If one mole of a monoatomic gas $({\gamma }_1=\frac{5}{3})$ is mixed with one mole of a diatomic gas $({\gamma }_2=\frac{7}{5}),$ the value of $\gamma$ for the mixture is $\frac{N}{10}.$ The value of $N$ is

$\text{ For monoatomic gas,}{\gamma }_1=\frac{5}{3},C_v=\frac{3}{2}R and C_p=\frac{5}{2}R,$

For Diatomic Gas, ${\gamma }_2=\frac{7}{5},C_v=\frac{5}{2}R and C_p=\frac{7}{2}R$

For mixture of gases,

$\frac{{\mathrm{n}}_1+{\mathrm{n}}_2}{{\gamma }_{\mathrm{m}}-1}=\frac{{\mathrm{n}}_1}{{\gamma }_1-1}+\frac{{\mathrm{n}}_2}{{\gamma }_2-1}$

$\Rightarrow \frac{1+1}{{\gamma }_{\mathrm{m}}-1}=\frac{1}{\frac{5}{3}-1}+\frac{1}{\frac{7}{5}-1}$

$\Rightarrow {\gamma }_{\mathrm{m}}=\frac{6}{4}=\frac{3}{2}$