The Maxwell-Boltzmann Speed Distribution:

The Ideal Gas

For many systems, including the ideal gas, the Maxwell-Boltzmann distribution provides the probability of finding a component in a particular state with a particular amount of energy. In the case of the gas, many different states have the same speed; a particle could be going left, right, up, down, back, front, or at any angle, and still be traveling the same speed. To find the probability of a particle having a particular speed, we must compute how many states have the same speed, and add up the probabilities of being in any of these states. The kinetic energy of a particle with mass m and speed u is

So the probability of finding the particle in any one state is

The number of states with the same speed is given by the surface area of a spherical shell,

(This is the same factor which enters when converting a three-dimensional Cartesian integral of a spherically-symmetric function to spherical coordinates.) Every state has a different velocity vector. All the states with the same speed have velocity vectors with the same length u, pointing in different directions. All these velocity vectors sweep out a spherical shell with radius u. Therefore, the probability of finding an atom with a particular speed is

The next step is to evaluate the normalization constant C. Since P(u) is a continuous distribution, we must integrate over all values of u (from 0 to infinity). This involves some advanced mathematics, and gives (see the section on finding average properties of the gas for a step-by-step derivation):

Exercises using the Maxwell-Boltzmann Speed Distribution

  1. Run the JAVA applet for 10 atoms and a variety of temperatures. Find the most probable speed for each temperature. How does the most probable speed change with temperature?
  2. Run the applet for 2, 10, and 20 atoms at 1000K.  Describe how the histogram of speeds changes as the simulation progresses.  What shape does the histogram attain after a long run?  How does the number of atoms affect the histogram?
  3. Vary the number of particles in the simulation for a fixed temperature. How does this affect the most probable speed?
  4. Which term in the Maxwell-Boltzmann Speed distribution gives the function its shape at low speeds (near u=0)?  Which term is most important for high speeds (u>1000m/s)?
  5. Thought question:  Suppose that a gas of atoms is trapped in a two-dimensional region (a box with length and width, but just about zero height).  How would the Maxwell-Boltzmann speed distribution be different for this case?

The Big Picture

    The Maxwell-Boltzmann speed distribution tells us the probability of finding an atom or molecule in the gas with any particular speed.  We can use this information to make powerful statements about the gas as a whole by computing average properties from the probability distribution.

© Andrew M. Rappe