Interpretation

1. Lattice Energy of Ionic Crystals
West 3.2.5
Nashiour Rohman
Sultan Qaboos University
College of Science
Department of Chemistry & Physics

2. It is possible to think of ionic crystals as typical 3D arrays of
point charges. The electrostatic forces that hold them
together can be determined by adding up all of the crystal's
electrostatic repulsions and attractions. The net potential
energy of the arrangement of charges that makes up the
structure is known as the lattice energy, or U. It is akin to the
energy needed to sublimate the crystal and change it into a
group of gaseous ions, for instance.

3. The crystal structure, the charge on the ions, and the
internuclear distance between anions and cations all affect
the value of U. Ionic crystal formations are determined by
two main types of force:
(i) Attraction and repelling electrostatic forces. An attractive
force, F, determined by Coulomb's law is experienced by
two ions, MZ+ and XZ, separated by the distance r:

4. The formula for their coulombic potential energy, V, is
(ii) Close-proximity repulsive forces, which are significant
when electron clouds of atoms or ions start to overlap. Born
asserted that the shape of this repulsive energy

5. By adding the energies of net electrostatic attraction and
Born repulsion, one can determine the lattice energy by
determining the internuclear spacing, re, which provides
the highest U value. The steps are as follows.Fig. 1.29(a)
shows the NaCl structure. Equation (3.11) describes the
electrostatic interaction between each pair of ions. We want
to determine the net attractive energy by adding up all of
these interactions in the crystal. Let's first calculate the
interaction between one specific ion—say, Na+ in the body
centre of the unit cell—and its neighbours. Six Cl ions are

6. 12 Na+ ions at edge centre locations are the next closest
neighbours, at a distance of 2r; this results in a repulsive
potential energy term.
Eight Cl ions at the cube corners, at a distance of 3r, are the
third closest neighbours; they are drawn to the central Na+
ion by

7. An infinite series provides the net attractive energy between
Na+ ion and all other ions in the crystal:
For 2N ions per mole of NaCl, this summing is performed
for each ion in the crystal. As a result, each interaction
between an ion pair is tallied twice, hence the final number
must be divided by 2, yielding

8. where the parenthesized total from equation (3.17) is the
numerical value of the Madelung constant, A. Only the
geometric configuration of the point charges determines the
Madelung constant. For all compounds with the rock salt
structure, it is equal to 1.748. Table 3.5 lists the values of A
for several additional forms.

9. Due to V 1/r, equation (3.18), and Fig. 3.5, the structure
would collapse in on itself if equation (3.18) were the sole
factor in the lattice energy. The mutual repulsion between
ions, regardless of charge, that occurs when they are
brought too close to one another and is described by
equation (3.13), prevents this disaster. In Fig. 3.5, a graphic
representation of how this repulsive force depends on r is
presented. By adding equations (3.18) and (3.13) and
differentiating with regard to r to determine the maximum U
value and equilibrium interatomic distance, re, the total

12. Six variables - A, N, e, Z, n, and re—determine the size of
U, four of which are fixed for a given structure. Only two
remain, the internuclear spacing (re) and the charge on the
ions (Z+Z-). Since the product (Z+Z-) has a significantly
wider range of change than re, charge is the more crucial of
the two. For instance, U should be four times larger for a
material with divalent ions than it would be for an
isostructural crystal with the same re but monovalent ions. A
decline in U is anticipated for isostructural phases with
constant Z values but rising re.

13. There is a link between U and the melting point since a
crystal's U is equal to its heat of dissociation. The
refractoriness of alkaline earth oxides in comparison to
alkali halides illustrates the impact of (Z+Z-) on the melting
point. MgO (2800 °C), CaO (2572 °C), and BaO (1923 °C)
are examples of isostructural series where the effect of re on
melting points may be observed.