Born-Haber Cycles Calculations

Now you know how to draw a Born-Haber cycle like the one shown below, what's next? Well, you can use it to calculate the lattice enthalpy of an ionic solid, or any other unknown enthalpy change shown in the cycle. How? Remember, Hess’ Law states that the enthalpy change of a reaction remains the same regardless of the route taken. This allows us to calculate the enthalpy change for any part of the cycle as long as we start and end at the same places.

This means:

$\mathbf{∆}{}_{\mathbf{f}}\mathbf{H}^{\mathbf{\Theta }}\mathbf{}\mathbf{=}\mathbf{}\mathbf{(}\mathbf{∆}{}_{\mathbf{at}}\mathbf{H}^{\mathbf{\Theta }}\mathbf{}\mathbf{(}\mathbf{K}\mathbf{\left)}\mathbf{\right)}\mathbf{}\mathbf{+}\mathbf{}\mathbf{(}\mathbf{∆}{}_{\mathbf{at}}\mathbf{H}^{\mathbf{\Theta }}\mathbf{}\mathbf{(}\mathbf{Cl}\mathbf{\left)}\mathbf{\right)}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\left(}{\mathbf{IE}}_{\mathbf{1}}\mathbf{\right)}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\left(}{\mathbf{EA}}_{\mathbf{1}}\mathbf{\right)}\mathbf{}\mathbf{+}\mathbf{}\mathbf{∆}{}_{\mathbf{LE}}\mathbf{H}^{\mathbf{\Theta }}$

If we rearrange this equation we can use it to calculate lattice enthalpy:

$\mathbf{∆}{}_{\mathbf{LE}}\mathbf{H}^{\mathbf{\Theta }}\mathbf{}\mathbf{=}\mathbf{}\mathbf{(}\mathbf{∆}{}_{\mathbf{f}}\mathbf{H}^{\mathbf{\Theta }}\mathbf{)}\mathbf{}\mathbf{-}\mathbf{}\mathbf{[}\mathbf{}\mathbf{(}\mathbf{∆}{}_{\mathbf{at}}\mathbf{H}^{\mathbf{\Theta }}\mathbf{}\mathbf{\left(}\mathbf{K}\mathbf{\right)}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{}\mathbf{(}\mathbf{∆}{}_{\mathbf{at}}\mathbf{H}^{\mathbf{\Theta }}\mathbf{}\mathbf{\left(}\mathbf{Cl}\mathbf{\right)}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{}\mathbf{(}{\mathbf{IE}}_{\mathbf{1}}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{}\mathbf{(}{\mathbf{EA}}_{\mathbf{1}}\mathbf{)}\mathbf{}\mathbf{]}$

Fig. 1 - Born-Haber cycle for potassium chloride

• In this article, you will practice calculating lattice enthalpies using Born-Haber calculations.
• We will look at how you can compare the lattice enthalpies of different substances and the factors that affect lattice enthalpy.
• At the end, we will see how physicists calculate lattice energy theoretically, and why these values sometimes differ from their corresponding experimental values.

Examples of Born-Haber calculations

You need to practice to get the hang of lattice enthalpy calculations, so let us try a few example calculations together!

Comparing lattice enthalpies

For your chemistry exams, you must be able to explain the differences between lattice enthalpies of different substances. You will also be expected to give a reason for the differences between the theoretical and experimental values of lattice enthalpy. Not to worry- we’ve got you covered!

Factors that affect lattice enthalpy

Two factors that affect lattice enthalpy are:

• The charges on the ions
• The radius of the ions

The charge on the ion

Compare the lattice enthalpies of sodium chloride and magnesium oxide below. Their crystal lattices have the same geometry, so why are their enthalpies different?

Fig. 4 - Lattice enthalpies of magnesium oxide and sodium chloride

Have a look now at their ions. Can you see a reason for the difference in their lattice enthalpies?

Fig. 5 - The ions of sodium chloride and magnesium oxide

Magnesium and oxygen ions have larger ionic charges than sodium and chloride ions. Ions with larger ionic charges have stronger electrostatic attraction between the ions. More energy is needed to overcome the attraction between the ions and break up the lattice. So the lattice dissociation enthalpy of magnesium oxide (MgO) is larger than that of sodium chloride (NaCl).

The radius of the ion

Magnesium ions and oxide ions are smaller than sodium and chloride ions. This means the ions in the magnesium oxide lattice are closer together. The strength of ionic attraction depends on the closeness of the centres of the attracted ions, so there is a stronger attraction between the magnesium and oxygen ions.

Ions with larger radii tend to have smaller lattice enthalpies, because the electrostatic attraction between larger ions is weaker.

You can observe this effect as you go down a group on the periodic table. As an example look at Group 1. As their positive ions get bigger, the lattice enthalpies of their chloride salts decrease.

Fig. 6 - Lattice enthalpies get smaller as you go down a group

Theoretical values for lattice enthalpy

Physicists use a theoretical method to calculate lattice enthalpies. This is based on the assumption that the substance is highly ionic with only electrostatic attraction between the ions. In other words, they consider the ions as point charges that bond together to make a lattice. They also take into account the geometry of the lattice and the distance between the atoms.

• A small difference, as in the case of NaCl, suggests that the assumption the compound is ionic is fairly accurate.

• A large difference, as in the case of AgCl, suggests that the compound is not purely ionic. Instead, the bonds show a covalent character.

Getting the hang of Born-Haber calculations takes practice! Review this article as many times as you need and use the flashcards in this section to strengthen your skills.