Calculations using Born-Haber Cycles (Cambridge International A-Level Chemistry)

Born-Haber Cycles Calculations

Calculations Using Born-Haber Cycles

• Once a Born-Haber cycle has been constructed, it is possible to calculate the lattice energy (ΔH_latt^ꝋ) by applying Hess’s law and rearranging:

ΔH _f^θ = ΔH_at^θ + ΔH _at^θ + IE + EA + ΔH_latt^θ

If we simplify this into three terms, this makes the equation easier to see:

• ΔH_latt^θ
• ΔH_f^θ
• ΔH₁^θ (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

ΔH _f^θ = ΔH₁^θ + ΔH_latt^θ

So, if we rearrange to calculate the lattice energy, the equation becomes

ΔH _latt^θ = ΔH_f^θ - ΔH₁^θ

• When calculating the ΔH_latt^θ, 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 ΔH_latt^θ of potassium chloride, KCl.

	ΔH atθ / kJ mol–1 **	IE / EA** / kJ mol –1 **
K	+90	+418
Cl	+122	-349
ΔH**fθ / 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:

  • ΔH**_latt^θ** = ΔH_f^θ**- **ΔH₁^θ
  • ΔH_latt^θ = ΔH_f^θ - [(ΔH_at^θ K) + (ΔH_at^θ Cl) + (IE₁ K) + (EA₁ Cl)]
  • Step 3: Substitute in the numbers:
  • ΔH_latt^θ = (-437) - [(+90) + (+122) + (+418) + (-349)] = –718 kJ mol^-1–1

Worked Example Using the data below, calculate the ΔH_latt^{**θ **}of magnesium oxide, MgO.

	ΔH**atθ** / kJ mol –1 **	IE1** / EA 1 / kJ mol –1 **	IE2** / EA 2 / kJ mol –1 **
Mg	+148	+736	+1450
O	+248	–142	+770
ΔH**fθ/ 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:

  • ΔH**_latt^θ_** = ΔH_f^θ_****- ΔH ₁^θ **
  • ΔH**_latt^θ = ΔH _f^θ - [(ΔH_at^θ Mg) + (ΔH _at^θ O) + (IE₁ Mg) + (IE ₂ Mg) + (EA ₁ O) + (EA ₂ O)]**
  • **Step 3: **Substitute in the numbers:
  • ΔH**_latt^θ** = (-602) - [(+148) + (+248) + (+736) + (+1450) + (-142) + (+770)]**

Exam Hint 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!