Physical Chemistry , 1st ed.

tal. It’s easy in theory, but not in practice. If you consider a simple system in

which you recede from a central ion by moving through a series of nested

spherical shells, the farther away you get from any one ion, the larger the num-

ber of ions in any spherical shell! The contribution of attraction or repulsion

from each successive shell does not decrease very quickly.

Once the Madelung constant is determined from simple structural argu-

ments, the lattice energy of an ionic crystal can be determined easily. These lat-

tice energies can be used in cycles to evaluate energies of difficult-to-determine

chemical processes. Such cycles are called Born-Haber cycles.

Table 21.6 lists some Madelung constants and repulsive range parameters 

for various ionic compounds. Madelung constants are unitless, whereas repul-

sive range parameters have units of distance. Because Madelung constants can

be determined from purely geometrical arguments, they are usually defined

only for the crystal that typifies the unit cell (for example, any crystal that has

the cesium chloride unit cell—simple cubic—has a Madelung constant of

1.7627). But because repulsive range parameters depend on the ion’s charges

as well as the unit cell dimensions, crystals having the same unit cell have dif-

ferent values for .

Example 21.12

Calculate the expected lattice energy of NaCl again, this time using equation

21.13 and using a Madelung constant of 1.748 and a repulsive factor of

0.321 Å. The distance between Naions and Clions is, again, 2.78 Å.

Compare it to a lattice energy of 769 kJ/mol. Compare your answer with

Example 21.11.

Solution

Recall that NaCl is a 11 ionic compound, so the greatest common divisor

variable Zequals 1. If we want to use standard units at the outset, we should

convert our 2.78 Å into meters to get 2.78  10 ^10 m. We will use

lattice energy 

NA

4

M





0

Z

r

(^2) e 2
 1 



r



and substitute for the various constants:

758 CHAPTER 21 The Solid State: Crystals

Table 21.6 Madelung constants and

repulsive range parameters

of some ionic crystals

Formula Madelung constant M (Å)

LiF 0.291

LiBr 0.330

NaCl 1.74756 0.321

NaBr 0.328

KCl 0.326

KBr 0.336

ZnS 1.6381 0.289

TiO 2 2.408 0.250

CsCl 1.7627 0.331

lattice energy   1 

0

2

.3

.7

2

8

1

Å

Å



(6.02  1023 /mol)(1.748)  12 (1.602  10 ^19 C)^2



4 [8.854  10 ^12 C^2 /(Jm)](2.78  10 ^10 m)

Note the slight inconsistency: in the second part of the equation, we still use

rin units of angstroms. This is because the parameter is given in units of

Å, and to keep units consistent we keep the Å unit for r. As a ratio, it is re-

quired that the units cancel; we could just as easily convert to units of me-

ters. You should satisfy yourself that the units in the first part of the expres-

sion cancel appropriately to yield units of J/mol. Solving numerically:

lattice energy 873,000 

m

J

ol

0.885

lattice energy 773,000 

m

J

ol

 773 

m

kJ

ol



Compared to an experimental value of 769 kJ/mol, we find that equation 21.13

does a much better job of predicting lattice energies than Coulomb’s law did.