Lattice Energy Calculator

Lattice energy is the enthalpy change for assembling one mole of a crystalline ionic solid from its constituent gaseous ions, and it is usually the largest single energy term in the thermodynamics of a salt. The Born-Lande expression treats the crystal as an array of point charges: the Coulomb term scales as z+ times z- divided by the interionic separation d0, so doubling both charges quadruples the energy while shrinking d0 raises it inversely. The Madelung constant M carries out the alternating attraction and repulsion sum over the entire infinite lattice, converging to 1.7476 for rock salt, 1.7627 for CsCl, 1.6381 for zinc blende and 2.4089 for fluorite. The factor (1 - 1/n) removes short-range Born repulsion, with the Born exponent n fixed by the noble-gas core of the ions: roughly 5 for helium-like, 7 for neon-like, 9 for argon-like, 10 for krypton-like and 12 for xenon-like configurations. Typical magnitudes run from about 600 kJ/mol for CsI to 3850 kJ/mol for MgO and beyond 15000 kJ/mol for Al2O3. Radii should come from one self-consistent set, most often the Shannon effective ionic radii of 1976, and must match the coordination number of the structure, since six-coordinate Na+ is 102 pm while the four-coordinate value is 99 pm. The principal limitation is the purely ionic assumption: for polarizable anions or partly covalent bonding, as in AgCl or ZnS, the calculated value falls 10 to 20 percent below the Born-Haber result. The model also omits van der Waals attraction and zero-point vibrational energy, each worth a few tens of kJ/mol.

The electrostatic model of ionic crystals took shape immediately after X-ray diffraction revealed their structures. Erwin Madelung published his lattice sum in 1918, supplying the geometric constant that now carries his name, and in the same year Max Born and Alfred Lande combined that sum with an inverse-power repulsion term to produce the equation still taught today. Born was pursuing a general dynamical theory of crystal lattices; Lande supplied the treatment of the repulsive ion core. In 1919 Born and Fritz Haber formulated the cycle now known as the Born-Haber cycle, which applies Hess law to extract lattice energy from measurable quantities such as sublimation enthalpy, ionization energy, electron affinity, bond dissociation energy and enthalpy of formation, giving the calculation an independent experimental check. Refinements followed as data improved. In 1932 Born and Joseph Mayer replaced the inverse-power repulsion with an exponential form, producing the Born-Mayer equation, which agrees better with measured compressibilities. Anatoli Kapustinskii, working in the Soviet Union during the 1940s and 1950s, observed that the ratio of the Madelung constant to the number of ions per formula unit is nearly constant across structure types, and published a simplified formula needing only charges and radii, making estimates possible for compounds of unknown structure. Later work by H. D. B. Jenkins and coworkers extended the idea into volume-based thermodynamics, deriving lattice potential energy directly from unit cell or molar volume, now common for complex salts and ionic liquids.