Mechanical Degrees of Freedom

Introduction to mechanical degrees of freedom

The mechanical degree of freedom of a system is defined as the total number of coordinates or independent quantities required to describe the configuration of the system completely. The gas molecule possesses energy due to three types of motion: Kinetic energy of translational motion, kinetic energy of rotational motion and kinetic energy of vibrational motion. In a system of N particles, if the particle possess k independent relations between them then the number of mechanical degrees of freedom is given by f = 3N – k.


Mechanical degrees of freedom for mono atomic gas molecules


A mono atomic gas molecule can absorb kinetic energy of translational motion along each of the three axes. So that the mono atomic gas molecules 3 degree of freedom corresponding to the translational motion. Because mono atomic gas molecule is a point mass so it cannot absorb kinetic energy of rotational motion and hence they do not have mechanical degree of freedom corresponding to the rotational motion. So in the case of mono atomic gas molecule are 3. Now according to the formula for the mechanical degrees of freedom f = 3 × 1 – 0 = 3 because here N = 1 and k = 0.

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Mechanical degrees of freedom for diatomic gas molecules


Let a molecule of diatomic gas which lying on the X-axis, so that its moment of inertia about Y-axis and Z-axis is non-zero but the moment of inertia is zero for X-axis. So it possess only three mechanical degrees of freedom corresponding to the translational motion. It also possesses the two mechanical degrees of freedom corresponding to the rotational motion. Therefore, the total mechanical degrees of freedom are 3 + 2 = 5. Now according to the formula for the mechanical degrees of freedom f = 3 × 2 – 1 = 5, because here N = 2 and k = 1. It may be remember that at very high temperature, the vibrational motion excites up for the gas molecules so that the mechanical degree of freedom for the diatomic gas molecules is 7, because of 2 degrees of freedom due to the vibrational motion.