••• (a) Use the curve in Fig. 12.12 to get a rough estimate of the force constant k for th...

••• (a) Use the curve in Fig. 12.12 to get a rough estimate of the force constant k for the molecule. (See the hint for Problem 1.) (b) Use your value of k to estimate the zero-point energy of vibration in . (Since the two nuclei have equal mass, you will need to use the reduced mass as in Problem 2.)

Problem 1

•• Figure 12.6 shows the energy E(R) for the Na+ and Cl− ions that make up the NaCl molecule. Near its minimum at R0, E(R) can be approximated by a parabola: . Use the information on the graph to make a rough estimate of the force constant k. [Hint: To the left of R0, the graph retains a reasonably parabolic shape as far as the point where it crosses the horizontal axis.]

Problem 2

•• Consider two classical atoms of mass m and m′, separated by a distance R, with their center of mass fixed at the origin. (a) Write down expressions for the atoms’ distances, r and r′, from the origin, in terms of m, m′, and R. (b) Suppose that the two atoms vibrate in and out along the internuclear axis. Show that their total kinetic energy can be expressed as , where μ is the reduced mass defined in Eq. (12.41) (Problem 3). This result proves for a classical molecule (what is true in quantum mechanics as well) that the vibrational motion of both atoms can be treated as if only one atom were moving, provided that we take its mass to be μ.

Problem 3

• Consider two atoms of masses m and m′, bound a distance R0 apart and rotating about their center of mass (as in Fig. 12.26). (a) Calculate their moment of inertia, I, and prove that it can be written as I = μR02, where μ is the reduced mass

This shows that one can treat the rotational motion of any diatomic molecule as if only one of the atoms were moving, provided that one takes its mass to be μ. (b) Prove that if m ≪ m′ then μ ≈ m.