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Proof of the Kinetic theory postulate " AVERAGE KINETIC ENERGY IS PROPORTIONAL TO TEMPERATURE"
The higher the temperature the faster the molecules move producing much more kinetic energy then normal.The lower the temperature the slower the molecules move and it produces less kinetic energy.


According to the kinetic equation of pressure of a gas:
 
P = r 1/3
  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)
 
P = 1/3 (mN / V )
P V= 1/3 (mN)
   But PV = nRT
putting the value of PV, we get,
 
nRT= 1/3 (mN)
  Since
number of mole (n) = molecules/Avogadro's number
number of mole (n) = N/NA
  Therefore,
 
[N/NA] R T= 1/3 (mN)
NA R T= 1/3 (m)
3 [NA R] T= (m)
But NA R = Boltzman's constant (K), 
3 K T= (m)
  Multiplying both  by 1/2
 
(3/2) K T= (1/2) m
 
(1/2) m = (3/2) K T
But (1/2) m = average translational kinetic energy of gas molecules = (K.E)av
(K.E)av = (3/2) K T
As (3/2) K is a constant term,
 
(K.E)av = (constant) T


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