This page contains...

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

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

- draw a frequency-speed graph according to the Maxwell-Boltzmann distribution and indicate on it different graphs for different temperatures
- explain why small temperature increases cause large increases in the number of particles possessing more than the activation energy

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 20°C) 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 e^{x} 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)mv^{2}.
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.

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 (0°C). You should notice
that there are few particles with low speeds, many with medium
speeds and few with high speeds. T**he 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 (100°C) 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.**

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

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

- Draw a graph showing the Maxwell-Boltzmann distribution
of speeds in an ideal gas. Indicate an approximate speed
scale in ms
^{-1}. - Explain how you would use such a graph to find the proportion of molecules capable of producing a transition state in a gaseous reaction.
- Explain why changing the temperature by (say) 20°C 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.

Last updated 25 April 1999.

Corrections and suggestions gratefully accepted.