Getting an equation for average kinetic energy of a molecule

  Molecule has 3 component, which is x, y and z value for velocity. We'll take a look at just for x value, which would be 'Vx' for this example

Let's say container has length(or width) of L meters. Time taken for molecules to travel from one side to another side would be L/Vx (Distance/velocity = time taken). From one side to another side and back would be 2L/Vx.

When this molecule strikes wall at velocity 'Vx', since it is assumed to be perfectly elastic, it would bounce off with velocity '-Vx'. Thus, every collision with a wall will end up change momentum by

Δpx = 2mVx

  Since this is done in time 2L/Vx

px = 2mVx/(2L/Vx) = 2mVx * Vx/2L = mVx²

∴ px = mVx²/L

There are N number of molecules, and pressure is force over area, L²

p = NmVx²/ V

  Applying this to both x,y and z.. sounds hard but it is actually easier than you might think. Let's say same molecule has velocity of c, and using Pythagoras' theorem to three dimensions would be c² = c₁² + c₂² + c₃² where c₁, c₂ and c₃ are x,y and z component. And since we are dealing with large # of molecules, average value for all three component would be pretty much same, thus <c₁²> = <c₂²> = <c₃²>, this gives idea that <c₁²> = ⅓<c²>

Now, equation might be

pV = ⅓Nm<c²>

Since average kinetic energy of a molecule is <Ek> = ½m<c²>   (KE = ½mv²)
pV = ⅓N *  2       *  ½m<c²>∴ pV = ⅔N<Ek>

From ideal gas equation, pV = nRT = NkT
This is important.. because now we can relate temperature of the gas to kinetic energy, as increase in pressure by temperature means rise in speed of molecules, thus we can relate kinetic energy with temperature. Thus
pV = ⅔N<Ek> = NkT

Making <Ek> subject
<Ek> = 3/2kT


Secondly, we can get average speed of the molecule with this equation.
We already had <Ek> = ½m<c²>
<Ek> = ½m<c²> = 3/2kT

Solving for <c²>
<c²> = 3kT/m

And
<c²>^½ = 3kT/m^½

Where <c²>^½ is root-mean-square speed or r.m.s speed (crms)