From equation (5) we have
PV = (1/3)Nmv_{mq}^{2}
where N is the number of molecules in the sample. Above equation can also be written as
PV = (2/3)N(1/2)Nmv_{mq}^{2} (7)

The quantity (1/2)Nmv_{mq}^{2} in equation (7) is the Kinetic energy of molecules in the gas. Since the internal energy of an ideal gas is purely kinetic we have,
E=(1/2)Nmv_{mq}^{2} (8)

Combining equation 7 and 8 we get
PV=(2/3)E

Comparing this result with the ideal gas equation (equation (4) ) we get
E=(3/2)K_{B}NT
or,E/N=(1/2)mv_{mq}^{2} =(3/2)K_{B}T(9)
Where, K_{B} is known as Boltzmann constant and its value is K_{B}=1.38 X 10^{-23} J/K

From equation (11) we conclude that the average kinetic energy of a gas molecule is directly proportional to the absolute temperature of the gas and is independent of the pressure , volume and nature of the gas.

Hence average KE per molecule is
(1/2)mv^{2}¯=(3/2)K_{B}T
from this since v^{2}¯=(v_{rms})^{2}, rms velocity of a molecule is
v_{rms}=√(3K_{B}T/m)(10)
This can also be written as
v_{rms}=√(3K_{B}NT/Nm)
=√(3RT/M)(11)
where, M=mN is the molecular mass of the gas.

Assignment
(1)Gas laws (Boyle's and Charle's law) and perfect gas equation can be derived using kinetic theory of gases. try to derive them.