The lattice energy of \({\rm{LiF}}\) is \({\rm{1023 kJ/mol}}\), and the \({\rm{Li - F}}\) distance is \({\rm{200}}{\rm{.8 pm}}\). \({\rm{NaF}}\) crystallizes in the same structure as \({\rm{LiF}}\) but with a \({\rm{Na - F}}\) distance of \({\rm{231 pm}}\). Which of the following values most closely approximates the lattice energy of \({\rm{NaF}}\): \({\rm{510, 890, 1023, 1175,}}\) or \({\rm{4090 kJ/mol}}\)? Explain your choice.

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.

Formula to calculate lattice energy is –

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

Where\({{\rm{R}}_{\rm{o}}}\), is the interatomic distance.

As far as charges are considered, this is same in both \({\rm{LiF}}\) and \({\rm{NaF}}\). Major difference is expected to be in interatomic distance i.e., \({\rm{2}}{\rm{.008}}\mathop {\rm{A}}\limits^{\rm{o}} \) vs \({\rm{2}}{\rm{.31}}\mathop {\rm{A}}\limits^{\rm{o}} \).

From the data for \({\rm{LiF}}\), with \({{\rm{Z}}^{\rm{ + }}}{{\rm{Z}}^{\rm{ - }}}{\rm{ = - 1}}\),

\(\begin{align}{\rm{C = }}\frac{{{\rm{U}}{{\rm{R}}_{\rm{o}}}}}{{{{\rm{Z}}^{\rm{ + }}}{{\rm{Z}}^{\rm{ - }}}}}\\{\rm{ = }}\frac{{{\rm{1023 \times 2}}{\rm{.008}}}}{{{\rm{ - 1}}}}\\{\rm{ = - 2054 kJ}}\mathop {\rm{A}}\limits^{\rm{o}} {\rm{mo}}{{\rm{l}}^{{\rm{ - 1}}}}\end{align}\)

Then, it is obtained that –

\(\begin{align}{{\rm{U}}_{{\rm{NaF}}}}{\rm{ = }}\frac{{{\rm{ - 2054 \times - 1}}}}{{{\rm{2}}{\rm{.31}}\mathop {\rm{A}}\limits^{\rm{o}}}}\\{\rm{ = 889 kJmo}}{{\rm{l}}^{{\rm{ - 1}}}}\end{align}\)

Therefore, value for lattice energy is obtained as \({\rm{889 kJmo}}{{\rm{l}}^{{\rm{ - 1}}}}\).