SHOW THAT THE AVERAGE TRANSLATIONAL KINETIC ENERGY OF THE MOLECULES OF A GAS IS DIRECTLY PROPORTIONAL TO ABSOLUTE TEMPERATURE

Starting from the expression P = r 1/3  show that the Average translational kinetic energy of the molecules of a gas is directly proportional to absolute temperature.

According to the kinetic equation of pressure of a gas:

But r = density of gas
                        r = density of gas = mass of gas / volume of gas
                        r = density of gas = mN / V
Putting the value of density (r)

 But PV = nRT
putting the value of PV, we get,

Since
number of mole (n) = molecules/Avogadro’s number
number of mole (n) = N/N_A Therefore,

But N_A R = Boltzman’s constant (K), thus,

Multiplying both sides by 1/2

But (1/2) m = average translational kinetic energy of gas molecules = (K.E)_av
therefore,

As (3/2) K is a constant term, therefore,