Maxwell–Boltzmann statistics. And limitations of Maxwell–Boltzmann statistics

Maxwell–Boltzmann statistics

Classical statistics is generally known as Maxwell–Boltzmann statistics because it got its birth from Maxwell’s law of distribution of molecular speed and Boltzmann’s theorem relating entropy and probability.

The classical statistics i.e. Maxwell-Boltzmann statistics explains successfully many observed phenomenon like temperature, pressure, energy etc. On the other hand, it fails to explain certain phenomenon like photo-electric effect, black body radiation etc., and thus a new branch of statistics known as quantum statistics came into existence.

It was first formulated by Dr. S.N. Bose in the deduction of Planck’s radiation law of purely statistical reasoning on the basis of certain fundamental assumptions which are entirely different from those of the classical statistics. Einstein in the same year used almost the same principle to evolve the kinetic theory of gases. Hence, the quantum statistics is called Bose-Einstein (B-E.) statistics.

Sr. no.	Symbol	Term Used as
1.		is the energy of the i-th energy level,
2.	Ni	is the average number of particles in the set of states with energy ,
3.	gi	is the degeneracy of energy level i, that is, the number of states with energy which may nevertheless be distinguished from each other by some other means.
4.	μ	is the chemical potential.
5.	k	is Boltzmann’s constant.
6.	T	is absolute temperature

Maxwell–Boltzmann statistics derivation

Consider an ideal gas contained in an enclosure of volume V. Let the number of molecules of the gas be very large i.e. n (say). These molecules go on colliding with each other and with the wall of the container.

As a result of these collisions, the momentum and energy of the gas molecules go on changing but the total energy of the whole system (i.e. gas) remains constant. The system under study is assumed to be an isolated system.

The total energy (U) of the system is divided into k intervals of magnitude u₁, u₂, u₃……. u_k respectively. Let these energy intervals be numbered as 1, 2, 3, …. k. Now n molecules of an ideal gas are to be distributed in k energy intervals.

Let the number of molecules lying in 1, 2, 3, …. k energy intervals respectively be n₁, n₂, n₃, ………., n_k. This is equivalent to the distribution of n particles in k compartments. The difference is only that here instead of compartments, we have energy intervals.

Further, suppose g₁, g₂, g₃, ……. g_k be the number of cells in phase space corresponding to the energy intervals 1, 2, 3 ….. k respectively. The total number of microstates (i.e., thermodynamical probability) of the macrostate (n₁, n₂, n₃…. n_k) is given by

Taking log of eqn. (1), we get

Since n and n_i are very large, therefore, Stirling’s formula log n ! = n log n – n can be applied.

Differentiating both sides, remembering that g_i and n are constant.

For most probable state, W = W_max

d log W = 0

Hence eqn. (3), becomes

Now, we introduce the condition of conservation of total number of molecules and total energy of the system.

Also

Multiplying eqn. (5) by α and eqn. (6) by β and subtract them from eqn. (4), we get

As variation dn_i are independent of each other.

Which is Maxwell Boltzmann statistics equation.

Here, n_i/g_i is called occupation index and it is equal to the number of particles per phase space cell in a compartment. In Maxwell Boltzmann’s distribution, the value of n_i/g_i< 1. That is, number of particles in a system is less than the number of phase cells in a compartment.

limitations of Maxwell Boltzmann statistics

• Maxwell–Boltzmann statistics is valid only in the classical limit.
• It applies well to an ideal gas.
• It also gives you the approx specific heat for solids in temperatures high enough.
• it fails to explain certain phenomenon like photo-electric effect, black body radiation etc.