From equation (5) we have
PV = (1/3)Nmvmq2
where N is the number of molecules in the sample. Above equation can also be written as
PV = (2/3)N(1/2)Nmvmq2 (7)
The quantity (1/2)Nmvmq2 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,
Combining equation 7 and 8 we get
Comparing this result with the ideal gas equation (equation (4) ) we get
or, E/N=(1/2)mvmq2 =(3/2)KBT (9)
Where, KB is known as Boltzmann constant and its value is KB=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
from this since v2¯=(vrms)2, rms velocity of a molecule is
This can also be written as
where, M=mN is the molecular mass of the gas.