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(C) For a mixture of $1$ mole of $H_2$ (diatomic,$C_v = 5R/2$) and $1$ mole of $He$ (monatomic,$C_v = 3R/2$):Total internal energy $U = n_1 C_{v1} T +

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$(A, B, D)$

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(C) For a mixture of $1$ mole of $H_2$ (diatomic,$C_v = 5R/2$) and $1$ mole of $He$ (monatomic,$C_v = 3R/2$):Total internal energy $U = n_1 C_{v1} T + n_2 C_{v2} T = 1(5R/2)T + 1(3R/2)T = 4RT$.Average energy per mole $= U / (n_1 + n_2) = 4RT / 2 = 2RT$. Thus,$(A)$ is correct.For the mixture,$C_{v,mix} = (n_1 C_{v1} + n_2 C_{v2}) / (n_1 + n_2) = (5R/2 + 3R/2) / 2 = 2R$. $C_{p,mix} = C_{v,mix} + R = 3R$. $\gamma_{mix} = C_{p,mix} / C_{v,mix} = 3R / 2R = 1.5 = 3/2$.Speed of sound $v = \sqrt{\gamma RT / M}$. For $He$,$\gamma = 5/3$ and $M = 4$. For mixture,$M_{mix} = (2+4)/2 = 3$. Ratio $v_{mix} / v_{He} = \sqrt{(\gamma_{mix} / M_{mix}) / (\gamma_{He} / M_{He})} = \sqrt{(1.5 / 3) / ((5/3) / 4)} = \sqrt{0.5 / (5/12)} = \sqrt{6/5}$. Thus,$(B)$ is correct.$RMS$ speed $v_{rms} = \sqrt{3RT/M}$. Ratio $v_{rms,He} / v_{rms,H2} = \sqrt{M_{H2} / M_{He}} = \sqrt{2/4} = 1/\sqrt{2}$. Thus,$(D)$ is correct.

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