According to law of equipartition of energy the molar specific heat of a diatomic gas at constant volume where the molecule has one additional vibrational mode is :-

Solution

Physics sir · 4 steps

A vibrational mode stores both kinetic and potential energy, so it counts twice, not once.

1. Rigid diatomic first. A diatomic molecule has 3 translational and 2 rotational degrees of freedom. By equipartition each one holds 1/2kT, which gives 5 terms of 1/2RT per mole.
2. Add the vibration. In one vibrational mode the two atoms oscillate along the bond like a spring. That mode has a kinetic energy term and a potential energy term, so it adds two terms of 1/2RT, a total of RT per mole.

   U = (3/2 + 2/2 + 2/2)RT = 7/2RT

3. Differentiate. C_V = dU/dT for one mole, so C_V = 7/2R.
4. So. The molar specific heat at constant volume is 7/2R, option (d).

The idea to keep. Each translational or rotational degree of freedom gives 1/2R to C_V. Each vibrational mode gives a full R, because it has both kinetic and potential energy.

Answer from the NTA final answer key. Solution by PrepWiser.