Reactivity 1.2.5 – The Born–Haber Cycle

What Is a Born–Haber Cycle?

A Born-Haber cycle is a thermochemical cycle based on Hess’s Law, used to analyze the formation of ionic compounds from their elements in standard states. It is particularly useful in calculating and understanding lattice enthalpies and electron affinity values, which cannot be measured directly in most cases.

Born–Haber cycles show the stepwise conversion of elements into an ionic solid, breaking down the formation process into distinct enthalpy changes. These individual steps follow Hess’s Law and allow indirect determination of otherwise inaccessible quantities.

Purpose and Application

• To determine the lattice enthalpy of an ionic compound.
• To verify or calculate the enthalpy of formation or electron affinity using other known data.
• To understand the relative contributions of different energy steps in forming ionic solids.
• To apply Hess’s Law to multistep ionic processes.

Key Enthalpy Changes in a Born–Haber Cycle

The formation of an ionic compound from its elements involves multiple distinct enthalpy changes. These steps reflect the energy required (or released) during the transformation of atoms from standard elemental forms to ions in a solid crystal lattice. Each enthalpy change corresponds to a specific physical or chemical process.

1. Enthalpy of Formation, \( \Delta H_f^\circ \)

• This is the total enthalpy change when 1 mole of an ionic compound is formed from its elements in their standard states under standard conditions (298 K, 100 kPa).
• It is the vertical step in the Born–Haber cycle connecting the elements directly to the ionic solid.

\( \text{Elements (standard state)} \rightarrow \text{Ionic compound (solid)} \)

\( \Delta H_f^\circ = \text{overall energy change of the cycle} \)

2. Enthalpy of Atomization, \( \Delta H_{\text{atom}} \)

• The energy required to convert elements into separate gaseous atoms.
• For metals (e.g., Na, Ca), this includes sublimation:

\( \text{Na}(s) \rightarrow \text{Na}(g) \)

• For nonmetals that are diatomic molecules (e.g., Cl₂, O₂), it includes bond dissociation:

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

• These processes are always endothermic (require energy input).

3. Ionization Energy, \( IE \)

• The amount of energy needed to remove electrons from gaseous atoms to form positive ions (cations).
• If the ion is multivalent (e.g., Mg²⁺, Ca²⁺), multiple ionization steps are needed:

\( \text{Mg}(g) \rightarrow \text{Mg}^+(g) + e^- \quad (IE_1) \)

\( \text{Mg}^+(g) \rightarrow \text{Mg}^{2+}(g) + e^- \quad (IE_2) \)

• Ionization energies are always positive (endothermic).

4. Electron Affinity, \( EA \)

• The enthalpy change when electrons are added to gaseous atoms to form negative ions (anions).
• The first electron affinity is usually exothermic (energy is released):

\( \text{Cl}(g) + e^- \rightarrow \text{Cl}^-(g) \quad EA_1 \)

• However, the second electron affinity (for ions like O²⁻ or S²⁻) is typically endothermic, because adding an electron to a negatively charged ion requires energy:

\( \text{O}^-(g) + e^- \rightarrow \text{O}^{2-}(g) \quad EA_2 \)

5. Lattice Enthalpy, \( \Delta H_{\text{latt}} \)

• The enthalpy change when 1 mole of an ionic solid is formed from its constituent gaseous ions:

\( \text{Na}^+(g) + \text{Cl}^-(g) \rightarrow \text{NaCl}(s) \)

• This value is always exothermic and large in magnitude because strong electrostatic forces hold the ions together in the crystal.
• It reflects the strength of the ionic bond; higher lattice enthalpy means a more stable ionic solid.

Summary Equation Using Hess’s Law:

A full Born–Haber cycle applies Hess’s Law to sum these changes:

\( \Delta H_f^\circ = \Delta H_{\text{atom}} + \sum IE + \sum EA + \Delta H_{\text{latt}} \)

Or, rearranged to solve for lattice enthalpy:

\( \Delta H_{\text{latt}} = \Delta H_f^\circ – (\Delta H_{\text{atom}} + \sum IE + \sum EA) \)

This formula allows calculation of any unknown value, provided the others are known from data sources like the IB Chemistry data booklet.

Energy Cycle Representation

In a Born–Haber cycle diagram:

• The vertical path represents the direct formation of the ionic compound from elements (enthalpy of formation).
• The horizontal and stepwise path involves all the intermediate steps (atomization, ionization, electron affinity, lattice enthalpy).
• Hess’s Law relates the total enthalpy changes around the cycle:

\( \Delta H_f^\circ = \Delta H_{\text{atom}} + IE + EA + \Delta H_{\text{latt}} \)

(Signs may vary depending on conventions used and direction of lattice enthalpy.)

Example Ionic Compound: Sodium Chloride (NaCl)

To understand the formation of NaCl(s), the following steps are considered in the cycle:

• \( \text{Na}(s) \rightarrow \text{Na}(g) \) —  sublimation (atomization)
• \( \text{Na}(g) \rightarrow \text{Na}^+(g) + e^- \) —  ionization energy
• \( \frac{1}{2} \text{Cl}_2(g) \rightarrow \text{Cl}(g) \) —  bond dissociation (atomization)
• \( \text{Cl}(g) + e^- \rightarrow \text{Cl}^-(g) \) —  electron affinity
• \( \text{Na}^+(g) + \text{Cl}^-(g) \rightarrow \text{NaCl}(s) \) —  lattice enthalpy

Interpreting and Determining Values from a Born–Haber Cycle

Using the Born–Haber Cycle to Calculate Unknown Values

A Born–Haber cycle uses known enthalpy values to determine unknown steps, such as:

• Lattice enthalpy (\( \Delta H_{\text{latt}} \))
• Electron affinity
• Enthalpy of formation (if not directly known)

To interpret the cycle, apply Hess’s Law by summing all enthalpy changes around the cycle. The sign and order of each term depends on the direction of energy flow and the definition of lattice enthalpy used (formation vs. dissociation).

Steps to Follow

1. List all known enthalpy changes provided in the cycle (atomization, ionization, electron affinity, etc.).
2. Insert numerical values with the correct signs and units (kJ/mol).
3. Use the cycle to solve for the unknown value using:

\( \Delta H_f^\circ = \sum \text{steps to form gaseous ions} + \Delta H_{\text{latt}} \)

1. Rearrange the equation if solving for lattice enthalpy or another unknown.