Question

If one mole of an ideal monoatomic gas $(\gamma =\frac{5}{3})$ is mixed with one mole of ideal diatomic gas $(\gamma =\frac{7}{5}),$ the value of $\gamma$ for the mixture will be

Answer 1

For mixture of gases -
$U_{\text{mixture}}=U_{\text{monoatomic}}+U_{\text{diatomic}}$
$\therefore \frac{(n_1+n_2)RT}{{\gamma }_{mix}-1}=\frac{n_{1}RT}{{\gamma }_1-1}+\frac{n_{2}RT}{{\gamma }_2-1}$
$\therefore \frac{2}{{\gamma }_{mix}-1}=\frac{1}{(\frac{5}{3}-1)}+\frac{1}{(\frac{7}{5}-1)}$
$\frac{2}{{\gamma }_{mix}-1}=\frac{3}{2}+\frac{5}{2}=4$
${\gamma }_{mix}-1=0.5$
${\gamma }_{mix}=0.5+1$
$\therefore {\gamma }_{mix}=1.5$