J. GUION, D. HENGSTENBERG, AND M. BLANDER
4620
Association in Mixed Alkali Halide Vapors by J. Guion,’ D. Hengstenberg, and M. Blander Science Center, Aerospace and Systems Group, North American Rockwell Corporation, Thousand Oaks, California 91960 (Received J u l y 1 , 1968)
Vapor density measurements on the six binary alkali halide mixtures NaBr-KBr, NaBr-RbBr, NaBr-CsBr, KBr-CsBr, NaCl-CsCl, and KC1-CsCl, were made in a liquid gold isotensiscope. Association constants for the vapor species NaKBrz,NaRbBrz, NaCsBrz, KCsBrz, NaCsC12,and KCsClz calculated from these measurements did not fit a relation derived from simple conformal theory (eq 2). Our preliminary calculations indicate that interactions not included in the conformal theory as cation-cation and anion-anion van der Waals interactions and high-order terms due to the ion-induced dipole interactions are probably significant. The energetic factors involved in forming these molecules appear to be more complex than we anticipated. This implies that the analogous factors in molten salt mixtures are also more complex and difficult to understand than is presently believed. Introduction In this paper we present vapor density measurements of the six binary alkali halide vapor mixtures, NaBrKBr, NaBr-RbBr, NaBr-CsBr, KBr-CsBr, NaC1CsC1, and KC1-CsC1. The vapor densities are consistent with the presence of alkali halide dimer molecules in the vapor as well as with the vapor species IYaKBrz, NaRbBrz, n’aCsBrz, KCsBrz, NaCsClz, and KCsCL. From our measurements we derive association constants for these six species, which are compared with a function of the interionic distances predicted from dimensional considerations. The thermodynamic study of compounds formed in vapors of ionic salts can provide data which will aid in understanding the fundamental interactions between ions. Because of the relative simplicity of the vapor species formed, one might expect that with association constants for a sufficient number of systems one would be able to unravel the different types of interactions which are significant in forming these associated molecules. In this paper we will give a preliminary discussion of this point. Many studies of one-component alkali halides have been madez-5 and have been discussed theoretically.2* These studies have given considerable insight into the significant interactions between ions. Few binary alkali halide mixtures have been studied q u a n t i t a t i ~ e l y . ~ ~Of~ the J six systems we have investigated, the two binary chloride systems NaCICsCl and KC1-CsC1 have been investigated previously.8 One of the purposes of this work was to test the validity of a relation derived in a dimensional analysis of associations in binary vapors.6 For the equilibrium
+
AzXz BzX2 the theory led to the relation
RT In (K/4) = M The Journal of Physical Chemistrg
2ABXz
(1)
(2)
where M is a function of temperature and of the anion. Mass spectrometric data for binary fluoride mixtures fitted eq 2 with M ‘v 100. I n this paper, we will compare our experimental results on chloride or bromide mixtures with this expression and with computer calculations of the energetics of these equilibria. Because of an apparent discrepancy in the dimerization constants for NaBr,3J we have repeated the measurements of Datz, Smith, and Taylor on NaBr, since these constants are necessary in our calculations. Experimental Method Our apparatus and technique were similar to those of Datz, Smith, and T a y l ~ rand , ~ our modifications of this method have been described previ~usly.~a~s A small weighed sample of salt is placed in a thermostated fused silica vessel of known volume which is connected to one side of a U tube containing liquid gold. The salt is completely vaporized at a fixed temperature and exerts a pressure on the liquid gold in one arm of the U tube. The liquid gold levels are equalized by introducing argon into the other arm of the U tube. The pressure (1) On leave from the Laboratoire D’Electrochimie et de Chimie Physique du Corps Solide, Strasbourg University, Strasbourg, France. Partial support was obtained in a grant from the North Atlantic Treaty Organization. (2) S. H. Bauer and R. F. Porter in “Molten Salt Chemistry,” M.Blander, Ed., Interscience Publishers, New York, N. Y., 1964. (3) S. Datz, W. T . Smith, Jr., and E. H. Taylor, J . Chem. Phys., 34, 558 (1961).
(4) (a) K. Hagemark, M. Blander, and E. B. Luchsinger, J . Phys. Chem., 70, 276 (1966); (b) K. Hagemark and D. Hengstenberg, ibid., 71, 3337 (1967). (5) I. G. Murgulescu and L. Topor, Rev. Chim., Acad. Rep. Papulaire Roumaine., 11, 1353 (1966). (6) M. Blander, J. Chem. Phys., 41, 170 (1964). (7) (a) R. C. Schoonmaker and R. F. Porter, ibid., 30, 283 (1959); (b) T. A. Milne and H. M. Klein, ibid., 33, 1628 (1960). (8) (a) A. V. Tarasov, L. A. Kuligina, G. I. Novikov, Vestn. Leningr. Univ., Ser. Fiz. Khim., 115 (1966); (b) A. V. Tarasov, A. B. Pospelov, G. I. Novikov, ibid., 101 (1965). (9) K. Hagemark, D. Hengstenberg, and M. Blander, J . Phys. Chem., 71, 1819 (1967).
4621
ASSOCIATION IN MIXEDALKALIHALIDE VAPORS of argon is measured and is equal to the pressure exerted by the salt. The deviations from ideal gas behavior are related to association of salt molecules so that one may derive association constants from these mea~urernents.~ The sources of starting materials and their stated purities are: NaCl and KC1; Baker Analyzed reagent, 99.9%; CsC1, Penn Rare Metals Division, Kawecki Chemical Co., 99.9%; NaBr and KBr, Mallinckrodt Analyzed reagent, 99.9%; and RbBr and CsBr, Electronic Space Products, Inc., 99.87%. The salts were dehydrated under vacuum a t moderate temperatures below the melting points. We used single crystals which yielded more reproducible results than powder. These were grown in an argon atmosphere from melts of dried salts and were stored over P205 in a vacuum desiccator. The originally clear NaBr crystals developed a fogged surface in storage at room temperature, probably due to water adsorption. Washing with dry methyl alcohol and storage at 180" eliminated this difficulty. No difficulties were encountered with the other salts.
of AX and BX sealed into the vessel and are related to the values of ni by the expressions
+ + nABXa nBXT = nBX + 212BZX2 + nABX2
%AXT
AX'^
then upon substituting for ~ A and ~ x P B~ ~ Xfrom ~ the expressions for the association constants KA and KB
and BX, we
p
- PAXid PBX
AX
2AXZAzX2
(KA)
(3)
2BXZBXz
(KB)
(4)
ABXz (KAB')
(5)
+ BX
=
PAX
+
KB = R*T(PB~X~/PBX~)
(13)
+
PBX
= 2p
- (PAX id + pBXid)
(14)
=
+ ( K B / R * T ) ~ B x-~ (KA/R*T)PAx~(15) B X we ,
obtain the quadratic equation
[(KB - KA)/R*TIPBX'4- {(~KA/R*T)X
- (PAXid + pBXid)] + 1]pBX ( K A / R * T ) [ ~-~ ( p . 4 ~+ ' ~ psxid)]' = 0 [2p
where the units of the association constants KA, KB, and KAB'are liter per mole. The presence of significant amounts of other species, such as trimers, can be ascertained from the internal consistency of the data a t different pressures and mole fractions of the two components and from mass spectrometric data? I n general, the measurements were made at pressures low enough so that trimers were not significant. If the only significant deviations from ideal gas behavior are due to associations, then the total pressure is the sum of the partial pressures of all vapor species
C pi
(12)
When solved for ~
and
p =
KA = R*T(PAaX2/PAX2) we obtain from combining eq 7 and 10-13 PAX
AX
(9)
2nAzXa
%AX
If we define the "ideal partial pressures" of AX and BX, and ~ B X as ~ ~the , partial pressures these salts would have if they behaved ideally, then
Thermodynamic Calculations For a binary mixture of the salts consider the equilibria
=
+
PA~X~
pBX
+ pBaXa +
PABXi
(7)
where the partial pressure, in millimeters, of a species i is given by pi = niR*T V
(8)
and where ni is the number of moles of i, V is in liters, T is in degrees Kelvin, and R* is 62.360 1. mm mol-' deg-l. nAXT and nBXT are the total number of moles
(16)
From eq 10, 11, and 14-16 and our measured values of nAXT, nBxT, V , T, and p, in addition to knownvalues of KA and KB, we can calculate PAX and PBX. From ~ ~ B using ~ x eq ~ 12 these we may calculate P A ~ Xand and 13. Using these partial pressures we may calculate ~ A B Xfrom ~ eq 10 or 11 and thus obtain KAB' and KAB,where KAB'= R*T(PABXa/PAXPBX)
(17)
PABX~'
KAB= --
(18)
PAzXzPBaXz
Table I: Values of Ai and Bi in the Equation Log Ki = A i + ( B i / T ) Used to Calculate the Dimerization Constants, Ki, of the Salt i Salt
Ai
Bi
Ref
NaBr KBr RbBr CsBr NaCl
-3.808 -3.407 -3.126 -2.226 -3.698 -3.434 -3.044
9,374 8,378 7,716 6,072 10,499 9,014 7,586
3 4a 4b 4b 3 3 3
KC1 CSCl
Volume 78, Number 13 December 1068
J. GUION,D. HENGSTENBERC, AND M. BLANDER
4622
Table I1 : Data Calculated from Vapor Density Measurements on the Six Binary Mixtures, NaBr-KBr, NaBr-RbBr, NaBr-CsBr,
lo",
mol/l.
3.737 (58.1% K.Br) 4.471 (78.6% KBr)
4.302 (50.0% KBr) 4.623 (57.0% KBr)
lO'p,
mol/l.
5 * 539 (76.1% RbBr)
TIOK
PXNaBri
1327 1338 1351 1363 1349 1355 1366 1367 1378 1387 1397 1350 1360 1378 1382 1399 1353 1354 1355 1361 1369 1378 1383 1392 1396
4.438 3.875 3.951 3.154 2.801 3.656 2.541 3.128 2.384 3.013 2.749 3,943 3.744 3.950 3.811 3 * 734 4 576 4.301 4,448 4.525 4.470 4.064 4.342 4.170 3.815
T,O
K
I
PNaRbBrz
1340 1348 1355 1356 1366 1380 1390 1352 1363 1364 1374 1380 1389 1394 1328 1341 1346 1357 1368 1383 1384 1328 1340 1354 1361 1379 1382
4.408 3.727 4.183 3.965 3.798 3.175 3.619 4.438 4.603 4.304 4.023 4.170 4.316 3.741 5.084 4.514 4.457 4,207 3.915 3.631 3.907 3.711 3.568 3.012 2.877 2.533 2.322
lo%, mol/I.
T,OK
PNaCaBra
3.914 (68.4% CsBr)
1334 1338 1349 1359
3.264 3.371 3.067 3.138
5.107 (57.5% RbBr)
4.887 (62.0% RbBr)
3.767 (48.4% RbBr)
The Journal of Physical Chemistry
PKBr
PKiBri
PNaBr
NaBr-KBr 11.131 11A49 12.168 13.099 20.442 20.246 21 * 575 21.247 22.344 22.396 23.099 12.016 12.440 12.760 12.970 13.457 14.583 14.813 14.746 14.904 15.224 15.833 16.795 16.224 16.628
1.207 1.204 1.094 1.109 3.160 2.896 2.909 2.790 2.735 2.493 2.383 1.079 1.034 0.892 0.882 0.792 1.537 1.568 1,536 1.468 1.401 1.373 1.295 1,240 1.248
6.604 7.210 7.447 8.257 4.565 3.954 4.958 4.498 5.215 4.791 5.110 10.546 11.014 11.491 11.713 12.311 9.390 9.592 9.530 9.658 9.928 10.459 10.413 10.785 11.146
PRbBr
PRbaBrz
PNnBr
NaBr-RbBr 24.598 25.545 25.657 25.867 26.576 27.834 28 071 17.536 17.798 18.065 18.632 18.719 18,901 19,531 16.751 17.672 17.906 18.511 19.043 19.909 19.726 10.177 10.529 11.258 11.506 12.135 12.371
3.104 3.076 2.884 2.901 2.762 2.628 2.419 1.390 1.277 1.303 1.251 1.189 1.108 1.126 1.638 1.585 1.544 1.470 1.404 1,305 1.268 0.604 0,569 0.561 0,548 0.505 0.509
5.535 6.152 5.938 6.114 6.399 7.098 6.915 10.425 10.693 10.914 11.428 11.530 11.722 12.273 7.717 8.420 8.593 9.053 9,545 10.142 9 * 975 8.946 9.412 10.194 10.502 11.290 11.523
PCniBri
0.775 0.750 0.726 0.681
I
PCeBr
NaBr-CsBr 17.456 17.466 18.002 18.187
PNarBri
0.950 0.983 0.889 0.941 0.343 0.238 0.327 0.266 0.313 0.237 0.239 1.806 1.738 1.518 1.503 1.356 1.404 1.447 1.411 1.345 1.287 1.281 1.195 1,151 1.173 PNazBra
Log K'
3.699 3.578 3.565 3.311 3.402 3.586 3.306 3.446 3.245 3.385 3.307 3.418 3.365 3.365 3.335 3.294 3.450 3.408 3.427 3.426 3.402 3.324 3.358 3.316 3.253 Log
K'
0.565 0.630 0.538 0.563 0.545 0.565 0.476 1.721 1.579 1,624 1.576 1.492 1.384 1.430 1.281 1.290 1.261 1.219 1.183 1.113 1.064 1.721 1.633 1.604 1.561 1.447 1.454
3.432 3.300 3.365 3.326 3.279 3.141 3.208 3.311 3.313 3.269 3.209 3.221 3.227 3.133 3.513 3.404 3.386 3 * 327 3.262 3.191 3.234 3.528 3.478 3.346 3.306 3.201 3.147
PNaBr
PNavBra
LOP K'
5.723 5.716 6.111 6.222
0.652 0.618 0.614 0.562
3.434 3.450 3.370 3.371
4623
ASSOCIATION IN MIXEDALKALIHALIDE VAPORS KBr-CsBr, NaC1-CsCl, and KCI-CsC1" 1 0 4 ~mol/l. ~
T,O
K
PNaCsBrz
PCsBr
NaBr-CsBr 18.838 18.783 18.772 19.226 19.463 15.442 16.453 16.825 16.617 16.877 17.250 17.738 18.388 18.178 18.903 20.933 20.861 21.701 22.290 22.229 22.397 22.663 23.360 23.458 24.071
PCszBrl
PNaQr
PNszBr2
Log K'
0.707 0.680 0.658 0.616 0.597 0.769 0.713 0.746 0.673 0.607 0.560 0.568 0.529 0.509 0.489 1.426 1.319 1.330 1.320 1.135 1.133 1.112 1.052 1.035 0.920
6.753 6.691 6.664 6.988 7.160 6.568 7.449 7.717 7.631 7.921 8.284 8.695 9.298 9.131 9.783 4.140 3.927 4.522 4.916 4.559 4.676 4.829 5.218 5.256 5.453
0.630 0.588 0.556 0.516 0.498 1.224 1.163 1.248 1.087 0.957 0.869 0,901 0.830 0.782 0.753 0.493 0.399 0.476 0.513 0.355 0.364 0.365 0.358 0.350 0.292
3.223 3.259 3.282 3.234 3.206 3.640 3.520 3.457 3.517 3.512 3,486 3,417 3.353 3.393 3.310 3.564 3.639 3.488 3.389 3.202 3.497 3.466 3.390 3.386 3.373
PKnBrz
Log K'
1363 1367 1371 1385 1392 1307 1330 1330 1339 1355 1370 1375 1393 1395 1410 1306 1314 1322 1329 1346 1348 1353 1367 1370 1391
2.502 2.678 2.801 2.664 2.581 5.426 4.922 4.484 4.991 5.139 5.116 4.697 4,435 4.720 4.291 3.898 4.355 3.661 3.240 4.031 3.911 3.791 3.512 3.511 3.573
lo%,mol/l.
T,OK
PCsKBrr
PCaBr
PCsrBn
PKBr
3.658 (58.6% CsBr)
1314 1322 1353 1364 1366 1385 1390 1322 1360 1371 1394 1400 1406 1429 1329 1342 1351 1369 1403 1428
2.095 1.928 1.774 1.578 1.767 1.712 1.569 3.325 2.978 2.437 2.279 2.565 2.293 2.351 3,669 2.737 2.536 3.336 2.590 2.497
KBr-CsBr 14.237 14.544 15,295 15.679 15.548 15.949 16.170 17.163 18.424 19.166 19.857 19.741 20,123 20.591 12.565 13.621 13.961 13.554 14.801 15.290
0.614 0.597 0.507 0.486 0.470 0.424 0.419 0.832 0.693 0.685 0.612 0,577 0.572 0.502 0.419 0.441 0.429 0.348 0.317 0.279
8.628 8.918 9.660 10.025 9.911 10.296 10.502 11.769 13.175 13.899 14.670 14.604 14.980 15.515 15.198 16.367 16.894 17.095 18.956 19.928
T,OK
PKCsCh
4.521 (60.8% CsBr)
4.507 (75.4% CsBr)
4.865 (55.2% CsBr)
4.867 (42.3% CsBr)
1040, rnol/l.
4.746 (36.1% CSC1) 3.983 (72.1% CSCI)
1368 1379 1382 1392 1417 1339 1350 1365 1381 1382 1396 1424
2.941 2.377 2,531 2.023 1.723 2.490 2.390 2.059 2.214 2.230 2,242 2.178
PCsCl
NeCl-CsCl 10.820 11.491 11.408 12,018 12.689 18.187 18.680 19.486 19.933 19.955 20.418 21.362
PCsrClr
0.435
0.440 0.421 0.423 0.372 1.657 1.559 1.455 1.298 1.288 1.176 0.987
PKC1
8.152 8.814 8.929 9.563 10.992 3.254 3.488 3.889 4.144 4.158 4.427 5.008
0.846 0.822 0.674 0.642 0.614 0.538 0.531 1.431 1.160 1.143 0.993 0.923 0.912 0.772 2.198 2.194 2.110 1.767 1.507 1.286
3.146 3.088 2.988 2.931 2.990 2.954 2.904 3.133 3.017 2.893 2.833 2.890 2.824 2.817 3.202 3.012 2.957 3.090 2.907 2.863
PKrCla
Log K'
7.383 7.437 7.331 7.363 7.034 1.762 1.733 1.750 1.601 1.589 1.497 1.336
3.454 3.305 3.331 3.184 3.038 3.546 3.490 3.364 3.363 3.365 3.334 3.257
Volume 76, Number 13 December 1068
J. GUION,D. HENGSTENBERG, AND M. BLANDER
4624 Table I1 (Continued) 104p, mol/l.
T,OK
4.647 (69.1% CSC1)
1352 1363 1385 1406 1428
2.777 2.669 2.291 1.759 1.769
20.587 21.132 22.316 23.553 24.355
1.855 1.746 1.564 1.421 1.235
4.930 (66.2% CSC1)
1353 1363 1380 1390 1397 1412 1423 1337 1350 1381 1388 1396 1418 1327 1349 1367 1384 1401 1425
3.543 3.061 3.274 3.436 3.679 3.860 3.604 2.795 2.557 2.609 2.489 2.612 2.771 2.024 2.011 2.002 2.090 2.081 2.081
KC1-CsC1 20.394 21.176 21.707 21.971 22.043 22.444 23,044 17.113 17.754 18.752 19.071 19.219 19.739 10.240 10.645 10.958 11.155 11.424 11.775
1.801 1.754 1.554 1.443 1.357 1.219 1.159 1.498 1.408 1.148 1* 109 1.042 0.891 0.596 0.511 0.451 0.395 0.351 0.297
3.882 (70.8% CSCl)
3.026 (53.7% CSCI)
0
PKCsClz
PCBCl
PCs2CIa
PKaClz
Log K’
4,272 4.569 5.247 6.002 6.603
2.529 2.484 2.432 2.415 2.208
3.425 3.371 3.228 3.038 2.877
7.968 8.519 8.806 8.934 8.927 9.125 9.547 5,300 5.664 6.096 6.287 6.306 6.473 6.919 7.438 7.846 8.148 8.498 8.954
1.273 1.290 1.129 1.035 0.954 0.843 0.817 0.685 0.667 0.535 0.525 0.482 0.397 1.321 1.164 1.044 0.923 0.827 0.703
3.265 3.159 3.169 3.181 3,212 3.220 3.163 3.410 3.331 3.294 3.255 3.273 3.283 3.374 3.330 3.298 3.298 3.273 3.244
PKCl
p is in millimeters.
The values of the dimerization constants we used were calculated from the equation log Ki = Ai
+ (Bi/T)
(19)
where Ai and Bi are given in Table I. Experimental Results The necessary values for the dimerization constants of all the salts in our mixtures are Because of the apparent discrepancy in the data for NaBr,a we have repeated the measurements for this salt a t three different densities. As has been previously di~cussed,~,4 the dimerization constants were calculated from the relations PAX
=
PAzXa
2p‘- PAXid = Pid
-P
K A = R*T(PA~x~/PAx~) The agreement with the reported values of Datz, Smith, and Taylora for log KA was good (iO.1) and there appears to be no discrepancy in the values reported in ref 3. NaBr exhibits the largest deviations from the predictions of the dimensional theory of any salt which has been measured.6 The data of Murgulescu and Topor5 are in agreement with these results. The measurements for the binary systems NaBrKBr, NaBr-RbBr, NaBr-CsBr, KBr-CsBr, NaC1-KC1, and NaC1-CsC1 are given in Table 11. For each binary system, we made at least three independent series of The Journal of Physical Chemistry
measurements at different compositions. Values of log KAB calculated from KAB’appear to be constant at different compositions within an uncertainty of lt0.15 and also differ little a t different temperatures. Because of the relatively narrow range of temperatures and the small temperature dependence, we could not establish a temperature dependence for KAB. This means that, within experimental error, the energy of forming the mixed compound from the monomers (eq 5) is the average of the energy of dimerization of the two monomers. I n all cases, except for the mixture of NaCl with CsC1, the values of KAB/4 are greater than unity. A value of greater than unity indicates that the tendency to form ABXz is stronger than the “average” for the two dimers AzXz and BZXZ. The experimental errors in KAB arise not only from errors in our measurements but also from errors in the dimerization constants of the two salts. We estimate that the cumulative errors in RT In (KAB/4) are about = k l kcal. Consequently, K A B / ~ for the NaC1-CsC1 system may be close to unity. In Figure 1 are plotted our measured values of RT In (KAB/4) US. [(I/&) - (l/dz)]’. The values for the fluorides which have been measured in the mass spectrometer’ lie close to a line plotted in the figure.
Discussion It can be seen in Figure 1 that the values of RT In (KAB/4) for the bromides are larger than for the fluorides or chlorides. If only coulomb interactions in-
ASSOCIATION IN MIXEDALKALI HALIDEVAPORS
-1
I
NaEr.CsEr
AL
4625
0
1 Equilibrium constants for the reaction1 A z X ~ + E ~SX ~PAEX,
Error'in points-fl.0 kcal. /mole
e
Figure 2. The model used in the calculations given in Table 111.
X
-f
2-
KBriCsBr
-It 0
I
2
3
4
5
NaCI-CsCI
I
7
0
6
the equilibrium shown in Figure 2. The energy change for this equilibrium was calculated taking into account only coulomb interactions and a repulsion representing the two extreme cases: (a) the soft inverse power potential k/r* and (b) a hard-core repulsion. The energy change is
AUAB = 2UAB
Figure 1. Plot of RT In (KAB/4) 21s. [ ( l / d ~ x ) (l/dsx)]a for six binary mixtures. The line labeled AU represents the coulombic energy change given in Table I11 which was calculated from the simple model.
-
fluenced the magnitude of K A B / ~then , there should be no difference between bromides, chlorides, and fluorides in a plot such as in Figure 1. Consequently, the differences indicate that other interactions are significant. Since vapor molecules are simple, fairly accurate calculations of the energetics are feasible. Thus, by comparing such calculations with our measurements, it is hoped that, in the future, we may gain insight into the magnitude of those interionic interactions which are important in mixtures. The implications of such calculations are significant not only for salt vapors but also in understanding the relative importance of different interactions in molten salt mixtures where simple calculations are not feasible. Some of the types of interactions which need to be considered may be deduced from previous calculations of the energies of formation of monomeric alkali halides and of the energies of association of the dimer molecules which have been made by several authors."J The general conclusion reached by most of these authors is that the three most significant interactions are the coulombic attractions and repulsions, the softcore repulsion between ions, and the ion-induced dipole interactions. I n ref 1Oc and d, the ion-induced dipole interactions were taken into account by the simple artifice of softening the repulsive potential. Because of the low coordination number of vapor molecules, the van der Waals (dispersion) interactions are relatively small in the monomers but may be significant in dimers. A preliminary idea of the influence of the coulomb interactions and the soft-core repulsions on KABmay be gauged by comparison of the data with calculations which include only these two interactions. Consider
- U A - UB
(20) where UABis the potential energy of the mixed alkali halide molecule, U Athat of one dimer, and U s that of the other. If the pair potential for the A+ and Xions is written as
where from the minimization of U , k = (c~Ax'/~)and d A X is the internuclear distance for the monomer, then for the dimer A S 2 (U,x/e2) =
-2.586(0.8750) dAx'
-2.586(0.8750) -2.126 =1.064307d~x dAx
(22)
and for BzX2
where dBx is the internuclear distance for the monomer BX. The factor 1.064 which is calculated by minimizing UAXarises because of the lengthening of the cation-anion distances in the dimer as compared with the monomer. For the mixed compound -UAB =----
e2
2 dAX"
2 dsx"
1 +
b+C
1 2k -+-+2a (dAx")'
+
2k (dBX")'
(24)
where k' = (d~x'/8). Core repulsions between ions of like sign were assumed to be negligible. Minimizing UABwith respect to the three variables a, b, and c (10) (a) E. S. Rittner, J . Chem. Phys., 19, 1030 (1951); (b) C. T. O'Konski and W. I. Higuchi, ibid., 23, 1175 (1955); (c) L. Pauling, Proc. Nat. Acad. Sci., India, A25 (1956); (d) T. A. Milne and D. Cubicciotti, J . Chem. Phys., 29, 846 (1958); (e) J. Berkowitz, ibid., 29, 1386 (1958); (f) G. M. Rothberg, ibid., 34, 2069 (1961); (g) Y.P.Varshni and R. C. Shukla, ibid., 35, 582 (1961). Volume 78, Number 18 December 1068
J. GUION,D. HENGSTENBERQ, AND M. BLANDER
4626
Table I11 : Energies for the Equilibrium AzX2 f BtXz F? 2ABXs Calculated from the Model in Figure 2 Utilizing the Anion-Cation Pair Potential U = - ( e $ / r ) + (kea/$) IOa [(l/dAX)
Salt pair
dAX
dBX
a
b
0
NaBr-KBr NaBr-RbBr NaBr-CsBr KBr-CsBr NaC1-CsC1 KC1-CsCl
2.5020 2.5020 2,5020 2.8207 2.3606 2.6666
2.8207 2.9447 3.0722 3.0722 2.9062 2.9062
1.9941 2.0330 2.0707 2.2125 1.9559 2.0923
1.7454 1.6934 1.6409 2.0136 1.5450 1.9028
2.2655 2.4154 2.5700 2.4241 2.4341 2.2939
led to three nonlinear equations which, when combined with the conditions
+ b2 = a2 + =
a2
c2
AX'')^ (dBXI')2
(25) (26)
were solved on a computer. The results are given in Table I11 for the salt pairs studied for the case in which the core repulsion was represented by a very soft inverse eighth power term. These calculations of AUAB,in effect, are the same as -RT In ( K A B / 4 ) at 0°K. The temperature dependence is unknown. Consequently, a direct comparison with our data is not possible. However, it is probable that the temperature coefficients are not very large, so that these calculations can provide us with considerable information. There are two points to be noted in these calculations. a. AU when plotted vs. the parameter [(l/dAX) (l/dBx) 12, which is suggested by the dimensional theory used to derive eq 2, yields an essentially linear plot. This linearity lends confidence in the validity of the derivation of eq 2 for coulomb attractions and a soft repulsion. Within small limits
which is plotted in Figure 1. A similar calculation for hard-core ions leads to a result which is smaller than this by about 9%. This relative insensitivity to the softness of the cores is an important result. b. When dAX' < dBXI,then in the mixed compound dAX'> dAXlland dBXf < dgx" (ie., the short interionic distances get shorter and the long ones longer). Although these idealized classical calculations are for the energetics at O'K, they are significant as a point of departure. The bromides lie above the line representing eq 27. This difference is, at least in part, related to the ion-induced dipole interactions. Calculation of the square of the electric intensity vectors, E2, for the ions using the data in Table I11 indicates that the changes in E2 on the anions are negative and approximately proportional to the parameter [ ( l / d ~ x ) ( I j d ~ x ) ] ~Thus, . to first order, the polarization of The Journal of Physical Chemistry
dAX"
dBX"
10a(A(l/e:)
-. (l/dsx)l'
2,6501 2.6459 2,6420 2.9917 2.4925 2.8281
3.0181 3.1571 3.3004 3.2820 3.1225 3.1048
-1.941 -3.509 -5.430 -0.893 -5.901 -0.960
2.039 3.610 5.503 0.842 6.325 0.956
the anions leads to a term of the form of eq 2, and the parameter M in eq 2 is more positive for salts with more polarizable anions. If the induced-dipole energy may be represented by (aE2/2),where a is a constant, then 330[1 we find as a first-order approximation M 0.3aanion]kcal/mol, where a is given in 10-24 cm3. This expression, of course, does not take into account the changes in internuclear distances which result from the inclusion of the polarization interactions. The magnitude of this interaction is, relatively, very large and leads to significant changes in the interionic distances. For example, the strong tendency of the more polarizable anions to be stabilized in regions of high field intensity leads to configurations in which the anion -anion distances are increased and the cation-cation distances are decreased. These changes will stabilize the dimers as well as the mixed compound. Our preliminary calculations show that, in general, the dimers are stabilized more than the mixed compound. Consequently, although the first-order contribution to M is positive, the change of the configuration of the molecules leads to higher order terms which are negative and may be large enough to lead to a net negative contribution to log (K/4) from the ion-induced dipole terms. Further calculations of this and of the influence of temperature will be necessary for a quantitative evaluation. Another interaction which may be of importance is the van der Waals (dispersion) interaction. A calculation of the magnitude of these interactions using parameters given by Mayer'l shows that the cesiumcesium interactions in cesium halide dimers and the halogen-halogen interactions in lithium and sodium halide dimers are significant and in reactions such as reaction 1 may lead to significant (e.g., -1-3 kcal/mol) positive contributions to the energy changes in the fluoride systems in which LiF or NaF is one component and in the NaC1-CsC1 system. These positive contributions result from the change of next neighbor cations of the cesium halide dimers and an increase in the distances between anions in the lithium and sodium halides in the other cases. The magnitudes of these van der Waals effects are very sensitive to the details of the calculation and are most sensitive to the softness N
(11) J. E.Mayer, J . Chem. Phys., 1, 327 (1933).
+
ASSOCIATION IN MIXEDALKALI HALIDE VAPORS
4627
of the core repulsions. A precise determination of this possibility must await more detailed calculations now in progress. The results for binary fluoride systems reported by Schoonmaker and Porter12 and for the NaC1-CsC1 system lie below the measured values of RT log (K/4) for the bromides. The higher order polarization terms and the van der Waals interaction terms are undoubtedly important for at least a part of these differences. I n addition, a calculation of the influence of temperature will be needed to correlate the absolute values of - A U with RT In (K/4) and thus go further than our present discussion of the relative contributions to these quantities. These conclusions are significant not only in understanding the energetics in salt vapors but also in understanding the corresponding effects in molten salt mixtures. Kleppa and Holmla have observed a difference between the heats of mixing in fluoride and bromide mixtures analogous to the one discussed here. We are now carrying out detailed calculations on the vapor molecules which we hope will aid in understanding the relative importance of different ionic interactions, especially of the higher order polarization and the anion -anion van der Waals interactions which have not been discussed previously.
where M is positive and is more positive the larger the polarizability of the anion. b. The ion-induced anion dipole interactions are so large that terms higher than first order are very significant, especially for polarizable anions in salts where d is small. These higher order terms lead to increased anion-anion distances and decreased cation-cation distances and generally make a negative contribution to RT log ( K A B / ~ ) . c. For lithium and sodium halides, the separation of next nearest neighbor anions is larger in the mixed compound than in the dimer which leads to significant negative contributions to RT In (KAB/4) from van der Waals interactions. This contribution has not been discussed previously and may have analogies in molten salt mixtures. d. For rubidium and cesium halides, the change of next nearest neighbor cations generally leads to signifi) a cant negative contributions to RT In ( K A B / ~ in manner analogous to molten salt mixtures.14~'6 A general conclusion is that the energetic factors involved in forming these molecules are much more complex than we anticipated. This implies that the analogous factors in molten salt mixtures are also more complex and difficult to understand than is presently believed.
Conclusions
Acknowledgments. The authors wish to thank Stephen T. Imrich and Dr. Jerome Spanier of this laboratory who carried out the computer solution of the nonlinear equations obtained from the model.
We may therefore conclude the following. a. Contributions to RT In (KAB/4) from the coulomb and ion-induced anion dipole interactions, to first order, depend upon the size parameters according to the relation /
1
1
\2
(12) R. C. Schoonmaker and R. F. Porter, J. Chem. Phys., 30, 283 (1959). (13) 0.J. Kleppa and J. Holm, to be submitted for publication. (14) M. Blander, J. Chem. Phys., 36, 1092 (1962); 37, 172 (1962). (15) J. Lumsden, Discussions Faraday SOC.,32, 138 (1961).
Volume 73,Number 18 December 1068
