LATTICE ENERGIES OF ALKALINEEARTH HALIDES
3611
The Lattice Energies of the Alkaline Earth Halides
by Thomas E. Brackett and Elizabeth B. Brackett Department of Chemistry, Colgate University, Hamilton, New York
(Received M a y 17,1965)
The Madelung constants and lattice energies of several alkaline earth halides are calculated. The lattice energies are found to compare favorably with experimental values for all salts except magnesium bromide and iodide and calcium iodide, in which cases the large discrepancies are difficult to explain.
Introduction The lattice energies of several groups of ionic salts have been successfully calculated. These treatments have usually been limited to salts such as the alkali halides, the alkaline earth oxides, and a few others which for the most part have simple highly symmetric structures. Now that computational procedures for calculating the electrostatic energy of more complicated structures are relatively straightforward, the alkaline earth halides provide an interesting group of compounds to study. It would be interesting to see, for example, if the calculation when applied to structures as diverse as these salts present would show fluctuations which could be correlated to structural differences. Also, it would be interesting to observe if and when the approximation breaks down when passing from salts such as BaF2, which is almost completely ionic, to salts such as MgIz,which is predominantly covalent. Previous lattice energy calculations involving some of these salts have included only those exhibiting the simple fluorite structure for which the Madelung constant is well known2 and the hexagonal structure for which the Madelung constant as determined by Hund3 is in error. The electron affinities of the halides are now fairly well established. We will utilize these and other thermodynamic data to calculate the lattice energy for comparison with the results calculated from
ular volume, and cij is a constant representing the van der Waals interaction between ions i and j . The terms represent the electrostatic attraction, the repulsion, and the van der Waals attraction, respectively. Two other terms (Dij/rij*)and the zero point energy are eval-
(
)
ij
uated for certain instances and found to be too small to justify their inclusion in this treatment.
The Electrostatic Attraction The Madelung constants of MgFz and CaC12 are taken from the report of T e m p l e t ~ n . ~(The value for SrBrz given there is calculated for an incorrect structure and is thus redone for this report.) All other Madelung constants except for the well-known fluorite structure are calculated for this report according to a method given by Wood,’ who discusses the method and its accuracy. Considering the uncertainties in the structure determination, in addition to the uncertainty in determining the lattice sum, the over-all uncertainty in determining the value of the Madelung constant is estimated to be less than 1%,except for SrBrz and MgC1,. The structure of SrBrt requires that n Sr2+ ions be statistically placed on 2n lattice positions. The Madelung constant for SrBrz was calculated by assuming the ions were equally divided between the two sites. That is, that half the ion occupied one of the 2n positions all the time. (The self-interaction was, of R. H. Wood, J . Chem. Phys., 32, 1690 (1960). F. Seitz, “The Modern Theory of Solids,” McGraw-HiU Book Co., Inc., New York, N. Y., 1940. (3) F. Hund, Z . Physik, 34, 833 (1925). (4) M. Born and J. E. Mayer, ibid., 75, 1 (1932). (5) Q. C. Johnson and D. H. Templeton, J . Chem. Phys., 34, 2004 (1)
(2)
based on the equation of Born and M a ~ e r . ei ~ is the charge on ion i; rij is the distance between ions i and j. R and g are constants, E is the cube root of the molec-
(1961). (6) R. L. Sass, T. E. Brackett, and E. B. Brackett, J . Phys. Chem., 67, 2862 (1963).
volume 69. iVumbsr 10 October 1966
3612
THOMAS E. BRACKETT AND ELIZABETH B. BRACKETT
course, not, included.) The procedure was found to Table I : Madelung Constants and Coulombic agree to within a few kilocalories per mole with other Attraction Energy possible k e d arrangements. The resulting Madelung Ref. constant for SrBrz is somewhat less certain than that Coulomb to for the other salts. energy, strucSalt Structural type f , K. A[ kcal./mole ture The structure of MgClz has one parameter, u,undetermined. Similar salts have values of u ranging from MgF2 Cassiterite 3.195 7 732 803.6 a MgC12 Cadmium chloride 4.034 7.489 616.5 b 0.250 to 0.260. The calculation for MgC12 is carried ( U = 0.26) out for both values of u and tabulated for compariMgC12 Cadmium chloride 4,034 6.957 572.7 b son. The comparison indicates that u is probably ( u = 0.25) closer to 0.260 than to 0.250. MgBrz Cadmium hydroxide 4.285 6,878 549.0 6 I n connection with an over-all evaluation of the unMgI2 Cadmium hydroxide 4.674 6.852 501.4 b CaF2 Fluorite 3.434 7.3306 709.0 certainty in the electrostatic energy term, it is interesta CaCh Cassiterite distorted 4.393 7.673 580.0 ing to note the agreement of the values 526.3 and 528.8 CaBr2 Cassiterite distorted 4.614 7.670 552.2 c kcal./mole for the electrostatic energy of BaC12. The CaIz Cadmium hydroxide 4.946 6,223 484.1 b former is calculated in this report, and the latter indeSrF2 Fluorite 3.653 7.3306 666.3 pendently from an independent determination of the SrC12 Fluorite 4.396 7.3306 553.7 d SrBrz Special structure 4.592 7.170 518.5 crystal structure by Sahl.' SrIz Unknown Because of the large differences in crystal types it was BaF2 Fluorite 3.906 7.3306 623.1 decided to express all the results in terms of the cube e BaC12 Fluorite 4.614 7.3306 527.8 root of the molecular volume, .$. The resulting values BaC12' Lead chloride 4.442 7.040 526.3 e of .$,the Madelung constant calculated to this basis BaBr2 Lead chloride 4.668 7.037 500.6 e Ba12 Lead chloride 5.020 7.022 464.5 e A,, and the electrostatic energy Ae2/.$are presented in Table I. a See ref. 5. * R. W. G. Wyckoff, "Crystal Structures," Vol. 1, Interscience Publishers, Inc., New York, N. Y., 1963. E. B. It will be noted that the values Ai for the three salts Brackett, T. E. Brackett, and R. L. Sass, J . Inorg. A-ucl. Chem., of the cadmium iodide structure (MgBrz, MgL, and 25,1295 (1963). See ref. 6. e E. B. Brackett, T. E. Brackett, CaIz) differ from those originally given by H ~ n d . ~ and R. L. Sass, J . Phys. Chem., 67, 2132 (1963). Cubic. That his values were in error was noted by Pinsker,* Orthorhombic. whose results compare favorably with those in this report. did not seem to vary greatly from one salt to another, The van der Waals Attraction and it is believed to be unrealistic to include it in this The coefficients Of the ' l r 6 term may be treatment since it is probably smaller than the uncerfrom the expression given by London tainty in the value of the l/r6 term. The low temperature specific heat measurements on 3 liljcuicrj MgFz and C a F P allow one to calculate the Debye c i j = 2 l i + Ij temperature from which the zero point energy may be where Q! and I refer to the polarizabilities and ionization estimated. The results for MgFz and CaFz are 2.2 and energies, respectively, of the ions involved. Mayerg 2.1 kcal./mole, respectively. These salts should have evaluated these coefficients for the alkali halide crystals the highest such contribution; it was decided not to using measurements of their optical properties. The include this term for any salt in this calculation. ionization energies of the halides used in this work were The Repulsive Energy the averages of the corresponding values calculated by I n accordance with accepted procedure, we assume Mayer for the alkali halides. The ionization energy that the repulsive energy between two ions due to the of the positive ion was assumed to be 75% of the ionioverlap of electron clouds is of the form e-gT, or sumzation potential (see ref. 9). The polarizabilities ming over all ions in the crystal used were those evaluated by Tessman, Kahn, and Shockley10 from measurements of refractive index. The (7) K. Sahl, Beitr. MineraZ. Petrog., 9, 111 (1963). coefficients appear along with the sums of l/r6 and the (8) Z.G.Pinsker, Acta Physicochim. U R S S , 18, 311 (1943). calculated van der Waals energy for each salt in Table (9) J. E.Mayer, J . Chem. Phys., 1, 270 (1933). 11. (10)J. R. Tessman, A. H. Kahn, and W. Shockley, Phys. Rev., 92, The l/r" term was investigated for some salts and was 890 (1953). found to contribute from 4 to 6 kcal./mole. This term (11)S. 8.Todd, J . Am. Chem. SOC.,71, 4115 (1949). The J O U Tof~Physical Chemistry
LATTICE ENERGIES OF ALKALINE EARTH HALIDES
3613
Table 11: van der Waals Energy 1
C,ergs A.6 x
Salt
++
MgClz ( U = 0.26) MgClz ( U = 0.25) MgBrz MgIz CaFz CaClz CaBrz CaIz SrFz SrClz SrBrz SrL BaFz BaC12" BaCl? BaBre BaIz
2.9 2.9 2.9 2.9 2.9 55.8 55.8 55.8 55.8 103 103 103 103 226 226 226 226 226
MgFZ
a
Cubic.
--
6.4 107 107 186 388 6.4 107 186 388 6.4 107 186 388 6.4 107 107 186 388
10-'
1
.2 z- p -11* A
--
f-
van der Waala energy, kcal./mole
0.00332 0.00084 0.00084 0.00108 0.00067 0.00210 0.00049 0,00036 0.00042 0.00148 0.00048 0.0043
0.0244 0.00630 0.00590 0.00518 0.00307 0.0203 0.00388 0.00287 0.00224 0,0142 0.00458 0.00336
0.0995 0.02909 0.02327 0.0227 0.0134 0.0501 0.0150 0.0110 0.00980 0.0353 0.0114 0.00845
7.1 15.0 13.4 19.0 21.4 15.4 20.3 20.6 28.4 15.1 23.9 25.2
0.00098 0.00036 0.00048 0.00035 0.00023
0.00953 0.00344 0.00418 0.00309 0 00200
0.0236 0.00851 0.00928 0.00684 0,00441
16.0 23.0 26.0 26.4 27.1
+-
++
3.3 12.5 12.5 15.5 21.8 16.4 64.6 79.7 110.2 22.9 88.7 114 156 35 135 135 172 239
I
' Orthorhombic. E,
=
-1 E'e-gij'ij 2
ij
where the indices on g express the fact that it is no% necessarily independent of r. In the case of a simple highly symmetric structure, the nearest neighbors all have the same value of T (and therefore g) and furthermore constitute a major fraction of the repulsive energy. Thus
E, 2 Be-Qr where B is some constant related to the number of nearest neighbors and r is the equilibrium nearest-neighbor distance. In some cases the interaction of next nearest neighbors has been included. In the case of more complex crystals, no simple relationship exists because, in general, numerous ions lie at different distances all fairly close to the central ion. I n such cases it is still desirable to express the repulsive energy in the simple two-parameter form E, = Be-Qi,where 4 should represent some sort of an average of the nearest-neighbor distances. We have chosen the cube root of the molecular volume to represent this average. Unfortunately, the compressibility data which allow a determination of g are very scarce. In fact, only for CaF2 has the compressibility been measured for a single crystal, and only SrF2 and BaF2 have been measured as powders. The approximate values of g
as determined from these experiments are 2.34, 2.15, and 2.11, r e ~ p e c t i v e l y . ~Because ~ ? ~ ~ of the scarcity of the independent determinations of g, it was decided to choose the single value of g which would give the best fit to all the data. The salts of magnesium bromide and iodide and calcium iodide have been left out of this determination because, as we shall see, they give totally unreasonable values. The value of g calculated in this way turned out to be 2.32, which agrees well with the measured values. B was determined by the criterion that the variation of the lattice energy with respect to 4 evaluated at the equilibrium value of 4 is zero. The Lattice Energy from Thermodynamic Data The lattice energy of a compound may be calculated from the equation L.E. =
-AHf,2980~.
f U . , ~ ~ S O fK .11f Iz f
The heats of formation, AHf,298011., and the heats of sublimation of the alkaline earth metals, h H s , 2 9 S o K . , as well as the dissociation energies of the halogens, D ( x ~ ) ~ ~ ~ o K . , were taken from the tables in Lewis, Randall, Pitzer, and Brewer.14 The ionization energies (IIand 14 are (12) P. W. Bridgman, Ed., Landolt-BGrnstein Tabellen, SpringerVerlag, Berlin 1937. (13) P. W.Bridgman, "Physics of High Pressure," The Macmillan Co., New York, N. Y., 1949.
volume 69,Number 10 October 1966
THOWAS E. BRACKETT AND ELIZABETH B. BRACKETT
3614
from Moore.]j The electron affinities E(x-, for F, C1, Br, and I were taken as 80, 86, 81, and 74 kcal./mole, respectively.I6 The integrated heat capacity of the solid is given in ref. 17. I n many cases it is estimated, but it is a very small term. The lattice energies thus obtained are listed in Table I11 under L.E. (exptl.). ~~~
Table I11 : Calculated and Experimental Lattice Energy (kcal./mole) Cou- v a n d e r lomb Waals Repulsion energy energy energy
Salt
MgFz MgClz
L.E. L.E.
(exptl.)
A
803.6 7 . 1 -114.3 616.5 15.0 -75.6
696.4 702.3 555.9 596.5
5.9 40.6
572.7 13.4
516.2 596.5
80.3
( u = 0.26)
MgClz
-69.9
( u = 0.25)
MgBrs MgIz CaFz CaClz CaBrz CaL SrFz SrClz SrBrz SrIz BaFs BaC12" BaClzb BaBrz BaIz
' Cubic.
549.0 501.4 709.0 580.0 552.2 484.1 666.3 553.7 518.5
19.0 -66.8 21.4 -58.2 15.4 -100.7 20.3 -68.9 20.6 -63.2 28.4 -57.1 1 5 . 1 -89.4 22.3 -67.5 23.5 -62.0
501.2 464.6 623.7 531.4 509.6 455.4 592.0 508.5 480.0
623.1 527.8 526.3 500.6 464.5
16.0 23.0 26.0 26.4 27.1
559.6 488.5 486.0 466.1 437.7
-79.5 -62.3 -66.3 -60.9 -53.9
573.0 547.1 623.4 532.1 509.4 487.4 591.6 508.3 487.4 463.6 557.1 484.8 484.8 465.6 441.0
71.8 82.5 -0.3 0.7 -0.2 32.0 -0.4 -0.2 7.4 -2.5 -3.7 -1.2 -0.5 3.3
* Orthorhombic.
Results and Discussion The three contributions to the energy and their sum, the lattice energy, appear along with the experimental value in Table 111. A brief survey of this table shows general agreement to within 1% for all cases except SrBrz, MgC12, and the three hexagonal salts. We feel that this agreement is surprisingly good considering the relatively crude approximation of assuming one value of g in the repulsive energy applies to all salts. As mentioned before, SrBrz involves exceptional difficulties
The Journal of Physical Chemistry
in the determination of the Madelung constant which, along with uncertainties in the structural parameters, could easily account for its discrepancy. The crystal structure of MgC1, is not well enough established to be certain that a real difference between calculated and experimental lattice energy exists for this case. Thus, it appears that only for iLlgBr2, MgIz, and C a b does a demonstrated distinct deviation of the calculated from the experimental lattice energy exist. Since these salts are expected to be more covalent than the others, one might be tempted to give this as a sufficient reason for the difference. On the other hand, this treatment is known to work well for other salts, for example, LiI which is by some measures as covalent as llIgBrz or CaIz. The assumption of a considerably different repulsive energy is of no use here since even reducing it to zero will not account for the experimental lattice energy. The remainder of the calculation is completely determined by the crystal structure parameters; thus, we are left with three possibilities: (1) the approximation breaks down for these salts; ( 2 ) the values going into the experimental value of the lattice energy are in error; or (3) the crystal structure determinations are in error. It would be interesting to do a more refined calculation when accurate compressibility data become available for these salts.
Acknowledgments. T. E. Brackett wishes to acknowledge the generous support of the Colgate Research Council, which contributed substantially to the completion of this project. We also thank Charles R. Dawson and Edward s. Macias for assistance with the machine calculations. Finally, we wish to thank Colgate University for their generosity in allowing us free time on the Colgate University computer. (14) G. N. Lewis, M. Randall, K. S. Pitzer, and L. Brewer, "Thermodynamics," 2nd Ed., McGraw-Hill Book Co., Inc., New York, N. Y., 1961. (15) C. E. Moore, U. S. National Bureau of Standards Circular No. 467 V, Vol. 1 and 3, U. S. Government Printing Office, Washington, D. C., 1949 and 1958. (16) L. Brewer, UCRL Report 9952, Nov. 1961. (17) L. Brewer, G. R. Somayajulu, and E. B. Brackett, Chem. Rev., 6 3 , 111 (1963).
