What is the relation between the average kinetic energy and pressure?

Explanation

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