Derivation of RMS Speed (and Pressure incidentally)

Suppose a particle in the box is bouncing between the blue wall and its opposite wall along distance 'a' and it has velocity v_{x} and mass m. It would then have momentum

Of course it takes time Dt to go between the two walls.

<= time to traverse distance 'a' at velocity v_{x} .

When the particle strikes the blue wall, it transfers its momentum to the wall. Thus the force going into the wall can be estimated as momentum / Dt.

In order to estimate the pressure that may derive from the force, we can utilize the area the wall, b x c.

pressure is force per unit area.

Also, if Dt is replace by a / v_{x} then we get

Now molecules are going in all directions, not just 'x', and the average velocity in all directions is

And since there is an equal probability for going all ways,

Further, instead of referring to one molecule, let's introduce 'n' to be the number of molecules.

<= A derived expression for pressure x volume.

And we know from the ideal gas law that PV = nRT so using 1 mole of molecules,

<= This is the root mean square velocity