The Lennard-Jones parameters that describe intermolecular interactions for a compound are ε=1 kJ/mol and σ=0.4 nm.
A) Determine the intermolecular distance at which potential energy is minimized.
B) Determine the potential energy when the intermolecular distance is 0.5 nm.
C) Determine the potential energy when the intermolecular distance is 0.39 nm.
D) Make a complete plot of potential energy in kJ/mol vs. intermolecular distance in nm.
A) We are looking for a minimum; a point at which:
\frac{\mathrm{d} \Gamma}{\mathrm{dr}}=0
We want to find the distance r that the following equation is minimized:
\begin{aligned}& \Gamma=4 \epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right] \\& \frac{\mathrm{d} \Gamma}{\mathrm{dr}}=4 \epsilon\left[-12 \sigma^{12}\left(\frac{1}{r}\right)^{13}+6 \sigma^6\left(\frac{1}{r}\right)^7\right]=0 \\& 12 \sigma^{12}\left(\frac{1}{r}\right)^{13}=6 \sigma^6\left(\frac{1}{r}\right)^7 \\& 2 \sigma^6=r^6 \\& r=\left[2(0.4 \mathrm{~nm})^6\right]^{1 / 6} \\&\bf r=0.449 ~nm\end{aligned}
B)
\begin{aligned}& \Gamma=4 \epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right] \\& \Gamma=4\left(1 \frac{\mathrm{kJ}}{\mathrm{mol}}\right)\left[\left(\frac{(0.4 \mathrm{~nm})}{(0.5 \mathrm{~nm})}\right)^{12}-\left(\frac{(0.4 \mathrm{~nm})}{(0.5 \mathrm{~nm})}\right)^6\right]=-\mathbf{0 . 7 7 4} \frac{\mathbf{k J}}{\mathbf{m o l}}\end{aligned}
C)
\begin{aligned}\Gamma & =4 \epsilon\left[\left(\frac{\sigma}{\mathrm{r}}\right)^{12}-\left(\frac{\sigma}{\mathrm{r}}\right)^6\right] \\\Gamma & =4\left(1 \frac{\mathrm{kJ}}{\mathrm{mol}}\right)\left[\left(\frac{(0.4 \mathrm{~nm})}{(0.39 \mathrm{~nm})}\right)^{12}-\left(\frac{(0.4 \mathrm{~nm})}{(0.39 \mathrm{~nm})}\right)^6\right]=\mathbf{0 . 7 6 4} \frac{\mathbf{k J}}{\mathbf{m o l}}\end{aligned}
D)
