G. J. Moody
and J. D. R. Thomas Welsh College of Advanced Technology Cardiff, Wales, U.K
I
I
Lattice Energy and Chemical Prediction Use of the Kapustinskii equations and the Born-Haber cycle
The chemist is fascinated to see the realizat'ion of chemical behavior previously predicted by t'heoretical reasoning. Mendeleev established the reasoning by the basic idea that "there must be some bond of union between mass and the chemical elements" ( I ) . In order that chemical analogies should be preserved, he left gaps in his periodic classification and predicted the properties of these missing elements with remarkable accuracy. More recently, crystal chemistry has provided a more quantitative basis for predicting chemical syntheses. An early example of this is provided by the work of Grimm and Herzfield (8) where use was made of lataticeenergies of neighboring stable compounds in the periodic table and, for some compounds, values calculated from the Born-Lande equation (3):
where U is the lattice energy, N is Avogadro's number, M is the Madelung constant, z+ and z- are the valencies of the ions, e is the charge on the electron, r is the interionic distance, and n is the electronic shells repulsion exponent. The various lattice energies were used to predict the stability of a number of hypothetical compounds. For example, it was predicted that the monohalides of the noble gases would be unstable in an ionic lattice and would decompose into the constituent elements ( 8 ) . The Born-Lande equation is dependent on an exact lcnowlcdge of crystal structure and hence of the Madelung constant. The same is also true of certain other equations employed for the calculation of lattice energies of compounds. One such equation is that of BornR l a ~ e r(41,
(6, 7) to give equat,ions (3) and (4). respectively.
I n both equations, v is the number of ions in the molecule and cr = [ M / ( v / 2 ) ] . The Rfadelung constant, M, is independent of cr, since it is proportional to the number of ions in the chemical molecule. However, or is not identical for different lattice types and Kapustinskii (5) found, empirically, that in passing from one lattice type to another, the change in u was proportional to the change in interionic distance. With these modifications and by taking the Goldschmidt ionic radii referred to the coordination number 6, giving (r+ r - ) for r , and the structural coefficient, u = 1.745, for roclr-salt type lattices, the same value is obtained for the lattice energy as by calculating it with the aid of r, derived by X-ray measurements, and the Madelung structural factor, M, corresponding to the given lattice. Several of the quantities in equation (3) may be grouped in a single proportionality coefficient sincen has the average value of 9 and Ne2 = 329.7 kcal per A. Substitution of these quantities gives rise to the eqnation'
+
U
=
256.luz+zkcal per mole (r+ r-)
which permits the calculation of the energy of any type of lattice with the aid of charges and radii only. Similar transformation of equation (4) which includes the factor p = 0.345 A, characterizing the quantum mechanical repulsion forces, gives (8) the equation2 U = 287.2vz+z(r+
an improved version of equation ( I ) , where p characterizes the quantum-mechanical repulsion forces acting between the electronic shells of the ions and has the di~nensionsof length. For most crystals p may be regarded as practically constant a t 0.345 A. The Kapustinskii Equations
For hitherto unknown conlpounds, it is imperative that a certain crystal structure be assumed before equat,ions of the type of (1) and (2) can be used. Kapustinskii (5) regarded such an assumption as unsatisfactory and proposed an equation with which it was possible to ext,end the sphere of calculations of lattice energies. The development of this equation may be traced by making a slight modification in equations ( 1 ) and ( 2 ) 204 / Journal of Chemicol Educofion
+
+ r-)
-
+ ] kesl per mole
0'345 (r+ r-)
(6)
An objection (7) to the Kapustinskii equations is the replacement of r, the interatomic distance, by ( I + r-). This is because there are many salts in which the unlike ions are not touching as, for example, in sodium iodide where the lattice spacing is determined by contacts between the iodide, I-, ions. Also, the assignment of ionic radii can only be regarded as approximate, even in the case of the alkali halides (6). I n this latter respect, it is interesting to recall the statement made by Gibb and Winnerman (9) that "the actual radius of an ion is as uncertain as the radius of a fluffy ball of cotton batting."
+
'Recalculation of the proportionality coefficient gives 25.5.7. Reealcul.%tion of the proportionality coefficient gives 287 6 .
Nevertheless, the lattice energies obtained by calculations using these equations compare favorably with those derived experimentally (by the Born-Haber cycle) and with those calculated from other equations which take into consideration the lattice types. This closeness is well illustrated by the dat,a for the alkali metal fluorides and iodides as shown in Table 1. Where comparisons can be made, agreement is also reasonable for the transition metal(I1) chalcogenides (Table 2). Of course, the above equations cannot be expected to produce values for the lattice energies that are as exact as those produced by more extended calculations. This is because all contributions to the lattice energy are ignored except for the Madelung term, U M ,arising from the electrostatic attraction of the ions, and the repulsive energy term, Un,allowing for the repulsive forces between the ions. The additional contributions to equation (2) are described in available literature (6, 7, 15) and generally allow for van der Waals energy and zero-point energy of the molecules in the crystal. However, lattice energies calculated from the refined equations do not differ greatly from those obtained by the more approximate methods (6, 7,15). Equation (6) has also been refined by Kapustinskii and Yatsimirskii (6,16) by the inclusion of an additional term to give
Table 1.
Lottice Energies (kcal per mole) of the Alkali Metal Fluorides and Iodides
Ex~eri- d a l c u l a t e d Values
Salt
(BornHsber Cycle) (Ref. 7)
LiF NaF KF RbF CsF LiI NsI KI RbI CsI
241 216 192 184 171 176 165 151 147 140
Table 2.
MCh MnO FeO COO NiO ZnO
Kapusaid tinskii Sherman HelmEq. (6) Eq. (1) holtz Huggins (Ref. 8) (Ref. 10) (Ref. 11) (Ref. 13) 228 212 189 182 170 171 161 147 141 135
239 214 189 181 172 170 160 148 143 135
240 215 190 182 174 174 164 151 145 140
244 215 193 183 176 176 164 152 148 143
Lattice Energies (kcal per mole) of SomeTransition Metollll) Chalcogenides
Calculated Values-Kmustinskii Ex~eri'Eas mental (Moddy & Hare & (BornThomas Brewer (Ref. 14) Sherman Haber Ref. 1.9) Cvele) 10) " . Eo.. (5) . . Eo.. (6) . . Eo.. (1) . . Eo.. (2) . . (Ref. . 911 937 954 974 964
931 953 967 976 957
880 899 908 914 901
916 937 958 974
890 907 917 934
912 944 950 968 9i7
although Yatsimirskii in a later publication, quotes a different quantity for the additional term in equation (7) which in fact is twice as great (I?), as shown in equation (8).
For calculation purposes, it is convenient to write equation (8) in the form
Here again, the additional t e r n does not alter the results appreciably; the effect of the doubled quantity in equations (8) and (9) being to increase the calculated lattice energy of metal(1) halides by 5 kcal per mole and of the much greater lattice energies of the met,al(III) halides by 30 kcal per mole. It is, however, difficult to have a clear idea of the accuracy of these expressions, since direct experimental computation of lattice energy has rarely been accomplished and the accuracy of the magnitudes of the BornHaber cycle of operations is also open to question, although these are gradually being improved. Kapustinskii (6) claims that there is very little point in further improvement of the expressions for lattice energy until the values of the ionic radii are known with greater accuracy, although a sacrifice in accuracy is more than compensated by the universal application of the equations to compounds, hypothetical or otherwise, without knowledge of crystal architecture. Compounds in their lattices vary from the ideal ionic state and Yatsimirskii has described proposals for correcting the low values for lat,ticeenergies given by the
Kapustinskii equations when compared with the experimental (Born-Haber) values (18,lQ). I n a subsequent discussion on the applicability of equations (8) and (9), Yatsimirskii (17) claims that these give, for triply-charged cations with eight electrons in the outer shell, values for the lattice energy that agree to within 3% of the experimental (BornHaber) values. Aluminum(II1) is an exception, the error reaching 9% for the iodide. For compounds of the tetravalent elements, the discrepancies are somewhat greater, but exceed 10% only for the iodides. Since many of the compounds studied have significant degrees of covalencies in their bonds, the low deviations are remarkable. The discrepancies for compounds formed by other types of cation are considerable and for such compounds, where the cations have the electronic confignrations d6, dl0, and d1°s2,Yatsimirskii (17) has suggested that the equation U = Ui+C+E.
(10)
is applicable. I n this equation, U is the lattice energy, U t is that calculated by equations (8) and (9), E , is the energy of extrastabilization by the crystalline field, and C is the energy of formation of a partially covalent bond in the crystal. The significant factor in the deviations between experimental (Born-Haher) values and those calculated by equations (8) and (9) are attributed by Yatsimirskii mainly to C which is Volume 42, Number 4, April 1965
/
205
shown to be a function of the ionization potential and electron &nity of the cations and anions, respectively, in the crystal lattice. Introducing corrections along these lines has brought the agreement between experimental and calculated lattice energies to +2%. Of course, these corrections for covalency introduced by Yatsimirskii, follows the practice of several authors of using the low calculated values of lattice energies compared with the high (Born-Haber) experimental values to indicate the deviations from heteropolarity (7). Applications of the Kapustinskii Equations
The applications of the ICapustinslui equations have been varied and apart from a check on the thermochemical data, the equations have opened up wider horizons for the extension of the evaluation of lattice energies to ionic crystals not yet investigated by X-ray measurements, and frequently to compounds not yet prepared. Before discussing these more obvious applications, there is the other use, where reliable thermal data are available, of calculating the radii of ions. Such a calculation is possible where the radius of either the cation or anion is lacking. The radii calculated in this way have heen termed "thermal" or "thermochemical radii" (6). An example of a radius thus evalu?ted is that of theSnZ+ion, found to be 1.04 =t0.02 A as determined from the experimental lat,tice energies of tin(I1) sulfide, oxide, and chloride (6). However, the lattice of these tin compounds vary from the ionic and, as noted above and as is apparent by inspection of Table 2, compounds with covalent tendencies give low lattice energies as computed from the Kapustinskii equations when compared with the experimental (Born-Haber) values. On making allowance for this feature, the value of 1.17 A calculated by Yatsimirskii is probably more realistic (19). Equation (6) has been used to calculate the thermochemical radii of a large number of complex ions and radicals. Values for a selection of anions, assumed to have spherical symmetry are shown in Table 3. There are, however, many ions where the structure is far from spherical. For these, "effective" ionic radii have been determined by equation (6) and the "effective thermochemical ionic radius" of the [Ba(H2O)IZ+ion has been found to be 1.61+0.01 A from various monohydrates of barium containing a range of anions of previously known dimensions, for example, the halides, nitrate, cyanide, and chlorate (6). Direct measurement of electron sffinities is difficult and has only been done for a few elements. To overcome this problem, an important application of calculated lattice energies is in the evaluation of electron affinities. Comparison of the values thus obtained for halogen atoms with those measured by comparing the rates of negative ion and electron emission from a heated surface in the halogen vapor, and in other ways, shows close agreement (15). The lattice energy of sodium telluride, determined from equation (5) above Table 3.
Ion Radius
206
/
(A)
504'-
2.30
CrOF 2.40
Journal of Chemical Education
has been used to evaluate the electron affinity of the telluride, TeZ-, ion (IS) : Te(,.)
+ 2e = Te2-l.,.l;
AHgrs= 64 kesl
The Born-Hober Cycle
Although proton affinities and the enthalpies of formation of complex ions can be evaluated, the principal use of the lattice energies derived from the Kapustinskii equations, both in the original and refined forms, has been in the prediction of the stability of various hypothetical compounds (IS, 20, 21, 22, 23). The various derivations from the lattice energies, including those of electron affinities, are made with the aid of the Born-Haber thermodynamic cycle of operations (24, 25). This is based on the first law of thermodynamics, and for a crystalline uni-univalent salt, M X , may he represented by the scheme (i)
Mx1.1
Mf(O1
+ X-(,) I
T
The first step, (i), is the complete breakup of the crystal, M X , into gaseous ions M + and X-, attended by an increase in the heat cont,ent which is the lattice enthalpy, U 2RT. For practical purposes, the quantity 2RT is neglected, being only about 1.2 kcal per mole a t 29S01
