- Born-Landé equation
The Born-Landé equation is a means of calculating the
lattice energy of a crystallineionic compound . In 1918 [ I.D. Brown, "The chemical Bond in Inorganic Chemistry", IUCr monographs in crystallography, Oxford University Press, 2002, ISBN 0198508700]Max Born andAlfred 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 incoulomb s, 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
coulomb s, 1.6022e|−19 C:epsilon_0, =permittivity of free space ::4pi epsilon_o = 1.112e|−10 C²/(J m) :r_0 = distance apartFor 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 crystalRepulsive term
Born and Lande suggested that a repulsion, between the ions would be proportional to1/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
Wikimedia Foundation. 2010.