CHEMICAL POTENTIAL IN CHARGE-UNSYMMETRIC MIXTURES OF MOLTEN SALTS
4383
Chemical Potential Interaction Parameters in Charge-Unsymmetric Mixtures of Molten Salts’
by J. Braunstein,* K. A. Romberger, and R. Ezell2 Reactor Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 87830
(Received April $0,1970)
The applicability of conformal ionic solution theory to the interaction parameters for chemical potential in charge-unsymmetric molten salt mixtures is demonstrated. The experimental interaction parameter, lim (palkali halideE/z2), evaluated for mixtures of alkali fluoride with alkaline earth fluoride or of alkali chloride x-0
with alkaline earth chloride, is shown to vary linearly, as predicted by theory, with the ion size parameter 6 = (dl - dz)/dldz, where dl and dz are the sums of cation and anion radii of the univalent and divalent salts, respectively. Mixing enthalpy dependence on 6 is rationalized by a simple physical model.
Introduction Kleppa, et ~ l . , have ~ ! ~ demonstrated the utility of the interionic distance parameter 6 = (dl - d2)/dld2 in correlating enthalpies of mixing in charge-unsymmetric molten salt mixtures, e.g., mixtures of alkali and alkaline earth salts with a common anion. dl and d2 are the sums of cation and anion radii in the pure univalent and divalent salts, respectively. The correlation has a theoretical basis in the conformal ionic solution theory (CIS), extended to charge-unsymmetric mixtures by Davis,6 predicting that the limiting molar excess free energy or enthalpy, A , should have the form lim’
a!+od1
AAE
- 2)
=
A2E(x
=
0) = [DJS
+u
(1)
where x is the mole fraction of divalent salt; D and U are combinations of unevaluated integrals occurring in the configurational partition’ function and its derivatives and contain the temperature and pressure dependence (and are different for free energy or enthalpy). For charge-symmetric salt mixtures, the first-order term in 6 vanishes, and CIS predicts that the molar excess thermodynamic quantities are proportional to the square of
lated to differences of ionic potentials for several chargeunsymmetric fluoride mixtures. plE is the excess chemical potential of solvent (univalent) and x is the mole fraction of solute (divalent). I n this paper we demonstrate for the first time the applicability of CIS to chemical potential interaction parameters in charge-unsymmetric systems. We show that experimental values of [plE/x2IZ= o vary linearly, as predicted by theory, with the conformal ionic solution parameter 6 for the alkali-alkaline earth fluoride systems considered by Holm, four additional ones, and several chloride systems. It is furthermore not necessary to assume the symmetric solution relation for the excess free energy, GE = az(1 - x), which would imply [p2E]z=0= [p1E/x2]Z,0, in order to derive the CIS correlation for the interaction parameter. Davis’ first-order expression for the composition dependence of the free energy of mixing, from which he derives eq 1 by a limiting process, contains quantities ( 0 1 2 and a12)relating to the configurational partition function and its derivatives, for the mixture and for the pure salts (see reference Sb, eq 2.25). By developing the Taylor series expansion of the first-order expression in powers of the solute concentration, it follows readily that [ p 1 E / s 2 ] x _ 0 should have the form of the right-
* To whom correspondence should be addressed. For the most part, curiously, the CIS correlation has been tested only with the excess enthalpy. This has been the case even when free energy data were available for a more general test and enthalpies were derived from temperature coefficients of emf rather than calorim e t r i ~ a l l y . ~Guionlsa. however, has demonstrated the fit of the charge-symmetric expression to excess free energies in binary univalent nitrate mixtures. Prior to Davis’ extension of CIS to charge unsymmet,ric systems, HolmEbcorrelated chemical potential interaction parameters, [p1E/x2]x,0, with an empirical parameter re-
(1) Research sponsored by the U. 8. Atomic Energy Commission under contract with the Union Carbide Corporation. (2) ORAU Summer Student Trainee from University of Arkansas, 1969. (3) J. Holm and 0. J. Kleppa, Inorg. Chem., 8 , 207 (1969). (4) 0. J . Kleppa and F. G . MoCarty, J. Phys. Chem., 70, 1249 (1966). (5) (a) K. D. Luks and H. T . Davis, Ind. Eng. Chem. Fundam., 6, 194 (1967); (b) H. T. Davis, J. Chem. Phys., 41, 2761 (1964). (6) H. Reiss, J. L. Katz, and 0 . J. Kleppa, ibid., 36, 144 (1962). (7) G. D. Robbins, T. FZrland, and T. Ostvold, Acta Chem. Scand., 22, 3002 (1968). (8) (81) J. Guion, J. Chim. Phys., 64, 1635 (1967); (b) J. Holm, Electrochm. Acta, 11, 351 (1966).
The Journal of Physical Chemistry, Vol. 74, No. $6,1970
J. BRAUNSTEIN, E(. A. ROMBERGER, AND R. EZELL
4384 hand side of eq 1. We also present in this paper a heuristic physical rationalization of the form of eq 1 for the enthalpy or energy.
Results Data on chloride mixtures were reported by Robbins, et ~ l . from , ~ emf measurements. Data on fluorides were taken mostly from the accurate cryoscopic measurements of Cantor9 with NaF as the solvent. Data on the LiF-BeF2 system were obtained by us from new measurements of the LiF liquidus which will be discussed more completely in a subsequent publication. The liquidus temperatures and compositions were obtained from discontinuities, on initiation of LiF precipitation, in the slopes of emf-temperature plots at constant overall composition, and of isothermal emf-composition plots. A concentration cell with transference, with beryllium electrodes, was, employed.
I LiFl
Be BeF2iBeF2 LiFi Be I I1 This cell, used in the determination of transference numbers, has been described previously.lo* The new liquidus data are presented in Table I and supplement previous data in this system.”JC Table I : Liquidus Temperature and Compositions in the LiF-BeFz System between 0.12 and 0.22 Mole Fraction BeFg Liquidus temperature,
Mole fraotion BeFz (zB~F~)
O C
Equilibrium Saturating Phase: 0.1188 0.1215 0.1371 0.1404 0.1574 0.1625 0.1853 0.1894 0.2053 0.2066 0.2140 0.2220
LiF 784.7 782.9 770.6 767.9 753.2 747.9 723.9 718.0 699.5 698.0 688.2 678.0
Excess chemical potentials of LiF ( p ~ ati tempera~ ~ ture T) were calculated from the data in Table I by means of the eq 3. p
~ = i-RT ~
C(T,
In~ (1 - x)
+L
- T + Tln
The heat of fusion, L, melting point, Tm, and heat The Journal of Physical Chemi;wtry, Vol. 74, No. $6,1970
capacity of solid and liquid LiF were taken from the JANAF tables.” The difference of heat capacities bT. was fitted by the expression CP1 - Cpa = C The interaction parameter was obtained by extrapolation to x = 0 of a plot of ~ I , ~ F ~us./ x2. x ~ This plot, which covered a broader range of compositions than is usual in the calculation of interaction parameters, appeared linear and showed less scatter than plots in which the abscissa was x, thus suggesting the form F L ~ F=~ ax2 bx4 over the rather extended range of the measurements. Graphical extrapolation was employed rather than a least-squares analysis in order to give higher weight to compositions close to the extrapolated limit. The interaction parameters of NaF in the systems reported by Cantor9 were obtained by short extrapolation to x = 0 of plots of pLiFE/x2 US. x or x2. (No apparent difference in linearity or scatter was evident in the plots US. x or x2 in this system, possibly because the range of the measurements was not sufficiently large to distinguish between cubic and quartic terms in pLipE. I n any case the uncertainty of the extrapolated interaction parameter was within the uncertainty of the correlation, about 0.5 kcal/mol.) The excess chemical potentials of alkali fluoride in CsF-BeF2, RbF-BeF2, and KF-BeF2 were estimated from phase diagrams,12thus having greater uncertainty than the cryoscopic results obtained at temperatures closer to the melting point, but they serve to extend the range of values of 6 in the correlation. The heat capacity terms in eq 2 were neglected, and the interaction parameter was taken as the value of pa1k.FE/X2 at about 10 mol % solute. For the chlorides, the interaction parameters were calculated from the reported coefficients in expressions for the excess free energy of mixing.7
+
+
Discussion Figure 1 shows the interaction parameters plotted vs. S for the chloride systems and for the fluoride systems. Pauling radii1*of the ions were used to calculate dl and d2. Thus the conformal ionic solution theory provides a useful correlation of excess thermodynamic functions in charge-unsymmetric molten salt mixtures, not only for enthalpies but for chemical potentials as well. The chlorides show considerably more scatter (9) S. Cantor, J . Phys. Chem., 65, 2208 (1961). (10) (a) K. A. Romberger and J. Braunstein, Inorg. Chem., 9, 1273 (1970); K. A. Romberger and J. Braunstein, MSRP Semiannual Progress Report, Feb 28, 1970, ORNL-4548, p 161; Aug 31, 1969, ORNL-4449, p 138; (b) H. A. Oye, Acta Chem. Scand., 18, 361 (1964). (c) R. E. Thoma, H. Insley, H. A . Friedman, and G. M. Hebert, J.Nucl. Mater., 27, 166 (1968). (11) “JANAF Thermochemical Tables,” D. R. Stull, Ed., PB 168 370, Clearing House for Federal Scientific and Technical Information, Springfield, Va., Aug 1965. (12) R. E. Thoma, Ed., “Phase Diagrams of Nuclear Materials,” ORNL Report No. 2548, 1959. (13) L. Pauling, “The Nature of the Chemical Bond,” 3rd ed, Cornell University Press, Ithaca, N. Y., 1960, p 514.
CHEMICAL POTENTIAL IN CHARGE-UNSYMMETRIC MIXTURESOF MOLTENSALTS 46
-
12
0
N
I
\
s
4
w :*
I
I
0
-4 -008
-004
0
004
008
6
0.12 (A-‘)
016
020
024
028
Figure 1. Interaction parameter for the limiting excess chemical potential of alkali halides in common anion mixtures of alkali and alkaline earth halides us. the ion size parameter. Arrows a t top and the symbols 0 correspond t o chlorides. Arrows a t bottom and the symbols 0 and 0 refer t o fluorides.
4385
Finally, we would like to point out that application of eq 1to the enthalpy of charge-unsymmetric mixtures can be rationalized physically in a simple if nonrigorous manner with the Fgrland mixing model.16 This simple Coulombic model suggested proportionality of enthalpy interaction parameters in charge-symmetric mixtures to a parameter related to a2, but has not been applied previously to charge-unsymmetric mixtures. l7-I9 Consider the change of Coulomb repulsion energy of a pair of cations separated by an anion for the mixing of a univalent salt A+D- with a divalent salt of the same anion, M2+Dz-. (A+D-A+)
+ (M2+D-M2+)+2(A+D-M2+)
The energy change per pair of mixed triplets for the simplified process illustrated in Figure 2 is
Setting the cation charges, Z1and Zz, equal to 1 and 2 and rearranging
Figure 2. Simplified representation of change of next-nearest neighbor cation coulomb repulsion energy on mixing two salts of differing charge type with a common anion.
than the fluorides, although a smaller range of values of 6 is covered and although the excess enthalpy data for the chlorides showed little scatter.’ Although a quantitative accounting of the apparently much higher slope of the chloride interaction parameters is not yet feasible, the greater polarizability of chloride (than fluoride) may be a contributing factor. The unusual properties of BeF2-rich mixtures,3*l o l l 4 related to the network structure of BeFz and its breakdown by alkali fluorides, are not manifested in the effect of BeFzin dilute solution on the chemical potential of the alkali fluoride solvent. By contrast, interaction parameters for transition metal fluorides (Cu2+,Fe2+,Ni2+,Co2+)16in iYaF show excess stability which may be related to the d electrons and ligand field effects.
The principal variation in this equation is in the term (dl - dz). The first term on the right-hand side is constant for a given univalent solvent, although its magnitude is much higher than the observed energies for mixtures of salts for which dl - dz = 0. The simple physical model thus predicts an excess energy and enthalpy of form similar to the result (eq 1) derived more rigorously from conformal ionic solution theory. (14) B. F. Hitch and C. F. Baes, Jr., Inorg. Chem., 9 , 201 (1969). (15) 8. Cantor and W. T . Ward, J . Phys. Chem., 67, 1868 (1963). (16) T. Fgrland, “Thermodynamic Properties of Fused Salt Sys-
tems,” in “Fused Salts,” B. R. Sundheim, Ed., McGraw-Hill, New York, N. Y., 1964, p 77; J. Braunstein, “Ionic Interactions: Dilute Solutions to Molten Salts,” S. Petrucci, Ed., Academic Press, New York, N. Y . , in press. (17) NOTEADDEDIN PROOF. A recent work18 containing an application of the ionic model has come to our attention. That work also discusses the polariaation energylg but fails to point out the relation of the model to the equations of conformal ionic solution theory. Also, it assumes regular solution behavior by neglecting the differencebetween enthalpy and free energy interaction parameters. (18) C. Vallet, Thesis, Universitk d’Aix Marseille, Marseille, France, 1970. (19) J. Liimsden, Discuss. Faraday Soc., 32, 138 (1961).
The Journal of Physical Chemistry, Vol. 74, No. 26,1970
