Isotopic substitution is employed to isolate features in a vibrational spectrum. For example, the C=O stretch of individual carbonyl groups in the backbone of a polypeptide can be studied by substituting ¹³C¹⁸O for ¹²C¹⁶O.

a. From quantum mechanics the vibrational frequency of a diatomic molecules depends on the bond force constant (κ) and reduced mass (μ) as follows:

v̅ = √K / µ

If the vibrational frequency of ¹²C¹⁶O is 1680 cm⁻¹, what is the expected frequency for ¹³C¹⁸O?

b. Using the vibrational frequencies for ¹²C¹⁶O and ¹³C¹⁸O, determine the value of the corresponding vibrational partition functions at 298 K. Does this isotopic substitution have a dramatic effect on qV?