# Boltzmann Distribution

*Edited by Pichet Adstamongkonkul, AP225, Fall 2011*

## Introduction

Widely known to be one of the most important fundamental concepts in Chemistry, Quantum Mechanics, Statistical Mechanics and Thermodynamics, the Boltzmann Distribution is an exponential distribution law that describes the probability distribution of states based on the underlying energy levels. For instance, if we are looking at the states of the configurations of a molecule, the Boltzmann distribution function will be the fraction of molecules that are in each conformation.

## Definition

Given a system with N particles, if we let <math>p_i = \frac{n_i}{N}</math> represent the fraction of a molecule in the state i, which has energy Ei associated with it, then the Boltzmann distribution has a general form of:

<math>\frac{n_i}{N}=p_i^*=\frac{g_i e^{\frac{-E_i}{k_BT}}}{Q}</math> , where <math>Q=\sum_{i=1}^N g_i e^{\frac{-E_i}{k_BT}}</math> is the partition function and gi is the degeneracy. KB is the Boltzmann’s constant, which equals to 1.38 x 10-23 J/K. N is a total number of molecules or particles in the system (<math>N=\sum_{i=1}^N n_i</math>)

The degeneracy of the state sometimes occurs when there are more than one *microstates* (the different states of molecules which have the same energy; each microstate is like a *snapshot* of the system) for each *macrostate* (a collection of microstates). The degeneracy becomes important in the partition of rotational energy of molecules.

For each molecule or particle, there is some *internal energy*, due to its intrinsic property, such as rotational or vibrational states of the molecule. The sum total of all the internal and interaction energies for a particular state i of the system gives the value of each energy level, <math>E_i</math>.

One can also find the relative populations of particles in two different microstates I and j at equilibrium by

<math>\frac{p_i^*}{p_j^*}=e^{\frac{-(E_i-E_j)}{kT}}</math>

The Boltzmann distribution only applies to particles at high enough temperature and low enough density that the quantum effects can be ignored. It predicts more particles, molecules, or configurations will be in the low energy states, while a few will have high energies, given that all energy levels are equivalent. If each molecule in the system takes a small fraction of energy of the system, there will be more ways to distribute the remaining energy to other molecules in the system.

## Application

The Boltzmann distribution can be applied to find the number of molecules, or the particle densities, in each layer of the atmosphere according to the gravitational field.

For the Newtonian particles with velocities and kinetic energy, the Boltzmann distribution provides the basis of another distribution function, the Maxwell-Boltzmann distribution, which gives a fundamental relationship between temperature and the velocities of the gas molecules.

<math>P(v_x)=\frac{m}{\sqrt{2 \pi kT}} e^{\frac{-m v_x^2}{2kT}}</math>

Classical particles with this energy distribution are said to obey Maxwell-Boltzmann statistics.

## References

[1] Ken A. Dill, and Sarina Bromberg. Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology. New York: Garland Science, 2003. 171-77.

[2] Wikipedia contributors. “Boltzmann distribution.” Wikipedia, The Free Encyclopedia. 9 Dec 2011.

[3] Edmund L. J. Tisko. “Boltzmann Distribution.” University of Nebraska at Omaha. Lecture.