Born-Haber Cycles (HL) (DP IB Chemistry): Revision Note

Born-Haber cycles

• A Born–Haber cycle is a specialised application of Hess’s Law used to analyse the formation of ionic compounds

• It allows us to determine the lattice enthalpy, a value that cannot be measured directly through experimentation

• When constructing a Born–Haber diagram, the key idea is to represent energy changes vertically:

  • Endothermic processes go upward (indicating an energy input)

  • Exothermic processes go downward (indicating energy release)

The basic principle of a Born-Haber cycle

Key definitions

Lattice enthalpy

• The lattice enthalpy, ΔH^Ꝋ_latt , is the energy change when one mole of an ionic compound is separated into its component gaseous ions

• This is an endothermic process so is a positive value

  • The arrow will therefore point upwards on a Born-Haber cycle 

CaF₂ (s) → Ca²⁺ (g) + 2F^- (g) ΔH^Ꝋ_latt = +2620 kJ mol^-1

First ionisation energy

• The first ionisation energy, ΔH^Ꝋ_ie, is the standard enthalpy change when one mole of gaseous atoms is converted into one mole of gaseous ions each with a single positive charge

Mg (g) →  Mg⁺ (g) + e^- ΔH^Ꝋ_ie1 = +738 kJ mol^-1

• First ionisation requires energy to remove an electron from an atom

• This means that ionisation is an endothermic process

  • The arrow will therefore point upwards on a Born-Haber cycle 

Second ionisation energy

• The second ionisation energy, ΔH^Ꝋ_ie2, is the standard enthalpy change one mole of gaseous 1+ ions is converted into one mole of gaseous 2+ ions

Mg⁺ (g) →  Mg²⁺ (g) + e^- ΔH^Ꝋ_ie2 = +1451 kJ mol^-1

• Second ionisation requires energy to remove an electron from a 1+ ion

• This means that second ionisation is an endothermic process

  • The arrow will therefore point upwards on a Born-Haber cycle 

Atomisation

• The enthalpy of atomisation, ΔH^Ꝋ_at, is the enthalpy change which accompanies the formation of one mole of gaseous atoms from the element in its standard state under standard conditions

Mg (s) → Mg (g) ΔH^Ꝋ_at = +148 kJ mol^-1

• Atomisation requires energy to change the physical state of an element

• This means atomisation is an endothermic process

  • The arrow will therefore point upwards on a Born-Haber cycle 

First electron affinity

• The first electron affinity, ΔH^Ꝋ_ea1, is the standard enthalpy change when one mole of gaseous atoms is converted to one mole of gaseous ions, each with a single negative charge

O (g) + e^- →  O^- (g) ΔH^Ꝋ_ea1 = -141 kJ mol^-1

• First electron affinity can require energy, release energy or be energetically neutral

• So, first electron affinity can be:

  • Exothermic - with a negative enthalpy change

  • Endothermic - with a positive enthalpy change

  • Neutral - with no enthalpy change

Second electron affinity

• The second electron affinity, ΔH^Ꝋ_ea2, is the standard enthalpy change when one mole of gaseous 1- ions is converted to one mole of gaseous 2- ions

O^- (g) + e^- → O^2- (g) ΔH^Ꝋ_ea2 = +798 kJ mol^-1

• Second electron affinity requires energy to overcome the repulsion between the negative 1- ion and the negative electron

• This means that second electron affinity is an endothermic process

  • The arrow will therefore point upwards on a Born-Haber cycle 

Drawing the cycle

• The steps for drawing the cycle for sodium chloride, NaCl, are outlined below

Drawing a Born-Haber cycle - Step 1

• Begin by placing the elements in their standard states, complete with state symbols, roughly one-third up the diagram

• This represents the reactants on the left-hand side of the overall formation equation

• Draw a horizontal line at this point to mark the initial energy level in the cycle

Drawing a Born-Haber cycle - Step 2

• Next, we need to create the gaseous ions

• Creating gaseous atoms is bond breaking, which is an endothermic process

  • So, the 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^ꝋ_at = +108 kJ mol^-1

• The enthalpy of atomisation (ΔH^ꝋ_at) of chlorine is

½Cl₂ (g) → Cl (g)       ΔH^ꝋ_at = +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 3

• Now, the gaseous atoms can be turned into ions

• The sodium atom loses an electron, so this energy change is the first ionisation energy (ΔH^ꝋ_IE) for sodium

Na (g) → Na⁺ (g) + e^–          ΔH^ꝋ_ie = +500 kJ mol^-1

• The ionisation of sodium is endothermic

  • So, the arrow is drawn upwards

• The chlorine atom gains an electron, so this is electron affinity (ΔH^ꝋ_EA)

Cl (g) + e^– → Cl^– (g)           ΔH^ꝋ_ea = -364 kJ mol^-1

• The electron affinity of chlorine is exothermic

  • So, the arrow is drawn downwards

  • The change is displaced to the right to make the diagram easier to read

Drawing a Born-Haber cycle - Step 4

• The two remaining parts of the cycle can now be added

  • Enthalpy of formation , (ΔH^ꝋ_f)

  • Lattice enthalpy, (ΔH^ꝋ_latt )

• The enthalpy of formation (ΔH^ꝋ_f) of sodium chloride is added at the bottom left of the diagram

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

• This is an exothermic change for sodium chloride

  • So, the arrow is drawn 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 an endothermic change

• The final change is lattice enthalpy (ΔH^ꝋ_latt ), which is shown as the change from solid to gaseous ions

NaCl (s) → Na⁺(g) + Cl^–(g)    ΔH^ꝋ_latt 

• 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