Q.For polyatomic molecules having 'f' vibrational modes, the ratio of two specific heats, $\dfrac{C_P}{C_V}$ is ______. (A) $\dfrac{1+f}{2+f}$
(B) $\dfrac{2+f}{3+f}$
(C) $\dfrac{4+f}{3+f}$
(D) $\dfrac{5+f}{4+f}$

Treating a water molecule (3 atoms) by the equipartition/Dulong-Petit-type argument, each atom contributes $3R$ to the molar specific heat.

Using the law of equipartition of energy extended to a solid-like/associated liquid model, each atom is treated as a three-dimensional oscillator with 6 degrees of freedom (3 kinetic + 3 potential), each contributing $\tfrac12 R T$ to the molar internal energy, i.e. $3R$ per …

Count total degrees of freedom (translational + rotational + $2\times$vibrational) and apply $C_V=\frac{n}{2}R$, $C_P=C_V+R$.

A general polyatomic (non-linear) molecule has:

• 3 translational degrees of freedom,
• 3 rotational degrees of freedom,
• and, for each vibrational mode, 2 degrees of freedom (one for kinetic energy of vibration, one for the associated potential energy of the vibrating bond) — so $f$ vibrational modes contribute $2f$.

Total degrees of freedom:

$n=3+3+2f=6+2f$

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