Q.The lattice enthalpy of an ionic compound is the enthalpy when one mole of an ionic compound present in its gaseous state, dissociates into its ions. It is impossible to determine it directly by experiment. Suggest and explain an indirect method to measure lattice enthalpy of NaCl(s).

Concept: Born–Haber cycle – an application of Hess's law that connects lattice enthalpy to experimentally measurable thermochemical quantities.

Lattice enthalpy cannot be measured directly because we cannot convert solid NaCl into isolated gaseous ions in one step. Instead, we construct a thermochemical cycle linking the formation of NaCl(s) from its elements to the formation of gaseous ions through alternative paths.

The Born–Haber cycle for NaCl:

1. Start with the standard enthalpy of formation: $\text{Na(s)} + \frac{1}{2}\text{Cl}_2\text{(g)} \to \text{NaCl(s)}$, $\Delta_f H^\circ$.

2. Break this into steps via gaseous ions:

   • Sublimation of sodium: $\text{Na(s)} \to \text{Na(g)}$, $\Delta_{\text{sub}}H$
   • Dissociation of chlorine: $\frac{1}{2}\text{Cl}_2\text{(g)} \to \text{Cl(g)}$, $\frac{1}{2}\Delta_{\text{diss}}H$
   • Ionization of sodium: $\text{Na(g)} \to \text{Na}^+\text{(g)} + e^-$, $\Delta_{\text{IE}}H$
   • Electron gain by chlorine: $\text{Cl(g)} + e^- \to \text{Cl}^-\text{(g)}$, $\Delta_{\text{EA}}H$
   • Lattice formation: $\text{Na}^+\text{(g)} + \text{Cl}^-\text{(g)} \to \text{NaCl(s)}$, $-U$ (where $U$ is lattice enthalpy)

3. By Hess's law: …

Why we need an indirect method

Lattice enthalpy is defined as the energy required for the process:

We cannot perform this reaction in a laboratory. Solid NaCl does not simply vaporize into separated ions when heated—it melts, then vaporizes as ion pairs, and eventually dissociates at extremely high temperatures where other decomposition pathways interfere. No calorimeter can isolate this single step cleanly.

The solution lies in recognizing that enthalpy is a state function: the total enthalpy change around any closed cycle is zero. If we construct a cycle connecting the same initial and final states through steps we can measure, we can solve for the one unknown step.

The Born–Haber cycle for NaCl

The cycle connects the formation of solid NaCl from its elements to the formation of gaseous ions through two different routes.

Route 1 (direct): Formation of NaCl(s) from elements in their standard states:

This is the standard enthalpy of formation, measured directly by calorimetry.

Route 2 (stepwise through gaseous ions): Break this into measurable steps:

1. Sublimation of sodium:

Energy needed to convert solid sodium metal to gaseous atoms.

2. Ionization of sodium:

First ionization energy of sodium.

3. Dissociation of chlorine:

Half the bond dissociation energy of $\text{Cl}_2$.

4. Electron affinity of chlorine:

Energy released when chlorine gains an electron (usually negative).

5. Lattice formation (reverse of lattice enthalpy):

This is the negative of what we want to find.

Applying Hess's law

Since enthalpy is a state function, the sum of enthalpies around the cycle equals zero:

Rearranging for lattice enthalpy:

Every term on the right can be measured experimentally:

• $\Delta H_\text{sub}$: heat solid sodium and measure energy absorbed.
• $\Delta H_\text{IE}$: spectroscopic measurement of ionization energy.
• $\Delta H_\text{diss}$: calorimetry or spectroscopy on $\text{Cl}_2$ molecules. …