- Email: [email protected]
Calculation model for the enthalpy of formation of multicomponent hydrides
Journal of the Less-Common
221
Metals, 105 (1985) 221 - 230
CALCULATION MODEL FOR THE ENTHALPY MULTICOMPONENT HYDRIDES
OF FORMATION
OF
A. L. SHILOV, M. E. KOST and N. T. KUZNETSOV Institute of General and Znorganic Chemistry Moscow 117071 (U.S.S.R.)
of the U.S.S.R.
Academy
of Sciences,
(Received February 22,1984)
Summary A semiempirical formula for calculating the enthalpy of formation of hydrides of inte~eta~ic compounds was developed from simple assumptions concerning the interaction between hydrogen and metals. The new model takes account of the electronegativities and electron densities of elements constituting the compounds, as well as the stoichiometry and the parameters of the unit cells of alloys. Comparison of the calculated and experimental values for hydrides of 62 binary, ternary, quaternary and more complex alloys shows good agreement (within 5 kJ (mol Hz)-“).
1. Introduction Assessment of the potential of hydrides of intermetallic compounds as reversible hydrogen storage materials and of many binary, ternary and more complex alloys as hydrogen absorbers requires the prediction of a number of their properties, of which the~odyn~i~ ~h~acter~tics are among the most important. In a continuation of the work of Wagner [l] and Beck [Z], attempts are made to correlate the thermal stability of hydrides with the geometrical characteristics of the crystal lattice of the parent metallic material. Although such an approach has been criticized by Burch and Mason [ 3 J, the role played by the geometrical factor in the energetics of hydrides is unquestionable. For hydrides of binary and ternary intermetallic compounds contain~g rare earth elements with CaCu,-type crystal st~c~res, some workers have shown a linear dependence of the free energy G of the cu-0 transition at a single temperature on the tetrahedral hole size of the metal matrix [ 41 or on the volume of its unit cell [53. However, for practical purposes a knowledge of G at a single temperature only is insufficient. Therefore we considered [6] the influence of the geometry of the crystal lattice of an intermetallic compound on the enthalpy of formation AEI of the hydride, a quantity which is almost independent of temperature. We chose as @ Elsevier Sequoia/Printed in The Netherlands
222
the variable defined as
parameter
the linear
dimension
a of the unit
cell which
is
where V is the volume of eight formula units of intermetallic compound of general formula RM, (R = rare earth, Ca, Ti, Zr, Hf is a hydride-forming element; M = Cr, MO, Fe, Co, Ni, Ru, Rh, Cu, Al, Si etc. is an element which does not react with hydrogen under normal conditions). An almost linear dependence of AiY on a was found for equistoichiometric intermetallic compounds of a rare earth with a common metal M [6]. The fact that data for isostructural intermetallic compounds with various M components can be approximated by a set of lines indicates the importance of such factors as electronegativity and electron density in addition to the geometric factor. The effects of electronegativity and electron density have been clearly demonstrated for ruthenium-containing ternary alloys [ 6 - 81.
2. Development
of the calculation
procedure
2.1. Choice of interaction model It follows from the above discussion that the energetics of the interaction between hydrogen and the intermetalhc compound is determined (i.e. the number of M mainly by the stoichiometry cM of the compound atoms per R atom), the dimension a of the unit cell of the intermetallic compound, and the electronegativity cp and the electron density n of the elements. As our aim is to consider the interaction of hydrogen with ensembles of various types of metal atoms, we introduce the average values of electronegativity and electron density which are defined as
ii =
C?liCi/CCi
where p, and ni are the values of the corresponding quantities for the elements and ci is the number of atoms of a given element (both M and R) per formula unit of RM, (i.e. ECi = CM+ 1). This procedure is similar to the introduction of the group orbital electronegativity in the work of Hinze et al. [9]. Data from various investigations show that M-H and H-H interactions which may be either attractive or repulsive and either short range or long range exist in metal-hydrogen systems. We have assumed that the enthalpy of formation of hydrides is determined mainly by the sum of a positive term due to the repulsion of the neighbouring hydrogen atoms and a negative term characterizing the H-M interactions. The first term depends on the
223
H-H distance (and consequently on the parameter a) and on the charge carried by the hydrogen atom. As in ref. 10, this value was assumed to be proportional to the difference Ap between the electronegativities of the hydrogen and the metal (A9 = $$r - g), with the relative hydrogen content in the hydride being ignored. The second term depends on both Ap and the of electron density difference An = nH -n”, as well as on the stoichiometry the intermetallic compound. In fact, the stoichiometry (ratio of M to R) is taken into account in calculating + and ii. Here we note the necessity of allowing for the difference between the formal stoichiometry of the intermetallic compound and the atomic ratio in the local environment of the hydrogen atom. Thus, when the enthalpy of molecular hydrogen dissociation Mti, is taken into account, the general form of the equation for M is as follows: AH=
EHH(&,
a)-E~d&,
An,
CM)
(1)
+ Lwdi,
2.2. Search for the form of the functional dependence and the determination of values of the constants in the equation For intermetallic compounds of a rare earth with the same M and CM, all the variables except a in eqn. (1) are almost constant. Therefore the form of the function Enn(Aq, a) was sought by inspection of data for these compounds. If EHH is assumed to be inversely proportional to c, we have AZY= Enn(Aq)e-’
+K
where K = -EHM + AH&,. Thus, from the slopes of the lines in Fig. 1, we can obtain the values of E~H(A(P) for various series of binary intermetallic compounds of the rare earths. These values were assumed to be functions of Ap. The values of Cpi were those given by Miedema [lo] and Bouten and
5050 .
,060,
oRRUz
I
ox_3
a425
11127
au9
n-!j-i
0.5
0.6
0.7
0.8
A’Q,V
Fig. 1. Dependence of the enthalpy of formation of the hydrides of RM3 (R = rare earth) on the dimension a of the unit cell of the intermetallic compound: 0, RNia; 0, RCo3; A, RFe3. Fig. 2. Em
us.
ACJJ for hydrides of various intermetallic compounds of the rare earths.
224
Miedema [ 111 corrected for a better description of the interactions in binary compounds. With some correction of the value for e (5.37 V instead of 5.2 V as reported in ref. 11) the value of Ear was found to be a linear function of Ap (Fig. 2): EHH(AP) = A AP where A = 6536.5 kJ A V-r (mol Hz)-l, and hence E~~(A~, a) = A Ape-’
(2)
Substituting eqn. (2) into eqn. (1) and using the above value of A and AH diss= 432.3 kJ (mol Hz)-’ [12] we calculated EWMfor hydrides of intermetallic compounds with R E rare earth, Ca, Ti, Zr, Hf from the formula -En&A@, An, cNI)= AHeXP-A
Aq u-l - B,,
As can be seen from Fig. 3, this quantity increases linearly with increase in the “enthalpy” A.@ of the H-M interaction, which is given by AI?=Ap’-FAn2 where F = 0.218 V2 (density units)- 2. In addition to correction for hydrogen (nn was taken as 4.6 density units), the values of cpi and ni given in ref. 10 were also corrected for copper (ncu was taken as 3.82 density units).
p%~
750 o
p aNi, PLrC”s
Fig. 3. Em us. &
0.4
0.0
for intermetallic
1.2 compounds
a:, v 2 with equivalent
stoichiometries.
The constants in the equation (3) EHM(AP, An, CM)= ki(A@ + fi) for the series of compounds considered are given in Table 1. However, it was necessary to obtain the explicit dependence of EHM on CMin order to be able to calculate the enthalpies of formation of those groups of compounds for which data for determining ki and fi were absent or insufficient. For example, all rare earth compounds with the composition R&o, are represented by a single point in Fig. 3. Therefore, assuming Iz is constant in eqn. (3), we attempted to relate f to CM:
225 TABLE 1 Parameters for eqn. (3) R
Stoichiometry of the intermetallic compound
ki (kJ VP2 (mol Hz)-‘)
fi (V2)
Rare earth, Ca
BMs BM3 BM2
700 446 431.2
0.9433 1.9312 1.9613
Ti, Zr, Hf
BM2 RM
412.3 448.3
2.3976 2.5191
EHM(&, An, cM) = him + f(cMM)l
(4)
As can beseen from Fig. 4 the dependence off on cM at Iz = 415 kJ VP2 (mol H2)-i can be represented by the relationship f(c&
=B--D
1gcM
(5)
where B = 2.8135 V2 and D = 1.4675 calculation is of the form H=AA~~-‘-~(AIJY~-FA~~+~)+AH~~
V2. Then the formula
for the finite (6)
and the accuracy of the calculation depends on whether the second term is calculated from eqn. (3) and Table 1 or from eqns. (4) and (5).
1.8
d
0.2
0.4
PRMs 0.6 ejcn
Fig. 4. Determination of the -dependence of the parameter f in eqn. (4) on the stoichiometry of the intermetallic compound.
The results of calculations for hydrides of binary intermetallic compounds are listed in Table 2 where it can be seen that the agreement with the experimental data is good. The maximum discrepancy is 6.2 kJ (6.6%) and the average discrepancy is 1.5 - 2.0 kJ. Equation (6) was derived mainly on the basis of data on ternary hydrides, but no restrictions were placed on the number of alloy components. Table 3 presents the results of calculations for quatemary and more complex hydrides. In general the agreement is similar to that obtained for
226 TABLE 2 Enthalpy of formation of the hydrides of binary intermetallic compounds Zntermetallic compound
-AH
(kJ (mol II,)-‘)
Reference
Calculated
Experimental 31.8 27.7 38.5 32.7 44.4 31.0 46.0 44.0
1131
CeNis
32.2 29.2 37.9 35.6 43.5 32.5 47.2 49.0
ErCoa GdCo3 ErFes ErFe2 ScFes ErRu2 ScMnz TiFe Tic0 TiNi Tic& TiMns
44.6 53.2 45.7 54.9 35.1 47.8 62.6 27.7 46.1 61.9 22.5 -1.0
44.1 53.4 43.1 55.0 35.4 45.8 63.0 28.2 44.4 60.0 26.2 -B
gf
ZrCo ZrNi ZrCra ZrMn2 HfVs
81.4 89.1 54.1 50.7 71.2
83.7 95.3 58.4 53.2 72.2
[=I
LaNis PrNis PrCOs SmCos LaGUs CaNis I3~2(-37
[I41 [151 [151 1161
1161 Cl71 1181
[211 161
1221 [61 t231 [241 [251 [261 [271
[291 [231
1301 1231
*No stable hydride.
the. ternary hydrides, but the values calculated for some of the compounds are 7 - 9 kJ less than the experimental results (AH is less exothermic), In this case, preferential occupation of the most energetically advantageous interstitial sites by hydrogen is likely to occur (e.g. for RM,_,M,’ interstiti~ sites of the same type may differ depending on whether M or M’ atoms are present in the local environment of the hydrogen atom). 2.3. Hydrides of non-stoichiometric compounds Some difficulties arise in applying the model proposed here to the calculation of the enthalpies of formation of hydrides of nont;toichiometric intermetallic compounds. For example, it is difficult to determine the parameters a and f for L&u4 (CaCu5-type structure), ZrMn3,s (MgZnz-type structure), TiCrl.s (k4gCuz-type structure) etc. If we take into account the fact that these compounds are substitution alloys [40] and the stoichiometry of the structural types in which they crystallize, we can write
221 TABLE 3 Enthalpy of formation of hydrides of multicomponent Intermetallic
compound
LaNi&u LaNi4.sCre.s LaNi4Fe LaNi&fn LaNi4.6A10.4
hNkdno.4
bNi4.6Si0.4 hNi4.6GeO.4 hNi4.6Sno.4
MmNis a MmNi4_6Si0.4a MmNi4.SSi0.sa ErFel.7Bue.s ErFel.lsBue.ss ZrVFe ZrVCo ZrCrFe ZrCrCo ZrMnCo ZrMnFe ZrMnl.6Eeo.4 Zr0.6m0.an2 Zr0.4mo.sMn2 ‘h.&rO.2Mn1.2CrO.s TiO.sZrO.2Mnl.gMoO.2 ‘I’iO.sZrO.lMn1.4QO.4VO.2 ~FeO.dbO.04
-AH
intermetallic compounds
(kJ (mol Ha)-‘)
Reference
Calculated
Experimental
38.8 35.2 35.7 52.5 31.2 41.5 34.1 34.1 40.6 27.8 28.5 28.0 46.1 42.7 45.3 33.6 45.1 48.0 36.4 37.7 42.2 37.9 25.6 21.5 22.1 20.2 27.9
36.1 35.3 38.0 48.6 38.0 39.7 35.7 34.4 39.1 26.4 28.1 27.7 49.2 43.2 46.2 49.3 49.4 40.2 34.8 35.8 42.1 36.0 25.7 23.9 29.3 29.3 26.6 b
[311 [311 [311 [321 [331 [331 1331 [331 1331 1341 [341 [341 [61 [61 [351 [351 [351 [351 [351 [361 [361 [301 [301 [371 [3f31 1331 [391
a Mm, misch metal. bCalculated from the isotherms of ref. 39.
5 LaCu4 = - La( Lao.-#uo.96)5 6 and ZrMns.s = 1.6(Zro.62sMno.&Mn2 In the case of LaCu4, for example, the parameters of the CaCu,-type lattice and the corresponding value of V are, in fact, those of La(Lao.&uo.96)s. Consequently, from the definition of a (see above) wei,h3”” aLacU,= . {(5/6)V) 1’3 . Similarly, for ZrMn3.s we obtain uZrMn,, = (1.6V) As for the determination off, we could not decide a priori whether the above compounds should be related to those of the structural types in which they crystallize or whether their actual stoichiometry should be used. Comparison of the calculations using each variant has shown that the second
228
assumption is correct. Consequently, the structural type of the intermetallic compound is less important in the hydride energetics than are the dimensional characteristics and chemical composition. The results obtained for some compounds are listed in Table 4. Figure 5 demonstrates the good agreement between the experimental and calculated data for increasingly complex non-stoichiometric intermetallic compounds. TABLE 4 Enthalpy of formation of hydrides of non-stoichiometric Intermetallic
compounds
Ldhl4
T%.8 ~Cro.92Mn0.62 mBMno.67 mCrl.73Mn0.09
ZrMn3.8 ZrJh0.u ZrMnd’eo.4 ZrMnz.6Feo.2
Zro.TLdW%.8
-AH
intermetallic compounds
(kJ (mol Hz)-‘)
Reference
Calculated
Experimental
43.2 20.2 19.2 15.9 20.4 13.5 26.1 19.1 24.1 16.7
40.7 19.8 25.1 25.1 20.1 17.0 24.7 * 22.0 15.0 14.1
[231 [411 [421 I421 [421 I431 r401 [441 [441 [451
a Calculated from the isotherms of ref. 40. -aH
[email protected]’
50 . 40 30
Fig. 5. Experimental (A) and calculated (0) values of the enthalpy hydrides of some zirconium-based nonstoichiometric compounds.
of formation
of
3. Conclusions Figure 6 presents a comparison of the calculated and experimental values of the enthalpy of formation of various multicomponent hydrides. As can be seen, few points lie outside the 5 kJ deviation limits. Since the exper-
229
o
i
’
20
40
60
*8o
-aHesp,_ KJ (md Hz)
‘
Fig. 6. Comparison of the calculated and experimental values of the enthalpy of formation of hydrides of intermetallic compounds: ---, limits of deviation (5 kJ (mol Hz)-“).
imental values of AFI obtained by various workers exhibit a mean deviation of 4 - 7 kJ (mol H&l [46], we may state that the formula derived gives a satisfactory description of a large range of compounds. Thus for the first time a model has been developed which enables the qu~ti~tive prediction of the enth~py of hydride formation and its variation when components are substituted. It should be noted that the calculation requires only the determination of the crystallographic characteristics of the alloy.
References
6 7 8 9 10 11 12 13 14
C. Wagner, 2. Phys. Chem. (Leipzig), 193 (1944) 407. R. L. Beck, Rep. LAR-55, 1961 (Denver Research Institute, Denver, CO). R. Burch and N. B. Mason, J. Less-Common Met., 62 (1979) 57. C. B. Magee, J. Liu and C. E. Lundin, J. Less-Common Met., 78 (1981) 119. J. C. Achard, A. Percheron-Gu&gan, H. Diaz, F. Briaucourt and F. Demany, Proc. 2nd Int. Congr. on hydrogen in Metals, Paris, June 1977, Pergamon, Oxford, 1978, Paper S.El2. A. L. Shilov, N. E. Efremenko and M. E. Kost, Russ. J. fnorg. Chem., 26 (1981) 1210. M. E. Kost, A. L. Shilov and M. V. Rayevskaya, Dokl. Akad. Nauk S.S.S.R., 252 (1980) 1135. A. L. Shilov, E. I. Yaropolova and M. E. Kost, Dokl. Akad. Nauk S.S.S.R., 252 (1980) 1397. J. Hinze, M. A. Whitehead and H. H. Jaffe, J. Am. Chem. Sot., 85 (1963) 148. A. R. Miedema, J. Less-Common Met., 32 (1973) 117. P. C. Bouten and A. R. Miedema, J. Less-Common Met., 71 (1980) 147. V. A. Kirillin, V. V. Sitchev and A. E. Sheindlin, Technical Thermodynamics, Energoizdat, Moscow, 1983, p. 388 (in Russian). H. H. van Mal, K. H. J. Buschow and A. R. Miedema, J. Less-Common Met., 35 (1974) 65. A. L. Shilov, Russ. J. Phys. Chem., 57 (1983) 1305.
230 15 F. A. Kuijpers, Philips Res. Rep., Suppl. 2 (1972) 1. 16 I. Shinar, D. Shaltiel, D. Davidov and A. Grayevsky, J. Less-Common Met., 60 (1978) 209. 17 A. Goudy, W. E. Wallace, R. S. Craig and T. Takeshita, Adu. Chem. Ser., 167 (1978) 312. 18 H. Oesterreicher, K. Ensslen, A. Berlin and E. Bucher, Mater. Res. Bull., 15 (1980) 275. 19 H. A. Kierstead, J. Less-Common Met., 78 (1981) 61. 20 H. A. Kierstead, J. Less-Common Met., 78 (1981) 29. 21 C. A. Bechman, A. Goudy, T. Takeshita, W. E. Wallace and R. S. Craig, Znorg. Chem., 15 (1976) 2184. 22 A. L. Shilov and M. E. Kost, Russ. J. Phys. Chem., 59 (1985), in the press. 23 A. L. Shilov, L. N. Padurets and M. E. Kost, Russ. J. Phys. Chem., 57 (1983) 555. 24 J. J. Reilly and R. Wiswall, Znorg. Chem., 13 (1974) 218. 25 0. de Poux and M. H. Lutz, in T. Veziroglu and W. Seifritz (eds.), Proc. 2nd WorM Hydrogen Energy Conf., Zurich, August 1978, Vol. 3, Pergamon, Oxford, 1979, p. 1525. 26 R. Burch and N. B. Mason,J. Chem. Sot., Faraday Trans. Z, 75 (1979) 561. 27 J. R. Johnson, J. Less-Common Met., 73 (1980) 345. 28 S. J. C. Irvine and I. R. Harris, J. Less-Common Met., 74 (1980) 33. 29 M. E. Kost, L. N. Padurets, A. A. Tchertkov and V. I. Mikheeva, Russ. J. Znorg. Chem., 25 (1980) 847. 30 H. Oesterreicher and H. Bittner, Mater. Res. Bull., 13 (1978) 83. L. A. Petrova and V. V. Burnasheva, Russ. J. Znorg. Chem., 28 31 K. N. Semenenko, (1983) 195. 32 C. Lartigue, A. Percheron-Guegan, J. C. Achard and F. Tasset, J. Less-Common Met., 75 (1980) 23. 33 D. M. Gruen, M. H. Mendelsohn and A. E. Dwight, Adu. Chem. Ser., 167 (1978) 327. 34 Y. Osumi, H. Suzuki, A. Kato and M. Nakane, J. Less-Common Met., 84 (1982) 99. 35 D. Shaltiel, I. Jacob and D. Davidov, J. Less-Common Met., 53 (1977) 117. 36 A. Suzuki, N. Nishimiya and S. Ono, J. Less-Common Met., 89 (1983) 263. and M. Asanuma, in A. F. Andresen and A. J. Maeland 37 Y. Mashida, T. Yamadaya (eds.), Proc. Znt. Symp. on Hydrides for Energy Storage, Geilo, August 14 - 19, 1977, Pergamon, Oxford, 1978, p. 329. N. Yanagihara, T. Yamashita and T. Iwaki, Proc. Hydrogen 38 T. Gamo, Y. Moriwaki, Energy Progress Conf., Tokyo, 1980, Vol. 4, Pergamon, Oxford, 1981, p. 2127. 39 Y. Sasaki and M. Amano, Proc. Hydrogen Energy Progress Conf., Tokyo, 1980, Vol. 2, Pergamon, Oxford, 1981, p. 891. 40 F. Pourarian and W. E. Wallace, Solid State Commun., 45 (1983) 223. 41 J. R. Johnson and J. J. Reilly, Znorg. Chem., 17 (1978) 3103. 42 Y. Osumi, H. Suzuki, A. Kato, K. Oguro, T. Sugioka and T. Fujita, J. Less-Common Met., 89 (1983) 257. 43 F. Pourarian, V. K. Sinha, W. E. Wallace and H. K. Smith, J. Less-Common Mtt., 88 (1982) 451. 44 F. Pourarian, V. K. Sinha and W. E. Wallace, J. Phys. Chem., 86 (1982) 4956. 45 V. K. Sinha and W. E. Wallace, J. Less-Common Met., 87 (1982) 283. 46 A. L. Shilov, L. N. Padurets and M. E. Kost, Russ. J. Phys. Chem., 59 (1985), in the press.
221
Metals, 105 (1985) 221 - 230
CALCULATION MODEL FOR THE ENTHALPY MULTICOMPONENT HYDRIDES
OF FORMATION
OF
A. L. SHILOV, M. E. KOST and N. T. KUZNETSOV Institute of General and Znorganic Chemistry Moscow 117071 (U.S.S.R.)
of the U.S.S.R.
Academy
of Sciences,
(Received February 22,1984)
Summary A semiempirical formula for calculating the enthalpy of formation of hydrides of inte~eta~ic compounds was developed from simple assumptions concerning the interaction between hydrogen and metals. The new model takes account of the electronegativities and electron densities of elements constituting the compounds, as well as the stoichiometry and the parameters of the unit cells of alloys. Comparison of the calculated and experimental values for hydrides of 62 binary, ternary, quaternary and more complex alloys shows good agreement (within 5 kJ (mol Hz)-“).
1. Introduction Assessment of the potential of hydrides of intermetallic compounds as reversible hydrogen storage materials and of many binary, ternary and more complex alloys as hydrogen absorbers requires the prediction of a number of their properties, of which the~odyn~i~ ~h~acter~tics are among the most important. In a continuation of the work of Wagner [l] and Beck [Z], attempts are made to correlate the thermal stability of hydrides with the geometrical characteristics of the crystal lattice of the parent metallic material. Although such an approach has been criticized by Burch and Mason [ 3 J, the role played by the geometrical factor in the energetics of hydrides is unquestionable. For hydrides of binary and ternary intermetallic compounds contain~g rare earth elements with CaCu,-type crystal st~c~res, some workers have shown a linear dependence of the free energy G of the cu-0 transition at a single temperature on the tetrahedral hole size of the metal matrix [ 41 or on the volume of its unit cell [53. However, for practical purposes a knowledge of G at a single temperature only is insufficient. Therefore we considered [6] the influence of the geometry of the crystal lattice of an intermetallic compound on the enthalpy of formation AEI of the hydride, a quantity which is almost independent of temperature. We chose as @ Elsevier Sequoia/Printed in The Netherlands
222
the variable defined as
parameter
the linear
dimension
a of the unit
cell which
is
where V is the volume of eight formula units of intermetallic compound of general formula RM, (R = rare earth, Ca, Ti, Zr, Hf is a hydride-forming element; M = Cr, MO, Fe, Co, Ni, Ru, Rh, Cu, Al, Si etc. is an element which does not react with hydrogen under normal conditions). An almost linear dependence of AiY on a was found for equistoichiometric intermetallic compounds of a rare earth with a common metal M [6]. The fact that data for isostructural intermetallic compounds with various M components can be approximated by a set of lines indicates the importance of such factors as electronegativity and electron density in addition to the geometric factor. The effects of electronegativity and electron density have been clearly demonstrated for ruthenium-containing ternary alloys [ 6 - 81.
2. Development
of the calculation
procedure
2.1. Choice of interaction model It follows from the above discussion that the energetics of the interaction between hydrogen and the intermetalhc compound is determined (i.e. the number of M mainly by the stoichiometry cM of the compound atoms per R atom), the dimension a of the unit cell of the intermetallic compound, and the electronegativity cp and the electron density n of the elements. As our aim is to consider the interaction of hydrogen with ensembles of various types of metal atoms, we introduce the average values of electronegativity and electron density which are defined as
ii =
C?liCi/CCi
where p, and ni are the values of the corresponding quantities for the elements and ci is the number of atoms of a given element (both M and R) per formula unit of RM, (i.e. ECi = CM+ 1). This procedure is similar to the introduction of the group orbital electronegativity in the work of Hinze et al. [9]. Data from various investigations show that M-H and H-H interactions which may be either attractive or repulsive and either short range or long range exist in metal-hydrogen systems. We have assumed that the enthalpy of formation of hydrides is determined mainly by the sum of a positive term due to the repulsion of the neighbouring hydrogen atoms and a negative term characterizing the H-M interactions. The first term depends on the
223
H-H distance (and consequently on the parameter a) and on the charge carried by the hydrogen atom. As in ref. 10, this value was assumed to be proportional to the difference Ap between the electronegativities of the hydrogen and the metal (A9 = $$r - g), with the relative hydrogen content in the hydride being ignored. The second term depends on both Ap and the of electron density difference An = nH -n”, as well as on the stoichiometry the intermetallic compound. In fact, the stoichiometry (ratio of M to R) is taken into account in calculating + and ii. Here we note the necessity of allowing for the difference between the formal stoichiometry of the intermetallic compound and the atomic ratio in the local environment of the hydrogen atom. Thus, when the enthalpy of molecular hydrogen dissociation Mti, is taken into account, the general form of the equation for M is as follows: AH=
EHH(&,
a)-E~d&,
An,
CM)
(1)
+ Lwdi,
2.2. Search for the form of the functional dependence and the determination of values of the constants in the equation For intermetallic compounds of a rare earth with the same M and CM, all the variables except a in eqn. (1) are almost constant. Therefore the form of the function Enn(Aq, a) was sought by inspection of data for these compounds. If EHH is assumed to be inversely proportional to c, we have AZY= Enn(Aq)e-’
+K
where K = -EHM + AH&,. Thus, from the slopes of the lines in Fig. 1, we can obtain the values of E~H(A(P) for various series of binary intermetallic compounds of the rare earths. These values were assumed to be functions of Ap. The values of Cpi were those given by Miedema [lo] and Bouten and
5050 .
,060,
oRRUz
I
ox_3
a425
11127
au9
n-!j-i
0.5
0.6
0.7
0.8
A’Q,V
Fig. 1. Dependence of the enthalpy of formation of the hydrides of RM3 (R = rare earth) on the dimension a of the unit cell of the intermetallic compound: 0, RNia; 0, RCo3; A, RFe3. Fig. 2. Em
us.
ACJJ for hydrides of various intermetallic compounds of the rare earths.
224
Miedema [ 111 corrected for a better description of the interactions in binary compounds. With some correction of the value for e (5.37 V instead of 5.2 V as reported in ref. 11) the value of Ear was found to be a linear function of Ap (Fig. 2): EHH(AP) = A AP where A = 6536.5 kJ A V-r (mol Hz)-l, and hence E~~(A~, a) = A Ape-’
(2)
Substituting eqn. (2) into eqn. (1) and using the above value of A and AH diss= 432.3 kJ (mol Hz)-’ [12] we calculated EWMfor hydrides of intermetallic compounds with R E rare earth, Ca, Ti, Zr, Hf from the formula -En&A@, An, cNI)= AHeXP-A
Aq u-l - B,,
As can be seen from Fig. 3, this quantity increases linearly with increase in the “enthalpy” A.@ of the H-M interaction, which is given by AI?=Ap’-FAn2 where F = 0.218 V2 (density units)- 2. In addition to correction for hydrogen (nn was taken as 4.6 density units), the values of cpi and ni given in ref. 10 were also corrected for copper (ncu was taken as 3.82 density units).
p%~
750 o
p aNi, PLrC”s
Fig. 3. Em us. &
0.4
0.0
for intermetallic
1.2 compounds
a:, v 2 with equivalent
stoichiometries.
The constants in the equation (3) EHM(AP, An, CM)= ki(A@ + fi) for the series of compounds considered are given in Table 1. However, it was necessary to obtain the explicit dependence of EHM on CMin order to be able to calculate the enthalpies of formation of those groups of compounds for which data for determining ki and fi were absent or insufficient. For example, all rare earth compounds with the composition R&o, are represented by a single point in Fig. 3. Therefore, assuming Iz is constant in eqn. (3), we attempted to relate f to CM:
225 TABLE 1 Parameters for eqn. (3) R
Stoichiometry of the intermetallic compound
ki (kJ VP2 (mol Hz)-‘)
fi (V2)
Rare earth, Ca
BMs BM3 BM2
700 446 431.2
0.9433 1.9312 1.9613
Ti, Zr, Hf
BM2 RM
412.3 448.3
2.3976 2.5191
EHM(&, An, cM) = him + f(cMM)l
(4)
As can beseen from Fig. 4 the dependence off on cM at Iz = 415 kJ VP2 (mol H2)-i can be represented by the relationship f(c&
=B--D
1gcM
(5)
where B = 2.8135 V2 and D = 1.4675 calculation is of the form H=AA~~-‘-~(AIJY~-FA~~+~)+AH~~
V2. Then the formula
for the finite (6)
and the accuracy of the calculation depends on whether the second term is calculated from eqn. (3) and Table 1 or from eqns. (4) and (5).
1.8
d
0.2
0.4
PRMs 0.6 ejcn
Fig. 4. Determination of the -dependence of the parameter f in eqn. (4) on the stoichiometry of the intermetallic compound.
The results of calculations for hydrides of binary intermetallic compounds are listed in Table 2 where it can be seen that the agreement with the experimental data is good. The maximum discrepancy is 6.2 kJ (6.6%) and the average discrepancy is 1.5 - 2.0 kJ. Equation (6) was derived mainly on the basis of data on ternary hydrides, but no restrictions were placed on the number of alloy components. Table 3 presents the results of calculations for quatemary and more complex hydrides. In general the agreement is similar to that obtained for
226 TABLE 2 Enthalpy of formation of the hydrides of binary intermetallic compounds Zntermetallic compound
-AH
(kJ (mol II,)-‘)
Reference
Calculated
Experimental 31.8 27.7 38.5 32.7 44.4 31.0 46.0 44.0
1131
CeNis
32.2 29.2 37.9 35.6 43.5 32.5 47.2 49.0
ErCoa GdCo3 ErFes ErFe2 ScFes ErRu2 ScMnz TiFe Tic0 TiNi Tic& TiMns
44.6 53.2 45.7 54.9 35.1 47.8 62.6 27.7 46.1 61.9 22.5 -1.0
44.1 53.4 43.1 55.0 35.4 45.8 63.0 28.2 44.4 60.0 26.2 -B
gf
ZrCo ZrNi ZrCra ZrMn2 HfVs
81.4 89.1 54.1 50.7 71.2
83.7 95.3 58.4 53.2 72.2
[=I
LaNis PrNis PrCOs SmCos LaGUs CaNis I3~2(-37
[I41 [151 [151 1161
1161 Cl71 1181
[211 161
1221 [61 t231 [241 [251 [261 [271
[291 [231
1301 1231
*No stable hydride.
the. ternary hydrides, but the values calculated for some of the compounds are 7 - 9 kJ less than the experimental results (AH is less exothermic), In this case, preferential occupation of the most energetically advantageous interstitial sites by hydrogen is likely to occur (e.g. for RM,_,M,’ interstiti~ sites of the same type may differ depending on whether M or M’ atoms are present in the local environment of the hydrogen atom). 2.3. Hydrides of non-stoichiometric compounds Some difficulties arise in applying the model proposed here to the calculation of the enthalpies of formation of hydrides of nont;toichiometric intermetallic compounds. For example, it is difficult to determine the parameters a and f for L&u4 (CaCu5-type structure), ZrMn3,s (MgZnz-type structure), TiCrl.s (k4gCuz-type structure) etc. If we take into account the fact that these compounds are substitution alloys [40] and the stoichiometry of the structural types in which they crystallize, we can write
221 TABLE 3 Enthalpy of formation of hydrides of multicomponent Intermetallic
compound
LaNi&u LaNi4.sCre.s LaNi4Fe LaNi&fn LaNi4.6A10.4
hNkdno.4
bNi4.6Si0.4 hNi4.6GeO.4 hNi4.6Sno.4
MmNis a MmNi4_6Si0.4a MmNi4.SSi0.sa ErFel.7Bue.s ErFel.lsBue.ss ZrVFe ZrVCo ZrCrFe ZrCrCo ZrMnCo ZrMnFe ZrMnl.6Eeo.4 Zr0.6m0.an2 Zr0.4mo.sMn2 ‘h.&rO.2Mn1.2CrO.s TiO.sZrO.2Mnl.gMoO.2 ‘I’iO.sZrO.lMn1.4QO.4VO.2 ~FeO.dbO.04
-AH
intermetallic compounds
(kJ (mol Ha)-‘)
Reference
Calculated
Experimental
38.8 35.2 35.7 52.5 31.2 41.5 34.1 34.1 40.6 27.8 28.5 28.0 46.1 42.7 45.3 33.6 45.1 48.0 36.4 37.7 42.2 37.9 25.6 21.5 22.1 20.2 27.9
36.1 35.3 38.0 48.6 38.0 39.7 35.7 34.4 39.1 26.4 28.1 27.7 49.2 43.2 46.2 49.3 49.4 40.2 34.8 35.8 42.1 36.0 25.7 23.9 29.3 29.3 26.6 b
[311 [311 [311 [321 [331 [331 1331 [331 1331 1341 [341 [341 [61 [61 [351 [351 [351 [351 [351 [361 [361 [301 [301 [371 [3f31 1331 [391
a Mm, misch metal. bCalculated from the isotherms of ref. 39.
5 LaCu4 = - La( Lao.-#uo.96)5 6 and ZrMns.s = 1.6(Zro.62sMno.&Mn2 In the case of LaCu4, for example, the parameters of the CaCu,-type lattice and the corresponding value of V are, in fact, those of La(Lao.&uo.96)s. Consequently, from the definition of a (see above) wei,h3”” aLacU,= . {(5/6)V) 1’3 . Similarly, for ZrMn3.s we obtain uZrMn,, = (1.6V) As for the determination off, we could not decide a priori whether the above compounds should be related to those of the structural types in which they crystallize or whether their actual stoichiometry should be used. Comparison of the calculations using each variant has shown that the second
228
assumption is correct. Consequently, the structural type of the intermetallic compound is less important in the hydride energetics than are the dimensional characteristics and chemical composition. The results obtained for some compounds are listed in Table 4. Figure 5 demonstrates the good agreement between the experimental and calculated data for increasingly complex non-stoichiometric intermetallic compounds. TABLE 4 Enthalpy of formation of hydrides of non-stoichiometric Intermetallic
compounds
Ldhl4
T%.8 ~Cro.92Mn0.62 mBMno.67 mCrl.73Mn0.09
ZrMn3.8 ZrJh0.u ZrMnd’eo.4 ZrMnz.6Feo.2
Zro.TLdW%.8
-AH
intermetallic compounds
(kJ (mol Hz)-‘)
Reference
Calculated
Experimental
43.2 20.2 19.2 15.9 20.4 13.5 26.1 19.1 24.1 16.7
40.7 19.8 25.1 25.1 20.1 17.0 24.7 * 22.0 15.0 14.1
[231 [411 [421 I421 [421 I431 r401 [441 [441 [451
a Calculated from the isotherms of ref. 40. -aH
[email protected]’
50 . 40 30
Fig. 5. Experimental (A) and calculated (0) values of the enthalpy hydrides of some zirconium-based nonstoichiometric compounds.
of formation
of
3. Conclusions Figure 6 presents a comparison of the calculated and experimental values of the enthalpy of formation of various multicomponent hydrides. As can be seen, few points lie outside the 5 kJ deviation limits. Since the exper-
229
o
i
’
20
40
60
*8o
-aHesp,_ KJ (md Hz)
‘
Fig. 6. Comparison of the calculated and experimental values of the enthalpy of formation of hydrides of intermetallic compounds: ---, limits of deviation (5 kJ (mol Hz)-“).
imental values of AFI obtained by various workers exhibit a mean deviation of 4 - 7 kJ (mol H&l [46], we may state that the formula derived gives a satisfactory description of a large range of compounds. Thus for the first time a model has been developed which enables the qu~ti~tive prediction of the enth~py of hydride formation and its variation when components are substituted. It should be noted that the calculation requires only the determination of the crystallographic characteristics of the alloy.
References
6 7 8 9 10 11 12 13 14
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