The Born–Mayer equation
المؤلف:
Peter Atkins, Tina Overton, Jonathan Rourke, Mark Weller, and Fraser Armstrong
المصدر:
Shriver and Atkins Inorganic Chemistry ,5th E
الجزء والصفحة:
ص88-89
2025-08-21
1042
The Born–Mayer equation
Key points: The Born–Mayer equation is used to estimate lattice enthalpy for an ionic lattice. The Madelung constant reflects the effect of the geometry of the lattice on the strength of the net Coulombic interaction. To calculate the lattice enthalpy of a supposedly ionic solid we need to take into account several contributions, including the Coulombic attractions and repulsions between the ions and the repulsive interactions that occur when the electron densities of the ions overlap. This calculation yields the Born–Mayer equation for the lattice enthalpy at T=0:
where d=r+ + r- is the distance between centres of neighbouring cations and anions, and hence a measure of the ‘scale’ of the unit cell (for the derivation, see Further information 3.1). In this expression NA is Avogadro’s constant, zA and zB the charge numbers of the cation and anion, e the fundamental charge, ε0 the vacuum permittivity, and d* a constant (typically 34.5 pm) used to represent the repulsion between ions at short range. The quantity A is called the Madelung constant, and depends on the structure (specifically, on the relative distribution of ions, Table 3.8). The Born–Mayer equation in fact gives the lattice energy as distinct from the lattice enthalpy, but the two are identical at T 0 and the difference may be disregarded in practice at normal temperatures.
■ A brief illustration. To estimate the lattice enthalpy of sodium chloride, we use z(Na+) 1, z (Cl-)=-1, from Table 3.8, A=1.748, and from Table 1.4 d=rNa++rCl- 283 pm; hence (using fundamental constants from inside the back cover):
or 756 kJ mol 1. This value compares reasonably well with the experimental value from the Born Haber cycle, 788 kJ mol-1.The form of the Born–Mayer equation for lattice enthalpies allows us to account for their dependence on the charges and radii of the ions in the solid. Thus, the heart of the equation is
Therefore, a large value of d results in a low lattice enthalpy, whereas high ionic charges result in a high lattice enthalpy. This dependence is seen in some of the values given in Table 3.7. For the alkali metal halides, the lattice enthalpies decrease from LiF to LiI and from LiF to CsF as the halide and alkali metal ion radii increase, respectively. We also note that the lattice enthalpy of MgO (|zA zB|=4) is almost four times that of NaCl (|zA zB|= 1) due to the increased charges on the ions for a similar value of d, noting that the Madelung constant is the same.
The Madelung constant typically increases with coordination number. For instance, A 1.748 for the (6,6)-coordinate rock-salt structure but A 1.763 for the (8,8) coordinate caesium-chloride structure and 1.638 for the (4,4)-coordinate sphalerite structure. This dependence reflects the fact that a large contribution comes from nearest neighbours, and such neighbours are more numerous when the coordination number is large. However, a high coordination number does not necessarily mean that the interactions are stronger in the caesium-chloride structure because the potential energy also depends on the scale of the lattice. Thus, d may be so large in lattices with ions big enough to adopt eightfold-coordination that the separation of the ions re verses the effect of the small increase in the Madelung constant and results in a smaller lattice enthalpy.
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