Discussion
If the interatomic potential energy function can be expressed as
\[ U(R) = -\frac{A}{R^{6}} + \frac{B}{R^{12}}, \]
where A and B are constants, the atomic spacing \(R_{0}\) for which the potential energy is a minimum is given by :
Answer: B
The equilibrium spacing is \(R_{0} = (2B/A)^{1/6}\).
The minimum of U(R) is where its derivative vanishes:
\[ \begin{aligned} \frac{dU}{dR} &= \frac{6A}{R^{7}} - \frac{12B}{R^{13}} = 0 \\ R^{6} &= \frac{2B}{A} \end{aligned} \]
so \(R_{0} = (2B/A)^{1/6}\). This is the Lennard-Jones form, with an attractive \(R^{-6}\) van der Waals term and a repulsive \(R^{-12}\) term. The other options do not have the dimension of a length; \(A^{2}/4B\) is the depth of the well, \(-U_{min}\).
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