You take on a research assistantship with a molecular physicist. She is studying the vibrations of diatomic molecules. In these vibrations, the two atoms in the molecule move back and forth along the line connecting them (see Figure $20.5 \mathrm{c}$ ). As an introduction to her research, she asks you to familiarize yourself with the Lennard-Jones potential (see Example 7.9 ), which describes the potential energy function for a diatomic molecule. She asks you to determine the effective spring constant, in terms of the parameters $\sigma$ and $\epsilon,$ for the bond holding the atoms together in the molecule for small vibrations around the equilibrium separation $r_{\mathrm{cq}} .$ After being stumped for a while, you ask her for a hint. She responds, "Example 7.9 provides the derivative of the potential energy function. Compare that to Equation 7.29 to find the force between the atoms. You want to show that $F$ is of the form $-k x$, and find $k .$ Let the separation distance $r=r_{\mathrm{eq}}+x,$ where $x$ is small and take advantage of the series approximations in Appendix Section B.5." Wow, that's several hints! You sit down and get to work.