Edexcel A
level Chemistry (2017)
Topic 13A:
Energetics (II): Lattice Energy
Here are
two learning objectives:
13A/4. To know
that lattice energy provides a measure of ionic bond strength
13A/5. To understand
that a comparison of the experimental lattice energy value (from a BornHaber
cycle) with the theoretical value (obtained from electrostatic theory) in a
particular compound indicates the degree of covalent bonding
Measuring
theoretical lattice energies
The Born Haber experimental cycle may give us a measure of the lattice
energy but the value needs to have a theoretical underpinning.
In other words, what is the model of bonding used to provide a lattice
energy?
If
we get good agreement between the theoretical model for lattice energy and the experimental value
of the lattice energy of an actual compound then it is likely that the model fits the real
compound.
It
means that the assumptions made to create the theoretical value from the model
are reasonable assumptions about .
Here
goes then with the theoretical model used to calculate lattice energy
First
let’s assume that there are two point charges separated by a distance r.
.← r →.
+ ^{—}
there
will be a force f between these point charges which is given by
Coulomb’s law:
where
e is the size of the charge on the
electron.
Now
the potential energy V needed to
separate these charges to an infinite distance is given by:
The
sign is negative because when the ions are infinitely separated the potential
energy between then is zero.
However
in a crystal of salt there is not just one pair of oppositely charged ions,
there are billions of pairs of ions.
Just
consider one ion X in a line of ions each separated by distance r:
.← r →.← r →.← r →.← r →.← r →.
+ ^{—} ^{ } + ^{—} + ^{—}
X
Then
the energy of an ion such as X, because of the presence of the other ions that
also exert forces of attraction on it, is:
But
ions exist not in 2D but 3D environments.
Take
the sodium chloride structure as a prime example.
Each
ion has six near neighbours, twelve next nearest and so on…. as the
illustration shows:
So
for the potential energy of the ion in grey Vg
is given by:
where r
is the distance between the centres of nearest neighbour ions.
The
quantity is brackets needs to be summed to infinity.
The
value is determined by the arrangement of the ions so each crystal form (NaCl,
CsCl, fluorite, zinc blende etc.) will have its own constant M.
The
following table shows crystal structures and their Madelung constant.
Structure

Illustration

Madelung
constant

NaCl

1.748


CsCl


1.763

CaF2

2.519


Zinc
blende

1.638


Wurtzite

1.641

In
summary, the expression for the coulombic forces within a crystal group is
given by:
where
the charges on the ions are represented by z
and the interionic distance is r
If
only the coulombic forces were to be considered then the crystal structure
would collapse under the se strong attractive forces.
We
must also realise that there are repulsive forces at work and the two forces in
tension allow the crystal structure to stabilise.
It
might seem unusual to think of two oppositely charged ions repelling each other
but we need to see that if the two ions were to come very close together their
electron shells would eventually repel each other.
We
can trace the two forces at work in this illustration:
At
very short distances, this repulsive force rises rapidly with distance.
The
coulombic force of attraction changes as a function of 1/r whereas you can see
that the interionic repulsive force alters as a function of 1/r^{n } where n is usually taken to be about 12.
Allowing
then for this additional term covering the repulsive forces in the crystal
form, the expression for the lattice energy per mole becomes:
But
note the assumptions in this model of ionic bonding.
The
ions are point charges, the electron shells of the ions do not interact with
each other and the charge on each ion is an integer not a fractional
charge.
Now
as we can see these assumptions are not the case in many supposedly ionic compounds
and that is because the experimentally determined lattice energy turns out to
be higher (that is more exothermic) than the theoretical value calculated from
this model and its equation.