Definitions

Lattice energy

The lattice energy of an ionic solid is a measure of the strength of bonds in that ionic compound. It is usually defined as the enthalpy of formation of the ionic compound from gaseous ions and as such is invariably exothermic.

Na+ (g) + Cl (g) → NaCl (s)

The experimental lattice energy of NaCl is −787 kJ/mol.

Some older textbooks define lattice energy as the energy required to convert the ionic compound into gaseous ions which is an endothermic process, and following this definition the lattice energy of NaCl would be +787 kJ/mol.

The experimental value for the lattice energy can be determined using the Born-Haber cycle.

Theoretical treatments

Born-Landé equation

In 1918 Born and Landé proposed that the lattice energy could be derived from the electrostatic potential of the ionic lattice and a repulsive potential energy term.
$E = -frac\left\{N_AMz^+z^- e^2 \right\}\left\{4 pi epsilon_o r_0\right\}\left(1-frac\left\{1\right\}\left\{n\right\}\right)$
where
$, N_A$ = Avogadros number
$, M$ = Madelung constant, relating to the geometry of the crystal.
$, z^+$ = charge of cation
$, z^-$ = charge of anion
$,e$ = electron charge in coulombs, 1.6022 C
$epsilon_0,$ = permittivity of free space
$,4pi epsilon_o$ = 1.112 C²/(J m)
$, r_0$ = distance to closest ion
$, n$ = Born exponent, a number between 5 and 12, determined experimentally by measuring the compressibility of the solid, or derived theoretically.
The Born-Landé equation gives a reasonable fit to the lattice energy
Compound Calculated Lattice Energy Experimental Lattice Energy
NaCl −756 kJ/mol −787 kJ/mol
LiF −1007 kJ/mol −1046 kJ/mol
CaCl2 −2170 kJ/mol −2255 kJ/mol

From the Born-Landé equation it can be seen that the lattice energy of a compound is dependant on a number of factors

• as the charges on the ions increase the lattice energy increases(becomes more negative),
• when ions are closer together the lattice energy increases (becomes more negative)

Barium oxide (BaO), for instance, which has the NaCl structure and therfore the same Madelung constant, has a bond radius of 275 picometers and a lattice energy of -3054 kJ/mol, while sodium chloride (NaCl) has a bond radius of 283 picometers and a lattice energy of -786 kJ/mol.

Kapustinskii equation

The Kapustinskii equation can be used as a simpler way of deriving lattice energies where high precision is not required.

References

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