If  γ  is the ratio of two specific heats of a gas, the number of degrees of freedom of a molecule of the gas is:

molar specific heat of a gas at constant volume is Cv=f.R2  here f is degree of freedom ,   R is universal gas constant  from difference of specific heatsCp=Cv+R  substitute the value of Cv =fR2+R=fR12+1f  Cp=fR(f+22f)specific heat ratio γ=CpCv , substitute the values of specific heats  =fRf+22ffR2 ⇒γ=f+2f ⇒γ=1+2f ⇒2f=γ-1 ⇒degree of freedom is f=2γ−1