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,
E=(1/2)Nmvmq2 (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)KBNT
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
(1/2)mv2¯=(3/2)KBT
from this since v2¯=(vrms)2, rms velocity of a molecule is
vrms=√(3KBT/m)(10)
This can also be written as
vrms=√(3KBNT/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.
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