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:
Table of contents |
2 Velocity Distribution in One Direction 3 Distribution of Speeds 4 Average Speed |
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 ax, ay, az are given by:
Maxwell-Boltzmann Velocity Distribution
where, nx, ny, and nz 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 (h2ni2/4ai2) corresponds to the square of the ith component of momentum, pi2 giving:
where q corresponds to the denominator in Equation 1. This distribution of Ni/N is proportional to the probability distribution function fp for finding a molecule with these values of momentum components, so:
It can be shown that:
For the case of a single direction Equation 8 can be reduced to:
Usually, we are more interested in the speed of molecules rather than the component velocities, where speed, v is defined such that:
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, vp, 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:
The mean speed,
The root mean square speed, vrms is given by
Velocity Distribution in One Direction
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.Distribution of Speeds
The corresponding speed distribution is:Average Speed
Most Probable Speed
Mean Speed
Substituting in Equation 11 and performing the integration gives:
Note that Root-mean-square Speed
Substituting for F(v) and performing the integration, we get
Thus,