Question

The lattice energy of $\mathrm{L}\mathrm{i}\mathrm{F}$ is $1023\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$, and the $\mathrm{L}\mathrm{i}-\mathrm{F}$ distance is $200.8\mathrm{p}\mathrm{m}$. $\mathrm{N}\mathrm{a}\mathrm{F}$ crystallizes in the same structure as $\mathrm{L}\mathrm{i}\mathrm{F}$ but with a $\mathrm{N}\mathrm{a}-\mathrm{F}$ distance of $231\mathrm{p}\mathrm{m}$. Which of the following values most closely approximates the lattice energy of $\mathrm{N}\mathrm{a}\mathrm{F}$: $510,890,1023,1175,$ or $4090\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$? Explain your choice.

Short answer

The value for lattice energy for $\mathrm{N}\mathrm{a}\mathrm{F}$ is $889\mathrm{k}\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$ and $890\mathrm{k}\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$ closely approximates to the result.

Step-by-step solution

Concept Introduction

The lattice energy is the change in energy that occurs when one mole of a crystalline ionic compound is formed from its constituent ions, which are considered to be in a gaseous state at the start. It's a metric for the forces that bind ionic solids together.

Calculation for Lattice Energy

Formula to calculate lattice energy is –

$\mathrm{U}=\mathrm{C}\left(\frac{{\mathrm{Z}}^{+}{\mathrm{Z}}^{-}}{{\mathrm{R}}_{\mathrm{o}}}\right)......(1)$

Where${\mathrm{R}}_{\mathrm{o}}$, is the interatomic distance.

As far as charges are considered, this is same in both $\mathrm{L}\mathrm{i}\mathrm{F}$ and $\mathrm{N}\mathrm{a}\mathrm{F}$. Major difference is expected to be in interatomic distance i.e., $2.008\stackrel{\mathrm{o}}{\mathrm{A}}$ vs $2.31\stackrel{\mathrm{o}}{\mathrm{A}}$.

From the data for $\mathrm{L}\mathrm{i}\mathrm{F}$, with ${\mathrm{Z}}^{+}{\mathrm{Z}}^{-}=-1$,

$\begin{array}{r}\mathrm{C}=\frac{\mathrm{U}{\mathrm{R}}_{\mathrm{o}}}{{\mathrm{Z}}^{+}{\mathrm{Z}}^{-}}\\ =\frac{1023\times 2.008}{-1}\\ =-2054\mathrm{k}\mathrm{J}\stackrel{\mathrm{o}}{\mathrm{A}}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}\end{array}$

Then, it is obtained that –

$\begin{array}{r}{\mathrm{U}}_{\mathrm{N}\mathrm{a}\mathrm{F}}=\frac{-2054\times -1}{2.31\stackrel{\mathrm{o}}{\mathrm{A}}}\\ =889\mathrm{k}\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}\end{array}$

Therefore, value for lattice energy is obtained as $889\mathrm{k}\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$.