Topic : LAW OF EQUIPARTITION OF ENERGY
What do you mean by the law of equipartition of energy? Find an expression for the total energy of a system.
Consider a molecule of gas of mass (m), moving with velocity (v). The translational kinetic energy of the molecule is given by:
$E = \frac{1}{2}mv^2 = \frac{1}{2}m(v_x^2 + v_y^2 + v_z^2)$
$\left( \therefore v^2 = v_x^2 + v_y^2 + v_z^2 \right) \quad \text{...(1)}$
$= \frac{1}{2}mv_x^2 + \frac{1}{2}mv_y^2 + \frac{1}{2}mv_z^2$
But the mean or average translational kinetic energy of a gas molecule is given by:
$< E > = < \frac{1}{2}mv_x^2 > + < \frac{1}{2}mv_y^2 > + < \frac{1}{2}mv_z^2 > = \frac{3}{2}kB T \quad \text{...(2)}$
Since all the three directions i.e., x-axis, y-axis, z-axis are equally preferred, so the average kinetic energy of the gas molecule along all the three directions is equal.
Thus, from equation (2):
$< \frac{1}{2}mv_x^2 > = < \frac{1}{2}mv_y^2 > = < \frac{1}{2}mv_z^2 > = \frac{1}{2}k_B T$
A gas molecule moving in space has three degrees of freedom. So, the energy associated with each gas molecule per degree of freedom is ( $\frac{1}{2}k_B T$). This fact is known as the law of equipartition of energy.
Definition of law of equipartition of energy:
For any system in thermal equilibrium, the total energy of the system is equally distributed among all the degrees of freedom. The energy associated with each degree of freedom per molecule is equal
to ( $\frac{1}{2}k_B T$).
Total energy of a system:
Consider a system having (N) molecules.
Let (f) be the degrees of freedom of each molecule.
Therefore, total degrees of freedom of the system = (Nf).
According to kinetic theory of gases, average kinetic energy per molecule is:
Average kinetic energy per molecule = $\frac{1}{2} m v^2 = \frac{1}{2} k_B T$
where ($k_B$) is Boltzmann constant and (T) is temperature.
Total energy of the system = total degrees of freedom × energy per degree of freedom
$E = Nf \cdot \frac{1}{2} k_B T$
If the system is 1 mole, then ($N = N_A$) (Avogadro number), and $(N_{A} k_{B} = R)$ (universal gas constant), so:
$E = f \cdot \frac{1}{2} R T$