15.6 Exercise 24

Consider the van der Waals potential $U \left(r\right) = U_{o} \left[\left(\frac{R_{o}}{r}\right)^{12} - 2 \left(\frac{R_{o}}{r}\right)^{6}\right]$, used to model the potential energy function of two molecules, where the minimum potential is at $r = R_{o}$. Find the force as a function of r. Consider a small displacement $r = R_{o} + r^{'}$ and use the binomial theorem:

$\left(1 + x\right)^{n} = 1 + n x + \frac{n \left(n - 1\right)}{2 !} x^{2} + \frac{n \left(n - 1\right) \left(n - 2\right)}{3 !} x^{3} + \hdots$,

to show that the force does approximate a Hooke’s law force.