## Kinetic Theory of Gases

The kinetic theory of gases is a model that explains gas behaviour – pressure, temperature and volume – using a model of the gas as hard little particles of no volume which exert no forces on each other. For many everyday situations these are very good approximations. Classical Newtonian mechanics is used to derive the equation of state which describes the relationship between Pressure Volumeand Temperaturefor an 'ideal gas'.

Suppose we have a cube full of N identical gas molecules of mass m inside a cube of side l. Imagine a gas molecule moving backwards and forwards between the walls 1 and 2. The collisions of the molecule with the walls of the box are perfectly elastic so no kinetic energy is lost, and the molecule always has speed v. We assume the wall to be fixed – this is not true, but they are so much heavier than the gas molecules that it is a very good approximation.

Each time the gas molecule collides with the wall (1) it's momentum changes by

Between each collision with wall 1 the molecule travels a distanceso takesseconds to travel from 1 to 2 and back, hence the molecule collides with wall 1times per second.

This means that the rate of change of momentum of the molecule is because of it's collisions with wall 1 isand we can equate this to the impulse exerted by the wall on the molecule and hence using Newton's Third Law, to the impulse exerted by the molecule on the wall.

(1) since

We add up terms such as (1) obtainingwhere N is the total number of molecules in the box, but now we have to take into account that not all the gas molecules are bouncing between walls 1 and 2: on average, only one third are, so we can introduce a factor ofto take account of this thenbecomes

using

The mean kinetic energy of gas molecules is given by

number of mols*number of molecules in each molwhereis Avagadro's number.

Finallythe Universal Gas constant:

This is the usually stated form of the ideal gas equation.