5.1.3 Born-Haber Calculations

Overview

Born-Haber calculations are used to determine the lattice enthalpy of ionic compounds, which cannot be measured directly. By applying Hess's Law in a Born-Haber cycle, we can calculate lattice enthalpy using known enthalpy values for other steps in the formation process of an ionic compound.

Required Data for Born-Haber Cycles

A typical Born-Haber cycle calculation involves the following thermodynamic data:

• Enthalpy of Formation ($ΔfH⁰$): Energy change when 1 mole of an ionic compound forms from its elements in their standard states.
• Ionisation Energy ($ΔiH⁰$): Energy required to remove electrons from gaseous atoms to form positive ions.
• Enthalpy of Atomisation ($ΔatH⁰$): Energy required to convert elements from their standard states to gaseous atoms.
• Bond Enthalpy ($ΔbondH⁰$): Energy required to break bonds in diatomic molecules, producing gaseous atoms.
• Electron Affinity ($ΔeaH⁰$): Energy released when electrons are added to gaseous atoms to form negative ions.

Steps for Calculating Lattice Enthalpy in a Born-Haber Cycle

To determine lattice enthalpy ($\Delta H_{latt}$) for a compound, like $NaCl$, the following steps are followed:

Step 1: Write the Enthalpy of Formation for the Compound

This step represents the enthalpy change when 1 mole of $NaCl(s)$ forms from sodium and chlorine in their standard states.

$$\text{Na(s) + } \frac{1}{2}\text{Cl}_2(g) \rightarrow \text{NaCl(s)}$$

Step 2: Atomisation of Elements

Atomisation of Sodium: Convert solid sodium ($Na$) to gaseous atoms.

$$\text{Na(s) → Na(g)}$$

Atomisation of Chlorine: Convert diatomic chlorine gas ($\text{Cl}_2$) to individual chlorine atoms.

$$\frac{1}{2} \text{Cl}_2(g) → \text{Cl(g)}$$

Step 3: Ionisation Energy of Sodium

Calculate the energy needed to remove an electron from $Na(g)$ to form $Na⁺(g)$

$$\text{Na(g) → Na}^+ \text{(g) + e}^-$$

Step 4: Electron Affinity of Chlorine

Calculate the energy change when an electron is added to $Cl(g)$ to form $Cl⁻(g)$

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

Step 5: Lattice Enthalpy Calculation

The lattice enthalpy ($ΔH₆$) can then be calculated by applying Hess's Law using the formula:

$$\Delta H_{latt} = \Delta H_{formation} - (\Delta H_{atomisation} + \Delta H_{ionisation} + \Delta H_{electron affinity})$$

Example Calculation for $NaCl$

Important Notes on Born-Haber Calculations

• The sign convention is important: exothermic values are negative, and endothermic values are positive.
• These calculations rely on accurate data and apply Hess's Law to ensure all enthalpy changes balance around the cycle.