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(C) The total internal energy $U$ of a gas mixture is the sum of the internal energies of its individual components.For a gas with $n$ moles,the inter

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$27$

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(C) The total internal energy $U$ of a gas mixture is the sum of the internal energies of its individual components.For a gas with $n$ moles,the internal energy is given by $U = n C_V T$,where $C_V$ is the molar heat capacity at constant volume.Argon is a monatomic gas,so its degrees of freedom $f_1 = 3$,and $C_{V1} = \frac{3}{2} R$.Oxygen is a diatomic gas,so its degrees of freedom $f_2 = 5$ (neglecting vibrational modes),and $C_{V2} = \frac{5}{2} R$.The total internal energy is $U = n_1 C_{V1} T + n_2 C_{V2} T$.Substituting the given values: $U = (8 	imes \frac{3}{2} R 	imes T) + (6 	imes \frac{5}{2} R 	imes T)$.$U = (12 RT) + (15 RT) = 27 RT$.

$A$ gas mixture consists of $8$ moles of argon and $6$ moles of oxygen at temperature $T$. Neglecting all vibrational modes,the total internal energy of the system is (in $RT$)

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