Physical Chemistry , 1st ed.

This is for a single pair of ions. For a mole of ions, we multiply this answer

by Avogadro’s number:

E(per mole) 4.995  105 J/mol

E499.5 kJ/mol

This answer suggests that the lattice energy of NaCl should be about 499 kJ.

The actual lattice energy is substantially higher than that, suggesting that the

two-ion model is not very good.

In fact, the two-ion (perhaps more generally, the single-formula-unit) model
is not very good because it ignores other surrounding ions. If you reconsider
the diagram of the unit cell of NaCl in Figure 21.28, it should be clear why the
two-ion model won’t work: each ion is actually surrounded by sixions of the
opposite charge! Shouldn’t the model take this into consideration? But there’s
more: around each oppositely charged ion are six ions of the same charge as
the central ion. These ions contribute a repulsive component to the overall
ionic interactions and contribute to an increasein the total potential energy of
the crystal. And around each of these like-charged ions are six oppositely
charged ions, contributing to a decrease in the total energy, and around
these... and so forth.
A proper model of lattice energy must take into account the layers of op-
positely charged and like-charged ions that compose a crystal. The model also
must take into account the repulsion between the electron clouds of all ions,
no matter what the magnitude or sign of their charges. In fact, it is the balance
between the attractions of opposite charges and the repulsions of electron
clouds that dictates the size of the unit cells.
Without derivation (which can be found in crystallography texts), one ex-
pression for the lattice energy of an ionic crystal is

lattice energy NA

4

M





0

Z

r

(^2) e 2
 1 
r
 (21.13)
In the above equation,Zis the greatest common divisor of the magnitudes on
the ions (that is, 1 for NaCl, Na 2 O, and so on, and 2 for MgO, TiO 2 , ZnS);eis
the charge on the electron;ris the distance between oppositely charged ions
(usually the closest or “nearest-neighbor” ions); and the 4 0 term is the con-
version between non-SI and SI units.NAis Avogadro’s number, so the lattice
energy has units of joules per mole (meaning joules per mole of ionic crystal
formula unit). There are two numerical parameters in equation 21.13:and
M. The repulsive range parameteris a distance parameter that relates to the
range of the repulsion between electron clouds. It is typically on the order of
0.1 times the value ofror less, showing that repulsive effects have a noticeable
but small contribution to the lattice energy. The repulsive range parameter 
has units of distance, typically Å.
The parameter Min equation 21.13 is called the Madelung constantfor the
crystal. The Madelung constant is the sum of the alternating coulombic at-
tractions and repulsions of successive spheres of alternately charged ions about
any single ion in an ionic crystal. These alternating attractions and repulsions
depend on the arrangement of the ions in the crystal (which is ultimately de-
termined from the crystal’s unit cell) and the unit cell parameters (that is, the
dimensional and angular parameters of the Bravais lattice). Because of this,
you might think that it is easy to calculate the Madelung constant for a crys-
21.8 Lattice Energies of Ionic Crystals 757