Chapter 15, Problem 71CP

Consider the van der Waals potential $U\left(r\right)=U_0\left[{\left(\frac{R_0}{r}\right)}^{12}-2{\left(\frac{R_0}{r}\right)}^6\right]$ , used to model the potential energy function of two molecules, where the minimum potential is at $r=R_0$ . Find the force as a function of r. Consider a small displacement $R=R_0+r\text{'}$ and use the binomial theorem:

   $\begin{array}{l}{\left(1+x\right)}^n=1+nx+\frac{n\left(n-1\right)}{2!}{x}^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}{x}^3+\cdot \cdot \cdot \\ \end{array}$ , to show that the force does approximate a Hooke’s law force.