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## Answers

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.