# Prove that the average kinetic energy of a molecule of an ideal gas in directly proportional to the absolute temperature of the gas.

Following is the deduction of kinetic theory in terms of pressure:

$p=frac{mnv^{2}}{3}$

Where,

m is the mass of the gas molecule

n is the number of molecules per unit volume

v is the rms speed

So, n = $frac{N}{V}$

Where N is the number of molecules

Substituting for n we get,

$pV=frac{mNv^{2}}{3}$ $frac{mv^{2}}{2}=E$ is the kinetic energy of the molecule

Therefore, $pV=frac{2NE}{3}$

From ideal gas equation,

$pV=mu RT$

Where $mu =frac{N}{N_{A}}$ $mu RT=frac{2NE}{3}$ $frac{NRT}{N_{A}}=frac{2NE}{3}$ $E=frac{3kT}{2}$

Where $k=frac{R}{N_{A}$

Therefore, it can be said that kinetic energy is proportional to temperature