Question

Assertion (A): The total internal energy of a diatomic gas at high temperature is higher than that of a diatomic gas at low temperature. Reason (R): At high temperatures, rotational and vibrational degrees of freedom also add to the internal energy of the gas.

The molecule of a monatomic gas has only three translational degrees of freedom. Thus, the average energy of a molecule at temperature T is (3/2) kB T. The total internal energy of a mole of such a gas is U= 3/2 RT, and Cv= 3/2 R. A diatomic molecule treated as a rigid rotator, like a dumbbell, has 5 degrees of freedom: 3 translational and 2 rotational. Using the law of equipartition of energy, the total internal energy of a mole of such a gas is U =5/2 RT, CV =5/2 R. A polyatomic molecule has 3 translational, 3 rotational degrees of freedom and a certain number (f) of vibrational modes. According to the law of equipartition of energy it is easily seen that one mole of such a gas has, CV = (3+f) R. Note: CP-CV=R is true for any ideal gas, whether mono, di or polyatomic. The ratio of specific heats γ = CP/CV