Ionic materials form crystalline lattices like metals but the lattice structure is determined by one ion (the larger, usually the anion) and the other ion fits into the holes. Since the smaller ion may not just right, lattice expansions can occur.
What holds an ionic lattice together?
A simple model: assume each ion is represented by a point charge, then sum up all the Coulomb interactions.
q+, q– are charges on the ion, d the distance between ions, ε0 = permittivity of free space, NA is Avogadro's number, A is the Madelung constant
The Madelung constant is a purely geometric factor that has been tabulated for many lattice types. It represents the potential energy associated with different relative geometries. d is the distance between ions so accounts for the different energy associated with different compounds with the same lattice type (hence, the same Madelung constant).
ECoul is negative, thus attractive, but maximizes at d = 0, i.e. when anions and cation occupy the the same point in space. Need a repulsive term (to account for nuclear-nuclear repulsion when the ions are close enough) which is done in an ad hoc fashion.
Born-Mayer: C, d* constants
Born-Landé: B, n constants
Then, Elat = ECoul + Erep graphically:
The total lattice energy has a minimum which is the equilibrium value for d (i.e., the observed distance between ions in a lattice). Erep is larger than ECoul at small distances (for large n or d*) but the reverse is true at short distances. We evaluate the constants by using experimental data at equilibrium and compressibility (this gives the slope up the low d part of the graph)
e = electronic charge, Z = ion charge in units of e (i.e., +1, +2, –1, –2, etc.)
Note: the lattice energy is only a slight modification of a simple dipole energy! (n ~ 5-10)
d0 = 2.814 Å = 2.814×10–10 m
A = 1.748
n = 7.66
experimental value = 787 kJ/mol
note the sign difference: Elat is usually written without the negative sign (thermodynamically wrong)