The Maxwell-Boltzmann distribution


Previous Next Back to the home page


This page contains...

  1. The objectives which you are trying to achieve from this page.
  2. The learning material itself.
  3. A review of the main points.
  4. A formative assignment to help you confirm what you have learned.

Objectives or back to the top

After working through this lesson, you should be able to ...

Content or back to the top

This lesson is about how fast the particles of a gas move about. Every particle of a gas is in motion, and therefore possesses some energy - kinetic energy, the energy due to motion. As a rough guide, the molecules of oxygen and nitrogen of the air at normal temperatures (say 293K, or 20C) move around at about 400ms-1 which is similar to the speed of sound in air. Of course, not all of these particles move at the same speed - some are fast and some slow. The Maxwell-Boltzmann distribution shows how many particles in a gas have the different speeds.

Why is this useful? Well, in the last lesson, you learnt that for a reaction to take place, molecules must collide with sufficient energy to create a transition state. This usually only happens for the fastest molecules, so if the Maxwell-Boltzmann distribution tells us how many molecules have energies or speeds above a certain level, that will be useful to us. Ludwig Boltzmann developed a very general idea about how energy was distributed among systems consisting of many particles. He said that the number of particles with energy E would be proportional to the value e(-E/kT) where exp represents the exponential function (often shown as ex on calculators), T is the absolute temperature and k is a universal constant known as Boltzmann's constant. Boltzmann's constant is very important in physics and chemistry, and an equation containing it was the epitaph on Boltzmann's tombstone following his tragic suicide.

James Clerk Maxwell was a famous English physicist who used Boltzmann's ideas and applied them to the particles of an ideal gas to produce the distribution bearing both men's names. He also used the usual formula for kinetic energy, that E=(1/2)mv2. The distribution is best shown as a graph which shows how many particles have a particular speed in the gas. It may also be shown with energy rather than speed along the x axis. This changes its shape a little, but not much. Here, we will plot speed along the x-axis. A graph is shown below.

Graph of Maxwell-Boltzmann distribution

In this graph there are three lines. These are drawn for the same number of particles but for different temperatures. For now concentrate on the one marked 273K (0C). You should notice that there are few particles with low speeds, many with medium speeds and few with high speeds. The area below the line represents the total number of particles in the gas. Suppose you want to know how many particles are moving fast enough to produce a transition state in some reaction. Let's assume that this is 1000ms-1. You would need to find the area under the line from the speed of 1000ms-1 upwards. The number of particles with speed enough to cause a reaction is the area under the graph from that speed upwards.

How does the distribution change as the temperature increases? Well, look at the 373K (100C) line. The total number of particles is the same so the total area under the line is the same, but more particles have the faster speeds. The line has to get lower because the area underneath it can't change.

More particles at faster speeds is what you would expect at a higher temperature. The important point about the graph is that there are more particles with speeds above 1000ms-1 (the critical speed needed for a transition state). You can see this is true because the 373K line is higher than the 273K line for speeds above 1000ms-1. It is also true for the 473K line. Notice that at the very highest speeds, the area under the line is very dependent on the temperature. For most reactions, it is only the very fastest molecules in the distribution which would have the activation energy. The number of these molecules increases very rapidly as the temperature is raised. Therefore, even small increases in temperature can cause a large increase in the number of particles with sufficient speed to make a transition state.

Review or back to the top

  1. The area below the line in the Maxwell-Boltzmann distribution represents the total number of particles in the gas.
  2. The number of particles with speed enough to cause a reaction is the area under the graph from that speed upwards.
  3. Even small increases in temperature can cause a large increase in the number of particles with sufficient speed to make a transition state.

Formative assignment or back to the top

This assignment should be done without looking at these notes or other notes about the Maxwell-Boltzmann distribution.

  1. Draw a graph showing the Maxwell-Boltzmann distribution of speeds in an ideal gas. Indicate an approximate speed scale in ms-1.
  2. Explain how you would use such a graph to find the proportion of molecules capable of producing a transition state in a gaseous reaction.
  3. Explain why changing the temperature by (say) 20C would make little difference to the general shape of the Maxwell-Boltzmann graph, but would make quite a big difference to the rate of reaction between two gases.

Previous Next Back to the home page


These lessons first compiled by Trevor Burton January 1996.
Last updated 25 April 1999.
Corrections and suggestions gratefully accepted.