Lattice Enthalpies and the Born-Lande Equation
When a crystalline solid is broken up and the gas phase ions formed, the enthalpy change accompanying the process is known as the Lattice Enthalpy. It takes energy to counteract the coulombic forces which hold the solid together, and so the lattice enthalpy is endothermic.
The enthalpy of reaction for this process is the lattice enthalpy, ΔH_{L}. |
In general, the crystal structure adopted by a given compound is the one which has the greatest lattice enthalpy.
Lattice enthalpies cannot be determined experimentally, due to the difficulty of preparing the plasma of ions. Instead, they are determined using a thermochemical cycle known as the Born-Haber cycle.
Born-Haber Cycle | |
The Born-Haber cycle shown is that for the KCl. The only unknown is the lattice enthalpy. Given the values of the other enthalpies, the lattice enthalpy, ΔH_{L}, may be calculated. |
The lattice enthalpy for KCl may be calculated from the cycle, as shown:
Theoretical values for the lattice enthalpy may be calculated using the Born-Lande equation (see below), and compared to the value obtained from the Born-Haber cycle. Good agreement suggests that the ionic model of bonding is a good one for the compound being considered, whilst poor agreement suggests that there are other important contributions which are not incorporated into the ionic model.
Derivation of the Born-Lande equation
The electrostatic interaction energy between two ions of opposite charges with an internuclear separation of r is given by Coulomb’s Law, with the magnitude of the charges on the ions z^{+} and z^{–}.
Coulomb’s Law: |
Take as a typical example the NaCl structure. Each Na^{+} ion has 6 Cl^{–} ions as nearest neighbours at distance r away, 12 Na^{+} ions as next nearest neighbours at distance 2^{0.5} away, and 8 Cl^{–} ions as next nearest neighbours as distance 2^{0.33} away, and so on. The total electrostatic interaction, for one mole of substance, is therefore, with N being Avagadro’s number:
Madelung Energy: |
The electrostatic interaction energy is known as the Madelung Energy. The sum in brackets depends purely on geometry and not on the identity of the ions. It is an infinite series which converges to a constant, known as the Madelung Constant, A. Madelung constants have been calculated for the common structures.
Now consider the environment of a Na^{+} ion. the attraction of the Na^{+} for the Cl^{–} neighbours increases as r decreases, but at a certain distance the core electrons of the ions begin to overlap and strong repulsion sets in. Born suggested that the short-range repulsive interaction can be expressed in terms of an inverse function of r, where B is a constant and the Born exponent n can be determined from compressibility data on the compound of interest (and n = 9 for NaCl).
Born short-range repulsive interaction: |
The total energy is then:
The value of U changes with r, and the equilibrium arrangement is when U is at a minimum. By considering when dU/dr = 0, we can eliminate the constant B from the expression, to give an expression for the total energy of the minimum energy structure, U_{0}. This is known as the Born-Lande equation.
The Born-Lande equation: |
The only variables in this expression are the charges on the ions, z^{+} and z^{–}, the internuclear separation at equilibrium, r, and the exponent n, and the Madelung constant, which varies from structure to structure.