One mole of ideal monatomic gas $$\left( {\gamma = \frac{5}{3}} \right)$$  is mixed with one mole of diatomic gas $$\left( {\gamma = \frac{7}{5}} \right).$$   What is $$\gamma $$ for the mixture ? $$\gamma $$ Denotes the ratio of specific heat at constant pressure, to that at constant volume

Solution :
$$\eqalign{ & \frac{{{n_1} + {n_2}}}{{\gamma - 1}} = \frac{{{n_1}}}{{{\gamma _1} - 1}} + \frac{{{n_2}}}{{{\gamma _2} - 1}} \cr & \Rightarrow \,\,\frac{{1 + 1}}{{\gamma - 1}} = \frac{1}{{\frac{5}{3} - 1}} + \frac{1}{{\frac{7}{5} - 1}} \cr & \Rightarrow \,\,\gamma = \frac{3}{2} \cr} $$