An Equation for Pressure  

You will not be required to derive this equation but you should be able to follow its derivation. The important point is that it is an equation derived from a mathematical model. The fact that it works very well for real gases tells us that our model is a good one.

Imagine a cube with length, height and width L which contains a very large number, N, of particles, each of mass m, moving randomly.

Consider one particle which has velocity c. The components of c are u, v and w in the x, y and z directions as shown.

Initially we just consider one of these dimensions, the x direction. What force does this particle exert on these walls?

 

Force due to 1 particle

Newton's 2nd law tells us that the average force on a wall due to this particle is equal to its rate of change of momentum

so 

Change in momentum per second   =   change in momentum per collision   x   number of collisions per second

The time between collisions = 2 L / u   ( v = d / t   so    t = d / v   and the distance travelled between collisions with a wall is 2L )

the average force due to 1 particle = 2mu  x  u / 2 L  = mu2 / L

Pressure due to lots of particles

as the area of the wall is L2 the average pressure ( p = F / A) due to one particle is mu2 / L3

There are lots of particles. Their velocities in the x direction are u1, u2, u3, ......... uN

so the total pressure on this wall due to N particles is = mu12 / L3 + mu22 / L3 + mu32 / L3 + ...... muN2 / L3

which equals m / L3  ( u12 + u22 + u32 + ........ uN2 )

which equals  Nm / L3   where = the average value of u2

An Equation for Pressure

so far we have just considered the x direction. If there are a large number of particles moving randomly then

now Nm is the mass of the gas and L3 is its volume so Nm / L3 =  the density of the gas,

and so we get the pressure on each wall of the container due to all particles