Lattice Energy and the Born-Haber Cycle

1
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 Hesss Law). In this case the cycle is known
as the Born-Haber Cycle
2
IE
EA
DHsub
½ BDE
Lattice Energy (U)
DHf
Na(s) ½ Cl2(g) NaCl(s)
3
From Hesss 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.
4
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 dont know DHf you can estimate U by
  several methods
• the Born equation
• the Born-Meyer equation
• the Kapustinski equation

5
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
• zs 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

6
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
7
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.
8

• 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)
9
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)