The mean translation kinetic energy of a perfect gas molecule at temperature T is (k = Boltzmann constant)

Explanation

We know that , pressure exerted by gas is given
`P=1/3 p C^2 =1/3 M/V C^2` or `PV=1/3 MC^2`……(i)
According to gas equation for one mole of gas `PV=RT`
From (i) and (ii) `1/3 MC^2=RT`
Since, mass of gas `M=mN` where N is avogadro number
`1/3 m NC^2=RT` o `1/3 mC^2=R/ N T`
Since `R/N=K` Boltzman.s constant therefore `1/3 mc^2=kT`
Multiplying both sides by `3/2` we get `1/2 mC^2=3/2 kT therefore 1/2 mC^2 prop T`
Hence , mean kinetic energy per molecule of a gas is directly proportional to the absolute temperature of the gas.