Born-Haber cycle is a process that allows one to understand and analyze energies in a chemical reaction. It is concerned with forming a stable ionic compound from reacting an alkali or alkaline earth metal with a nonmetal like a halogen or oxygen. The process enables one to understand the overall reaction progress through steps by visually representing the energies involved in these steps ^{[1 – 4]}.

The Born-Haber cycle is named after two German scientists, Max Born and Fritz Haber, who developed it in 1919.

Born-Haber Cycle and Lattice Energy

Born-Haber cycle is used primarily to calculate lattice energy, which cannot typically be measured directly. Lattice energy is the energy released when one mole of an ionic compound is formed from its constituents in their gaseous state. It is due to the electrostatic force of attraction and is responsible for keeping the anions and cations of the compound in fixed positions ^{[1 – 4]}.

Born-Haber cycle uses Hess’s Law to find the lattice energy exclusively. The formation of a solid ionic compound from the elemental state of the constituent atoms goes through several steps and involves enthalpy in each step. Hess’ law states that the enthalpy change in a chemical process is independent of the path the process takes. When summed over the individual steps, the total enthalpy change over a closed cycle is zero.

How to Use Born Haber Cycle

The Born-Haber cycle takes advantage of the enthalpy change to determine the lattice energy of ionic compounds indirectly. It utilizes the known thermodynamic quantities like ionization energy and electron affinity. Let us understand how to use Born-Haber cycle to determine the lattice energy of sodium chloride (NaCl) ^{[1 – 4]}.

Heat of Formation

Formation involves the reaction between the elements in their standard states to form the ionic compound. The heat of formation is the enthalpy change when forming a compound from its elements. Usually, this value is negative as ionic compounds are stable. It also indicates that the process of formation is an exothermic one.

The heat of formation (ΔH_f⁰), also known as enthalpy of formation, is 411 kJ/mole for sodium chloride.

Na (s) + ½ Cl₂ (g) → NaCl (s)                        ΔH_f⁰ = -411 kJ/mol

The formation of ionic sodium chloride from solid sodium metal and gaseous chlorine is not a single-step process. Instead, it goes through several steps. Energy changes in each step except the lattice energy can be experimentally measured. These steps are given below.

Step 1: Sublimation Energy and Disassociation Energy

The elements involved in the reaction should be in their gaseous forms. Atomization is the formation of gaseous atoms from their naturally occurring elements. Metals in nature occur as single atoms. On the other hand, nonmetals exist in polyatomic form. Hence, energy is added to nonmetals to disassociate into atoms.

1. A solid sodium atom undergoes sublimation to form a gaseous atom by absorbing energy. Sublimation energy is the energy required to change the phase of an element from solid to gas. The sublimation energy (ΔH_sub) of a solid sodium atom is 107 kJ/mol.

Na (s) → Na (g)                 ΔH_sub = +107 kJ/mol

2. Diatomic gaseous chlorine breaks into two individual atoms by absorbing energy. Dissociation energy is the energy required to break apart a compound. The chlorine molecule’s disassociation energy (ΔH_bond) is 242 kJ/mol. Each chlorine atom absorbs half of this bond energy.

½ Cl₂ (g) → Cl (g)                             ½ ΔH_bond = +121 kJ/mol

Step 2: Ionization Energy and Electron Affinity

Ionization involves the removal of electrons from the gaseous metal to form an ion. In order to accomplish this, energy must be added to the gaseous atom. Ionization energy is the energy required to remove an electron from a neutral atom or ion. On the other hand, energy is released by a gaseous nonmetal atom when electrons are added to form an ion. Electron affinity is the energy released when an electron is added to a neutral atom or an ion.

1. A gaseous sodium atom absorbs 502 kJ/mol of energy (ΔH_IE) and releases one electron to form a gaseous sodium ion.

Na (g) → Na⁺ (g) + e^–                       ΔH_IE = +502 kJ/mol

2. Each gaseous chlorine atom releases 355 kJ/mol of energy (ΔH_EA) and absorbs one electron to form a gaseous chlorine ion.

Cl (g) + e^– → Cl^– (g)                          ΔH_EA = -355 kJ/mol

Step 3: Lattice Energy

The last step is forming the ionic compound from its constituent gaseous ions. The gaseous sodium ion and gaseous chloride ion combine to form a solid sodium chloride molecule and release energy (exothermic) equivalent to the lattice energy (ΔH_LE), which we shall determine.

Na⁺ (g) + Cl^–(g) → NaCl (s)                            ΔH_LE = ?

Determining Lattice Energy from Hess’ Law

From Hess’s Law, the overall change in enthalpy of a process can be determined by breaking the process down into steps and then adding the enthalpy changes of each step. So, summing the enthalpy change of all the steps from 1 to 3 gives the enthalpy of formation of solid crystalline sodium chloride from sodium and chlorine in their natural forms of solid and gas, respectively. It should be equal to the experimentally measured enthalpy of formation of solid sodium chloride.

Equation

ΔH_f⁰ = ΔH_sub + ½ ΔH_bond + ΔH_IE + ΔH_EA + ΔH_LE

Or, ΔH_f⁰ – (ΔH_sub + ½ ΔH_bond + ΔH_IE + ΔH_EA + ΔH_LE) = 0 ( => ΔH_cycle = 0)

Or, -411 kJ/mol – (107 kJ/mol + 121 kJ/mol + 502 kJ/mol – 355 kJ/mol + ΔH_LE) = 0

Or, -411 kJ/mol – 375 kJ/mol – ΔH_LE = 0

Or, ΔH_LE = -786 kJ/mol

How to Draw Born Haber Cycle ^[1,5]

1. Draw a vertical axis denoting the enthalpy axis.
2. Draw a horizontal line denoting the energy level of the ionic compound.
3. Construct and connect the various energy levels involved in the abovementioned steps with increasing energies.
4. First, construct and connect the levels that require energy on the left.
5. Then, construct and connect the levels that release energy on the right.
6. This diagram will lead to a closed-loop

Using the method described above, let us look at the Born-Haber cycle diagram for sodium chloride.

In this way, the Born-Haber cycle can be constructed for other ionic compounds like lithium fluoride (LiF), calcium fluoride (CaF₂), potassium chloride (KCl), magnesium chloride (MgCl₂), and magnesium oxide (MgO).