Lattice Energies and Unit Cell Volumes of Complex Ionic Solids

We develop a powerful new limiting relation between lattice potential energy, UPOT, 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: UPOT = AI(2I/Vm)1/3. It is capable of estimating lattice energies in the range 5000 < UPOT/kJ mol-1 ≤ 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∑nizi2, where ni is the number of ions of type i per formula unit, each bearing the charge zi, 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) Vm, the volume of the formula unit (the “molar” or “molecular” volume). The equation provides estimates of UPOT 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., UPOT/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.