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

ΔH_f^??= ΔH_at^??+ ΔH_at^??+?IE?+?EA?+ ΔH_latt^?

• If we simplify this into three terms, this makes the equation easier to see:
  • ΔH_latt^?
  • ΔH_f^?
  • ΔH₁^??(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??= ΔH₁??+ ΔH_latt?

So, if we rearrange to calculate the lattice energy, the equation becomesΔH_latt^??= ΔH_f^??- ΔH₁^?

• When calculating the ΔHlatt?, 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: Calculating the lattice energy of KCl

Answer

• Step 1:?The corresponding Born-Haber cycle is:

• Step 2:?Applying Hess’ law, the lattice energy of KCl is:

ΔH_latt^??= ΔH_f^??- ΔH₁^?

ΔH_latt^??= ΔH_f^??- [(ΔH_at^??K) + (ΔH_at^??Cl) + (IE₁?K) + (EA₁?Cl)]

• Step 3:?Substitute in the numbers:

ΔH_latt^??= (-437) - [(+90) + (+122) + (+418) + (-349)]?= -718 kJ mol^-1

Worked example: Calculating the lattice energy of MgO

Answer

• Step 1:?The corresponding Born-Haber cycle is:

• Step 2:?Applying Hess’ law, the lattice energy of MgO is:

ΔH_latt^??= ΔH_f^??- ΔH₁^?

ΔH_latt^??= ΔH_f^??- [(ΔH_at^??Mg) + (ΔH_at^??O) + (IE₁?Mg) + (IE₂?Mg) + (EA₁?O) + (EA₂?O)]

• Step 3:?Substitute in the numbers:

ΔH_latt^??= (-602) - [(+148) + (+248) + (+736) + (+1450) + (-142) + (+770)]= -3812 kJ mol^-1