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`
From the equation (1), the pressure of the gas is equal to two thirds of internal energy per unit volume or internal energy per unit volume or internal energy density ( u = UV)
Writing pressure in terms of mean kinetic energy density using equation
` P = 1/3 nmv^(bar2) = 1/3 pv^(bar2)`
Where p= nm = mass density ( Note n is number f density)
Multiply and divide R.H.S of equation (2) by 2, we get
` P = 2/3 ( P/2 bar(v^(2)))`
` P = 2/3 K E `
From the equation (3) , pressure is equal to 2/3 of mean kinetic energy per unit volume.