the vibrational frequency of I₂ is 214.5 cm^-1.

(i) Using the harmonic oscillator model, evaluate the vibrational partition function and the mean vibrational energy of I₂ at 1000K.

(ii) What is the characteristic vibrational temperature of I₂?

(iii) At 1000K, assuming high-temperature approximation, evaluate the vibrational partition function and the mean vibrational energy of I₂.

(iv) Comparing (i) and (iii), is the high-temperature approximation good for I₂ at 1000K?