R1.2.5 Born-Haber Cycle (Higher Level Only)

Application of Hess’s Law: Born-Haber Cycles and Lattice Enthalpy

What is a Born-Haber Cycle?

It connects the energy changes from individual steps involved in forming an ionic lattice from its elements in their standard states. These steps include:

1. Enthalpy of atomization– converting elements into gaseous atoms.
2. Ionization energy– removing electrons from metal atoms to form cations.
3. Electron affinity– adding electrons to non-metal atoms to form anions.
4. Lattice enthalpy– forming the ionic lattice from gaseous ions.

Hess’s Law states that the total enthalpy change for a reaction is the same, regardless of the pathway taken.

This principle allows us to construct the Born-Haber cycle, where the sum of all energy changes equals the enthalpy of formation of the ionic compound.

Key Components of a Born-Haber Cycle

Let’s break down each step of the Born-Haber cycle using sodium chloride $NaCl$ as an example.

1. Enthalpy of Atomization ($\Delta H_{\text{at}}$)

• The enthalpy of atomization is the energy required to convert one mole of an element in its standard state into gaseous atoms.
• For sodium, this involves sublimation:

$$\text{Na(s)}\to \text{Na(g)}\mathrm{\Delta }{H}_{\text{atomization}}=+107{\text{kJ mol}}^{-1}$$

• For chlorine, the process involves bond dissociation, as chlorine exists as diatomic molecules:

$$\frac{1}{2}\text{Cl}2(g)\to \text{Cl(g)}\mathrm{\Delta }H\text{dissociation}=+122{\text{kJ mol}}^{-1}$$

2. Ionization Energy (IE)

• Ionization energy is the energy required to remove an electron from a gaseous atom to form a cation. For sodium:

$$\text{Na(g)}\to {\text{Na}}^{+}(g)+e^{-}\text{IE}=+496{\text{kJ mol}}^{-1}$$

3. Electron Affinity (EA)

• Electron affinity is the energy change when an electron is added to a gaseous atom to form an anion. For chlorine:

$$\text{Cl(g)}+e^{-}\to {\text{Cl}}^{-}(g)\text{EA}=-349{\text{kJ mol}}^{-1}$$

4.Lattice Enthalpy (ΔH_\text{lat}$)

• Lattice enthalpy is the energy required to separate one mole of an ionic solid into its gaseous ions. For NaCl:

$$\text{NaCl(s)}\to {\text{Na}}^{+}(g)+{\text{Cl}}^{-}(g)\mathrm{\Delta }{H}_{\text{lattice}}=?$$

Constructing the Born-Haber Cycle for NaCl

1. The Born-Haber cycle combines these steps with the enthalpy of formation $\Delta H_x$ of NaCl.
2. The enthalpy of formation is the energy change when one mole of NaCl is formed from its elements in their standard states:$$\text{Na(s)}+\frac{1}{2}\text{Cl}2(g)\to \text{NaCl(s)}\mathrm{\Delta }H\text{formation}=-411{\text{kJ mol}}^{-1}$$
3. Using Hess’s Law, we calculate the lattice enthalpy by summing the other energy changes in the cycle: $$\mathrm{\Delta }{H}_{\text{formation}}=\mathrm{\Delta }{H}_{\text{atomization}}(\text{Na})+\mathrm{\Delta }{H}_{\text{dissociation}}(\text{Cl}2)+\text{IE}(\text{Na})+\text{EA}(\text{Cl})+\mathrm{\Delta }H\text{lattice}$$
4. Rearranging to solve for lattice enthalpy: $$\mathrm{\Delta }{H}_{\text{lattice}}=\mathrm{\Delta }{H}_{\text{formation}}-[\mathrm{\Delta }{H}_{\text{atomization}}(\text{Na})+\mathrm{\Delta }{H}_{\text{dissociation}}({\text{Cl}}_2)+\text{IE}(\text{Na})+\text{EA}(\text{Cl})]$$
5. Substitute the values: $$\mathrm{\Delta }{H}_{\text{lattice}}=-411-[107+122+496-349]$$ $$\mathrm{\Delta }{H}_{\text{lattice}}=-411-376=+787{\text{kJ mol}}^{-1}$$

Thus, the lattice enthalpy of NaCl is $+787{\text{kJ mol}}^{-1}$.

Interpreting the Born-Haber Cycle

The Born-Haber cycle provides critical insights into ionic compounds:

1. Magnitude of Lattice Enthalpy: A higher lattice enthalpy indicates stronger ionic bonds, leading to higher melting points and lower solubility in water.
2. Stability of Ionic Compounds: A negative enthalpy of formation suggests that the compound is thermodynamically stable.
3. Trends in Ionic Size and Charge: Smaller ions with higher charges result in stronger electrostatic forces, increasing lattice enthalpy.

Common Mistakes in Born-Haber Cycles

Understanding the Born-Haber cycle requires attention to detail. Here are common errors to avoid:

Reflection and Broader Implications