Equipartition of Energy

The theorem of equipartition of energy states that molecules in thermal equilibrium have the same average energy associated with each independent degree of freedom of their motion and that the energy is


The equipartition result serves well in the definition of kinetic temperature since that involves just the translational degrees of freedom, but it fails to predict the specific heats of polyatomic gases because the increase in internal energy associated with heating such gases adds energy to rotational and perhaps vibrational degrees of freedom. Each vibrational mode will get kT/2 for kinetic energy and kT/2 for potential energy - equality of kinetic and potential energy is addressed in the virial theorem. Equipartition of energy also has implication for electromagnetic radiation when it is in equilibrium with matter, each mode of radiation having kT of energy in the Rayleigh-Jeans law.

For the translational degrees of freedom only, equipartition can be shown to follow from the Boltzmann distribution.

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Average thermal energyDefine constants
Specific heats of solidsSpecific heats of gases
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Thermal Energy

The average translational kinetic energy possessed by free particles given by equipartition of energy is sometimes called the thermal energy per particle. It is useful in making judgements about whether the internal energy possessed by a system of particles will be sufficient to cause other phenomena. It is also useful for comparisons of other types of energy possessed by a particle to that which it possesses simply as a result of its temperature.
Define constants

If the temperature is C = F = K = x10^ K,

the average "thermal energy" is
3kT/2 = eV = MeV = GeV
3kT/2= x 10^ joules = x10^ eV.

For an atom or molecule of mass amu,

this corresponds to an effective or rms speed of

vrms = m/s = mi/hr.



Molecular speed distribution for ideal gas
Fraction of particles above a certain energy
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Internal Energy for Ideal Gas

Internal energy in general includes both kinetic energy and potential energy associated with the molecular motion. But the potential energy is associated with intermolecular forces which are presumed to be zero in an ideal gas except for the perfectly elastic collisions between molecules. Therefore the internal energy of an ideal gas is entirely kinetic energy.
While steam at 100 degrees Celsius is not strictly an ideal gas, the diagram illustrates the fact that the phase change to the gaseous state leaves only the kinetic portion of the internal energy. Fora monoatomic ideal gas this internal energy is given by

If rotation and vibrational kinetic energies are involved (polyatomic molecules) then

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