The Kinetic Model of Gases

The kinetic model is based upon 3 assumptions;

  1. the molecules of the gas are in ceaseless random motion.

  2. they have no size

  3. there is no potential energy between them, ie they don’t interact except in perfectly elastic collisions

In these collisions, the molecules may exchange kinetic energy, ie one may speed up and the other slow down. Thus molecules in a sample of gas do not all travel at one particular speed, but at a range of different speeds. The distribution f(x) of speeds over this range is known as the Maxwell Distribution, after the man who derived its precise form;

And this gives a graph of the form:

The formula can be broken into 3 parts,

  • constant (1st term)

  • v2 (2nd term)

  • exp -(kv2) (3rd term)

So we could consider the graph a competition between the 2nd (increasing) and 3rd (decreasing) terms.

The graph gives rise to 3 particularly useful molecular speeds;

  1. Most probable speed, c*, obviously the speed that particles are most likely to be travelling at. It corresponds to the maximum point in the above graph.

  2. Mean speed, c. This is the average of all the speeds and is to the right of the maximum since the graph is asymmetric and “tails off” that way.

  3. Root Mean Square speed, crms. Whilst the most probable and mean speeds have clear physical significance, the root mean square speed is purely a function of all the molecular speeds. We use is because we find that we can relate macroscopic properties, such as temperature and pressure, to it. Simply, crms = ><v2> 1/2.

If we simplify the situation to a 2-dimensional case, we could consider a pool table where the balls lose no kinetic energy in collisions with the walls and KE is preserved between collisions. In this case, the pool balls would never stop moving – they don’t lose energy – and the velocity of one particular ball will be affected by collisions with other balls in the same way as intermolecular collisions in a sample of gas determine molecular velocities. We would find that the distribution of velocities of the pool balls follows the Maxwell distribution.

We say that when we consider a gas to be limited by these assumptions and results, it is behaving ideally or is perfect.