Question

Use the following to calculate \(\Delta H_{\text {latice }}^{\circ}\) of \(\mathrm{NaCl}:\) \(\begin{array}{ll}\mathrm{Na}(s) \longrightarrow \mathrm{Na}(g) & \Delta H^{\circ}=109 \mathrm{~kJ} \\ \mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{Cl}(g) & \Delta H^{\circ}=243 \mathrm{~kJ} \\ \mathrm{Na}(g) \longrightarrow \mathrm{Na}^{+}(g)+\mathrm{e}^{-} & \Delta H^{\circ}=496 \mathrm{~kJ} \\ \mathrm{Cl}(g)+\mathrm{e}^{-} \longrightarrow \mathrm{Cl}^{-}(g) & \Delta H^{\circ}=-349 \mathrm{~kJ} \\\ \mathrm{Na}(s)+\frac{1}{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{NaCl}(s) & \Delta H^{\circ}=-411 \mathrm{~kJ}\end{array}\) Compared with the lattice energy of LiF (1050 kJ/mol), is the magnitude of the value for \(\mathrm{NaCl}\) what you expected? Explain.

Short answer

The lattice energy of NaCl is \(-788.5\, \text{kJ/mol}\). This is less than the 1050 kJ/mol of LiF, as expected, due to the smaller ionic radius of Li+ compared to Na+.

Step-by-step solution

Sublimation of Sodium

Convert solid sodium (Na) to gaseous sodium (Na). This step involves the sublimation of sodium. The enthalpy for this process is \( \Delta H^{\circ} = 109 \, \text{kJ} \).

Dissociation of Chlorine Molecule

Convert one mole of chlorine gas (\( Cl_2 \)) into two moles of chlorine atoms (Cl). This step involves the dissociation of chlorine molecules. The enthalpy for this process is \( \Delta H^{\circ} = 243 \, \text{kJ} \).

Ionization of Sodium

Convert gaseous sodium (Na) into sodium ions (\( Na^+ \)) by removing an electron. This step involves the first ionization energy of sodium. The enthalpy for this process is \( \Delta H^{\circ} = 496 \, \text{kJ} \).

Electron Affinity of Chlorine

Convert chlorine atoms (Cl) into chloride ions (\( Cl^-\)) by adding an electron. This step involves the electron affinity of chlorine. The enthalpy for this process is \( \Delta H^{\circ} = -349 \, \text{kJ} \).

Formation of Sodium Chloride

Form solid sodium chloride (\(NaCl\)) from sodium and chlorine gas. This step involves the formation enthalpy of sodium chloride. The enthalpy for this process is \( \Delta H^{\circ} = -411 \, \text{kJ} \).

Hess's Law Application

Use Hess's Law to find the lattice enthalpy \( \Delta H_{\text{lattice}}^{\circ} \). Combine the enthalpies calculated in the previous steps: \[ \Delta H_{\text{lattice}}^{\circ} = \Delta H_{\text{formation}}^{\circ} - (\Delta H_{\text{sublimation}}^{\circ} + \frac{1}{2}\Delta H_{\text{dissociation}}^{\circ} + \Delta H_{\text{ionization}}^{\circ} + \Delta H_{\text{electron affinity}}^{\circ}) \]Substitute the values: \[ \Delta H_{\text{lattice}}^{\circ} (\text{NaCl}) = -411 - (109 + 0.5(243) + 496 - 349) \]Calculate the enthalpy: \[ \Delta H_{\text{lattice}}^{\circ} (\text{NaCl}) = -411 - (109 + 121.5 + 496 - 349) = -411 - 377.5 = -788.5 \, \text{kJ/mol} \]

Comparison with LiF

The lattice energy of NaCl is calculated to be -788.5 kJ/mol, while the lattice energy of LiF is 1050 kJ/mol. We expect the lattice energy of NaCl to be less than that of LiF because the Li+ ion is smaller than the Na+ ion, which results in a stronger attraction between Li+ and F- ions in LiF compared to Na+ and Cl- ions in NaCl.

Key concepts

Hess's Law

Hess's Law is a fundamental principle in chemistry that states the total enthalpy change of a reaction is the same, regardless of the pathway it takes. This is important because it allows us to calculate the lattice energy of compounds by breaking down complex reactions into simpler steps whose enthalpy changes are known. Essentially, Hess's Law leverages the fact that enthalpy is a state function, which means the change in enthalpy for a system depends only on the initial and final states, not the journey in between. This makes it easier to calculate otherwise challenging energy changes, like the lattice enthalpy of NaCl.

Sublimation Enthalpy

Sublimation enthalpy refers to the energy needed to convert a substance from solid to gas without passing through the liquid phase. In our exercise, this step involves converting solid sodium (\text{Na}) to gaseous sodium (\text{Na}). The given sublimation enthalpy \(\text{ΔH}^{\text{circ}} = 109 \text{kJ/mol}\).

This transformation is crucial in the lattice energy calculation, as it provides the necessary energy to bring sodium atoms into a gaseous state, which is a prerequisite for further ionization and reaction processes.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom or ion in the gaseous state. In our problem, this step describes the conversion of gaseous sodium (Na) into sodium ions (\text{Na}^+) by removing one electron. The ionization energy given is \(\text{ΔH}^{\text{circ}} = 496 \text{kJ/mol}\).

This step is essential because it changes the sodium atom into a positively charged ion, which can then participate in ionic bonding with chloride ions (\text{Cl}^-), forming the ionic compound NaCl.

Electron Affinity

Electron affinity measures the energy change when an electron is added to a neutral atom in the gaseous state to form a negative ion. In this exercise, chlorine gas (\text{Cl}) accepts an electron to become a chloride ion. The given electron affinity is \(\text{ΔH}^{\text{circ}} = -349 \text{kJ/mol}\).

This step is crucial because it provides the energy change associated with the gain of an electron by a chlorine atom, resulting in the formation of a negatively charged ion. The negative value means that this process releases energy, contributing positively (or rather, subtractively) to the overall energy changes in the lattice energy calculation.

Formation Enthalpy

Formation enthalpy is the energy change when one mole of a compound is formed from its constituent elements in their standard states. For NaCl, the formation enthalpy given is \(\text{ΔH}_{\text{formation}}^{\text{circ}} = -411 \text{kJ/mol}\).

This value indicates that energy is released when sodium and chlorine react to form sodium chloride. Overall, this value is crucial for using Hess's Law to determine lattice energy since it represents the total energy change directly related to forming the ionic compound from its elemental components.