Mixture Of Gases And Internal Energy

Computing the mixture's energy applies the internal energy expressions from NCERT Class 11, Chapter 13 (Kinetic Theory), which give the internal energy of $n$ moles as $U=\frac{f}{2}nRT$, where $f$ is the number of degrees of freedom. For the monatomic component with $f=3$, the energy is $U_1=\frac{3}{2}\times 2\times RT=3RT$. For the diatomic component with $f=5$ at room temperature, the energy is $U_2=\frac{5}{2}\times 2\times RT=5RT$. Since internal energy is additive, the total is $U=U_1+U_2=3RT+5RT=8RT$. The option 5RT counts only the diatomic contribution. The option 10RT wrongly treats both gases as diatomic. The option 6RT wrongly treats both as monatomic. The additivity works because internal energy is an extensive property, so the energies of independent components in thermal equilibrium at the same temperature simply add together. A plausibility check confirms the total: the diatomic gas, having more degrees of freedom, must store more energy than the monatomic gas at the same temperature and mole number, and indeed $5RT>3RT$, with their sum $8RT$ lying sensibly between the all-monatomic value $6RT$ and the all-diatomic value $10RT$.