Lattice Energy (3) Measuring Theoretical Lattice Energies

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 Born-Haber 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. 

Crystal structures are large networks of ions held in place by electrostatic forces of attraction. 

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.

This is called the Madelung constant. 

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/rⁿ  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. 

You can find a discussion of the experimental lattice energies and theoretical lattice energies here.