Lattice Energy and the Born-Haber Cycle For a reaction such as Na(s) + ½ Cl2(g)  NaCl(s) we want to decide if the compound will be stable as an ionic salt. The customary way of doing this is to use a thermodynamic cycle (an application of Hess’s Law). In this case the cycle is known as the Born-Haber Cycle

Na+(g) Cl-(g) + IE EA Na(g) Cl(g) DHsub ½ BDE Lattice Energy (U) DHf Na(s) + ½ Cl2(g) NaCl(s)

DHf = DHsub + IE + ½ BDE + EA + U From Hess’s Law: DHf = DHsub + IE + ½ BDE + EA + U Energy Economics: exothermic endothermic EA (usually) DHsub ½ BDE IE What about U? If DHf is to be negative (needed if it is to be a stable compound), U had better be negative.

How Do We Get the Value of the Lattice Energy? if know all of the other terms, including DHf, then you can calculate U from U = DHf – (DHsub + IE + ½ BDE + EA) if you don’t know DHf you can estimate U by several methods the Born equation the Born-Meyer equation the Kapustinski equation

Start by thinking of U as a purely electrostatic term Start by thinking of U as a purely electrostatic term. Since crystal lattices of ionic compounds have ions of opposite charge next to one another, energy should be released when the crystal is formed. The general form of an electrostatic interaction is: where z’s are the charges on A and B e is the charge on the electron 4 is one greater than 3 e0 is the permittivity of the vacuum rAB the internuclear distance in the crystal lattice

To get the total lattice energy you need to sum all of the electrostatic interactions experienced by a given ion. Consider the linear crystal shown here: focus on the red atom - assume it is a cation - then the green atoms are anions and the blue cations, the electrostatic interaction is given by

The total electrostatic energy is then given by: where the term M, called the Madelung constant, corrects for the geometry of the crystal lattice The value of M varies from lattice type to lattice type. For NaCl the value is 1.748 - the next slide shows part of that calculation.

look at central Cl: 6 Na at distance r 12 Cl at 21/2r 8 Na at 31/2r etc The geometric Madelung constant is therefore: (may be slow to converge)

This approach treats the ions as hard spheres, and assumes 100% ionic character to the bonds. Born and Meyer modified the approach to include additional interactions (primarily dispersion forces and covalency). The result was an equation of the form: where r* is a parameter that is approximately 34.5pm if rAB is in pm. Kapustinski further modified this equation by noting that M/r did not change much from lattice to lattice. His equation for estimating U is: n = ions per formula unit (2 for NaCl)