5.1.3 Constructing Born-Haber Cycles

• A Born-Haber cycle is a specific application of Hess's 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 direction of the arrows in Born-Haber cycles indicates if a reaction is exothermic or endothermic

• 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^θ = +108 kJ mol ^-1

• The enthalpy of atomisation of chlorine is

½Cl₂ (g) → Cl (g)       ΔH_at^θ = +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^θ = +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^θ = -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^θ = -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^θ 

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