5.3.2 Born-Haber Cycles

• A Born-Haber cycle is a specific application of Hess’ Law for ionic compounds and enable us to calculate lattice enthalpy which cannot be found by experiment
• The basic principle of drawing the cycle is to construct a diagram in which energy increases going up the diagram

The basic principle of a Born-Haber cycle

• The cycle shows all the steps needed to turn atoms into gaseous ions and from gaseous ions into the ionic lattice
• The alternative route to the ionic lattice begins from the enthalpy of formation of the elements in their standard states

Drawing the cycle for sodium chloride

• A good starting point is to draw the elements with their state symbols about a third of the way up the diagram
• This is shown as the left hand side of the equation for the process indicated
• The location is marked by drawing a horizontal bar or line which represents the starting energy level

Drawing a Born-Haber cycle step 1

• Next, we need to create the gaseous ions
• This is a two step process of first creating the gaseous atoms and then turning them into ions
• Creating gaseous atoms is a bond breaking process, so arrows must be drawn upwards
• It doesn't matter whether you start with sodium or chlorine
• The enthalpy of atomisation of sodium is

Na (s) → Na (g)           ΔH_at^ꝋ = +108 kJ mol ^-1

• The enthalpy of atomisation of chlorine is

½Cl₂ (g) → Cl (g)       ΔH_at^ꝋ = +121 kJ mol ^-1

• We can show the products of the process on the horizontal lines and the energy value against a vertical arrow connecting the energy levels

Drawing a Born-Haber cycle step 2 - creating the gaseous atoms

• Now the ions are created
• The sodium ion loses an electron, so this energy change is the first ionisation energy for sodium

Na (g) → Na⁺ (g) + e^–          ΔH_ie^ꝋ = +500 kJ mol^-1

• The change is endothermic so the direction continues upwards
• The chlorine atom gains an electron, so this is electron affinity

Cl (g) + e^– → Cl^- (g)           ΔH_ea^ꝋ = -364 kJ mol^-1

• The exothermic change means this is downwards
• The change is displaced to the right to make the diagram easier to read

Drawing a Born-Haber cycle step 3 - creating the gaseous ions

• The two remaining parts of the cycle can now be completed
• The enthalpy of formation of sodium chloride is added at the bottom of the diagram

Na (s) + ½Cl₂ (g) → NaCl (s)            ΔH_f^ꝋ = -411 kJ mol ^-1

• This is an exothermic change for sodium chloride so the arrow points downwards
• Enthalpy of formation can be exothermic or endothermic, so you may need to show it above the elements ( and displaced to the right) for a endothermic change
• The final change is lattice enthalpy, which is usually shown a formation. For sodium chloride the equation is

Na⁺(g) + Cl^-(g) → NaCl (s)  ΔH_latt^ꝋ 

 

Drawing a Born-Haber cycle step 4 - completing the cycle

• The cycle is now complete
• The cycle is usually used to calculate the lattice enthalpy of an ionic solid, but can be used to find other enthalpy changes if you are given the lattice enthalpy

Answer

Answer

• 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^ꝋ = ΔH_at^ꝋ + ΔH_at^ꝋ + IE + EA + ΔH_latt^ꝋ

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

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

ΔH_latt^ꝋ = ΔH_f^ꝋ - ΔH₁^ꝋ

• When calculating the ΔH_latt^ꝋ, 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

Answer

Step 1: The corresponding Born-Haber cycle is:

 

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

ΔH_latt^ꝋ = ΔH_f^ꝋ - ΔH₁^ꝋ

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

Step 3: Substitute in the numbers:

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

Answer

Step 1: The corresponding Born-Haber cycle is:

 

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

ΔH_latt^ꝋ = ΔH_f^ꝋ - ΔH₁^ꝋ

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

Step 3: Substitute in the numbers:

ΔH_latt^ꝋ = (-602) - [(+148) + (+248) + (+736) + (+1450) + (-142) + (+770)]

= -3812 kJ mol^-1