Internal Energy Of A Gas Mixture

The internal energy of a gas mixture is the sum of the internal energies of its components, each evaluated from its own degrees of freedom at the common temperature, since the gases are non-interacting and share a single equilibrium temperature. For the monatomic gas, $U_1=\frac{3}{2}{n}_{1}RT$; for the diatomic gas, $U_2=\frac{5}{2}{n}_{2}RT$. With $n_1=2$ mol and $n_2=4$ mol at $T=300$ K, the total is $U=\frac{3}{2}(2)RT+\frac{5}{2}(4)RT=(3+10)RT=13RT$. Substituting $R=8.314$ J/mol\cdotK and $T=300$ K gives $U=13\times 8.314\times 300\approx 3.24\times 10^4$ J. The value $2.49\times 10^4$ J wrongly treats every mole as diatomic. The value $4.99\times 10^4$ J counts the diatomic component as having seven degrees of freedom by including frozen vibration. The value $1.50\times 10^4$ J omits the diatomic contribution entirely. This uses the NCERT principle that energy is additive across non-interacting gas components, each obeying equipartition independently with its own characteristic share. Because the diatomic species supplies four of the six moles and carries more energy per mole, it dominates the total internal energy of the mixture. As a plausibility check, the energy per mole works out to $(13/6)RT=2.17RT$, lying between the monatomic value $1.5RT$ and diatomic value $2.5RT$ and skewed toward diatomic since it dominates the mixture, confirming the result.