Law of Equipartition of Energy State

The law of equipartition of energy was first proposed by Sir John James Waterston in 1843. Sir Waterston shared a refined version of the same in 1845 with more accuracy. The equipartition of energy theorem states that “the total energy of a dynamic system residing in thermal equilibrium is shared equally among all the degrees of freedom. The energy possessed by each particle/molecule in every degree of freedom is equally divided in every mode.” This theorem of equipartition of energy touches upon concepts in the field of quantum mechanics and the degree of freedom of a system. Degree of freedom is the concept in quantum mechanics that defines the state and configuration of a molecule or system in 3-Dimensional space. According to the kinetic theory of gases, the average kinetic energy of a molecule is half the product of mass and root-mean-square velocity. Different theories and explanations were put forward by different physicists, both supporting and criticising the law. However, the law is now largely accepted across the world, albeit after various iterations.

Later, in 1859, Sir James Clerk Maxwell put forward the idea that the kinetic energy of gas due to an increase in internal energy is cut into two linear and rotational energies. However, the experimental observations of the specific heat capacity of gases questioned the validation of this law. The correct explanation for this failure in the experimental setup was brought up by Ludwig Boltzmann in 1876 by expanding the principle by showing the average energy of the molecule is equally divided among all independent components of motion in the particular system. Boltzmann also used this law to support the ideas of Dulong-Petit’s law for specific heat capacities of solids. In 1906, Albert Einstein supported this law of equipartition of energy by putting forward the main cause of the anomalies in the specific heat of solids. These reasons were specifically due to the quantization of energy in the elastic modes of the solid. This argument was proved again by W.H. Nernst’s measurement of specific heat at lower temperatures. These contributions by various physicists over the years have led to major refinement of the law that is now widely accepted.

Even though the equipartition of energy theorem provides accurate predictions in most conditions, it is found to be inaccurate when quantum effects are significantly changed, especially at low temperatures. When the thermal energy is less than the difference in quantum energy level in a particular degree of freedom, the average energy and heat capacity of this degree of freedom as predicted by the equipartition of energy theorem is higher than the actual observed values. Such a degree of freedom is called “frozen out” when the thermal energy is much smaller than their difference in respective energy levels.

Degree of freedom:

The degree of freedom of a physical system refers to an independent physical parameter that defines its configuration or state. The molecular degree of freedom of a molecule is the total number of ways a molecule in the gas phase can move, vibrate or rotate in 3-dimensional space.

Example: A monatomic gas has three degrees of freedom i.e. it has linear, rotational and vibrational energy.

Applications:

The law of equipartition of energy has several applications as follows:

1. Equipartition to derive the classical ideal gas law from Newtonian mechanics.
2. The equipartition theorem is used to derive the energy and pressure of non-ideal gases.
3. Equipartition theorem is also used in astrophysics.
4. Helps in deriving the Brownian motion of a particle.

Limitations:

1. The law of equipartition of energy collapses when the thermal energy is significantly lesser than the difference between energy levels. Equipartition of Energy becomes invalid due to haphazardly distributed energy in different modes. 
2. The law of equipartition is valid only for ergodic systems in thermal equilibrium, which signifies that all states with the same energy are equal and likely to be populated as well. A general example is where energy is not shared among its various different forms and where equipartition of energy does not hold in the system of coupled harmonic oscillators. If the system is in an isolated condition, then the energy in each normal mode is constant and does not vary with respect to time. Thus, equipartition does not hold for such systems; the amount of energy in each normal mode is fixed at its initial value. 

Conclusion:

The law of equipartition of energy was first proposed by Sir John James Waterston in 1843, and then later in 1845 with more accuracy. The equipartition of energy theorem states that “the total energy of a dynamic system residing in thermal equilibrium is shared equally to all the degrees of freedom. The energy possessed by each molecule/particle in every degree of freedom is equally divided in every mode.” This theorem also gave a rough idea in the field of quantum mechanics and the degree of freedom of a system. The contribution of several other physicists to this theorem, explaining the flaws in a scientific manner, led it to wide acceptance.