If  $\text{γ}$  is the ratio of two specific heats of a gas, the number of degrees of freedom of a molecule of the gas is:

$\begin{array}{rcl}{\text{molar specific heat of a gas at constant volume is C}}_{\text{v}}& =& \text{f.}\frac{\text{R}}{\text{2}}\\ & & here f is degree of freedom , \\ & & R is universal gas cons\tan t\\ & & from difference of specific heats\\ {\text{C}}_{\text{p}}& =& {\text{C}}_{\text{v}}+\text{R}\\ & & substitute the value of {\text{C}}_{\text{v}}\\ & =& \frac{\text{fR}}{\text{2}}\text{+R}\text{=fR}\left(\frac{{\displaystyle \text{1}}}{{\displaystyle \text{2}}}\text{+}\frac{{\displaystyle \text{1}}}{{\displaystyle \text{f}}}\right)\\ & & {\text{C}}_{\text{p}}\text{=fR}\left(\frac{\text{f+2}}{\text{2f}}\right)\\ specific heat ratio \text{γ}& =& \frac{{\text{C}}_{\text{p}}}{{\text{C}}_{\text{v}}} , substitute the values of specific heats\\ & & \text{=}\frac{\text{fR}\left(\frac{\text{f+2}}{\text{2f}}\right)}{\frac{\text{fR}}{\text{2}}}\\ & \Rightarrow & \gamma =\frac{\text{f+2}}{f}\\ & \Rightarrow & \gamma =1+\frac{2}{\text{f}}\\ & \Rightarrow & \frac{2}{\text{f}}=\text{γ-1}\\ & \Rightarrow & degree of freedom is \text{f}=\frac{2}{\text{γ}-1}\end{array}$