Q.Construct a Born-Haber cycle for $\text{NaCl}(s)$ and calculate its lattice enthalpy from the following data: enthalpy of sublimation of $\text{Na}(s) = +108.4\ \text{kJ mol}^{-1}$; first ionization enthalpy of $\text{Na}(g) = +496\ \text{kJ mol}^{-1}$; bond dissociation enthalpy of $\text{Cl}_2(g) = +242\ \text{kJ mol}^{-1}$; electron gain enthalpy of $\text{Cl}(g) = -349\ \text{kJ mol}^{-1}$; standard enthalpy of formation of $\text{NaCl}(s) = -411.2\ \text{kJ mol}^{-1}$.

The Born-Haber cycle, named after Max Born and Fritz Haber, is a specialized application of Hess's law that decomposes the formation of an ionic solid from its elements into a sequence of individually measurable enthalpy steps, in order to calculate lattice enthalpy — the energy released forming (or, equivalently, needed to separate) one mole of ionic solid from its gaseous ions — a quantity that cannot be measured directly, since gaseous ions cannot practically be combined and observed forming a crystal under controlled calorimetric conditions.

For a general ionic solid $\text{MX}(s)$, the cycle links the elements $\text{M}(s)$ and $\text{X}_2(g)$ to the compound $\text{MX}(s)$ through a closed thermochemical loop of five conceptual steps: sublimation of the solid metal to gaseous atoms; ionization of the gaseous metal atoms to gaseous cations; dissociation of the gaseous non-metal molecule into atoms; electron gain by the gaseous non-metal atoms to form gaseous anions; and finally lattice formation, combining the gaseous ions into the solid ionic crystal.

Because enthalpy is a state function, the sum of these five step-enthalpies must equal the directly measurable standard enthalpy of formation of the compound, $\Delta_fH^\circ$:

$\Delta_fH^\circ = \Delta_{sub}H + (\text{ionization enthalpy}) + \tfrac{1}{2}(\text{bond dissociation enthalpy}) + (\text{electron gain enthalpy}) + \left[-\Delta_{lattice}H\right]$ …