Concise Physical Chemistry

c11 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

178 LIQUIDS AND SOLIDS

This enthalpy is the negative of the enthalpy of formation of the crystal from its

gaseous ions. Please recall from Chapter 4 that this is not the standard enthalpy of

formation because Na+(g)+I−(g) is not the standard state of elemental sodium and

iodine. The lattice energy for NaI and can be calculated from the standard state value

(^) fHo(NaI) by a procedure called theBorn–Haber cycle, which is nothing more than
an enthalpy of formation calculation like the one shown in Fig. 4.1 except that it is
a little more complicated because of the nonstandard state of the products. Example
11.1 is not only an illustration of the Born–Haber cycle, but it also should serve to
review and drive home the difference between standard and nonstandard states in
thermochemistry.
The lattice energies of crystals show logical regularities. For example, the alkali
metal iodides fall between 600 and 775 kJ mol−^1 and decrease gradually from the
lithium salt (which has a short ionic bond) to cesium iodide (which has a long one).
The lattice energies of the doubly charged alkaline earth salts are very much larger
than those of the alkali metals.
Lattice energies can also be calculated from a theoretical model in which the
energy of the ionic crystal is supposed to be a function entirely of electrostatic forces.
The model is fairly successful, but it involves some infusion of empirical data. All
electrostatic attractions and repulsions can be calculated over the distances separating
ions of opposite charge or the same charge. Knowing the exact geometry of the unit
cell, these interionic distancesaican be calculated precisely, not only for ions in
the same unit cell but also for those in distant cells. The total Coulombic energy
calculated from electrostatic theory in this way is
U(r)=

NAZ+Z−

r

e^2

4 πε 0

[

1

2

(∑

Zi+

ai+

+

∑Zi−

ai−

)]

+Be−r/ρ

where the summed terms can be either positive for repulsion of like charges or

negative for attraction between unlike charges. The constantsBandρcan be assigned

reasonable values by an essentially empirical method (Barrow, 1996). Application to

NaI gives 682 kJ mol−^1 by comparison to the Born–Haber value of 658 kJ mol−^1.

The difference is about 3.6%.

PROBLEMS AND EXERCISES

Exercise 11.1 The Born–Haber Cycle

Find the lattice energy (enthalpy) of NaI by the Born–Haber cycle.

Solution 11.1 Note that the small distinction between energy and enthalpy is often

ignored in calculations involving large energies. We shall need several pieces of

information to begin. First, we need the transition enthalpy of sodium metal in the

solid form at 298 K and 1 bar, which is its standard state, to the state of the gaseous ion.

This is calculated in two steps, first vaporization and then ionization, even though the

steps may not be differentiated in the real process. We need not worry about the actual