Born Haber Cycle

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Based on Hess's law Born and Haber in 1919 developed a simplified and cyclic method to correlate lattice energies of ionic crystals to other thermodynamic data. For example, the net energy changes during the formation of sodium chloride from metallic sodium and chlorine gas can be represented by DH_f·

The overall process is thought to take place in following intermediate steps:

Intermediate steps

(i) Metallic sodium into gaseous sodium atom

The energy required per mole of sodium is 'enthalpy of sublimation' which is represented by (Hs). This step is energy consuming process.

(ii) Dissociation of chlorine molecule into chlorine atoms

The energy required per mole of chlorine is 'enthalpy of dissociation' represented by (_d)

(iii) Gaseous sodium atom into gaseous cation

The energy required in this process is called Ionization energy (IE).

(iv) Gaseous chlorine atom into gaseous anion

This step involves the release of energy referred as Electron Affinity (EA).

(v) Combination of oppositely charged gaseous ions to form solid crystal

This involves the release of energy referred as lattice energy (U).

The various energy changes in different steps are as shown:

Born Haber Cycle for NaCl

The sum of the energy changes taking place during various steps is equal to DH_f i.e., heat of formation of NaCl_(s) according to Hess' Law.

Various values for NaCl are as follows:

IE of sodium (IE) = 495.8 kJ mol^-1

EA of chlorine (EA) = -349 kJ mol^-1

Lattice energy of NaCl (U) = -769.8 kJ mol^-1

Substituting these values in equation (v) we get

= 393.0 kJ mol^-1

Applications of Born Haber Cycle

Lattice energy of ionic solids

Born Haber Cycle helps us to calculate the lattice energy of ionic solid, provided other thermodynamic data is known. For example, the lattice energy of magnesium fluoride (MgF₂) can be calculated when the sublimation energy (S) of Mg = 146.4 kJ mol^-1; IE₁ and IE₂ values of Mg=737 and 1449 kJ mol ^-1 respectively; Dissociation energy (D) of fluorine = 158.8 kJ mol ^-1; EA of fluorine = - 328 kJ mol ^-1 and DH_f of MgF₂ =- 1096.5 kJ mol ¹. Born Haber Cycle for MgF₂ is as shown:

Born Haber Cycle for MgF₂

DH_f = DH_s + DH_d + (IE₁ + IE₂) + 2 EA₁+ U

U = DH_f - DH_s - DH_d - (IE₁ + IE₂) - 2 EA₁

= -1096.5 - 146.4 - 158.8 - (737 + 1449) - 2(-328)

= -2931.7 kJmol^-1

Born Haber cycle can help us to calculate the values of DH_f for unknown compounds. From the calculated values of DH_f one can predict whether the compound is stable or not. If DH_f value is negative, the compound is stable. If DH_f is positive, the formation of compound is highly unfavourable. For example, to calculate the value of DH_f for hypothetical compound ArCl the data given is: IE₁ for Ar = 526.3 kJ mol^-1, Dissociation energy of chlorine (D) = 243 kJ mol^-1; EA of chlorine is - 349 kJ mol^-1 ; Lattice energy (U) of ArCl_(s) is - 703 kJ mol ^-1. In the cycle below:

Born Haber cycle for ArCl_(s)

The + ve value of DH_f indicates that net energy is required for this process. Hence, formation ArCl is energetically unfavourable.

Electron affinities

The Born Haber cycle can be used for the calculation of electron affinities of some elements that are otherwise very difficult to measure. Heat of formation of a compound may be expressed as:

S, DH_f, D, and IE are experimentally determined and lattice energy, U may be calculated by using other equation (Born Lande equation). Using the above equation, electron affinity may be calculated.

Proton affinities

The Born Haber cycle can also be used to calculate the proton affinities (PA) of some bases. The proton affinity of a species X is defined as the energy released in the reaction:

To calculate the proton affinity of NH₃ using Born Haber Cycle for the process,

PA = DH - DH_d - IE - EA - U ...(i)

The thermodynamic data is as:

DH = - 144.5 kJ mol^-1; IE = 1312 kJ mol^-1; EA = -349.0 kJ mol^-1

DH_d = 433.0 kJ mol^-1; U = -649.0 kJ mol^-1

Substituting the values,

PA = - 144.5 - (433.0) - (1313.0) - (-349.0) - (-649.0) = -891.5 kJ mol^-1