Equipartition theorem: Rotational motion

We begin by deriving the classical expression for the energy of a body rotating about an axis (see diagram below).

The body has an angular velocity  of

where  is the period, i.e. the time taken for the body rotate an angle of  radians about the -axis.

Multiplying eq47 on both sides by the radius , we have:

where is the tangential velocity of the body with respect to the -axis.

The energy of the rotating body is also purely kinetic:

Substituting the definition of rotational inertia or moment of inertia about the -axis, , in eq48,

In three dimensions, the total rotational energy of the body is:

If the body is a molecule, its moment of inertia about an axis is the sum of the moments of the inertia of atoms making up the molecule about that axis:

where  is the perpendicular distance from the -th atom to the rotational axis, which passes through the centre of mass of the molecule.

For a non-linear molecule (see above diagram), its total rotational energy is:

In other words, there are three degrees of freedom of rotational motion for a non-linear molecule that are associated with three rotational energy components. For a linear molecule, its moment of inertia about one of the axes (arbitrarily taken as the -axis) is zero because . Hence, there are only two degrees of freedom that are associated with two energy components for a linear molecule:

According to the equipartition theorem mentioned in the previous article, each kinetic energy component of a molecule has an average energy of . Hence, for a system of molecules, the average rotation energy of a linear molecule and the average rotation energy of a non-linear molecule are  and respectively.


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