# Derivation of Born-Lande Equation

The Born-Lande equation is used to calculate the lattice energy (or lattice enthalpy) of an ionic crystal (ionic solid). In 1918 Max Born and Alfred Lande proposed that the lattice energy could be derived from the electrostatic interaction between ions in the crystal lattice. This equation helps to understand the stability and properties of ionic crystals.

When ionic solids are vaporized, they form either covalent or atomic species. Hence the direct use of Born Haber cycle to determine the lattice energy is not possible. Lattice energy is calculated by considering the two oppositely charged ions say M^{+z} cation and X^{−z} anions at a distance r.

Then the attraction energy-

where, k is 1/4πε_{o} and e is electronic charge.

Under the influence of attraction energy, gaseous cation and anion come close together. The repulsion is set between them due to repulsion of valence electrons of each other. This repulsion does not allow them to fuse together. Born calculated this repulsion energy as B/r^{n}.

Where, n is Born exponent which depends upon the principal quantum number of the electrons and hence the electronic configuration of the ions.

Valence Shell Electrons | Born Exponent |
---|---|

He (1s^{2}) |
5 |

Ne (2s^{2} 2p^{6}) |
7 |

Ar (3s^{2} 3p^{6})Cu ^{+} (2s^{2}3p^{6} 3d^{10}) |
9 |

Kr (4s^{2}4p^{6})Ag ^{+} (4s^{2}4p^{6} 4d^{10}) |
10 |

Xe (5s^{2}5p^{6})Au ^{+} (5s^{2}5p^{6} 5d^{10}) |
12 |

Therefore, the lattice energy (U) of the ion pair is-

These two terms have opposite signs, the plot of U vs r will show a minima at equilibrium intermolecular distance(r_{o}).

When r = r_{o}, then-

dU/dr = 0 and then-

Putting the value of B in equation:1 we get-

A crystal provides more coulombic interactions than those present in the ion pair for example in NaCl crystal, for each Na^{+} ion, there are six Cl^{−} ions are at r_{o}, 12Na^{+} ions at √2r_{o} and eight more Cl^{−} ions at √3r_{o} etc. Hence, the actual coulombic energy-

Where A is Madelung constant and is the summation of interactions which arises due to geometrical arrangement of ions. Therefore, the energy of the ion pair in lattice-

And the total lattice energy for one mole of an ionic crystal is-

Where, N is Avogadro number.

This is the desired **Born-Lande equation** for the lattice energy of ionic compounds.

In 1933, Kapustinskii observed that the ratio of Madelung constant (A) and number of ions (v) i.e. A/v is simply 0.874.

∵ A/v = 0.874

∴ A = 0.874 v

Hence the Born-Lande equation becomes-

Where, v value for BeCl_{2}, Li_{2}O etc. is 3 and r_{o} is the sum of ionic radii of the cation and anion i.e. r^{+} + r^{−}. This is called **Kapustinskii equation**. The Kapustinskii equation is an approximation for the lattice energy of an ionic solid, especially useful when the crystal structure and the Madelung constant are not known.