Calculate the cohesive energy per ion-pair of LiCl and KCl...

Calculate the cohesive energy per ion-pair of LiCl and KCl crystals using the following data: $\begin{array}{lll} & \mathrm{LiCl} & \mathrm{KCl} \\ \text { Madelung constant } & 1.748 & 1.748 \\ \mathrm{Li}^{+}-\mathrm{Cl}^{-} / \mathrm{K}^{+}-\mathrm{Cl}^{-} \text {spacing } & 2.57 \AA & 3.14 \mathrm{~A} \\ \text { lonization energy of } & 5.4 \mathrm{eV} & 4.34 \mathrm{eV} \\ \mathrm{Li} / \mathrm{K} & & \end{array}$ Assume that the repulsive potential energy is negligibly small.

Calculate the cohesive energy per ion-pair of LiCl and KCl crystals using the following data:
$\begin{array}{lll} & \mathrm{LiCl} & \mathrm{KCl} \\ \text { Madelung constant } & 1.748 & 1.748 \\ \mathrm{Li}^{+}-\mathrm{Cl}^{-} / \mathrm{K}^{+}-\mathrm{Cl}^{-} \text {spacing } & 2.57 \AA & 3.14 \mathrm{~A} \\ \text { lonization energy of } & 5.4 \mathrm{eV} & 4.34 \mathrm{eV} \\ \mathrm{Li} / \mathrm{K} & & \end{array}$ Assume that the repulsive potential energy is negligibly small.

Key Concepts

Simplifying Assumptions in Lattice Energy Calculations

In many lattice energy or cohesive energy calculations, certain contributions, such as the repulsive short-range interactions, are assumed to be negligible to simplify the analysis. This allows the focus to be on the dominant Coulombic attractive forces reflected by the Madelung constant and ionic separations. Such assumptions are particularly useful when estimating the net cohesive binding energy in ionic solids.

Ionization Energy

Ionization energy is the amount of energy required to remove an electron from a neutral atom to form a positively charged ion. In the context of forming ionic compounds, it is an important quantity in the overall energy balance of the process, as it contributes to the energy cost of producing ions that then gain stabilization through lattice formation.

Ionic Spacing

Ionic spacing, or the distance between oppositely charged ions in a crystal, directly affects the electrostatic interaction energy calculated via Coulomb's law. Smaller separations lead to stronger attractions and therefore higher (more negative) lattice energies. Accurate values of ionic separation are essential for estimating the cohesive energy of the ionic crystal.

Cohesive Energy

Cohesive energy in an ionic crystal refers to the energy released when isolated gaseous ions come together to form a crystalline solid. It is essentially the negative of the lattice energy and quantifies the strength of the binding in the ionic lattice, serving as a measure of the stability of the crystal. In calculations, it reflects the net energy gained when ionic bonds are formed.

Madelung Constant

The Madelung constant is a dimensionless sum that accounts for the geometry of the ionic crystal lattice and quantifies the net electrostatic interactions between a reference ion and all other ions in the crystal. It depends on the arrangement of ions and is crucial for accurately determining the lattice (or cohesive) energy since it scales the Coulombic potential energy in an infinite lattice.

Coulomb's Law in Ionic Solids

Coulomb's law describes the electrostatic interaction between two charged particles. In the context of ionic crystals, it is used to calculate the attractive energy between oppositely charged ions. The potential energy is inversely proportional to the distance between the ions; therefore, the separation (or spacing) of ions is a critical factor in determining the magnitude of the cohesive energy when combined with the charge values.

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