One mole of mono atomic gas $\left(\gamma =\frac{5}{3}\right)$  is mixed with one mole of diatomic gas   $\left(\gamma =\frac{{\displaystyle 7}}{{\displaystyle 5}}\right)$. What will be the value of $\gamma$ for the mixture.

$\begin{array}{rcl} & & formula to find {\gamma }_{\text{mix}} is \frac{{\text{n}}_{\text{1}}{\text{+n}}_{\text{2}}}{{\gamma }_{\text{mix}}\text{-1}}\text{=}\frac{{\text{n}}_{\text{1}}}{{\text{γ}}_{\text{1}}\text{-1}}\text{+}\frac{{\text{n}}_{\text{2}}}{{\text{γ}}_{\text{2}}\text{-1}}\text{---(1)}\\ & & here {\text{n}}_{\text{1}}{\text{\&n}}_{\text{2}}{\text{=number of moles; γ}}_{\text{1}}{\text{ \&γ}}_2\text{= specific heat ratio}\\ given {\text{n}}_{\text{1}}{\text{=n}}_{\text{2}}& =& 1;{\gamma }_{\text{1}}\text{ =}\frac{5}{3}{\text{;γ}}_2\text{=}\frac{7}{5}\\ & & substitute given values in \left(1\right)\\ \frac{1+1}{{\gamma }_{\text{mix}}\text{-1}}& =& \frac{1}{\frac{5}{3}-1}+\frac{1}{\frac{7}{5}-1}\\ \frac{2}{{\gamma }_{\text{mix}}\text{-1}}& =& \frac{1}{\frac{2}{3}}+\frac{1}{\frac{2}{5}}\\ \frac{2}{{\gamma }_{\text{mix}}\text{-1}}& =& \frac{3}{2}+\frac{5}{2}\\ & \Rightarrow & \frac{2}{{\gamma }_{\text{mix}}\text{-1}}=4\\ {\gamma }_{\text{mix}}\text{-1}& =& \frac{1}{2}\\ & \Rightarrow & {\gamma }_{\text{mix}}=1.5\\ & & \end{array}$