Prove that the average kinetic energy of a molecule of an ideal gas is direcly proportional to the absolute temperature of the gas.

Explanation

We know the average kinetic energy of a molecule is directly proportional to absolute temperature of the gas.
Let one gram mole of an ideal gas occupy a volume V at temperature T
If m is the mass of each molecule of the gas, then
`M=m times N_A`[Where `N_A` is the Avogadro.s number]
IF c is the rms speed of the gas molecules then the pressure exerted by the ideal gas is `p=1/3 pc^2=1/3 M/V c^2` or, `pV=1/3 Mc^2`
Now from the equation pV=RT.
`1/3Mc^2=RTor,Mc^2=3RTor1/2pc^2=3/2RT`
or,`1/2mN_Ac^2=3/2RT(because M=mN_A)`
or,`1/2mc^2=3/2(R/N_A)T=3/2k_BT[becausek_B=R/N_A]`
Here , `k_B` is the Boltzmann constant
So average kinetic energy of tranlation per molecule of a gas
`=1/2mc^2=3/2k_BT`
Hence, average kinetic energy of a molecule is directly proportional to the absolute temperature of the gas.