# Maxwell-Boltzmann distribution

The

**Maxwell-Boltzmann distribution** is an important relationship that finds many applications in physics and chemistry. It forms the basis of the

kinetic theory of gases, which accurately explains many fundamental gas properties, including pressure and diffusion. The Maxwell-Boltzmann distribution also finds important applications in electron transport and other phenomena.

The Maxwell-Boltzmann distribution can be derived using statistical mechanics (see the derivation of the partition function). It corresponds to the most probable energy distribution, in a collisionally-dominated system consisting of a large number of non-interacting particles. Since interactions between the molecules in a gas are generally quite small, the Maxwell-Boltzmann distribution provides a very good approximation of the conditions in a gas.

In many other cases, however, the condition of elastic collisions dominating all other processes is not even approximately fulfilled. That is true, for instance, for the physics of the ionosphere and space plasmas where recombination and collisional excitation (i.e. radiative processes) are of far greater importance: in particular for the electrons. Not only would the assumption of a Maxwell distribution yield quantitatively wrong results, but even prevent a correct qualitative understanding of the physics involved.

The Maxwell-Boltzmann distribution can be expressed as:

- (1)

where N

_{i} is the number of molecules at equilibrium temperature T, having energy level E

_{i}, N is the total number of molecules in the system and k is

Boltzmanns constant. Essentially Equation 1 provides a means for calculating the fraction of molecules (N

_{i}/N) that have energy E

_{i} at a given temperature, T. Because velocity and speed are related to energy, Equation 1 can be used to derive relationships between temperature and the speeds of molecules in a gas.

For the case of an "ideal gas" consisting of non-interacting atoms in the ground state, all energy is in the form of kinetic energy. From the Particle in a box problem in Quantum mechanics we know that the energy levels for a gas in a rectangular box with sides of lengths a_{x}, a_{y}, a_{z} are given by:

- (2)

where, n

_{x}, n

_{y}, and n

_{z} are the quantum numbers for x,y, and z motion, respectively. However, for a macroscopic sized box, the energy levels are very closely spaced, so the energy levels can be considered continuous and we can replace the sum with an integral. Furthermore, we can recognize that (h

^{2}n

_{i}^{2}/4a

_{i}^{2}) corresponds to the square of the ith component of momentum, p

_{i}^{2} giving:

- (3)

where q corresponds to the denominator in Equation 1. This distribution of N

_{i}/N is

proportional to the probability distribution function f

_{p} for finding a molecule with these values of momentum components, so:

- (4)

The constant of proportionality, c, can be determined by recognizing that the probability of a molecule having

*any* momentum must be 1. Therefore the integral of equation 4 over all p

_{x}, p

_{y}, and p

_{z} must be 1.

It can be shown that:

- (5)

so in order for the integral of equation 4 to be 1,

- (6)

Substituting Equation 6 into Equation 4 and using p

_{i}=mv

_{i} for each component of momentum gives:

- (7)

Finally recognizing that the velocity probability distribution, f

_{v} is proportional to the momentum probability distribution function as

we get:

- (8)

Which is the Maxwell-Boltzmann velocity distribution.

For the case of a single direction Equation 8 can be reduced to:

- (9)

This distribution has the form of a Gaussian error curve. As expected for a gas at rest, the average velocity in any particular direction is zero.

Usually, we are more interested in the speed of molecules rather than the component velocities, where speed, v is defined such that:

- (10)

The corresponding speed distribution is:

- (11)

## Average Speed

Although Equation 11 gives the distribution of speeds or in other words the fraction of molecules having a particular speed, we are often more interested in quantities such as the average speed of the particles rather than the actual distribution. In the following subsections we will define and derive the **most probable speed**, the **mean speed** and the **root-mean-square** speed.

The **most probable speed**, v_{p}, is the speed most likely to be possessed by any molecule in the system and corresponds to the maximum value or mode of F(v). To find it, we calculate dF/dv, set it to zero and solve for v:

- (12)

- (13)

### Mean Speed

The **mean speed**, , or average speed can be calculated using the expression:

- (14)

Substituting in Equation 11 and performing the integration gives:

- (15)

Note that and v_{p} differ by a constant factor (4/π)^{1/2}.

The root mean square speed, v_{rms} is given by

- (16)

Substituting for F(v) and performing the integration, we get

- (17)

Thus,