In class, Prof. Miller introduced the "rigid rotor harmonic oscillator" approximation for a diatomic molecule. This approximation assumes (among other things) that the vibrational motion can be separated from the rotational motion. This means that the total energy for the system is the sum of the vibrational energy and the rotational energy.

Prof. Miller also discussed that the spacing between vibrational energy levels is substantially larger than the spacing between low rotational energy levels. So for this bonus problem, we're going to explore that idea in more detail. We'll use N₂ as an example system. We were given hν = 2360 cm^-1 (that's also h-bar omega, which doesn't render in HTML) and B = 2 cm^-1. Representing the states with the vibrational quantum number v and the rotational quantum number J by the ordered pair (v, J), we'll try to answer the following questions:

1. For a given vibrational state, how many rotational states have energy less than the next vibrational state? That is, find J such that the energy of the state (v, J) is greater than or equal to the energy of the state (v + 1, 0). Note that since the vibrational energy level spacing is even (from the harmonic oscillator approximation), this will be the same for any vibrational level v.
2. After what rotational level does the rotational spacing become larger than the vibrational spacing?

Doing the math for these problems is a good start, but do think a little about what it means physically. Compare these spacings to the average thermal energy, about 200 cm^-1 at room temperature. Would you expect many molecules to be in excited vibrational states? What about excited rotational states? (You'll learn how to quantify how many molecules are expected to be in each state in 120B.)