Q.(a) State and explain equipartition of energy. OR

This law says that in thermal equilibrium, a gas's total internal energy is spread out equally, in shares of (1/2)k_BT per molecule, over every independent quadratic 'degree of freedom' — translational, rotational, or vibrational — that the molecule has.

Statement of the Law of Equipartition of Energy: For a system in thermal equilibrium at absolute temperature T, the total energy is distributed EQUALLY among all its degrees of freedom, and the average energy associated with EACH degree of freedom that appears as a quadratic term in the total energy expression (e.g., (1/2)mv_x², (1/2)Iω²) is (1/2) k_B T per molecule, where k_B is Boltzmann's constant.

Explanation with examples:

1. Translational degrees of freedom: A gas molecule moving in 3D has velocity components vx, vy, vz, each contributing a kinetic energy term (1/2)mv². These are 3 independent quadratic terms (3 translational degrees of freedom), so average translational energy = 3 × (1/2)k_BT = (3/2)k_BT per molecule.

2. Rotational degrees of freedom: A rigid diatomic molecule can additionally rotate about 2 independent axes (perpendicular to the bond axis), each contributing a rotational kinetic energy term (1/2)Iω². This gives 2 more degrees of freedom, contributing 2 × (1/2)k_BT = k_BT.

   So total average energy of a rigid diatomic molecule = (3/2)k_BT (translational) + k_BT (rotational) = (5/2)k_BT.

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