Born Haber Cycles

What is lattice enthalpy?

We call the measure of the strength between the bonds of the ions in an ionic lattice, lattice enthalpy.

However, you can also say:

Why do we have two definitions for lattice enthalpy? Remember, enthalpy changes not only when bonds break, but also when they form. We consider the bonds in an ionic compound completely broken only when the ions are in a gaseous state. The particles are so far apart we view them as having negligible interactive forces. So in an enthalpy diagram that would look like this:

Fig. 1 - Lattice enthalpy

One definition considers the formation of an ionic bond from gaseous ions, and the other one looks at breaking an ionic bond to make gaseous ions. So we don’t get confused, we say this instead:

And also:

Lattice enthalpy helps scientists predict how soluble an ionic compound might be in water. We cannot directly measure change in lattice enthalpy. Since it’s impossible to measure lattice enthalpies, we say they are experimental values. This is because we calculate lattice enthalpy using enthalpy changes we can measure. Let us discuss these enthalpy changes and how we use them in a Born-Haber cycle.

Enthalpy changes

Have a look at the Born-Haber cycle below. How many different enthalpy changes can you spot?

Fig. 2 - A Born-Haber cycle

You may have spotted the following enthalpy changes:

• Enthalpy of formation (${∆}_{\mathrm{f}}{\mathrm{H}}^{\mathrm{\Theta }}$)

• Ionisation energy

• Bond enthalpy

• Electron affinity

When we draw a Born-Haber cycle, we aim to fill in as many of these values as we can. By using Hess’ Law we can use them to calculate lattice enthalpy as long as we start at the same place in the cycle:

Lattice enthalpy (direct route) = Enthalpy of formation + Enthalpy of atomisation - Ionisation energy - Bond enthalpy - Electron affinity (indirect route)

In summary, Born-Haber cycles use enthalpy changes that can be measured to calculate lattice enthalpy, a change that can’t be measured. We have covered three of these enthalpies already: ionisation energy, electron affinity and bond enthalpy. In order to calculate Born-Haber cycles, you will also need to know about enthalpy change of formation, as well as enthalpy of atomisation.

What is enthalpy change of formation?

What is standard enthalpy of atomisation?

Before you can form gaseous ions in a Born-Haber cycle, you will need to atomise the elements that make up the compound. That means you take the elements in their standard states and turn them into monatomic gases as shown below:

½Cl2(g) → Cl(g) 𝚫Hat𝛉 = +122 kj mol-1

Ionisation energy and electron affinity

As you know, atoms become ions by losing or gaining electrons. They do this to achieve a complete valence shell. You have also learned that the energy required to remove one electron from the outer shell of an atom is called the first ionisation energy. Similarly, we call the energy released when an atom gains an electron, electron affinity.

You must include the change in energy when an atom loses an electron (ionisation energy) and the change in energy when an atom gains an electron (electron affinity) as individual steps in a Born-Haber cycle. Here’s an example:

Fig. 3 - You must show the ionisation energy and electron affinity as individual steps in a Born-Haber cycle

Bond enthalpy

How to draw a Born-Haber cycle

As you can see, there are various enthalpy changes involved when an ionic lattice is split into its gaseous ions. When we draw Born-Haber cycles we must show these enthalpy changes in the following order:

1. The enthalpy of formation of the compound.
2. Enthalpy of atomisation of each element.
3. The first ionisation energy of the metal.
4. Subsequent ionisation enthalpies if appropriate.
5. First electron affinity of the non-metal.
6. Subsequent electron affinities if appropriate.

Why do we draw the energy changes in that order? You will remember that ionisation energy turns gaseous atoms into gaseous ions. So, the first ionisation energy cannot come before atomisation enthalpy. Similarly, electron affinity must come after ionisation energy. For the non-metal to gain an electron, the metal has to lose one first.

That sounds like a lot to remember! To help you see how it all comes together, let us draw a Born-Haber cycle.

We will construct a Born-Haber cycle for the lattice formation enthalpy of potassium chloride (KCl). We start with KCl, and go round the cycle filling in all the different enthalpies until we arrive at the beginning again. We use downward arrows for exothermic enthalpies and upwards arrows for endothermic enthalpy changes.

Step 1

Separate potassium chloride (KCl) into the atoms of the elements using enthalpy of formation.

$\mathrm{KCl}\to {\mathrm{K}}_{\left(\mathrm{s}\right)}+\frac{1}{2}{\mathrm{Cl}}_{2\left(\mathrm{g}\right)}{∆}_{\mathrm{f}}{\mathrm{H}}^{\mathrm{\Theta }}$

Fig. 4 - Separate the ionic solid into the atoms of the elements

Step 2

Atomise potassium (K) using enthalpy of atomisation

${\mathrm{K}}_{\left(\mathrm{s}\right)}+\frac{1}{2}{\mathrm{Cl}}_2\to {\mathrm{K}}_{\left(\mathrm{s}\right)}+\frac{1}{2}{\mathrm{Cl}}_{2\left(\mathrm{g}\right)}∆{}_{\mathrm{at}}\mathrm{H}^{\mathrm{\Theta }}\left(\mathrm{K}\right)$

Fig. 5 - Step 2: Atomise the metal element

Step 3

Atomise chlorine (Cl) using enthalpy of atomisation.

Fig. 6 - Atomise the non-metal element

Step 4

Ionise potassium using first ionisation energy.

Fig. 7 - Ionse the metal element using the ionisation energy

Step 5

Ionise chlorine using first electron affinity.

${{\mathrm{K}}^{+}}_{\left(\mathrm{g}\right)}+{\mathrm{Cl}}_{\left(\mathrm{g}\right)}+{\mathrm{e}}^{-}\to {{\mathrm{K}}^{+}}_{\left(\mathrm{g}\right)}+{{\mathrm{Cl}}^{-}}_{\left(\mathrm{g}\right)}{\mathrm{EA}}_1$

Fig. 8 - Ionise the non-metal using the electron affinity

Step 6

Complete the cycle with the lattice enthalpy.

Fig. 9 - Complete the cycle

Well done! You’ve completed the Born-Haber cycle for potassium chloride. Notice how we followed the steps from enthalpy of formation to the last electron affinity. The steps to draw a Born-Haber cycle are always the same. As a recap here is the order of enthalpy changes:

1. The enthalpy of formation of the compound.
2. Enthalpy of atomisation of each element.
3. The first ionisation energy of the metal.
4. Subsequent ionisation enthalpies if appropriate.
5. First electron affinity of the non-metal.
6. Subsequent electron affinities if appropriate.