TEWE~ S T V O L D
1616 calculated molar volumes of the liquid alkali metals a t various temperatures agree almost exactly with the observed values. The most difficult part of applying the significant structure theory to calculate the thermodynamic properties of liquids is to find the equilibrium vapor pressure and the molar volume of the liquid. By using the tangent method mentioned earlier, the equilibrium vapor pressure and the molar volume of the liquid are obtained. The calculated molar volume is then used to calculate all thermodynamic properties. Vilcu and Misdolea omitted this important part and used the experimental molar volume of the liquid alkali metals to carry out their calculations. They in all probability would not have obtained the observed volumes which they introduced had they used their theoretical Helmholtz free cnergy to calculate these volumes.
There are two key steps to take account of in the theory of metals. (1) The volume dependence of the solidlike structure of liquid metals must be taken into account. (2) Because the metal ions move independently of the conductance electrons, they require only about a third as much volume increase as the atoms would to give an alternative position. The model presented goes far beyond anything developed heretofore in giving a proper picture of the important way volume enters into the behavior of solid and liquid metals .
Acknowledgment. The authors wish to thank the Kational Institutes of Health, Grant GM 12862, Kational Science Foundation, Grant G P 28631, and the Army Research-Durham, Contract DA-ARO-D-31124-72-G15, for support of this work.
Thermochemistry of Fused Halide Systems. Enthalpies of Mixing of the Alkaline Earth Halides with the Alkali Halides by Terje 4lstvold Institute of Physical Chemistry, The University of Trondheim, Trondheim, Norway
(Received September 9, 1971)
Publication costs borne completely by The Journal of Physical Chemistry
The enthalpies of mixing of the fused alkali halides except the fluorides with the alkaline earth halides (common anion) have been measured. The results are compared with similar data reported by Kleppa and McCarty for the magnesium chloride-alkali chloride melts and discussed with respect to the following points: (1) comparison with the conformal solution theory of Davis; (2) influence of the common anion on the enthalpy of mixing; (3) concentration dependence of the enthalpy of mixing; and (4) temperature dependence of the enthalpy of mixing. It can be concluded from the present investigation that the enthalpies of mixing at constant temperature and volume, calculated by the conformal solution theory in its simplest form, do not compare with the enthalpies of mixing measured at constant temperature and pressure. A very recent conformal solution theory developed by Davis for charge-unsymmetrical fused salt systems at constant temperature and pressure show that the enthalpy of mixing should be a linear function of the difference in the cationanion distances for the two salts, dl - da, as long as one salt is kept as a common salt. This is in agreement with the present experimentalobservations. The variations in slopes and intercepts at 812 = (di - d2)/dld2 = 0 observed when AHobad is plotted us, Sl2 is explained in terms of the very simple Fplrland and Lumsden models where the change in cation-cation Coulomb and cation-anion polarization energies by the process of mixing, respectively, were considered.
Introduction Since 1960 a great deal of new information relating to thermodynamic properties of binary mixtures of simple fused salts has become available. A significant part of this information has become available by the Work Of “leppa and On the Of AHM, of charge-symmetrical and chargeunsymmetrical fused salts. The Journal of Physical Chemistry, Yol. 76, No. 11, 1972
Among the charge-symmetrical systems studied are the binary mixtures formed by the alkali nitrates,’ alkali fluorides,2 alkali chlorides and alkali bromidq8 alkali iodide^,^ alkali sulfates,6 and the mixtures of (1) 0.J. Kleppa and L. S, Hersh, J . Chem. Phys., 34, 351 (1961). (2) J. L.Holm and 0.J. Kleppa, ibid., 49,2425 (1968). (3) L.5. Hersh and 0. J. Kleppa, ibid., 42, 1309 (1966).
THERMOCHEMISTRY OF FUSED HALIDESYSTEMS silver and thallium chlorides and nitrates with the corresponding alkali salts.E-8 Recently the binary mixtures formed among the alkaline earth chloridess and among the transition metal dichlorides and the binary mixtures of the transition metals with calcium and magnesium chloride were investigated. lo Most recently the binary anion mixtures formed by the alkali halides1' and the alkali metaphospateslZwere investigoted calorimetrically. The enthalpy of mixing data were interpreted in light of the conformal solution theories for fused salts developed by Reiss, Katz, and Kleppa (RKK),13 Blander,14 and Davis and Rice (DR).16 It is, however, interesting to note that the enthalpy of mixing of several of the above mentioned binary mixtures deviates significantly from the theoretical predictions. The difference between theoretical predictions and actual data is most pronounced when the simple theories are applied to calculate the enthalpy of mixing. This is not a t all surprising since these theories neglect the contribution to the enthalpy of mixing from the well known polarization interactions. I n the D R theory, however, these interactions are included as a pertubation to the Coulomb potential. The basic assumption in the D R theory is that the Coulomb part of the pair potential is of significantly larger magnitude than the contribution from the particular short-range interactions under consideration. For the alkali-common halide mixtures it turns out that polarization and van der Waals interactions amount to a significant part of the total enthalpy of mixing. I n their study of the mixtures formed by the alkali sulfates, IQ)stvold and Kleppas studied the effect on the enthalpy of mixing of the doubly charged sulfate anion. They found that the results could be interpreted in terms of the D R theory suitably modified to take into account the higher charge on the anion. Mixtures formed among the alkaline earth halides containing magnesium or mixtures formed among the transition metal chlorides show a much more endothermic enthalpy of mixing than what should be expected from the above-mentioned theories. Papatheodorou and Kleppag~loexplained these positive contributions to the enthalpy of mixing as a consequence of the covalent character of the magnesium and the transition metal chloride melts. Several charge-unsymmetrical fused salt systems have also been investigated by Kleppa and coworkers. Among the systems studied are the binary mixtures of the alkaline earth nitrates with the alkali nitrate^,^^^'^ lead and magnesium chloride with the alkali chlorides, l8,l9 beryllium fluoride with the alkali fluorides,20 and the transition metal chlorides (CdC1z,21MnClZ, FeClZ,C O C I ZNiClZz3) ,~~ with the alkali chlorides. The enthalpy of mixing data were for some of the systems interpreted in terms of the conformal solution theory for charge-unsymmetrical fused salts by Davis.24 The Davis theory is based on Coulomb interactions only
1617 between the ions of the salt mixture. As a first approximation the Davis theory predicts a linear relationship between the interaction parameter, X = AHM/ X I X ~ , and the distance parameter, 812 = (dl - dz)/d&, at constant temperature, volume, and composition. XI and xz are the mole fractions of the two components and dl and dz are the cation-anion distance for the twocomponent salts. The alkaline earth-alkali nitratez4 mixtures and the magnesium-alkali and lead-alkali ch10ride'~~'~ mixtures all give approximate straight lines when the interaction parameter, X, is plotted us. 812 for systems having a common salt. For the transition metal dichloride-alkali chloride mixtures the limiting interaction parameter, X ( X M C ~ ~= 0) was changing linearly with a12 while the limiting interaction parameter X ( X M C ~ ~= 1) deviated somewhat from a straight line relationship for common MClzsystems.22~2a The concentration dependence of the interaction parameter, however, shows a much more complex behavior in the magnesium and transition metal dichloride melts than in the simple charge-unsymmetrical nitrate, chloride, and bromide mixtures involving the calcium, strontium, and barium salts. This is due to the additional forces which are present between the ions in the magnesium and transition metal dichloride mixtures. These forces are covalent in nature and sufficiently strong to modify the thermodynamic properties of the salt mixture as well as the local arrangement of the ions to a significant degree. The beryllium-alkali fluoride melts studied by Holm and Kleppa20 also show a much more complex concentration dependence than the simple charge-unsymmetrical fused salt systems. (4)M. E. Melnichak and 0. J. Kleppa, J . Chem. Phys., 52, 1790 (1970). (5) T.astvold and 0. J. Kleppa, Acta Chem. Scand., 25, 919 (1971). (6) 0.J. Kleppa, R. B. Clarke, and L. S. Hersh, J . Chem. Phys., 35, 175 (1961). (7) 0.J. Kleppa and L. S. Hersh, ibid., 36, 544 (1962). (8) L. S. Hersh, A. Navrotsky, and 0. J. Kleppa, ibid., 42, 3752 (1965). (9) G. N. Papatheodorou and 0. J. Kleppa, ibid., 47, 2014 (1967). (IO) G.N. Papatheodorou and 0. J. Kleppa, ibid., 51, 4624 (1969). (11) M. T.Melnichak, unpublished results. (12) H. C. KO and 0. J. Kleppa, Imrg. Chem., 10, 771 (1971). (13) H.Reiss, J. L. Katz, and 0. J. Kleppa, J . Chem. Phys., 36, 144 (1962). (14) M. Blander, ibid., 37, 172 (1962). (15) H.T.Davis and S. Rice, ibid., 41, 14 (1964). (16) 0.J. Kleppa and L. 8. Hersh, Discuss. Faradag Soc., 32, 99 (1962). (17) 0.J. Kleppa, J . Phys. Chem., 66, 1668 (1962). (18) F. G.McCarty and 0. J. Kleppa, ibid., 68, 3846 (1964). (19) 0.J. Kleppa and F. G. McCarty, ibid., 70, 1249 (1966). (20) J. L.Holm and 0. J. Kleppa, Imrg. Chem., 8 , 207 (1969). (21) G.N. Papatheodorou and 0. J. Kleppa, ibid., 10, 872 (1971). (22) G.N. Papatheodorou and 0. J. Kleppa, J. Inorg. Nucl. Chem., 33, 1249 (1971). (23) G. N. Papatheodorou and 0. J. Kleppa, ibid., 32, 889 (1970). (24) H. T.Davis, J . Chem, Phys., 41, 2761 (1964). The Journal of Physical Chemistry, Vol. 76, No. 11, 107.2
1618
TERJE~JSTVOLD
The purpose of the present work is first of all to present new reliable enthalpy of mixing data for binary mixtures of charge-unsymmetrical fused salts. The second task is to compare these data with the present theories describing charge-unsymmetrical fused salt systems to see if some of the properties of these mixtures can be described within the framework of these theories.
Experimental Section Apparatus. All calorimetric experiments reported in the present work were performed in a single unit microcalorimeter suitable for work up to 1100". Apart from its single (rather than twin) construction, this apparatus is similar to the one used by Hersh and KleppaS at temperatures up to 800". I n the absence of a twin construction, the furnace surrounding the calorimeter is equipped with a Leeds and Northrup proportional temperature controller giving a stable temperature (.tO.l") in the furnace system. The calorimeter assembly is heavily lagged with respect to the furnace so as to avoid as far as possible short-term drifts resulting from slight variations in the controlled temperature. The temperature-sensing device of this calorimeter thermoconsists of a 54 54 junction Pt-Pt-l3%Rh pile, the output of which is amplified by means of a Leeds and Northrup 9835-B DC amplifier and recorded on a Leeds and Northrup Type H-Azar recorder. The emf vs. time curves were integrated by means of an Ott precision planimeter. I n this way the area between curve and base line, which is proportional to the total heat, could be determined with a precision of about 0.3%. All experiments were performed in fused silica containers under an excess pressure of 10 mm of pure dry nitrogen to prevent moisture from entering the system. The container was maintained in the calorimeter inside a fused silica envelope of about 24 mm o.d., 32 in. long. The lower 10 in. of this envelope was reduced in diameter to 22 mm 0.d. to fit snugly inside the calorimeter proper. At the top of the envelope was a gas inlet and a female standard-taper 29/42 ground joint. The male part of this joint supported a "charging and mixing device." This consisted of an outer fused silica tube of 9 mm 0.d. with six blown radiation shields and a central manipulation tube of 6 mm 0.d. The latter was used to move the break-off tube up and down. Its bore served as a channel for the platinum-drop calibrations and as a gas inlet. The fused silica break-off tubes, which were connected to wire, the manipulation tube with a thin wcre E 1 0 mm 0.d. and 5-7 in. long. At the bottom of each there was a fine breal
1 1 2 d ~ e 1 ~2d~e11x (4)
where e is the electronic charge and d's with suffixes are the interatomic distances. In Table X estimated values of AEMx,' and AEMx?/A8lz are given for different reactions of the above type. It is apparent from these values that the changes in Coulomb interaction when one alkali ion is substituted with another in a binary alkali-alkaline earth halide mixture will tend to give the Davis plot (A vs. 812) a negative slope, that is a negative value of the constant B in eq 3. This slope will get more negative when the radius of the common alkaline earth ion is decreasing. Figure 10. Interaction parameter, AHM/x(1 - z), in the alkali-alkaline earth chloride (bromide) mixtures us. the distance parameter, SIz, at zhtx2 = 0.333. The data for the magnesium-alkali chloride melts are from a paper by Kleppa and McCarty.
Table X : The Change in Coulombic Energy Occurring in a Binary Alkali-Alkaline Earth Halide Mixture When One Alkali Ion Is Substituted by Another Alkaline earth chloride MClz
MgCh
CaClz
SrCll
BaClz 6,2.102,~-'
Alkali1 chloride MerCl
Alkali11 chloride MerICl
LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl
NaCl KCl RbCl CSCl NaCl KCl RbCl CSCl NaCl KCl RbCl CSCl NaCl KC1 RbCl CSCl
AEI\IOI~~I B~A~IZ
-0.029 -0.60 -0.071 -0.085 -0.022 -0.046 -0.056 -0.068 -0.019 -0.041 -0.050 -0.061 -0.016 -0.036 -0.042 -0.052
-0.55 -0.62 -0.64 -0.66 -0.42 -0.48 -0.50 -0.53 -0.36 -0.44 -0.45 -0.48 -0.30 -0.37 -0.38 -0.40
Figure 11. (See caption to Figure 10 for details.)
energy following the substitution due to the change in cation-cation repulsion may be illustrated by a model introduced by F ~ j r l a n d . Let ~ ~ us consider the following substitution reaction
In addition to the change in Coulomb interaction occurring when an alkali ion is substituted by another according to the above-mentioned reaction, there will also be a change in polarization of the common anion due to the change in the cation environment. F@rland2' and Lumsden31 pointed out that an anion Xwould become polarized by the unsymmetrical electric field due to the different size and charge of the two cations on opposite sides of the anion (Figure 12). In the group (M*+X-Me+) the anion X- is subjected to an electric field
F According to the FZrland model illustrated in Figure 12 the change in Coulombic energy following the above substitution may roughly be described by the equation The Journal of Physical Chemistry, Vol. 76,No. 11, 1973
=
e ( 2 / d ~-~l/dMeX2) ~
If a is the polarizability of the anion, the polarization energy of a (M2+X-Me+) group is (31) J. Lumsden, Discuss. Faraday Soc., 32, 138 (1961,).
1625
THERMOCHEMISTRY OF FUSED HALIDE SYSTEMS
E’ = -(rF2/2
=
---( 4%~- d M x ) 2 ( d z x -k dMx (ye2
2
dM eX&X
dnn d M x
The change in polarization energy following the above reaction can thus be approximated by the expression AEMXZ’N - ffe2{ 1/d&reI1x4- 1/dMe1X4 ~ / ~ M x ’ ( I / ~M l/dMe11X2) ~Ix~ ]
(5)
Let us now consider the reaction
+
2 (Me +X-R!tI +)mix (1) (MII +X-M 11 +)pure (1) = 2(Me+X-M1l2+)mix(l) (R/I12 +X-M12+)pure(l)
+
with the change in energy
- ~ E M ~ -I-X EM~XMI M I - EMIIXMII
UhleX = ~EM~XMII
According to the Fqirland model illustrated in Figure 12 the change in Coulombic energy following the substitution may roughly be described by the equation
The polarization energy of the pure salts can be neglected. In Table XI estimated values of the change 4 in polarization energy, AEMX?, following the sub‘v e2{dMax dMIrx stitution are given together with the contribution of 4 this energy change to the Davis plot slope, A E M X ~ / A ~ E . + - -4It is apparent from the data in Table X I that the dMeX f dMIx MIX ~ ~ M I I X(6) change in polarization energy following the substituwhere e is the electronic charge and d’s with suffixes tion tend to give a negative Davis plot slope. This are the interatomic distances. Estimated values of slope is increasing with increasing size on the common A E M e X C and AE~~x~/A812 for different alkali-alkaline alkaline earth ion. earth chloride mixtures show that the changes in Coulomb intereaction when substituting one alkaline earth ion by another in a binary alkali-alkaline earth Table XI : The Change in Polarization Energy Occurring halide melt will tend to give negative Davis plot slopes in a Binary Alkali-Alkaline Earth Halide Mixture When which are decreasing when the size of the common One Alkali Ion Is Substituted by Another alkali ion is increasing. Alkaline The change in polarization energy which follows the earth Alkali1 Alkali11 substitution of one alkaline earth halide by another chloride chloride chloride AEMCl~/aeZ, AEMCIP/ MCla MeICl MeIICl A-1 ae*A&e can be estimated in the same way as outlined previ- 0.0146 ously. The change in polarization energy following NaCl -0.28 MgClz LiCl -0.0274 KCI -0.28 LiCl the above reaction can be approximated by the exRbCl -0.0315 LiCl -0.28 pression
1
CaClz
SrClz
BaClz
LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl LiCl
CSCl NaCl KC1 RbCl CSCl NaCl KC1 RbCl CSCl NaCl KC1 RbCl CSCl
- 0.0367
-0.0084 -0.0166 -0.0195
- 0.0231 - 0.0065
- 0.0134 -0.0157 - 0.0188 - 0.0040 - 0.0089
-0.0107 0.0132
-
-0.28 -0.16 -0.18 -0.18 -0.18 -0.12 -0.14 -0.14 -0.14 -0.08 -0.10 -0.10 -0.10
From the above calculations we can conclude that the Davis plot slopes for the common alkaline earth systems should increase in the sequence Mg, Ca, Sr, Ba. This is indeed experimentally observed. Figures 8 and 9 show the interaction parameter, A, a t X M X ~= 0.333 as a function of the distance parameter, 812, for the alkali-alkaline earth chloride and bromide mixtures. Except for the lithium-alkaline earth halide systems, the data are well represented by a Davis plot for each common alkaline earth halide subsystem. As observed from these figures the Davis plot slopes are increasing, that is getting more positive, when the radius of the common alkaline earth ion is increasing.
AEM~x’N -ae2{4/d~,,x4- 4 / d ~ ~-x ~
MIX^))
~ / ~ M ~ x ~ (~ / ~ M I I x(7) ~
Estimated values of A E M ~ ~ Xand ’ AEMeXP/A612 for different substitutions of the above type show that the change in polarization energy following the substitution will tend to give negative Davis plot slopes which are decreasing with increasing size of the common alkali ion. In Figures 10 and 11 the interaction parameter, A, is plotted vs. 812 at X M X ~= 0.333. As observed from
AEc
E
4
e IdMeX -+ dMX
- - -I
2dMeX
-2dMX I4
Figure 12. Change in cation-cation repulsion energy, AEC, by the process of mixing. The Journal of Physical Chemistry, Val. 76, N o . 11, 1 W B
TERJEPSTVOLD
1626 these figures the Davis plot is a good approximation for each common alkali halide subsystem. For the common alkali salt mixtures the slopes of the lines are decreasing with increasing size on the common cation in agreement with the above calculation. From Figures 8-11 we can observe that the enthalpy interaction parameter a t 612 = 0 for the common-salt subsystems is increasing when the radius of the common-salt cation is increasing. The only exceptions are the common lithium and common magnesium systems. Let us consider the change in energy due to Coulomb repulsion between next nearest neighbors and due to changes in the polarization of the common anion for the mixing process Rile+X-Me+
E -6.0 -
3
+ M2+X-M2+ = 2Me+X-M2+
When dMex = dMx = d, the change in Coulomb energy according to the Fqjrland model will be
0 20 40 6.0 6,ilOz,A”
-6.0 -40 -2,O
-
Figure 13. Interaction parameter, AHM/x(l z),in liquid mixtures of strontium-alkali halides us. the distance parameter, &a, at xsrxz= 0.5.
and the change in polarization energy according to Lumsden will be
(9)
It is apparent from these equations that the energy of formation of the binary reference mixture will become more negative when the common-salt cation radius is decreasing. At small cation radii, however, anionanion repulsion will become significant and contribute to a positive enthalpy of mixing. This is probably the case for the common lithium and common magnesium systems where an increase in the enthalpy of mixing of the reference mixture (812 = 0) compared to the enthalpy of mixing of the common sodium and common calcium systems, respectively, is observed. Influence of the Common Anion on the Enthalpy of Mixing. When we change the anion in a strontiumalkali halide mixture from chloride to bromide and from bromide to iodide, the enthalpy of mixing is decreasing. For all systems the enthalpy interaction parameter, X, is a linear function of 812 with the usual exceptions for the lithium containing mixtures. I n Figure 13 the interaction parameter at the 50-50 composition is plotted us. 8~ for the strontium containing melts. Two important features with this plot should be noted: (1) the enthalpy of mixing is decreasing in the sequence common chloride, common bromide, common iodide; (2) the slope of the A us. 812 plot is also decreasing in the above sequence. The entropies of mixing for mixtures involving strontium and barium do not deviate much from the ideal entropy of m i ~ i n g . 3 ~The relatively simple nature of these melts therefore indicates that the variation in the enthalpy of mixing with the common anion The Journal of Physical Chemistry, Vol. 76, No. 11, 1978
ought to be explainable by a model taking into account Coulomb and polarization interactions between the ions in a fairly simple manner. Let us consider the energy change which follows the substitution of one alkali ion by another in the liquid strontium-alkali halide melts. Following the arguments presented previously, the results given in Table XI1 are obtained. It is apparent from these results that the slope of the Davis plot lines should decrease with increasing polarizability of the common anion. The change in size of the common anion does not change the value of the constant B to any significant extent. This is true also for the Davis plot intercept, A . According to eq 8 and 9 this intercept should be roughly equal to a constant plus a term proportional to the polarizability of the common anion. The following values of X a t = 0 are taken from Figure 13: XCI(612 = 0) = -2.4 kcal/mol, X ~ ~ - ( 6 ~=2 0) = -2.7 kcal/mol, and = 0) = -3.3 kcal/mol. These values show, as predicted, a straight line relationship when plotted vs. the polarizability of the common anion. The dependence of the enthalpy of mixing on the common anion is more complex for the magnesium and calcium mixtures. This is not surprising since these mixtures contain cations with considerable difference in properties. The anion-cation association in the magnesium halide-alkali halide melts, for example, will undoubtedly be dependent on the size of the common anion. This size effect will contribute to a change in the enthalpy of mixing going from one common anion system to another. (32) T.Idstvold, J . High Temp. Sei., 4, 61 (1972).
THERMOCHEMISTRY OF FUSED HALIDESYSTEMS
1627
Table XII: The Change in Coulombic and Polarization Energy Occurring in a Binary Strontium-Alkali Halide Mixture When One Alkali Ion Is Substituted by Another Common salt, SrXn
Alkali1 halide MeiX
Alkali11 halide MeIIX
SrCln
LiCl
SrBr,
LiBr
SrL
LiI
NaC1, KC1 RbC1, CsCl KaBr, KBr RbBr, CsBr NaI, KI RbI, CsI
AECsrx21 2AEPs,xzl elA61n ezAh (average values)
-0.43
-0.41
-0.4.5
-0.56
-0.48
-0.64
We will not at the present stage try to put forward any detailed analysis of how the common anion influences the enthalpy of mixing in these systems. With additional structural information, like X-ray and/or Monte Carlo radial distribution functions, however, a possible explanation for the difference between the chlorides and the bromides might be found. The Concenbration Dependence of the Enthalpy Interaction Parameter. I n a previous paper,32we discussed the concentration dependence of the enthalpy and entropy of mixing in the binary magnesium chloride-alkali chloride, a few calcium chloride-alkali chloride, and a few strontium and barium chloride-alkali chloride melts. It was apparent from this discussion that no simple quasi-lattice models could account quantitatively for the concentration dependence of these thermodynamic functions. The interaction parameter, A, was changing with concentration, and the change was dependent on the properties of the ions constituting the binary mixture. The enthalpy interaction parameter was varying considerably with concentration in systems where “complexing” occurs, but when the ionic potentials of the two cations are comparable the concentration dependence of X was approaching the regular solution behavior, namely a constant. For most of the binary charge-unsymmetrical systems investigated, however, there seems to be at least some regularity when we compare one system with another. For 28. systems among the 32 charge-unsymmetrical fused alkali halide-alkaline earth halide mixtures investigated, the partial enthalpy of mixing of an alkali halide in the pure alx 0), is more positive kaline earth halide, ARM.x( ~ ~ 1 .--t than the partial enthalpy of the alkaline earth halide in ( ~ M3x 0). ~ Note that the pure alkali halide, An&tx2
A1g1(x1+-0) = lim AH‘/x1x2
(10)
Xl+O
The only exceptions are the {Mg-Cs(Rb) \Cl and the { Ca-Cs(Rb) ]Br mixtures. We can also observe from Figures 1-6 that the interaction parameter, A, is increasing with increasing content of alkaline earth halide in binary mixtures where the cation properties are not too different. These effects may be elucidated by the following argument.
The volumes of most fused salt mixtures are roughly determined by the anions. The density of cations in charge-unsymmetrical salt mixtures will therefore decrease with a decrease in concentration of lower-valent cations. This may be described as a change in the anion-cation nearest neighbor coordination number. From the assumption that the anion-cation coordination number is determined by the number of anions follows that the coordination number of nearest cations in pure MeX is two times the coordination number of nearest cations in pure MXz. Let us further assume that in a mixture the number of nearest cations, z , varies with mixture composition according to the expression
where zo is the coordination number of nearest cations in pure MeX, and n, is the number of moles of component i. The enthalpy of mixing is roughly equal to the difference between the repulsion energies in the mixture and the repulsion energies in the pure components. On the basis of these very questionable assumptions, it can easily be shown that
+
AHMN ( 2 n ~ ~ z% M e X ) X ’ ~ I x & ’ M e x b
(12)
where the primed x denotes equivalent fraction. Since b is negative for almost all the mixtures investigated, we can easily verify that
ARM,X(ZM~X +0 ) > ARMX~(XMX~ +0 ) (13) From eq 12 we can further observe that AHM/x1x2 is increasing when x~ is decreasing. The Temperature Dependence of the Enthalpy of Mixing. For a few binary charge-unsymmetrical fused-salt systems the temperature dependence of the enthalpy of mixing was investigated. Some variation with temperature was observed in the CaClz(Brz)CsCl(Br) , CaCl2-RbC1, and SrBrz-CsBr mixtures (see Figures 2, 3, and 5) while no significant change with temperature was found in the enthalpy of mixing for the MgBr2-CsBr and SrBrz-KBr(RbBr) mixtures. Papatheodorou and KleppaZ2 discussed the observed temperature dependence in the enthalpy of mixing for the alkali chloride-transition metal chloride melts in terms of the covalent character of the cation-anion bonds. They found a satisfactory agreement between observed and predicted temperature dependences. The present investigation shows enthalpies which are changing with temperature for the CaC12(Brz)-RbCl(Br) and CaCl?(Brz)-CsCl (Br) systems mainly. These systems have an intermediate compound of the type CaMeXs with a melting point higher than the melting point of the pure salts. The vapor pressures of the rubidium and cesium salts are increasing fairly rapidly with temperature. The range of temperature variation The Journal of Physical Chemistry, Vol. ‘76,No. 11, 1978
1628 was accordingly below the melting point of the intermediate compound. When mixing the pure salts, CaRleXa crystals might form. Due to the experimental arrangements these crystals might not dissolve in the fused mixture resulting in an incorrect enthalpy of mixing. Even so we believe that the difference between the enthalpies of mixing measured at the various temperatures reported in the present investigation is significant. For the calcium-cesium and calcium-rubidium halide melts the tendency for association between the divalent cation and the anion is much “weaker” than in the analogous magnesium mixtures, but definitely stronger than in the corresponding strontium and barium melts. We can observe from Figures 5 and 2 that some of the calcium melts exhibit weak minima in X around the 50-50 composition. These minima indicate a tendency toward a special stability of the melts at this composition. The association or ordering of the ions around this composition is so “weak,” however, that an increase in temperature, resulting in higher thermal energy, will tend to randomize the ions in the mixture and thus increase the enthalpy of mixing. When the tendency for association is even “weaker” than in these calcium mixtures, no temperature dependence of the enthalpy of mixing will be observed. When the tendency for complexing is greater as in the magnesium case, the association equilibrium will not be shifted by a small change in thermal energy since the energy of association or complexing is much greater than RT thus resulting in no variation in the enthalpy of mixing with temperature. These arguments are in
The Journal of Physical Chemistry, Val. 76,N o . 11, 1072
TERJE~ S T V O L D fair agreement with the experimental observations. However, since the present data are encumbered with the above-mentioned experimental uncertainties and since no systematic information is available on the temperature dependence of enthalpies of mixing of simple fused-salt systems, we do not want to carry these arguments any further. I n summary, the enthalpies of mixing of several alkali-alkaline earth halide mixtures have been experimentally determined and compared with conformal solution theory calculations performed by Davis. It can be concluded that the enthalpy of mixing is changing linearly with the distance parameter, 812, in agreement with theoretical predictions for a series of binary mixtures with one common salt. The variations in slopes and intercepts at 612 = 0 of these linear relationships are elucidated in terms of the very simple Ffirland and Lumsden models where the change in cation-cation Coulomb and cation-anion polarization energies by the process of mixing, respectively, were considered.
Acknowledgments. This work has been performed in the laboratories of Professor 0. J. Kleppa at The James Franck Institute, University of Chicago. I want to thank Professor Kleppa not only for providing the necessary facilities during my one-year stay at the institute, but as much for his interest in my work and his guidance. This work has been supported by the National Science Foundation Grants (GP-7782 and GP-14064) given t o Professor Kleppa. The spectrochemical analyses were performed by Miss R4. C. Bat chelder .
