The oxygen molecule has a vibrational energy that is listed by spectroscopists as 1580 cm ¹. What they're actually describing is the inverse wavelength of the photons with the corresponding energy. Multiplying by hc, the vibrational energy spacing comes out to ε ≈ 0.2 eV. a. Derive the vibrational partition function of a generic harmonic oscillator. Simplify it as much as possible. b. Evaluate the vibrational partition function at room temperature for oxygen at room temperature. Compare c. Calculate what fraction of oxygen molecules are in the first excited vibrational state at room temperature. d. Taking the vibrational ground state to be energy zero, calculate the average energy of an oxygen molecule, both in eV and in proportion to KT.