In using statistical mechanics to describe the thermodynamic properties of molecules, high-frequency vibrations are generally not of importance under standard thermodynamic conditions since they are not populated to a significant extent. For example, for many hydrocarbons the C―H stretch vibrational degrees of freedom are neglected. Using cyclohexane as an example, the IR-absorption spectrum reveals that the C―H stretch transitions are located at ~2850 cm^–1.

a. What is the value of the vibrational partition function for a mode of this frequency at 298 K?

b. At what temperature will this partition function reach a value of 1.1?