J O U R N A L OF THE AMERICAN CHEMICAL SOCIETY (Registered in U. S. Patent Office)
VOLUME
(Copyright, 1955, by t h e American Chemical Society)
KOVEMBER 11, 1955
77
NUMBER 21
PHYSICAL AND INORGANIC CHEMISTRY [CONTRIBUTIOS FROM THE
LEWIS FLIGHT PROPULSION LABORATORY OF THE
x A T I O S A I . ADVISORY COMMITTEE FOR
AERONAUTICS]
Lattice Energies and Related Thermodynamic Properties of the Alkali Metal Borohydrides and of the Borohydride Ion1 BY AUBREYP. ALTSHULLER RECEIVED JUXE 6, 1955 A theoretical calculation has been made of the lattice energies of S a B H 4 ,KBHa, RbBHa and CsBH4. The lattice energy of LiBH4 and the heats of formation of KBHa, EbBH, and CsBH4 have been estimated. The polarizability of BHa- in the solid state has been calculated t o be 3.9 f 0.1 A . 3 A heat of formation for BHa-(g) of -23 i- 5 kcal./mole has been obtained. The heat of the reaction BH3 H - + BH4- is -75 f 5 kcal./mole, while the heat of the reaction '/2B2He H - +. BH4- is -61 + 5 kcal./mole. The thermodynamic functions C,' ( N o - Ho")/T, - ( F " - H o " ) / T ,and S o have been computed for BH4-(g) in the temperature range from 200 to 1000°K. Combining the above data with other entropy and free energy data available in the literature, an entropy and free energy of -25.7 cal./deg./mole and -67 kcal./mole have been calculated for thereaction BH3 f H--+ BH4-. The entropy and free energy have been calculated for the reaction 1/2B2He H - + BHa- as -8.6 cal./deg./mole and -58 kcal./mole. The heat and entropy of hydration of BH4- have been found to be -72 =I= 5 kcal./mole and -13 e.u., respectively.
+
+
+
The recent measurements of the lattice constants of the alkali metal borohydrides2p3make possible a theoretical calculation of the lattice energies of XaBH4, KBH4, RbBH4 and CsBH4. These lattice energies can be combined with the available thermochemical data to obtain the heats of formation of BH4-(g), of KBHl(c), RbBHl(c) and CsBHl(c), and the lattice energy of LiBH4. The heats of the gas reactions of the hydride ion with borane and diborane t o form borohydride ion can also be evaluated. The thermodynamic functions, Cg, (H" H : ) / T , - ( F " - H ; ) / T and So,can be computed for BH4-j.g) by the usual statistical thermodynamic methods. The entropies and free energies of the reactions of BHa(g) and B2H6(g)with H-(g) to form BH4-(g) can also be calculated. These various thermodynamic properties are obtained in the present investigation. Since the compressibilities of the alkali metal borohydrides have not been reported, and the "main frequency" energy,4 E , is not known for HH4-, the lattice energies are calculated in part for a face-centered cubic lattice2s3 (NaC1 type) from (1) Presented (in part) before t h e Division of Physical a n d Inorganic Chemistry a t t h e 127th meeting of t h e American Chemical Society, Cincinnati, Ohio, April, 1955. (2) S. C. Abrahams and J. Kalnajs, J . Chcm. P h y s . . 2 2 , 434 (1954). (3) A. M. Soldate, THISJOURNAL, 69, 987 (1947). (4) J. E . Mayer, J . Cheiiz. P h y s . , 1, 270, 327 (1933).
the simple Born e q ~ a t i o n . ~
where S is Avogadro's number, A is the lfadelung constant, e is the electronic charge, r o is the smallest cation-anion distance and n is a constant for a particular ion type. In the lattice energy calculations for the alkali metal borohydrides, the repulsive exponent n values are those employed formerly for the alkali metal fliiorides.6 These n values are chosen because the BHI- united atom (ion) is isoelectronic with F- and is thus of the Ne ion type.6 In addition to the contributions from the coulonibic and repulsion energy terms which are included in the Born equation, the van der Waals energy term must be included in a t least an approximate manner. The van der Waals interactions in the crystal depend on the lattice type which is the came for the Na, E(,Kb and Cs borohydrides as for the alkali metal halides (except CsC1, CsBr and CsI), on the polarizabilities of the ions, and the "main frequency" energies of the ions. The polarizability of the borohydride ion niay be ( 5 ) M Born, Bev deut phystk G e s , 21, 13 (191C4) (6) J Sherman, C h i n R e u s , 11, 03 (1932)
VOl. 77
AUBREYP. -ILTSHL'LLEK
5456
evaluated from the refractive indices' of NaBH4, TABLE 111 KBH4, KbBH4 and CsBH4 by the method used by LATTICEENERGIESOF T H E ALKALIMETALBOROHYDRIDES Tessman, et al., to evaluate the polarizabilities of Van der - Uc Waals ions in crystals.* The polarizability of BH4- iii (Born e q . ) , energy, - Ul i . ~ 4., ~ ii kcal./mole kcal./mole kcal./mole the crystal is given in Table I along with the quan- Conipd. 7.0 tities Vm = u3/4, where a is the lattice constant, NaBH, 3.082 161.4 6.3 168 and cym = 3Vm(n2- 1)/4r(n2 2 ) = ah^+ ~ B H ; . KBHi 3.304 8.0 150.9 8.4 159
+
+
TABLE I POLARIZABILITY OF BH4Compd.
vm,4.3
NaBHa KBHd RbBHa CsBHa
58.56 76.14 86.82 102.1
am,
A.a
( A = NaD)
A.a
a ~ + ,
(A = N ~ D ) '
aBH;,
A.s
RbBH4 8.5 3.515 145.7 9. 0 C S B H ~ 3.715 9.5 139.9 10.4 a From the lattice constants a of reference 2 .
155 150
( A = NaD)b
all the terms in the energy equation should be niinimized a t the equilibrium distance. T o estimate the uncertainty resulting from this approximation, the same approximate procedure was used for the alkali metal halides. The approximate lattice energies proved to be several kcal./mole higher than the very accurate lattice energies for the alkali metal halides calculated by Huggins.lo The total uncertainty in the lattice energies for the alkali metal boroThepolarizability of B H - of 3.9 A.3is only slightly hydrides resulting from the use of the Born equasmaller than the polarizability for Br- of 4.158 A,3 tion, the approximate nature of the van der LVaals energy calculations and the addition procedure, is (cr(Cl-) = 2.960 A.3,cr(I-) = 6.431 A.9, The polarizabilities of the alkali metal ions most estimated to be about =t5 kcal./niole. The heat of formation of BH*-(g) can now be applicable to crystal lattice calculations are those given by Tessman, et al.,5 and listed in Table I. calculated to be -23 + 5 kcal./mole from a AH? The "main frequency" energies of the alkali metal (298'K.) for NaBHj(c)" of -45.5 kcal. 'mole, ions, E + , are taken as 0.9 of the second ionization AH? (298OK.) of Na+(g)'? of 146.0 kcal./niole and potential of the elements4 The "main frequency" the lattice energy of 168kcal./mole for l\'aEH4(c). The lattice energy of LiBH4(c)13can be calcuenergy E- for BH4- is assumed to be close to the Elated to be 186 kcal./mole from the heats of formafor Br-.9 The contribution from the dipole-dipole interactions, C/ro6, to the van der Waals energy can tion of LiBHd(c),ll of Lif(g)12and of BH4-(g). The heats of formation of KBHh(c), RbBH*(c) now be calculated by &layer's method* using the above data. The quantities involved in the evalua- and CsBH*(c),may be estimated also from the reaction MBHd(c) + bI+(g) BH4-(g). The heats of tion of the energy resulting from the dipole-dipole these reactions are the lattice energies already calinteractions are listed in Table 11. culated. The heat of formation of BH4- has been TABLE I1 calculated in this paper and the heats of formation of K7(g), Rb+(g) and Cs+(g) are available.12 QUASTITIESI S V O L V E D IS ~ V A L U A T I O i iO F YAK D E R [I'AALS ENERGYFROM DIPOLE-DIPOLEIXTERACTIOSS IS LATTICE From these da-ta,the heats of formation of KBH4(c), RbBH,(cj and CsBHq(c)a t 298°K. are estimated to Ergs/molecule 10'2 C c, t C- C+ Compd. bc - 5S, - -59 and - 6.3 kcal. 'inole, respectively. lG0 27.9 ;Kj(i SaBH4 8.0 The heats of the reactions .. 83.4 740 KBHa (XI.6 BHa(g) H-(g) --+ B K - ( g ) (1) .. 119 1036 RbBHa 116 and 0.408 4.02 1.334 3.92 1.979 3.98 3.335 3.81 Av. 3 . 9 i 0 . 1 a Reference 8. These values are in excellent agreement with the values for a obtained by Stockmayer, Rice and Stephenson, THISJOURNAL, 77, 1980 (1955). 4.43 5.25 5.96 7.14
+
CsBH4
28 1
..
193
1671
The contributions from the dipolc quadrupole interactions, D/ro8, are approximated as 2,'1~ the contributions from the dipole-dipole interactions. The fraction 2,'15 is chosen from the relative magnitudes of the two terms for the alkali metal halides. -I The values of ro and n, the lattice energy contributions calculated from the Born equation, the total van der \.Z'aals energies, and the total lattice ener, listed in Table 111. qies, - l - ~are Thc addition of the van der It-aals cncrgics to the coulonibic and repulsion energies from the Born equation is an approximate procedure. ,Ictually, (7) AT. D . Banus, R. \V, Bragdon and A . A. Hinckley, THISJ O U K N A L , 76, 3849 (1954). (8) J. R. Teisman, A . 11. K a h n and W. Shockley, P h y s . Reo., 9 2 , 890 (1953). (9) I f , instead of 6- for BEIa- being close t o t- for D r - it were actually close t o c for F - , then t h e \.an der Waals energies would be in creased by about 2 kcal./mole. Since t h e range from e- for B r - t u t h a t for F - probably represents t h e range of uncertainty in e- for BHt, about 1 2 kcal./mole would represent t h e uncertainty in thc lattice cuergies resulting from this approximation.
+ 1/2&I-16(g) + H-(g) +BHa-(g)
(2 )
may now be calculated also. Using the heats of formation of BH8(g),11x1*of B2H6(g),11of H-(g)12 and of BH4-(g), the heat of reaction for (1) is -75 rt 5 kcal./mole and the heat of reaction for (2) is -61 + 5 kcal.,'mole. The thermodynamic functions, C,", ( H o - IT:)/ T , -- (F"- H'),/T and S o can be calculated by statistical thermodynamic methods for BH,-(g). The B--H distance in BH4- has bcen obtained from nuclear magnetic resonance measurements on alkali metal b~rohydrides.'~The r(B-H) in BHI- is (IO) LiI. H u g g i n s , J. Chein. P h y s . , 6 , 143 (1937). (11) D. D. Wagman, T. R. Munson, W. H. Evans and E. J. Prosen, h-BS Report 3456 (Aug. 30, 1954). (12) F. D. Rossini, et a!., "Selected Values of Chemical Thermodynamic Properties," N B S Circular 500, 1952. (13) T h e LiBHt lattice is orthorhombic (P. M. Harris and E. P. Rfeibohm, THISJ O I J R N A L , 69, 1231 (1947)) and a theoretical treatment leading to its lattice energy would be difficult. (14) A . Shepp and S. H. Bauer, ibid., 76, 265 (1954). (1.5) Private communication from Dr. R . E. Richards of Oxford. T o be published in the Discs. Furaduy SOL.
Nov. 5, 1955
LATTICEEKERGIES OF ALKALI METALBOROHYDRIDES
1.255 f 0.02 A.16 Therefore, the moment of inertia, I , for BH4- is (7.03 =t0.22) g. The BHI- ion is a spherical top with the observed vibrational frequencies,16-18 v 1 (1) = 2290 cm.-l l8; v 3 (3) = 2299 em.-' 17; v4 (3) = 1123 cm.-l.17 The frequency v2(2) has not yet been observed. This frequency, v2, can be approximated as 1180 cm.-l from the valence force field expressionlg for tetrahedral molecules, XYd: v ~ v ~ / v=~ v[2/3~ (1 4my/mx)]1/p. However, for first row elements 4my/m,) ]1/2 are appreciably the values of [ 2 / 3 ( 1 larger than the v ~ v ~ / vvalues. ~ v ~ I n particular, for the first row hydrides, XH4, the calculated values average about 5% higher than the values obtained from v3v4/ v1v2. Consequently, the vibrational frequency values for v2 obtained from the valence force field equation has been corrected by 5% to give a value for v 2 of 1240 =t50 cm.-l. From the combination band v2 v4 = 2398 cni.-l l7 a value for vz of 1275 cm.-' is obtained. The average of these two values, v 2 = 1260 cm.-' is used in calculating the thermodynamic functions. The thermodynamic functions as calculated by the rigid rotator-harmonic oscillator approximation are given below for BH4- in the ideal gaseous state a t one atmosphere pressure.
5457
The uncertainties in the moment of inertia, I , and the vibrational frequencies, along with the use of the rigid rotator-harmonic oscillator approximation, limit the accuracy of Ci to f 0 . 2 unit and the accuracy of (HO - H ; ) / T , - ( F o - H;)/T and So to *0.1 unit a t 300°K. and below and to *0.2 unit a t higher temperatures. Near 1000OK. the neglect of the effects of anharmonicity may decrease the accuracy of the thermodynamic functions even more than is estimated. The entropies and free energies can now be evaluated for reactions (1) and (2). The entropy of BHs(g) a t 300°K. is 44.9 c a l . / d e g . / r n ~ l e ,while ~ ~ the entropy of B2He(g) is 55.5 cal./deg./mole.ll The entropy of H-(g) from the Sackur equation is 26.04 cal./deg./mole, and the entropy of BHd-(g) a t 300'K. is 45.23 cal./deg./mole. Consequently, the entropy of reaction (1) is -25.7 cal./deg./mole and that of reaction (2) is -8.6 cal./deg./mole. Using the heat of reaction for reaction (1) of - 75 f 5 kcal./mole with its entropy of reaction, one ob5 kcal./mole. tains a free energy for (1) of -67 Similarly, for reaction (2) the free energy of the reaction is - 58 5 kcal./mole. Thus, both reactions a t standard state conditions will tend to result in borane and diborane reacting with hydride ion almost completely to form borohydride ion. TABLEIV The heat of formation, AHf(aq) and the entropy, THERMODYNAMIC FUNCTIONS FOR BHa- I N THE I D E A L for the aqueous borohydride ion have recently GASEOUSSTATEAT ONE ATMOSPHERE PRESSURE(IN been reported.20 These values, AHf(aq) = 12.4 CAL./DEG./MoLE) kcal./mole and so = 25.5 f 1 e.u., can be comHo - HoO F 0 - Heo ___ bined with the heat of formation and the entropy of T T CP T,OK. the gaseous borohydride ion given above to calcu41.84 8.00 33.84 200 8.11 late the heat and entropy of hydration of the BHd43.69 8.06 35.62 8.49 250 ion. The heat of hydration, - A H h y d . , of BHd- is 45,23 8 . 1 8 37.05 9 . 0 8 298.16 72 5 kcal./mole. The entropy of hydration (cor37.10 45,29 9.11 8.19 300 rected for the entropy change resulting from volume 39.50 48.13 10.76 8 . 6 3 400 contraction, 6.35 e.u.) is - 13 e.u. The heat of hy41.48 50.72 12.55 9.24 500 53.16 dration of BH4- is about the same as A g h y d . for a 9.94 43.22 600 14.26 10,67 44.80 55.47 halide-like ion of the same size (2.03 A.) which 15.78 700 46.27 57.67 would be about 77 kcal./mole.21 The entropy of 17.12 11.40 800 47.65 59.75 hydration of BH4-, - 13 e.u., ( - A S H + = 0) is only 18.27 12.10 900 48.96 61.73 slightly more negative than the 3" for a halide-like 19.24 12.77 1000 ion of the same size which would be - 11 e.u.*l
+
+
+
*
*
so,
S O
*
(16) W. C. Price, J . Chcm. Phys., 17, 1044 (1949). (17) T. H. Walnut, Jr., Ph.D. Thesis, Brown Univ., 1951. (18) W. J. Lehmann, P h . D . Thesis, St. Louis Univ., 1954. (19) G. Herzberg. "Molecular Spectra and Molecular Structure.
Infrared and Raman Spectra of Polyatomic Molecules," Van Nostrand Co., New York, N. Y..1945, pp. 181-188.
11.
D.
CLEVELAND, OHIO ( 2 0 ) W. H. Stockmayer, D. W. Rice and C. C. Stephenson, THIS 77, 1980 (1955). (21) W. M . Latimer, J. C h o n . P h y s . , 23, 90 (1955).
JOURNAL,
