3. Vibrating Diatomic Molecules (20 points) Consider the following two diatomic molecules described by the harmonic oscillator model: HF ($k = 970 \text{ N/m}, \mu = 0.95 \text{ amu}$) and HI ($k = 320 \text{ N/m}, \mu = 0.99 \text{ amu}$).
(a) (4 points) Without doing any explicit numerical calculations, which molecule do you expect to absorb a higher infrared frequency and why? (4 points)
(b) (10 points) Sketch the potential energy $V(x)$, $x = R - R_e$, for both HF and HI, illustrating any differences between the potentials (6 points). Sketch at least 4 energy levels on each diagram to illustrate your answer to part (a) (2 points). Draw an arrow to indicate the transition observed in the infrared spectrum for each molecule (2 points).
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(c) (6 points) Real diatomic molecules can dissociate if they are in a highly excited vibrational state with large $n$. Draw a more realistic potential energy function $V(x)$ for a vibrating diatomic molecule, including at least 4 energy levels (4 points). How does the energy level spacing change compared to the harmonic oscillator model (2 points)?
