenological in the sense that no attempt is made to deduce the magnitude of w from molecular properties. More recent measurements have been made on the similar systems (Li-Ag)C16 and (Na-Ag)C15 by Panish, et ul. Wider temperature ranges were studied for these materials and analysis of the temperature coefficients of the e.m.f. consistently gives positive partial molar entropies of AgC1, although the precision is not good. Application of the regular solution theory to these systems shows that the interaction energy w does not remain constant on determining the best straight line a t various temperatures. Table I shows-the agreement of the experimental results of F1 - Fi with the calculated n2w. Although there is considerable scatter in the data, it is clear that a single temperature independent interaction energy cannot be used to describe the systems. It is also possible to calculate excess partial molar entropy from the partial molar entropies given by Panish, et u L , ~ from , ~ the relation
SIE= SI - 3,’
=
Sl
+ R log nl
(3)
These values are also given in Table I. It should be noted that the values are all positive. Thus it mould be of little use to attempt to extend the regular solution theory with any of the orderdisorder theories such as the quasi-chemical theory15 or the expansion method of Kirkwood,16 since these extensions would lead to negative values of the excess partial molar entropy and would result in poor agreement with experiment. In any event, these corrections to the regular solution theory are small since the interaction energy is not large compared to RT. It would appear, from the few systems studied so far, that the behavior of fused (Ag-M)X systems is (15) See, for example, R. Fowler and E. A. Guggenheim, “Statistical Thermodynamics,” Cambridge University Press, New York. N. Y., 1939, Chapter T’III. (16) J. G. Kirkwood, J . Phy8. Chem., 43, 97 (1939).
GEORGE K. BLOMGREN
1502
Vol. 66
TABLE I PARTIAL MOLAR Excmss FREE ENERGIES AND ENTROPIES FOR (Ag-Li)Cl
(Ag-Na)Cl SOLUTIONS~
AND
(Ag-Li)CP BOOC 2240 cal./mole Plg(obstl.) FiE(oalrd.) w =
ni
0,0286 .I05 ,191 ,238
2 14 1.74 1.39 1.20 1.18 0.30 .02 - .05 0
,252
,585 .815 .905 1.000
2.12 1.79 1.46 1.30 1.26 0.38 .08 .02 0
700' 2140 cal./mole FiE(obsd.J PiE(calcd.) o =
2.05 1.74 1.39 1.18 1.14 0.30
8000 2060 cal./mole PIE(obsd.) PIE(oalcd.) w =
2.02 1.72 1.40 1.24 1 20 0.37 .07 .02 0
.04
.01 0
1.95 1.73 1.38 1.14 1.17 0.24 .05 .05 0
9000
1940 cal./mole PIE(ohsd.) PIE(calcd.) w =
1.95 1.65 1.35 1.20 1.15 0.35 .07 .02 0
1.85 1.74 1.36 1.10 I 08 0.25 04 .06 0
&"(e.u.)(ohstl.)
1.83 1.55 1.27 1 13 1 08 0
1.16 0.16 .29 .46 .52 .46 I13 -- .20 0
aa
.07 02
0
(Ag-Na)CP ni
0.0313 .0978 ,209 ,351 .505 ,615 1.000
" Subscript 1
."I
=
700' w = 1120 cal./mole ftE(obsd.) PiE(calcd.)
.. .. ..
0.46 .28 .08 0
800°
9003
w = 940
oal./mole PiE(obsd.) WE ( d c d . )
..
0.91 $76 .62
.. .. 0.47 .27
.23
.01
0
PIE(ob8d.)
0.88 .76 .59 .39 .23 .I4 0
.34
.I7
= 750 cal./mole
0
PIE(ealcd.j &"(o.u.) (obsd.)
0.77 .66 .43 .20 .10 - .06 0
0.70 .61 .47 .31
1.48 1.10 0.75 1.36 0.74 .57 0
.18 .I1 0
AgC1.
1
Fig. 1.-Excess volume in the (Ag-?u'a)NOz system as a function of mole fractions of AgNOa.
fairly insensitive to the nature of the anion and that the trends in the behavior with respect to the other cation are apparently independent of the anion. This can be seen in Table 11, where the interaction energies from regular solution theory are compared for various bromide, chloride, and nitrate solutions. Another measurement of interest concerning silver salt solutions is the density determination of (Ag-Sa)N03 solutions by Byrne, Fleming, and Wetmore.10 The result of converting these meas-
urements into excess volume quantities is shown in Fig. 1. The system shows a rather large positive excess volume, and since the regular solution theory assumes a geometric mean for the volume behavior and the ideal volume is given by an arithmetic mean, the excess volume predicted by regular solution theory is of the opposite sign from the observed and is much too small. The disagreement of excess entropy and volume quantities with regular solution theory for silver salt solutions is not unexpected. This situation is often encountered in non-electrolyte solutions and is generally explained1' in terms of effects arising from the dependence of particle oscillations on the surroundings, which is entirely ignored in the regular solution theory. This sort of explanation, however, presupposes some knowledge of the intermolecular interactions in the system and, a t least in the case of silver salts, very little is known about these interactions. Thus, silver salts are poor examples from the theoretical point of view, although they are very good ones from the experimental electrochemical point of view. TABLEI1 INTERACTION ENERGIES FROM REGULAR SOLUTION THEORY FOR SILVER SALTSOLUTIONS Li w
(Ag- h1)BBr'
w w
(Ag-;M)Cl (Ag-M)N&*
1880
Na
IC
1050 -1480 1940-224W 750-11006 --10004 840
Hh
-2580
Thermal measurements probably are the best means of determining thermodynamic excess properties of simple salts such as the alkali halides, for (17) See I. Prigoginr, "The LIoleculal Theory of Solutiuns," Interscience Publishers I n c , New York, N. Y . , 19b7.
THEORIES OF FUSED SALTSOLUTIONS
August, 1962
1503
which some hope exists for a molecular theory of N A D = NA, NBC:= Nc and A'BD = NB - Xc solutions. The recent work of Kleppa and Hersh' (5) extends the above phenomenological approach, including first-order correction to the regular solution are the number of molecules of the salt AD, etc., theory, in order to explain their heat of solution added to the mixture, since any concentration of data on dkali nitrate solutions. An asymmetry the ions A, B, C, and D can be specified by mixtures in the heat of solution us. mole fraction curre re- of the salts AD, BC, and BD. By definition mains and is empirically taken into account. All 01 = -(WAC WBD - WAD - WBC) of the heats are negative, however, while the heats of the corresponding solid solutions are all positive. A possible explanation of this phenomenon will be discussed in a later section. B. Ternary Solutions.-Ternary or reciprocal salt solutions mill be taken to mean solutions of three salts involving two different cations and two different anions. Blander and Braunstein13+14 have developed a theory based on the quasi- It is clear from eq. 4 that the system has been dechemical approximation which is applicable to scribed in terms of short range interaction only dilute solutions, but this theory can be developed and the long range coulomb forces have not been in more general terms to include interactions not introduced. If the molar volumes of the pure salts considered by those authors for application to con- are equal and if there is no volume change in the centrated 3olutions. In this discussion the nota- system on mixing, there will be no change in the tion of Fowler and Guggenheim15 will be largely lattice const,ant on mixing and the coulomb part followed. First of all, the interactions to be con- can be added on, since sidered will be limited to first neighbors (cationanion interactions) and second neighbors (cationNwc = (NAD hTBc NBD)(JC cation and anion-anion interactions). Thus we can write a table of numbers of pairs and energies where N is the total number of molecules in the for any given configuration of the system where x1 system (half the number of ions) and wc is the is the first coiirdination number of a cation or coulomb energy of a system of point charges. anion, x2 is the second co6rdination number of a Then eq. 4 becomes cation or anion, A and B are the two cations, C and D are the anions, N A is the number of A ions, W = NADXADNBCXBC NBDXBD and WAC/I~, etc., are the (short range) interaction XlWl XZWZ x3w; (7) energies piyr AC, etc., pair for given and fixed interionic separ,btions. The results are given in Table where XAD, etc., are the lattice energies of the pure 111. The total energy of the configuration is given salts defined by
+
+
+
DEFIKITION~ FOR Pair
Energy per pair
No. of pairs 21x1
AD
z i ( i V ~ - SI) Zi(.\TC - XI)
13n bA BB AB
zl(\'B
-
1C
-t
-WAC/Zl -OAD'ZI -lOWgG/Z1
XI)
-wBD/Zl
I/ZZZ(L~A
- s2) 1 / 2 2 2 ( i L ' ~ - Xz)
-WAA/ZZ
z2x2
-W.4B/Z2
- X3) i/2m(V13 - BJ)
- WBB/ZZ
cc
i/Z21(VI:!
-WCG/ZZ
DD CP
22x3
-wDD/Z2 -lUCD/ZZ
+
+ +
Total energy of pans -XIWAG
-(KA -(BC -(SB
- X~)WAD - Sl)OBC - +
x'l)WllD
2\'C
-"/zL\A - ' / ~ ( N B-
XZ)WAA XZ)WBB
-XZWAB -'/z(Nx -'/?(L\'D --3WCD
Xa)wcc Xd)wDD
-
by the sum of the terms of the last column of Table 111, as
XlWl
where
+
TABLE I11 PAIRINTERACTIONS FOR TERKARY SALT SOLCTIONS
AC
1:c
+
+
x2wz
4X , . W 3
(4)
The configurational part of the partition function is given by ,
GEORGEE. BLOMGREN
1504
and the integrations for A and B are carried out over the cation space and those for C and D over the anion space. Some interesting results are obtained from the condition w1
=
Equation 9 becomes Q(T)
=
=
w2
(J3
=
+
0
model for the ideal solution. For the general case where noneof wl, w2, or o3 is equal to zero, new quantities Xi can be defined for mathematical convenience by the equation e-(T,wx
f
+
Vol. 66
+ F ~+PT'x,ws) f . . .
..J'
,-(XIUS
+
XZWZ
(drA)NA
A 4 XswSf/kT JJ
exp [(XADNAD XBCNBC XBDNBD )/kTl N A!NB!Arc!Na!
X f
* *
f
(16)
(d7B)NB(d7c)NC (d7D)ArD From the left hand side of eq. 16 the free cnergy of
(d7A)"A
+
i- XBCNBC XBDNBD) lkT1 - eXp [ (XADNAD N.~!NB!Nc!ND! + NB
"O
'
(10) where Voat and V A are ~ the volumes of the cation and anion space, respectively, defined by ( V c a t ) NA
(d 78) N A
A
(VAn)
solution is given by
A.4 = kT
ND
+ V A= ~ NCVC+ NDVD
Voat = NAVA NBVB
A
NA N A In AT
+ x,wl + x 2 w 2+ Tbwa (17)
By differentiating both sides of eq. 16 with respect to temperature, we find
where VA, etc., are the ionic free volumes. If the ionic free volumes are not very different, the relat'ions =
(VCaJN" + (VAn)Nc+ ND
(NA (Nc
+ N B ) ~+A
N~
V A ~ A U B ~ B
+ ND)Arc4- ND vCNc V ~ N D (11)
are nearly true. From Stirling's approximation
+ + (Nc + ND)Nc+
(NA
NB)NA
=
eNA
= eNc
+NB)! (Nc + ND)!
where by definition 8,f . . . J e - (xiwl + xzwz -k
A
.
f.. f Xle-(X~w~f
[(VB
+
f
ND
(12)
Equation 18 can be simplified to b(Tl/T) b(l/T)
wT2/
oD)e2exAD]NAD
+
NAkT In ____NA N A
NBkT In
x (13)
f NB
+
+ N B + NckT In Nc + ND
lVB
NA
O2
T)
W T ) WlXl
where the last three terms of the product are conventional free volume terms for the pure salts, AD, BC, and BD. Thus, the free energy of solution and entropy of solution are given by =
I-J (dTA)NA (19)
V C ) ~ ~ ~ X [(VB ~ ~ ] ~ VCD B) ~ ~ ~ X B D ] ~ B D
AA = A s
XPWZ X d / k T
A
a1 ____
+
I'I (dTA),vA=
+ Ng ( N A
Substituting eq. 11,12, and 5 into eq. 10
[(VA
Xawa)/kT
Arc
a (23/T)
+w3-----=
+
W T ) wzx2
+us3
(20)
Equatiop 20 can be integrated t o give an equatio_n for the Xi,if an approximate expression for the Xi can be given. An obvious solution occurs if three separate quasi-chemical conditions are used. l5 This mill not be quite correct, since the mixing of cations will be influenced by the cation-anion interaction and the anion-anion interaction and so on, but the approximation may not be too severe. Thus, we have
+
The partial molar free energy of solution is given by dAA WAD where N A is~ Avogadro's number. Equations 14 and 15 give the same results as the Temkin'l
~ AAAD = N A __ ~ - R T l n n ~ n (15)
We could have set the left hand side of any or all of eq. 21 equal to unity and obtained the zeroth or regular solution result. We proceed to integrate eq. 20. Since
THEORIES OF FUSEDSALTSOLUTIONS
August, 1962
8;
1 T
- -+ 0 ase- -+ 0 T
we have, on changing the partial to total differentials
The integrations on the right hand side of eq. 22 have already been done, the first by Blander and Braunstei~il~ and the second and third are the standard quasi-chemical integrations.I5 Performing the integrations and substituting the result into eq. 3 7 gives the free energy of solution. Differentiating with respect to N A D gives the partial molar free energy of solution -
AAAD = RT In n A n D
[
______ (1 - J’:L/NA)] nn
+
+ zlRT In XBRT
[8 2 + nA - nB nA(b
where n A , Ctc., are the ion fractions and
and the activity coefficient is given by
Ag AbrrX\JOj(n’~~)
NaNOo,
+ 1)
I+
1505
which gives for any reasonable energy, and a concentration of NaCl to about mole fraction, a contribution of very nearly unity. It would be interesting to test eq. 25, but unfortunately no system has yet been studied for which the equation would be applicable. Although experimental information is lacking for testing the results of the theoretical development of this section, some insight into some of the approximations involved in applying the conventional lattice theories to molten salt solutions has been gained. It has been shown that, in order to use a theory of nearest neighbor or next nearest neighbor lattice statistics, the molar volumes of the pure liquids and all solutions must have the same value, or else serious difficulties are encountered with the coulomb energy. This result is valid for binary as well as ternary solutions. It also has been shown that the use of concentration cells of the above type with nearly equal silver ion concentrations on either side fortuitously simplifies the theoretical analysis of the data.
111. Molecular Theory of Simple Salt Solutions It %as mentioned in the Introduction that the heat of solution is an interesting property of solutions. This is particularly so from a theoretical point of view. The oscillations of particles in the lattice and the changes which take place in solution affect the excess entropy and volume properties to a much greater degree than the heat of solution. In fact, if one uses a simple square well for the particle potentia,l energy, no effect is manifested in the heat of so1ution.l’ Thus the rigid lattice model can be applied with greater confidence to the heat of solution than to the other properties. Blanderlg recently has calculated the effect on the coulomb potential of one molecule of solute in a linear chain of solvent molecules, implicitly assuming a hard sphere repulsion. While the model gives results in qualitative agreement with experiment, it is difficult to extend it to two or three dimensions and, of course, indicates nothing about the effect of short range repulsive forces. Another approach to the problem is to extend the solid solution theory of Durham and Nawkins20 i o the case of liquid solutions. The only difference between the model of the liquid and the solid is the magnitude of the lattice constant. Hence, we can apply the theory directly by choosing appropriate lattice constants for the solutions. The (Li-K)Cl system was chosen for the calculation for several reasons. First, there are already some crude measurements of the heat of solution in the literatures with which semiquantitative comparisons can be made. The densities of the entire solution range have been accurately measured by Van Artsdalen and Yaffee and the molar volume has been found to be a linear function of concentration to within experimental error. Also, the interionic potential constants for the pure liquids are known from the work of Mayer and Huggins,21-2a (19) M. Blander, J . Chem. Phgs., 34, 697 (1961). (20) G. S. Durham and J. A. Hawkins, zhzd., 19, 149 (1951). (21) J. E. Mayer, ihid., 1, 270 (1933). (22) M. L. Huggins and J. E. Mayer, tbzd., 1, 643 (1933). (23) M. L. Huggins, e’hid., 6, 143 (1937).
GEORGEE. BLOMGREN
1506
Vol. 66
rvhere the cation-anion distance ii = 3.246 A. was obtained by linear interpolation of the molar volume curve. The displacements and displacement energies for repulsion only then were calculated for each configuration and these results are summarized in Table V, where the notation of configurations is the same as Durham and Hawkins'.20 The total displacement energy is calculated to be 2.16 kcal. and subtracting this from the unrelaxed heat of solution gives AH = 1.44 kcal. The experimental value of the heat of solution is, holyever, approximately -0.6 kcal. and thus disagrees in both sign and magnitude lated value. The effect of the co not yet been considered and this may be considerable since the displacements are relatively large (values for the solid solution20 are about 0.03 ?. whereas our displacements are of the order of 0-1 F ) . Durham and Hawkiizszoconsidered thc coulomb effect for anions displaced individually with all other ions a t normal lattice positions and found the effect to be negligible. This effect is also calculated to be small for the liquid solution. However, the neglect of coulomb interactions among displaced particles is not really justified, especially with large displacements, since on the average the chloride ions mill be slightly closer to each lithium ion, and therefore to each other, and slightly farther from each other around the potassium ions. The effect of cation displacements also could be considerable in the presence of large anion displacements. Both of these effects are difficult to account for in concentrated solutions, but the situation is considerably simplified in Ihe dilute solution. In a dilute solution (takcii as 1 mole q0 IiC1 for this calculation), the probability of two lithiun i o i i h lying close together is very small. Hww, t h ~ lithium ions and their environments caii be t r r a t d as being independent. I n general, then, one \ ~ o u l d expert that the environment of each lithium ioii should be somewhat contracted about the lithium. as shown in Fig. 2. It is likely that first neighbor displacements d l iiot be sufficient to describe thc AH = AI? = E, - n L I C l EI,,c~ - ~ I Z EC K~C l system, and for this calculation x e will coiisidw first through fourth neighbor displacements. Tho TABLEI V program for the calculation is to calculate thc hri-1 iirighbor displacements from repulsive interactions, POTWATIALC o ~ ~ a roR s ~(Li-K)Cl s Su~c~nous then the second neighbor ones with the first already 1 = L I - , ~= K + , 3 = C1displaced, and so on through the fourth neighbors. Dipole-dipole21 This has been done using the same repulsion (miCii = 0.073, C',z = 1.61, Czz = 24 3, Cia = 2 0, CB = 48, stants as for the con( tcd solilt ion and ill1 C33 = 118 X erg average anion-cation ce from the molar Dipole-quadrupoleZ1 T-olume ciirve of F = 3.4322 A., with the result that 0.03, dlz 0.85, d22 * 21, d u = 2.1, d n = 73, d ~ z= the first neighbor displacement is 0.13F with an dn 236 X lopi6erg emis energy of -0.055 x 10-12 erg, sccoiid and third Repulsive radii93 neighbor displacemeiits are ncgligiblc (Icss tha71 r1 = 0.570, r2 = 1.235, T ? = 1.435 A,, o = 3 00 -I 0.00.5F) and the fourth neighbor displawrnt~ill is Citiou ltiiioii distarico 0 1Oii with ail energy of -0.038 x IO-" crg rl'hv 4 ' L I C I * 2.909, tkC1 = :3 43'; ' probability factor is now given approximately by 6n~= i 0.06 for both the first and fourth neighFor a 4Q-50 mole % (Li-K)C! solution. me calcu- bors, so that the total relaxation energy is -0.006 x erg, The energy of solution foT the u n late relaxed Inttirc is c~alrnlatedto b t +0.010 X lo-'" A.If = -172.10 U.l(193.!N) 3. O.O(lO4.23) = crg, where cad aid d33 arc iiou tukcti as 1 hc \ a l ~ ~ s crg c m G 3.GO lical./ mole for piire 1
