Obtain the equation for $\gamma = \frac{C_P}{C_V}$ in terms of the degree of freedom $f$.

(D) For a gas with $f$ degrees of freedom,the internal energy $U$ of $1 \text{ mole}$ of gas is given by:
$U = f \times \frac{1}{2} k_B T \times N_A = \frac{1}{2} f RT$ (since $k_B N_A = R$).
From the definition of molar specific heat at constant volume:
$C_V = \frac{dU}{dT} = \frac{d}{dT} \left( \frac{1}{2} f RT \right) = \frac{1}{2} f R$.
Using Mayer's relation for molar specific heat at constant pressure:
$C_P = C_V + R = \frac{1}{2} f R + R = \left( \frac{f}{2} + 1 \right) R$.
The adiabatic index $\gamma$ is defined as the ratio of specific heats:
$\gamma = \frac{C_P}{C_V} = \frac{(\frac{f}{2} + 1) R}{\frac{1}{2} f R}$.
Simplifying the expression:
$\gamma = \frac{f + 2}{f} = 1 + \frac{2}{f}$.