Calculations Using Born-Haber Cycles
- Once a Born-Haber cycle has been constructed, it is possible to calculate the lattice energy (ΔHlattꝋ) by applying Hess’s law and rearranging:
ΔHfθ = ΔHatθ + ΔHatθ + IE + EA + ΔHlattθ
If we simplify this into three terms, this makes the equation easier to see:
- ΔHlattθ
- ΔHfθ
- ΔH1θ (the sum of all of the various enthalpy changes necessary to convert the elements in their standard states to gaseous ions)
- The simplified equation becomes
ΔHfθ = ΔH1θ + ΔHlattθ
So, if we rearrange to calculate the lattice energy, the equation becomes
ΔHlattθ = ΔHfθ - ΔH1θ
- When calculating the ΔHlattθ, all other necessary values will be given in the question
- A Born-Haber cycle could be used to calculate any stage in the cycle
- For example, you could be given the lattice energy and asked to calculate the enthalpy change of formation of the ionic compound
- The principle would be exactly the same
- Work out the direct and indirect route of the cycle (the stage that you are being asked to calculate will always be the direct route)
- Write out the equation in terms of enthalpy changes and rearrange if necessary to calculate the required value
- Remember: sometimes a value may need to be doubled or halved, depending on the ionic solid involved
- For example, with MgCl2 the value for the first electron affinity of chlorine would need to be doubled in the calculation, because there are two moles of chlorine atoms
- Therefore, you are adding 2 moles of electrons to 2 moles of chlorine atoms, to form 2 moles of Cl- ions
Worked example
Using the data below, calculate the ΔHlattθ of potassium chloride, KCl.
| ΔHatθ / kJ mol–1 | IE / EA / kJ mol–1 | |
| K | +90 | +418 |
| Cl | +122 | -349 |
| ΔHfθ / kJ mol–1 | ||
| KCl | -437 | |
Answer:
- Step 1: The corresponding Born-Haber cycle is:
- Step 2: Applying Hess’ law, the lattice energy of KCl is:
- ΔHlattθ = ΔHfθ - ΔH1θ
- ΔHlattθ = ΔHfθ - [(ΔHatθ K) + (ΔHatθ Cl) + (IE1 K) + (EA1 Cl)]
- Step 3: Substitute in the numbers:
- ΔHlattθ = (-437) - [(+90) + (+122) + (+418) + (-349)] = –718 kJ mol-1–1
Worked example
Using the data below, calculate the ΔHlattθ of magnesium oxide, MgO.
| ΔHatθ / kJ mol–1 | IE1 / EA1 / kJ mol–1 | IE2 / EA2 / kJ mol–1 | |
| Mg | +148 | +736 | +1450 |
| O | +248 | –142 | +770 |
| ΔHfθ / kJ mol–1 | |||
| MgO | –602 | ||
Answer:
- Step 1: The corresponding Born-Haber cycle is:
- Step 2: Applying Hess’ law, the lattice energy of MgO is:
- ΔHlattθ = ΔHfθ - ΔH1θ
- ΔHlattθ = ΔHfθ - [(ΔHatθ Mg) + (ΔHatθ O) + (IE1 Mg) + (IE2 Mg) + (EA1 O) + (EA2 O)]
- Step 3: Substitute in the numbers:
- ΔHlattθ = (-602) - [(+148) + (+248) + (+736) + (+1450) + (-142) + (+770)]
Examiner Tip
Make sure you use brackets when carrying out calculations using Born-Haber cycles as you may forget a +/- sign which will affect your final answer!
