$\gamma_{\text {mixture }}=\frac{\left(\frac{\mu_{1} C p_{1}+\mu_{2} C p_{2}}{\mu_{1}+\mu_{2}}\right)}{\left(\frac{\mu_{1} C v_{1}+\mu_{2} C v_{2}}{\mu_{1}+\mu_{2}}\right)}$

$\gamma_{\text {mixture }}=\frac{\mu_{1} C p_{1}+\mu_{2} C p_{2}}{\mu_{1} C v_{1}+\mu_{2} C v_{2}}$

$=\frac{\mu_{1}\left(\frac{\gamma_{1}}{\gamma_{1}-1}\right) R+\mu_{2}\left(\frac{\gamma_{2}}{\gamma_{2}-1}\right) R}{\mu_{1}\left(\frac{R}{\gamma_{1}-1}\right)+\mu_{2}\left(\frac{R}{\gamma_{2}-1}\right)}$

or $\gamma_{\text {mixture }}=\frac{\mu_{1} \gamma_{1}\left(\gamma_{2}-1\right)+\mu_{2} \gamma_{2}\left(\gamma_{1}-1\right)}{\mu_{1}\left(\gamma_{2}-1\right)+\mu_{2}\left(\gamma_{1}-1\right)}$

It is given, $\gamma_{\text {mixture }}=\frac{19}{13}$

$\gamma_{1}=\frac{5}{3}, \gamma_{2}=\frac{7}{5}$

$n_{1}=1, n_{2}=n$

Substituting values, we have

$\frac{19}{13}-\frac{\frac{5}{3}\left(\frac{7}{5}-1\right)+n \frac{7}{5}\left(\frac{5}{3}-1\right)}{\left(\frac{7}{5}-1\right)+n\left(\frac{5}{3}-1\right)}$

$\Rightarrow \quad \frac{19}{13} =\frac{\frac{5}{3} \times \frac{2}{5}+n \frac{7}{5} \times \frac{2}{3}}{\frac{2}{5}+n \frac{2}{3}}$

$\Rightarrow 19\left(\frac{2}{5}+n \frac{2}{3}\right) =13\left(\frac{2}{3}+n \frac{14}{15}\right)$

$\Rightarrow \quad n =2$