The linear generalized equation described in this paper provides a further dimension to the prediction of lattice potential energies/enthalpies of ionic solids. First, it offers an alternative (and often more direct) approach to the well-established Kapustinskii equation (whose capabilities have also recently been extended by our recent provision of an extended set of thermochemical radii). Second, it makes possible the acquisition of lattice energy estimates for salts which, up until now, except for simple 1:1 salts, could not be considered because of lack of crystal structure data. We have generalized Bartlett's correlation for MX (1:1) salts, between the lattice enthalpy and the inverse cube root of the molecular (formula unit) volume, such as to render it applicable across an extended range of ionic salts for the estimation of lattice potential energies. When new salts are synthesized, acquisition of full crystal structure data is not always possible and powder data provides only minimal structural information-unit cell parameters and the number of molecules per cell. In such cases, lack of information about cation-anion distances prevents use of the Kapustinskii equation to predict the lattice energy of the salt. However, our new equation can be employed even when the latter information is not available. As is demonstrated, the approach can be utilized to predict and rationalize the thermochemistry in topical areas of synthetic inorganic chemistry as well as in emerging areas. This is illustrated by accounting for the failure to prepare diiodinetetrachloroaluminum(III), [I(2)(+)][AlCl(4)(-)] and the instability of triiodinetetrafluoroarsenic(III), [I(3)(+)][AsF(6)(-)]. A series of effective close-packing volumes for a range of ions, which will be of interest to chemists, as measures of relative ionic size and which are of use in making our estimates of lattice energies, is generated from our approach.

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This paper is one of a series exploring simple approaches for the estimation of lattice energy of ionic materials, avoiding elaborate computation. The readily accessible, frequently reported, and easily measurable (requiring only small quantities of inorganic material) property of density, rho(m), is related, as a rectilinear function of the form (rho(m)/M(m))(1/3), to the lattice energy U(POT) of ionic materials, where M(m) is the chemical formula mass. Dependence on the cube root is particularly advantageous because this considerably lowers the effects of any experimental errors in the density measurement used. The relationship that is developed arises from the dependence (previously reported in Jenkins, H. D. B.; Roobottom, H. K.; Passmore, J.; Glasser, L. Inorg. Chem. 1999, 38, 3609) of lattice energy on the inverse cube root of the molar volume. These latest equations have the form U(POT)/kJ mol(-1) = gamma(rho(m)/M(m))(1/3) + delta, where for the simpler salts (i.e., U(POT)/kJ mol(-1) < 5000 kJ mol(-1)), gamma and delta are coefficients dependent upon the stoichiometry of the inorganic material, and for materials for which U(POT)/kJ mol(-1) > 5000, gamma/kJ mol(-1) cm = 10(-7) AI(2IN(A))(1/3) and delta/kJ mol(-1) = 0 where A is the general electrostatic conversion factor (A = 121.4 kJ mol(-1)), I is the ionic strength = 1/2 the sum of n(i)z(i)(2), and N(A) is Avogadro's constant.

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Thermochemical radii (1-3) can be used in the Kapustinskii equation (1) for binary salts or in the more recent Glasser generalization of this equation (4 ) for more complex salts, to predict lattice potential energies and stabilities of new inorganic materials (see, for example, refs 5, 6 ). They also provide parameters of molecular size to correlate with other ion properties (see, for example, ref 7).Our previous "reappraisal" (3) of these magnitudes is widely cited in the literature and is quoted in inorganic textbooks (8-11) and many others. Our present work offers the largest self-consistent set of thermochemical radii yet produced. These tables include (i) ions previously not considered, (ii) estimates for complex ions of recent and evolving topical interest, and (iii) estimates to update the values of the radii for the more conventional ions where necessary. Estimation of Thermochemical RadiiBartlett et al. (12) demonstrated a linear correlation of lattice enthalpy against the inverse cubic root of the volume per molecule, V, for simple MX salts. We have generalized this correlation and have studied crystals containing complex anions partnered with alkali-metal counter ions of known radius, and have extended the correlation (13) to include complex salts of the type M p X q . Thus for a crystal M p X q containing pM q+ ions and q complex anions X p ᎑ whose thermochemical radius is to be assigned, we use the unit cell parameters (a, b, c, α, β, and γ) derived from the crystal-structure data for the salts to calculate the volume per molecule, V : V = abc 1 -cos 2 α -cos 2 β-cos 2 γ + 2cos α cos β cos γ z where z is the number of molecules per unit cell.

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Standard absolute entropies of many inorganic materials are unknown; this precludes a full understanding of their thermodynamic stabilities. It is shown here that formula unit volume, V(m)(), can be employed for the general estimation of standard entropy, S degrees 298 values for inorganic materials of varying stoichiometry (including minerals), through a simple linear correlation between entropy and molar volume. V(m)() can be obtained from a number of possible sources, or alternatively density, rho, may be used as the source of data. The approach can also be extended to estimate entropies for hypothesized materials. The regression lines pass close to the origin, with the following formulas: For inorganic ionic salts, S degrees 298 /J K(-)(1) mol(-)(1) = 1360 (V(m)()/nm(3) formula unit(-)(1)) + 15 or = 2.258 [M/(rho/g cm(-)(3))] + 15. For ionic hydrates, S degrees 298 /J K(-)(1) mol(-)(1) = 1579 (V(m)()/nm(3) formula unit(-)(1)) + 6 or = 2.621 [M/(rho/g cm(-)(3))] + 6. For minerals, S degrees 298 /J K(-)(1) mol(-)(1) = 1262 (V(m)()/nm(3) formula unit(-)(1)) + 13 or = 2.095 [M/(rho/g cm(-)(3))] + 13. Coupled with our published procedures, which relate volume to other thermodynamic properties via lattice energy, the correlation reported here complements our development of a predictive approach to thermodynamics and ultimately permits the estimation of Gibbs energy data. Our procedures are simple, robust, and reliable and can be used by specialists and nonspecialists alike.

We develop a powerful new limiting relation between lattice potential energy, U POT , and unit cell volume, V (hence, also, density), applicable to some of the most complex ionic solids known (including minerals, and superconductive and even disordered, amorphous or molten materials). Our equation (which has a correlation coefficient, R ) 0.998) possesses no empirical constants whatsoever, and takes the following form: U POT ) AI(2I/V m ) 1/3 . It is capable of estimating lattice energies in the range 5000 < U POT /kJ mol -1 e 70 000 and extending toward 100 MJ mol -1 . The relation relies only on the following: (i) an ionic strength related term, I (defined as 1 / 2 ∑n i z i 2 , where n i is the number of ions of type i per formula unit, each bearing the charge z i , with the summation extending over all ions of the formula unit); (ii) a standard electrostatic conversion term, A/kJ mol -1 nm ) 121.39 (the normal Madelung and electrostatic factor as found in the Kapustinskii equation, for example); and (iii) V m , the volume of the formula unit (the "molar" or "molecular" volume). The equation provides estimates of U POT to certainly within (7%; in most cases, estimates are significantly better than this. Examples are provided to illustrate the uses of the equation in predicting lattice energies and densities; the calculations require minimal data and can be performed easily and rapidly, even on a pocket calculator. In the lower lattice energy range (i.e., U POT /kJ mol -1 < 5000, corresponding to the simpler compounds and to many inorganic salts possessing complex ions), our recently published linear correlation is more accurate. The linear equation, though empirically developed, is consistent with and can be rationalized following the approach developed here. Lattice potential energy (U POT ) is a dominant term in the thermodynamic analysis of the existence and stability of ionic solids. Direct experimental determination is generally not possible since, in practice, the crystalline solid dissociates into atoms and not into gaseous ions, as is required in the lattice energy evaluation. Therefore, its indirect experimental determination, computation, or estimation is of considerable interest in modern materials science; indeed, whenever the energetics of condensed-state materials are studied, the chemical processes under consideration may be rationalized if the appropriate lattice energy steps can be incorporated into the thermochemical cycle.A variety of estimation methods for lattice energies is available. These include the Born-Haber-Fajans thermochemical cycle 1 (which requires ancillary thermodynamic data) and modern computational methods (which generally require knowledge of the lattice constants and the coordinates of the ions as well as needing an established force field). The computational methods range from direct energy calculational procedures 2 through to programs that produce lattice energies in the course of their modeling of the solid. 3-5 Quantum mechanical procedures are also available, but are ...