Law of equipartition of energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by all the degrees of freedom. The energy associated with each degree of freedom per moelcule is ? kT. where k is the Boltzmann?s constant.

**Law of equipartition of energy**

Law of equipartition
of energy states that for a dynamical system in thermal equilibrium the total
energy of the system is shared equally by all the degrees of freedom. The
energy associated with each degree of freedom per moelcule is ? kT. where k is the Boltzmann?s constant.

Let us consider one
mole of a monoatomic gas in thermal equilibrium at temperature T. Each molecule
has 3 degrees of freedom due to translatory motion. According to kinetic theory
of gases, the mean kinetic energy of a molecule is 3 /2 kT.

Let C_{x} , C_{y}
and C_{z} be the components of RMS velocity of a molecule along the
three axes. Then the average energy of a gas molecule is given by

? mC^{2} = ?
mC_{x}^{2} + ? mC_{y}^{2}
+ ? mC_{z}^{2}

? mC_{x}^{2}
+ ? mC_{y}^{2} + ? mC_{z}^{2} =3/2 kT

Since molecules move
at random, the average kinetic energy corresponding to each degree of freedom
is the same.

? mC_{x}^{2}
+ ? mC_{y}^{2} + ? mC_{z}^{2}

? mC_{x}^{2}
+ ? mC_{y}^{2} + ? mC_{z}^{2}= 1/2 kT

∴ Mean kinetic energy per molecule per degree of freedom is 1/2
kT

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