Engineering & Science
Kapustinskii & Born-Landé Ionic Crystal Lattice Energy Formula (kJ/mol)
Estimate ionic crystal lattice enthalpy U_L = ((-120,200 * nu * |z+ * z-|) / (r+ + r-)) * (1 - 34.5 / (r+ + r-)) in kJ/mol from Shannon/thermochemical ionic radii (pm) without needing crystal structure!
Explain why MgO melts at 2,852 °C while NaCl melts at 801 °C!
Born-Landé's exact electrostatic equation computes ionic lattice energy from crystal-specific Madelung constants M (1.74756 for rock-salt NaCl, 2.51939 for fluorite CaF2)—but what if you synthesize a brand-new ionic salt or ionic liquid whose crystal symmetry and Madelung constant are still unknown? Anatoli Kapustinskii (1956) noticed that dividing Madelung's constant by the number of ions per formula unit (M / nu) is nearly invariant (~0.87–0.88) across ALL 3D ionic lattices!
That insight yields the universal Kapustinskii Equation (for ionic radii r+ and r- entered in picometers, pm): U_L(kJ/mol) = ((-120,200 * nu * ABS(z_plus * z_minus)) / (r_plus + r_minus)) * (1 - (34.5 / (r_plus + r_minus)))! Compare rock-salt NaCl (nu = 2 ions, |+1 * -1| = 1, r_Na+ = 102 pm, r_Cl- = 181 pm -> r0 = 283 pm) at U_L = -745.9 kJ/mol against refractory periclase MgO (nu = 2, |+2 * -2| = 4, r_Mg2+ = 72 pm, r_O2- = 140 pm -> r0 = 212 pm) at a colossal U_L = -3,797.7 kJ/mol—over 5x stronger ionic bonding!
| Ionic Crystal Formula Unit | Ions nu & Charges |z+*z-| | Radii Sum r+ + r- (pm) | Kapustinskii Lattice Energy U_L |
|---|---|---|---|
| Sodium Chloride (NaCl) | nu = 2, |+1 * -1| = 1 | 102 + 181 = 283 pm | -745.9 kJ/mol (Exp ~ -787) |
| Magnesium Oxide (MgO) | nu = 2, |+2 * -2| = 4 | 72 + 140 = 212 pm | -3,797.7 kJ/mol (Refractory!) |
| Calcium Fluoride (CaF2) | nu = 3, |+2 * -1| = 2 | 100 + 133 = 233 pm | -2,637.6 kJ/mol |
Kapustinskii Ionic Lattice Energy U_L (kJ/mol, Exothermic Formation Convention)
Evaluates ((-120200 * nu * ABS(z+ * z-)) / (r+ + r-)) * (1 - 34.5 / (r+ + r-)) for radii in picometers.
=IF(OR(B2 < 2, B3 = 0, B4 = 0, (B5 + B6) <= 34.5), "Invalid Ionic Crystal Inputs", ROUND(((-120200 * B2 * ABS(B3 * B4)) / (B5 + B6)) * (1 - (34.5 / (B5 + B6))), 1))Interionic Center-to-Center Equilibrium Distance r0 = r+ + r- (Angstroms)
Sums cation and anion radii (pm) and divides by 100 pm/Angstrom.
=IF(OR(B5 <= 0, B6 <= 0), "Invalid Ionic Radii", ROUND((B5 + B6) / 100, 3))Try the example
Use the original values or change them to test your understanding.