Born-Landé equation

The Born-Landé equation is a means of calculating the lattice energy of a crystalline ionic compound. In 1918 [ I.D. Brown, "The chemical Bond in Inorganic Chemistry", IUCr monographs in crystallography, Oxford University Press, 2002, ISBN 0198508700] Max Born and Alfred Landé proposed that the lattice energy could be derived from the electrostatic potential of the ionic lattice and a repulsive potential energy term.David Arthur Johnson, "Metals and Chemical Change",Open University, Royal Society of Chemistry, 2002,ISBN 0854046658]

:$E\; =\; -frac\{N\_AMz^+z^-\; e^2\; \}\{4\; pi\; epsilon\_o\; r\_0\}(1-frac\{1\}\{n\})$ (Joules/mol)where:$,\; N\_A$ = Avogadros number:$,\; M$ = Madelung constant, relating to the geometry of the crystal.:$,\; z^+$ = charge of cation in electron units:$,\; z^-$ = charge of anion in electron units:$,e$ = electron charge in coulombs, 1.6022e|−19 C:$epsilon\_0,$ = permittivity of free space::$,4pi\; epsilon\_o$ = 1.112e|−10 C²/(J m) :$,\; r\_0$ = distance to closest ion in meters:$,\; n$ = Born exponent, a number between 5 and 12, determined experimentally by measuring the compressibility of the solid, or derived theoretically. [Cotton, F. Albert; Wilkinson, Geoffrey; (1966). Advanced Inorganic Chemistry (2d Edn.) New York:Wiley-Interscience.]

Derivation

The ionic lattice is modelled as an assembly of hard elastic spheres which are compressed together by the mutual attraction of the electrostatic charges on the ions. They achieve the observed equilibrium distance apart due to a balancing short range repulsion.

Electrostatic potential

The electrostatic potential,$E\_c$, between a pair of ions of equal and opposite charge is:-

:$E\_c\; =\; -frac\{z^+z^-\; e^2\; \}\{4\; pi\; epsilon\_o\; r\_0\}$where:$,\; z^+$ = charge of cation:$,\; z^-$ = charge of anion:$,e$ = electron charge in coulombs, 1.6022e|−19 C:$epsilon\_0,$ = permittivity of free space::$,4pi\; epsilon\_o$ = 1.112e|−10 C²/(J m) :$,\; r\_0$ = distance apart

For a lattice the interactions between all ions needs to be summed to give $E\_M$, sometimes called the Madelung energy:-

:$E\_M\; =\; -frac\{z^2\; e^2\; M\}\{4\; pi\; epsilon\_o\; r\_0\}$where:$,\; z$ = charge of ions :$,e$ = 1.6022e|−19 C:$,4pi\; epsilon\_o$ = 1.112e|−10 C²/(J m) :$,\; M$ = Madelung constant, which is related to the geometry of the crystal

Repulsive term

Born and Lande suggested that a repulsion, between the ions would be proportional to$1/r^n$(where r is the distance between the ions) so that the repulsive energy term,$E\_R$, would be expressed:-:$,E\_R\; =\; frac\{B\}\{r^n\}$where:$,B$ = constant :$,r$ = distance apart:$,n$ = Born exponent, a number between 5 and 12

Total energy

The total energy of the lattice can therefore be expressed as the sum of the attraction and repulsion potentials :-

:$E\; =\; -frac\{z^+z^-\; e^2\; \}\{4\; pi\; epsilon\_o\; r\_0\}\; +\; frac\{B\}\{r^n\}$

and the minimum energy at the equilibrium separation is using standard calculus:

:$E\; =\; -frac\{N\_AMz^+z^-\; e^2\; \}\{4\; pi\; epsilon\_o\; r\_0\}(1-frac\{1\}\{n\})$

Calculated lattice energies

The Born-Landé equation gives a reasonable fit to the lattice energy

References