Question

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

Answer 1

The internal energy of the gas 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), 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

P = `1/3"nm"bar("v"^2) = 1/3ρbar("v"^2)` ..........(2)

where ρ = nm = mass density (Note n is number density)

Multiply and divide R.H.S of equation (2) by 2, we get

P = `2/3(ρ/2bar("v"^2))` ........(3)

From equation (3), pressure is equal to `2/3` of mean kinetic energy per unit volume.