Internal Energy And Equipartition

An ideal diatomic gas at moderate temperature has five active degrees of freedom—three translational and two rotational—so its internal energy is $U=\frac{5}{2}nRT$ by the law of equipartition. Each degree of freedom contributes $\frac{1}{2}RT$ per mole, and for an ideal gas the internal energy depends only on temperature, not on volume or pressure. The two rotational modes correspond to tumbling about the two axes perpendicular to the molecular bond, while rotation about the bond axis carries negligible energy. Substituting $n=2$ mol, $T=400$ K, and $R=8.314$ J/mol\cdotK: $U=\frac{5}{2}\times 2\times 8.314\times 400=16,628$ J $\approx 1.66\times 10^4$ J. The value $9.98\times 10^3$ J treats the gas as monatomic with only $\frac{3}{2}nRT$. The value $2.49\times 10^4$ J wrongly assumes seven degrees of freedom by including vibration, which is inactive at this temperature. The value $3.33\times 10^4$ J doubles the correct count of moles. This directly uses the NCERT equipartition theorem applied to a rigid diatomic molecule, and it is worth noting that the same temperature would give a monatomic gas a smaller internal energy because it lacks the rotational storage modes. As a magnitude check, the internal energy of a few moles of gas at several hundred kelvin should be of order $10^4$ J, which matches our result and confirms the calculation is dimensionally and numerically sound.