Equipartition Of Energy

As we discussed the equipartition of energy, we can say that there are some laws related to the equipartition of energy. The law of equipartition states that when the total energy for the system is divided among the degrees of freedom, they only need three degrees of freedom to know about its position. So, first, we have to know what the degrees of freedom are.

Degree of freedom

The normal components (sq. value) of velocity of a gas molecule c are equally distributed in thermal equilibrium, i.e., along three axes X, Y, and Z include u2 = v2 = w2 as per the kinetic theory of gases. The average value of the components of velocities along the X, Y, and Z axes is u, v, and w, respectively.

½ mu2 = ½ mv2 = ½ mw2

But c2 = u2 + v2 + w2 and u2 = v2 = w2

so, ½ mu2 = ½ mv2 = ½ mw2 = (⅓) x  ½ mc2

Again, we know the average kinetic energy of each molecule.

½ mc2 = 3/2 KT

Then, ½ mu2 = ½ mv2 = ½ mw2 = ½ x (3/2) KT = ½ KT

As a result, the average energy for each degree of freedom is calculated which is ½ KT.

Again, in the case of an oscillating particle, kinetic energy makes up half of the total energy and potential energy makes up the other half. As a result, the total energy per degree of freedom equals kinetic energy plus potential energy, which equals ½ KT + ½ KT = KT.

Law of equipartition of energy formula:

According to the law of equipartition of energy, the total energy of any dynamic system in thermal equilibrium is evenly distributed among the degrees of freedom.

A single molecule’s kinetic energy along the x, y, and z axes is given as

Along with the x-axis → ½mvx²

Along with the y-axis → ½mvy²

Along with the z-axis→ ½mvz²

The average kinetic energy of a molecule, according to the kinetic theory of gases, is given by

½ mvrms ²=(3/2)K T

where vrms is the root-mean-square velocity of the molecules, K is the Boltzmann constant, and T is the temperature of the gas

Because a monatomic gas has three degrees of freedom, the average kinetic energy per degree of freedom is

KEx= ½KT

If a molecule is free to move in space, it needs three coordinates to define its location, meaning that it has three degrees of freedom in translation. It has two translational degrees of freedom if it is confined to travel in a plane, and one translational degree of freedom if it is constrained to move in a straight line. A triatomic molecule has six degrees of freedom. The kinetic energy of the gas per molecule is calculated as follows:

6×N×½ KT = 3×(R/N)NKT = 3RT

Equipartition of energy 

The average energy of degree of freedom is ½ kT. But if the equation is non-quadratic, then U = Nf( 1/2kT) (where Nf is number of particles). It’s mainly not the total energy but the energy stored in chemical bonds. A gas in the gravitational field (mgz) is not quadratic. With complex molecules, mainly internal energy is taken, but there is an easy way for the energy change: the translational motion of the centre of mass. For example, an atom of a gas has three degrees of freedom (the three spatial, or position, coordinates of the atom) and will, therefore, have an average total energy of 3/2kT. 

For an atom in a solid, vibratory motion involves potential energy and kinetic energy, and both modes will contribute a term of 1/2 kT, resulting in average total energy of 3kT. The equipartition theorem states that energy is shared equally amongst all energetically accessible degrees of freedom of a system. This is not a particularly surprising result and can be thought of as another way of saying that a system will generally try to maximise its entropy (i.e. how spread out the energy is in the system) by distributing the available energy evenly amongst all the accessible modes of motion. The degree of freedom can be viewed as the minimum number of coordinates required to specify a configuration. Applying this definition, we have:

•For a single particle in a plane, two coordinates define its location, so it has two degrees of freedom;

• A single particle in space requires three coordinates, so it has three degrees of freedom.

• Two particles in space have a combined six degrees of freedom.

• If two particles in space are constrained to maintain a constant distance from each other, such as in the case of a diatomic molecule, then the six coordinates must satisfy a single constraint equation defined by the distance formula. This reduces the degree of freedom of the system to five because the distance formula can be used to solve for the remaining coordinate once the other five are specified.  

The equipartition theorem

It relates the temperature of a system with its average energies. The original idea of equipartition was that in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in the translational motion of a molecule should equal that of its rotational motions. At temperature T, the average energy of any quadratic degree of freedom is 1/2 kT. Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, such as at low temperatures. When the thermal energy kT is smaller than the quantum energy spacing in a particular degree of freedom, the average energies and heat capacities of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is “frozen out” when the thermal energy is much smaller than this spacing. 

Conclusion

In this article, we have learnt about equipotential energy. It’s the kinetic energy equally distributed among degrees of freedom, and its principle and laws state that the average energy of degree of freedom is ½ kT. We derive the equation of equipotential of energy and the formula of a degree of freedom.