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(C) At temperatures below $100\, K$,the vibrational degrees of freedom for both $H_2$ and $N_2$ are frozen. Both gases behave as diatomic molecules wi

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$7/5$

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(C) At temperatures below $100\, K$,the vibrational degrees of freedom for both $H_2$ and $N_2$ are frozen. Both gases behave as diatomic molecules with only translational and rotational degrees of freedom. Thus,the degrees of freedom for both gases is $f = 5$.The molar heat capacity at constant volume for both gases is $C_V = \frac{fR}{2} = \frac{5R}{2}$.For a mixture with equal number of moles $(n_1 = n_2 = n)$:$(C_V)_{	ext{mix}} = \frac{n_1 C_{V1} + n_2 C_{V2}}{n_1 + n_2} = \frac{n(\frac{5R}{2}) + n(\frac{5R}{2})}{2n} = \frac{5R}{2}$.The molar heat capacity at constant pressure is $(C_P)_{	ext{mix}} = (C_V)_{	ext{mix}} + R = \frac{5R}{2} + R = \frac{7R}{2}$.The ratio of specific heats is $\gamma = \frac{(C_P)_{	ext{mix}}}{(C_V)_{	ext{mix}}} = \frac{7R/2}{5R/2} = \frac{7}{5} = 1.4$.

$A$ gaseous mixture contains an equal number of hydrogen $(H_2)$ and nitrogen $(N_2)$ molecules. Specific heat measurements on this mixture at temperatures below $100\, K$ would indicate that the value of $\gamma$ (ratio of specific heats) for this mixture is

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