Derivation of Kinetic Energy using Kinetic Molecular Theory:

From Interpretation of Pressure on Kinetic Theory of Gases we know that:

P=(1/3)ρv_rms²

since ρ is density which is equal to mass per unit volume:

ρ=mass/volume
ρ=mN/V because mass=molecular mass x total number of molecules.

Substituting ρ=mN/V in above equation:

P=(1/3)(mN/V)v_rms²
mv_rms²=3PV/N
Since PV=nRT
mv_rms²=3nRT/N
We know that n=N/N_A
mv_rms²=3(N/N_A)RT/N
mv_rms²=3RT/N_A

R/N_A is known as Boltzmann constant and has a value of 1.38x10^-23Jmole^-1K^-1

mv_rms²=3kT

Multiplying both sides with 1/2,

(1/2)mv_rms²=(3/2)kT
We know that (1/2)mv_rms²=K.E, So
K.E=(3/2)kT