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To find out degree of freedom, the correct expression is :
A
$f = \dfrac{2}{\gamma - 1}$
B
$f = \dfrac{\gamma + 1}{2}$
C
$f = \dfrac{2}{\gamma + 1}$
D
$f = \dfrac{1}{\gamma + 1}$
Detailed Solution
For a gas with f degrees of freedom: $C_V = \dfrac{f}{2}R$ and $C_P = \left(\dfrac{f}{2} + 1\right)R$
$\because \gamma = \dfrac{C_P}{C_V} = 1 + \dfrac{2}{f}$
$\Rightarrow \dfrac{2}{f} = \gamma - 1 \Rightarrow f = \dfrac{2}{\gamma - 1}$
Check: for a monatomic gas, $\gamma = \dfrac{5}{3}$ gives f = 3.
$\because \gamma = \dfrac{C_P}{C_V} = 1 + \dfrac{2}{f}$
$\Rightarrow \dfrac{2}{f} = \gamma - 1 \Rightarrow f = \dfrac{2}{\gamma - 1}$
Check: for a monatomic gas, $\gamma = \dfrac{5}{3}$ gives f = 3.