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Kinetic interpretation of temperature




Kinetic interpretation of temperature


  • 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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