Question

Can anyone help me with this. Prove of law of equipartition of energy.

Answer 1

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Answer 2

The law states that: “In thermal equilibrium, the total energy of the molecule is divided equally among all Degrees of Freedom of motion”. Before delving into the calculations, let’s understand the law better. If a molecule has 1000 units of energy and 5 degrees of freedom (which includes translational, rotational and vibrational movements), then the molecule allocates 200 units of energy to each motion.

Now, let us look at some equations!

Kinetic Energy of a single molecule: KE = 1/2 mv2. A gas in thermal equilibrium at temperature T, the average Energy is:

Eavg = 1/2 mvx2 + 1/2 mvy2 + 1/2 mvz2  = 1/2KT + 1/2 KT + 1/2 KT = 3/2 KT

where K = Boltzmann’s constant. In case of a monoatomic molecule, since there is only translational motion, the energy allotted to each motion is 1/2KT. This is calculated by dividing total energy by the degrees of freedom:

3/2 KT ÷ 3 = 1/2 KT

In case of a diatomic molecule, translational, rotational and vibrational movements are involved. Hence the Energy component of translational motion= 1/2 mvx2 + 1/2 mvy2 + 1/2 mvz2. Energy component of rotational motion= 1/2 I1w12 + 1/2 I2w22  {I1 & I2 moments of inertia. w1 & w2 are angular speeds}

And, the energy component of vibrational motion= 1/2 m (dy/dt)2+ 1/2 ky2. Where k is the force constant of the oscillator and y is the vibrational coordinate. It is important to note here that this has both kinetic and potential modes.

According to the Law of Equipartition of Energy, in thermal equilibrium, the total energy is distributed equally among all energy modes. While the translational and rotational motion contributes ½ KT to the total energy, vibrational motion contributes 2 x 1/2KT = KT since it has both kinetic and potential energy modes.

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