1. Equations for ideal gases
The ideal gas equation and its variant for N molecules
Relates pressure to the density of a gas and root mean square velocity.
You may have to manipulate this one
Relates absolute temperature to average molecular energy

2. Particle in a Box
Deriving

3. The particle in a box
The idea is to derive an expression for to relate the pressure and volume of a gas to the movement of the many individual molecules which make it up
We are building a MATHEMATICAL MODEL of what is happening and to do this we will need to make several assumptions

4. Assumptions
Intermolecular forces are negligible except during collisions.
All collisions are perfectly elastic
The volume of the molecules themselves can be neglected compared with the volume occupied by the gas.
The time for any collision is negligible compared with the time spent between collisions.
Between collisions molecules move with uniform velocity.
There is a sufficiently large number of molecules for statistics to be meaningfully applied.
These conditions imply that all of the internal energy of the gas is kinetic.

5. We also start with the knowledge that :
Pressure = Force / Area
Force = rate of change of momentum

6. The molecule has mass m and velocity u so its momentum is mu
We start with one molecule and take the simplest case, that I is moving only in the x direction heading for face A A m u y z x
The molecule has mass m and velocity u so its momentum is mu

7. So its momentum has changed from mu to -mu
The molecule collides with A and bounces back in the opposite direction. Its momentum velocity has changed from u to -u A u m y z x
So its momentum has changed from mu to -mu

8. A 2mu This means that its change in momentum is u m y z x
So its momentum has changed from mu to -mu

9. its rate of change of momentum is given by:
this is equal to the force acting on the particle
The time (∆t) between successive collisions is the time it takes to travel from wall A to the far wall of the box and back.
The distance it has to travel is 2x.
It does this travelling at velocity u y A m u x

10. remembering that velocity is distance travelled / time taken
So the time taken between collisions (∆t) is

11. so the rate of change of momentum of the moleculeIs ie
Now this is equal to the force of the wall which acts on the particle to change its momentum and by Newton’s third law this must be equal to the
force of the particle acting on the wall.

12. So the force on the wall from this one particle is
As pressure is force / area
and the area of wall A is y x z
The pressure on wall A due to this one molecule is
A= y x z y z

13. This is an expression for the pressure due to 1 molecule of the ideal gas moving in the x direction.
BUT there are N molecules in the gas and they have individual velocities: u12,+ u22+ u32,……uN2 l
(u12,+ u22+ u32,……uN2)
Or where N is the number of molecules and u2rms is the mean square of the velocities in the x direction:

14. But there are three components of velocity
u in the x direction , say v in the y direction and w in the z direction v w u
Then taking c2rms = u2rms+ v2rms+w2rms
So u2rms = 1/3c2rms