5.1.2 Born-Haber Cycles

Overview

The Born-Haber cycle is an application of Hess's Law, which states that the enthalpy change of a reaction is independent of the path taken. This approach is particularly useful for calculating lattice enthalpies, which cannot be measured directly.

A Born-Haber cycle allows us to indirectly determine the lattice enthalpy of an ionic compound by considering a series of enthalpy changes leading to the formation of the compound from their elements in their standard states.

Key Components of a Born-Haber Cycle

The Born-Haber cycle for an ionic compound includes several enthalpy changes:

Enthalpy of Formation ( $ΔfH°$ ) – Enthalpy change when one mole of an ionic compound is formed from its elements in their standard states.
Enthalpy of Atomisation ( $ΔatH°$ ) – Enthalpy change when one mole of gaseous atoms forms from the element in its standard state.
Ionisation Enthalpy ( $ΔieH°$ ) – Enthalpy change when electrons are removed from one mole of gaseous atoms to form ions.
Electron Affinity ( $ΔeaH°$ ) – Enthalpy change when electrons are added to one mole of gaseous atoms to form anions.
Lattice Enthalpy ( $ΔlattH°$ ) – Enthalpy change when one mole of an ionic compound is formed from its gaseous ions. This can be calculated indirectly using the Born-Haber cycle.

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Example: Born-Haber Cycle for Sodium Chloride ( $NaCl$ ) To illustrate, let's apply these steps in constructing a Born-Haber cycle for $NaCl$

Step 1: Atomisation of Sodium ( $ΔatH°$ )

Na(s) → Na(g)

Step 2: Atomisation of Chlorine ( $ΔatH°$ )

\frac{1}{2}Cl₂(g) → Cl(g)

Step 3: Ionisation of Sodium ( $ΔieH°$ )

Na(g) → Na⁺(g) + e⁻

Step 4: Electron Affinity of Chlorine ( $ΔeaH°$ )

Cl(g) + e⁻ → Cl⁻(g)

Step 5: Formation of $NaCl$ ( $ΔfH°$ )

Na(s) + \frac{1}{2}Cl₂(g) → NaCl(s)

Step 6: Lattice Enthalpy ( $ΔlattH°$ )

Na⁺(g) + Cl⁻(g) → NaCl(s)

Note:

Each step is represented by an enthalpy change in the cycle, and Hess's Law allows for the calculation of the lattice enthalpy when these enthalpies are summed.

Born-Haber Cycles with Group 2 Elements

Born-Haber cycles for compounds containing Group 2 elements, such as magnesium chloride ( $MgCl₂$ ), require additional steps compared to Group 1 compounds due to differences in ion charges.

Second Ionisation Energy of Mg

Since Mg forms $Mg²⁺$ ions, the second ionisation energy must be considered:

\text{Mg}^+ \rightarrow \text{Mg}^{2+} + e^-

Atomisation of Chlorine and Electron Affinity

Because two moles of $Cl⁻$ ions are formed for each mole of $Mg²⁺$ , both the atomisation of chlorine and the electron affinity of chlorine are multiplied by 2:

Atomisation of Cl

\text{Cl}_2(g) \rightarrow 2\text{Cl}(g)

Electron Affinity

2 \times (\text{Cl}(g) + e^- \rightarrow \text{Cl}^-(g))

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Summary of Steps in a Born-Haber Cycle

Atomisation: Converts elements to gaseous atoms.
Ionisation: Converts gaseous atoms to positive ions.
Electron Affinity: Converts gaseous atoms to negative ions.
Formation and Lattice Enthalpy: Combines ions to form the ionic lattice.

Born-Haber Cycles Simplified Revision Notes for A-Level AQA Chemistry