Boltzmann factor and equipartition principle

[YFrancis]

Could someone help me understand the Boltzmann factor?

Edited April 20, 2015 by YFrancis

[studiot]

The Boltzman factor and the equipartition system refer to different things.

 

First the equipartition principle

This should not be confused with a partition function Z, which may be related to the Boltzman factor.

 

When a system has different modes of accepting and storing thermal energy, translational KE, PE, rotational, vibrational etc of its molecules, the EP asserts that the total energy is divided equally between all the modes. Each mode has an average that represents it, so there is an average molecular velocity that represents the average spped of the molecules at any given temperature and so on.

 

But, of course, the actual velocities are distributed about this average.

Further since the kinetic energy depends upon the square of the speed the distribution will not be symmetrical ie there will be many more slow molecules than the fast ones to balance the average.

 

A partition function, Z, is an equation or function that describes this distribution.

 

A particular partition function is the Boltzman distribution, where the Boltzman factor comes in to play.

 

It is assumed that there are a finite number of molecules and a finite number of energy levels available to the molecules.

 

The partition function describes the distribution between these levels.

 

If there are only two levels (1) and (2) or between any pair of levels we label (1) and (2) the Boltzman factor applies.

This is the ratio of the population or number of molecules in each level N₁ and N₂, in terms of the energies of those levels, E₁ and E₂ and the absolution temperature.

 

[math]\frac{N_{2}}{N^{_{1}}} = e^{-\beta \left ( E_{2} -E_{1}\right )}[/math]

 

Where

[math]\beta = \frac{1}{kT}[/math]

 

T is the absolute temperature and k is Boltzman's constant.

 

The total can be reckoned by summing over all pairs.

 

It should be noted that these calculations assume that the system is in thermal equilibrium.

Edited April 20, 2015 by studiot

[YFrancis]

Studiot: Great! Many thanks for your explanation. It is very clear.