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Kinetic Theory of Gases and Equipartition Theorem

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Kinetic Theory of Gases and Equipartition Theorem Results of Kinetic Theory KE of individual particles is related to the temperature of the gas: mv2 = 3/2 kT Where ... – PowerPoint PPT presentation

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Title: Kinetic Theory of Gases and Equipartition Theorem


1
Kinetic Theory of Gases and Equipartition Theorem
2
Results of Kinetic Theory
  • KE of individual particles is related to the
    temperature of the gas
  • ½ mv2 3/2 kT
  • Where v is the average velocity.

3
Boltzmann Distribution
  • Demonstrated that there is a wide range of speeds
    that varies with temperature.

4
Equipartition of Energy
  • The Kinetic Theory result tells us that the
    average translational KE is proportional to the
    temperature.
  • Types of allowed motion are referred to as
    Degrees of Freedom (DOF)
  • Linear motion in three directions (x,y,z) is
    three DOF.

5
Equipartition of Energy
  • Each degree of translational freedom takes ½ kT.
  • KEx KEyKEz ½ kT ½ kT ½ kT
  • KEtotal 3/2 kT
  • This is true for single point masses that possess
    no structure.

6
Equipartition of Energy
  • For molecules, i.e. multi-atom particles, there
    are added degrees of freedom.

7
Equipartition of Energy
  • Each new DOF requires ½ kT of energy.
  • Each new DOF contributes ½ kT to the total
    internal energy of the gas. This is the
    Equipartition Theorem.

8
Internal Energy, U
  • The internal energy of a gas is the sum of all
    the kinetic and potential energies.
  • U KE PE.
  • For an ideal gas we allow no PE of interaction,
    so U is comprised solely of the various type of
    kinetic energies, i.e. translational, rotational
    and vibrational.

9
Internal Energy of Di-Atom
  • Three translational DOF 2 rotational DOF 5
    DOF.
  • Each DOF contributes ½ kt, so the internal energy
    of a diatomic gas is,
  • U 5/2 NkT,
  • For a gas of N molecules.
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