A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas $(\gamma=5 / 3)$ and one mole of an ideal diatomic gas $(\gamma=7 / 5)$. Here, $\gamma$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of $66$ Joule when heated at constant pressure. The change in its internal energy is. . . . . . .Joule.
IIT 2023, Easy
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At constant pressure
$W=n R \Delta T=66$
$\Delta U=n\left(C_V\right)_{\text {mix }} \Delta T$
$\left(C_v\right)_{\text {mii }}=\frac{n_1 C_{v_1}+n_2 C_{v_2}}{n_1+n_2}$
$\left(C_v\right)_{\text {mii }}=\frac{2 \times \frac{3}{2} R+1 \times \frac{5}{2} R}{3}$
$\left(C_v\right)_{\text {min }}=\frac{11}{6} R$
$\Delta U=\frac{11}{6}(n R \Delta T)$
$\Delta U=\frac{11}{6} \times 66=121 J$
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