Derive the relation between pressure and mean kinetic energy.

Explanation

The internal of the gas is given by
`U = 3/2 NkT`
The above equation can also be a written as
`U=3/2 PV`
Since , PV = NkT
`P = 2/3 U/V = 2/3 u " " …(1)`
From the equation (1), we can state that the pressure of the gas is equal to two thirds of internal energy per unit volume or internal energy density `(u = U/V)` .
Writing pressure in terms of mean kinetic energy density using equation,
`P =1/3nmbar(v^2) =1/3rhobar(v^2)" " ...(2)`
where `rho` = nm = mass density (Note n is number density)
Multiply and divide R.H.S of equation (2) by 2 , we get
`P = 2/3 (rho/2bar(v^2))`
`P = 2/3 bar(KE)`