Interpreting Lennard-Jones parameters. Intermolecular interactions are often described by the Lennard-Jones potential $u(r)$, which gives the internal energy of interaction between two molecules as a function of intermolecular separation:
$$
u(r)=4 \varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]
$$
where $\varepsilon$ and $\sigma$ are characteristic energy and bond length parameters.
(a) At low temperatures, entropy is relatively unimportant, and the free energy is minimized at the intermolecular separation of the molecules at which the potential energy is minimized. At what separation does that occur? What is the energy of that state?
(b) It's often convenient to divide energies $\varepsilon$ by Boltzmann's constant $k$ to give them in units of temperature $(\mathrm{K})$. In these units, the constants in Table $24.4$ have been found. What physical properties of these systems can you deduce from this information?
Table 24.4
\begin{tabular}{lcc}
\hline Molecule & $\varepsilon / k(\mathrm{~K})$ & $\sigma(\mathrm{A})$ \\
\hline He & $10.22$ & $2.58$ \\
Ethane & 205 & $4.23$ \\
Benzene & 440 & $5.27$ \\
\hline Source: PW Atkins, Physical Chemistry, 6 th
\end{tabular}