1251
COHESIVE ENERGIES OF THE ALKALIHYDRIDES AND DEUTERIDES
Cohesive Energies of the Alkali Hydrides and Deuterides by R. C. Bowman,Jr. Monsanto Research Corporation, Mound Laboratory,‘ Miamiaburg, Ohw
46348
(Received December 81, 1970)
Publication costa assisted by the Monaanto Research Corporation
The effect of isotopic substitution on cohesive energies of alkali hydrides has been estimated using BornMayer equations. Second-neighbor interactions and van der Waals terms were included in the calculations. Vibrational contributions were obtained from Debye lattice theory, Theoretical cohesive energies are in excellent agreement with experimental values. Differences between calculated hydride and deuteride cohesive energies are slightly larger than experimental differences. Estimates of compressibilities and Debye temperatures are also given.
Introduction
calculations, special attent’ion was given to include systematically all relevant isotopic contributions. Besides the cohesive energies, the compressibilities and Debye temperatures have been estimated for the alkali hydrides and deuterides. These results are compared, whenever possible, with values deduced by other workers. A Digital PDP-8/1 computer was used for the actual computations.
Since the hydride ion possesses a noble gas electron configuration, the alkali hydrides can be considered members of the alkali halide family. I n fact, these hydrides have many ionic properties similar to those found in the alkali halides. However, some past studies on LiH as discussed by Pretzel, et uL12 and Magee3 indicate that there may be extensive covalent bonding in the alkali hydrides. The Born theory of solids provides a means of estiTheoretical Considerations mating covalency in “ionic” crystals. When the bonds The cohesive energy of an ionic crysta,l is the sum of have large covalent contribution^,^ the cohesive enerthe lattice energy, W L ,and the internal energy assogies determined from the Born model are significantly ciated with lattice vibrations, WVIB. I n terms of the smaller than values deduced from thermochemical WL is expressed as a sum of two-body Born theory, cycles. Although this analysis is not very sensitive in interaction energies using adjustable parameters dedetecting departures from ideal ionicity, it does predict duced from crystal data. For cubic solids, which inlower limits of the covalent contributions to alkali hyclude all alkali hydrides, these parameters are found by dride cohesion. fitting the Born latt’ice energy model to the solid-state While the crystal properties of alkali hydrides have equation of state and its volume d e r i ~ a t i v e . ~In~ the been estimated for a variety of Born-type m ~ d e l s , ~ ~ ~ - ~ present study, the thermodynamical condition of the the majority of these calculations are based on simple alkali hydride crystals are specified as room temperamodels which only include electrostatic static interture and atmospheric pressure. W V 1 ~which , represents actions between point charges and repulsive forces bethe sum of zero point and heat capacity energies, can be tween nearest neighbors (n”).The important conestimated“ from well-known Debye models. t r i b u t i o n ~ of ~ ~ next-nearest-neighbor (NNN) repulsions, van der Waals (VDW) potentials, and lattice vi(1) Mound Laboratory is operated by Monsanto Research Corp. brations have been consistently omitted. The alkali for the U. 5.Atomic Energy Commission under Contract No. AThydride cohesive energies reported in this paper were 33-1-GEN-53. determined from a Born-Rlayer model4bwith NNN(2) F. E. Pretzel, G. N. Rupert, C. L. Mader, E. K. Storms, G. V. Gritton, and C. C. Rushing, J. Phys. Chem. Solids, 16, 10 (1960). VDW terms. The most recent data were used in the (3) C. G. Magee in “Metal Hydrides,” W. H. Mueller, J. P. Blackcalculations where the VDW coefficients were obtained ledge, and G. G. Libowitz, Ed., Academic Press, New York, N. Y., from crystal optical absorption spectra. The lattice 1968, Chapter 6. vibrational energies were estimated using the Debye (4) (a) M. F. C. Ladd and W. H . Lee, J . Inorg. Nucl. Chem., 11, 264 (1959); (b) M.P. Tosi, Solid State Phys., 16, 1 (1964). theory of lattices. (5) J. Sherman, Chem. Rev., 11, 93 (1932). Because of a large mass ratio betw-een hydride and (6) E. C. Baughan, Trans. Faraday SOC.,55, 736 (1959). deuteride ions, alkali hydride cohesive energies should (7) T. R. P. Gibb, Jr., in “Progress in Inorganic Chemistry,” Vol. be particularly sensitive to isotopic substitution. Al111, F. A. Cotton, Ed., Interscience, New York, N . Y., 1962, pp 315-511. though calorimetric measurements’O show a small, but (8) L. Dass and J, C. Saxena, J . Chem. Phys., 43, 1747 (1965). discernible, isotopic effect for the alkali hydrides, no (9) D. W. Hafemeister and J. D. Zahrt, ibid., 47, 1428 (1967). previous Born-type calculation has explicitly consid(10) S. R . Gunn and L. G. Green, J . Amer. Chem. SOC., 80, 4782 ered the role of isotopic substitution. I n the present (1958). The Journal of Physical Chemistry, Vol. 76, N o . 9,1971
1252
R. C.BOWMAN, JR.
When VDW and NNN terms are included in the Born-Mayer model, the lattice energy is given by
7
=
p+- exp
+
WL = - (a,e2/r) - (C,/r6) - (Dr/r8)
Here, a r is the Madelung constant, e is the electronic charge, r is the interionic separation between nearest neighbors, C, and D, are dipole-dipole and dipolequadrupole VDW coefficients, respectively, p12 are constants of order unity which are qualitatively related to the electronic configuration of the ions, r+ and r- are ionic radii, b and p are Born-Alayer repulsive parameters, and sr is the separation between KNN ions. Since all alkali hydrides crystallize in a NaCl lattice structure, and s equal 1.75756 and 4,respectively. The present calculations are based upon the "static lattice" approximation of equation of state in which the Born-Mayer parameters can be determined using only the lattice parameter, 2ro, and isothermal compressibility, K . A trivial, but important, reason for choosing this version of the equation of state is the nearly complete absence of reported crystal data (other than lattice parameters) for all alkali hydrides except LiH and LiD. A more fundamental difficulty arises from the very high Debye temperatures possessed by the alkali hydrides.2 Detailed thermodynamic ana1ysislz has shown that conventional Hildebrand or Nie-Gruneisen formulations of the equation of state are invalid in temperature regions much lower than the Debye temperature. Since room temperature lies within this temperature range for the alkali hydrides, the use of these more involved formulations cannot be justified. Therefore, the static lattice version has been used for the alkali hydrides. The calculated cohesive energies are expected t o be reasonable as careful studies on the alkali halides by Tosi4b indicate the energies determined from the static lattice approximation agree within a couple per cent with values obtained from more elaborate models. Methods for evaluating the equation of state and its volume derivative from a given Born-Mayer model have been adequately described e l ~ e w l i e r e . ~ ~Only J~ the final equations are presented here. When b is eliminated from eq 1 by use of the static lattice equations, the lattice energy can be rewritten as
(" + 'b- - ro
The Born-Mayer parameter p is obtained by solving the equation K-1 = -l[$ 18r03
(1
- xz) +
where
The VDW coefficients Cr and D, represent summations over two-body VDW interactions between the various ions. When the ionic crystal is comprised of two Bravais lattices, the van der Waals coefficients are given by14
+ l/z(c++ + C--)SFP(O) (9) D R = d+-sRc0(r-) + '/z(d++ + d--)S~@)(o) (10) c, = C + S R @ ) ( P - )
where the lattice sums, SR('Q(r)are known and have been tabulated by T o ~ i . The ~ ~ two-body VDW coefficients c12 and dlz are determined from the bIayer14 equations ciz = 3aiazEiEz/2(Ei
+ Ez)
(11)
Here, E is the average excitation energy, a is the ionic polarizability, and n is the number of outer-shell electrons. (11) L. D.Landau and E. M. Lifshitz, "Statistical Physics," Addison-Wesley, Reading, Mass., 1968, pp 187-190. (12) F. G. Fumi and M. P. Tosi, J. Phys. Chem. solids, 23, 396
where
The Journal of Physical Chemistry, Vol. 76, N o . 9,1971
(1962). (13) M. Born and K. Huang, "Dynamical Theory of Crystal Lattices," Clarendon Press, Oxford, 1954, Section 3. (14) J. E.Mayer, J . chem. Phys., 1, 270 (1933).
1253
COHESIVE ENERGIES OF THE ALKALIHYDRIDES AND DEUTERIDES Table 1: Experimental Data" LiH yo
(10-8 em)
(10-8 cm) (10-8 cm) K (1O-I2 dyn/crnz)
T+ T-
01+
(10-24
01113)
(10-24 cm3) E+ (10-l2 erg) E- (10-12 erg) c++ (10-60 erg om5) c-- (10-50 erg erne) c+- (10-50 erg cm5) C, (10-60 erg cm5) d++ (10-75 erg em8) d-- (10-78 erg cm8) d+erg cm8) D, ( l O W 7 6 erg em8) 01-
2.042 2,034 0.90 1.30 2.88 2.85 0.029 1.86 118.4 10.7 0.075 22.3 0.828 25.7 0.025 45.2 0.98 24.1
NaH
KH
2.440 2.434 1.21 1.30
2.854 2.848 1.51 1.30
3.024
3.194
1.65 1.30
1.80 1.30
0,258 1.86 107.8 13.9 5.26 17.3 7.03 66.8 3.53 35.0 9.48 73.7
1,201 1.86 63.3 11.8 68.5 20.3 31.5 288.0 127 41.2 61.1 442
1.797 1.86 71.0 10.3 180 23.2 48.8 505 535 46.9 125 1000
RbH
CsH
3.137 1.86 66.0 9.36 487 23.2 85.6 1030 2460 51.8 304 2870
and K , the upper values are for the hydrides while the lower values pertain to the deuterides. For the other 5 In the rows for rows, single values represent both hydrides and deuterides.
Data The data used in the cohesive energy calculations are summarized in Table I. The ro values are from the compilation by 1Clagee.3 The alkali radii Y+ are taken from the work of Fumi and Tosi15 on the alkali halides. The hydride (deuteride) radius r- is the average deduced from differences between yo and the Fumi-Tosi r+ for all alkali hydrides. The only experimental compressibilities reported for the alkali hydrides pertain to LiH and LID. The recent values obtained by Stephens and Lilley16are used in the present calculations. The van der Waals coefficients, as ell as the polarizibilities and excitation energies required to determine them, are also listed in Table I. The alkali polarizibilities a+ are from the study by Tessman, et d , l 7 while a- was calculated from LiH refractive index data1* using the familiar Clausius-RIossotti equation. Following the procedure developed by R1ayer,l4 E- is estimated for each salt from the exciton band in the corresponding ultraviolet absorption s p e c t r ~ m . ~In,~~ stead of the arbitrary alkali energies E+ suggested by Mayer, semiquantitative values were determined from free ion polarizibilitiesZ0using an analysis due to Ruffa.21 Deuteride VDW coefficients are taken equal to the corresponding hydride values since nuclear mass substitution has negligible effect upon a or E values. PIZcoefficients cannot be determined from the traditional P a ~ l i n gformula ~ ~ , ~ which ~ predicts p-- = 0 and implies repulsion between hydride ions vanishes. Recognizing the general property that overlap repulsive forces between two cations with high electron densities are stronger than repulsive forces between two anions with diffuse electron distributions, the pl2 coefficients are set equal to the following values: p++ = 1.25,
p+- = 1.0, and p-- = 0.75 in the present calculations. A careful examination of the results for LiH, KaH, and KH indicates that WL is insensitive ( ~ 0 . 1 % ) for a wide choice of plz coefficients (excluding the Pauling values which gave AWL = 0.5%). Cohesive Energy The cohesive energy of alkali hydrides has been determined from thermochemical data by a number of workers3+8 using the familiar Born-Haber cycle. Traditionally, the cohesive energy has been described as the energy difference between the dispersed ions and the crystal lattice at absolute zero. Because room temperature crystal data were used in the present Born$layer calculations, the experimental cohesive energy W B H is found from a modified Born-Haber cycle in which the energy difference is between the crystal lattice at room temperature and dispersed ions at absolute zero. In this case, W B H is given by
WBH = Q - X - I -
'/z
D -E
+ 2AHo
(13)
Here, Q is the enthalpy of formation of alkali hydride (deuteride) crystals from alkali metal and hydrogen (deuterium) molecules; X is enthalpy of sublimation of (16) F. G. Fumi and M. P . Tosi, J . Phys. Chem. Solids, 25, 45 (1964). (16) D. R. Stephens and E. M. Lilley, J . Appl. Phys., 39, 177 (1968). (17) J. R. Tessman, A. H. Kahn, and W. Shookley, Phys. Rev.,92, 890 (1953). (18) E. Staritzky and D. I. Walker, Anal. Chem., 28, 1055 (1956). (19) W. Rauoh, 2. Phys., 111, 650 (1939). (20) L.Pauling, Proc. Roy. Soe., Ser. A, 114, 181 (1927). (21) A. F. Ruffa, Phys. Rev.,130, 1412 (1963). (22) L. Pauling, 2. K r i s t . , 67, 377 (1928). The Journal of Physical Chemistry, Vol. '76,No. 9,1971
R. C. BOWMAN, JR.
1254 the alkali metal; I is the ionization potential of the metal; D is the dissociation enthalpy of the hydrogen (deuterium) molecules; E is the electron affinity of the hydrogen (deuterium) atom; and AH" represents the enthalpy change in cooling monatomic alkali and hydrogen gases from room temperature to absolute zero and is taken equal to -5RAT/2. Reliable values of these quantities are available for most of the alkali hydrides. The S and I values are the same as those used by T ~ s for i ~ the ~ alkali halides. Calorimetric values of Q for LiH, NaH, KH, and the corresponding deuterides are reported by Gunn and Greenlo while Her01d~~ obtained Q for RbH and CsH from dissociation-pressure measurements. Values for E and D were also taken from Gunn and Green.1° The resultant experimental cohesive energies WBHare presented in Table 11, column 2. Since uncertainties in the various data amount to a few tenths kilocalories per mole, the total uncertainty in WBHshould not be larger than -1-2 kcal/mol. However, the uncertainties between corrcsponding hydride and deuteride WBHare expected to be about a factor of 10 smaller.
Table I1 : Cohesive Energies" of the Alkali Hydride and Deuterides A =
WBH
LiH LiD NaH NaD KH KD RbH CsH a
-216.2 -217.2 -189.5 -190.3 -167.1 -167.4 -160.6 -153.7
WL
-224.4 -225.1 -195.2 -195.7 -172.7 -173.1 -165.5 -159.6
WVIB
wc
WBH Wc
6.0 5.0 5.3 4.5 4.9 4.3 4.8 4.7
-218.4 -220.1 -189.9 -191.2 -167.8 -168.8 -160.7 -154.9
2.2 2.9 0.4 0.9 0.7 1.4 0.1 1.2
All quantities are in kcal/mol.
tion that p determined from LiH (LiD) data represents all the alkali hydrides (deuterides). Determination of the alkali hydride vibration energies WVIBlisted in column 4 of Table I1 requires" knowledge of the corresponding Debye temperatures OD. Since experimental Debye temperatures are not available for the alkali hydrides, the well-known BlackmanZ4equation 8D =
&(5?'0/Kp)''~/?€
(14)
is used t o determine 8D from the compressibility and reduced mass 1.1. Although experimental compressibilities are not available for most alkali hydrides, K can be calculated by solving eq 6 for p values determined from lithium hydride data. The theoretical K and 8D values used to estimate WVIB are givep in Table I11 along with results from other workers. Table I11 : Compressibilities and Debye Temperatures of the Alkali Hydrides -K,
10-12 om$/dyn---
Present study" ( e q 6)
Shermanb
Hafemeister and ZahrtC
LiH
2.88d
2.32
3.396
LiD NaH NaD KH KD RbH CsH
2.85d 4.55 4.51 6.75 6.71 7.88 8.92
3.78
5.42
6.16
7.90
6.67 7.83
9.15 10.60
---Present (eq 14)
1190 895 991 702 869 627 827 793
BD, OK--
Experimental
815,' (920),f 8508 1091' (614, 744)'
Theoretical K determined using p values obtained from experimental LiH compressibility in footnote d of this table. * Reference 5. Reference 9. d Reference 16. e Calculated using eq 32 and p data from Hafemeister and Zahrt (ref 9). f Reference 2. 8 V. N. Kostryukov, Russ. J. Phys. Chem., 35, @
0
865 (1961).
Before WL can be calculated, p must be obtained by The theoretical cohesive energy W C ,which equals the solving eq 6 using compressibilities determined indesum of WL and W V I B , is listed in column 5 of Table 11. pendently. As mentioned previously, experimental The differences between Wc and WBHare summarized compressibilities among the alkali hydrides are availin column 6 of Table 11. able for only LiH and LiD. To circumvent this difficulty in the present study, the alkali hydrides (deuDiscussion terides) are assumed to be described by a constant p As is evident from Table 11, Wc is in excellent agreedetermined from knownl6 LiH (LiD) compressibility. ment with WBHfor all alkali hydrides and deuterides Although this important approximation has not been where A is much smaller than combined experimentaland rigorously justified, previous independent ~ t u d i e s ~ J - ~theoretical uncertainties (-3% j. These results indistrongly support the use of a constant p for the alkali cate a constant p determined from LiH (LiD) compreshydrides. From the data in Tab12 I, p is determined to sibility data and give an adequate description of alkali be p~ = 0.463 and pD = 0.462 A for the hydrides and hydride (deuteride) short-range forces when NNN and deuterides, respectively. The WL values calculated VDW terms are included in the Born-Alayer model. with these p choices are given in column 3 of Table 11. Careful analysis of the calculation procedure indicates (23) A. Herold, Ann. Chim. (Paris),6 , 536 (1951). (24) M. Blackman, Proc. Roy. SOC.,Ser. A, 181, 68 (1942). an uncertainty of 1-2% arises in W Lfrom the assumpThe Journal of Physical Chemistry, Vol. 76, No. 9,1071
THEENTROPY OF MIXING
1255
A
nt 0.5 .
T
?I
-
I
0
I
Electrostatic
Since extensive covalent bonding leads to large negative A’s,4 the present slight positive A’s substantiate the basic ionic nature of the alkali hydrides. This conclusion agrees with recent analysisz6of electrical conductive studies on molten LiH. The effect of isotopic substitution on alkali hydride cohesive energies is shown in Figure 1 where the percentage differences R between hydride and deuteride energies are given for Li, Na, and K salts. The electrostatic differencesresult from changed lattice parameters, while differences in WC and WBH include effects in short-range forces and vibrational energies. The
isotope effect is small (
