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The Nature, Use and Proof of the
Equipartition of Energy Theorem

Perfectly spherical molecules have three degrees of freedom; one for each coordinate axis. In a gas made up of such molecules one third of the energy is embodied in motion in each of the three directions. Such molecules could also spin but there is no way for translational movement to be converted into and outof molecular spin. The term degrees of freedom is misleading. It is not potential degrees of freedom that is important; it is effective degrees of freedom; i.e., effectively interacting energy modes.

For a gas made up of elongated but cylindrically symmetric molecules like hydrogen H2 there are an additional two degrees of freedom for rotation about two axes. There is no way translational or spinning motion about the other axes can be converted into spinning about the symmetry axis perpendicular to the circular cross sections.

For triatomic molecules with a bend like those for water (H2 or CO2) the bent nature of the molecule allows energy to be transferred to and from rotation about the long axis of the molecule. Thus for these types of molecules the effective number of rotational degrees of freedom would be three.

A molecule may have internal structure and that internal structure may have vibrations. However if the energy required to initiate a vibration is vastly greated than the average energy involved in collisions of the molecules then the internal structure may not share in the equipartition of energy. For example, the average energy involved in molecular collisions at room temperature is about 1/40 of an electron voltl (ev); the energy required to move an electron from one state to another in an atom is several electron volts. Likewise the energy involved in vibrations in the separation distance or angle between atoms in a molecule may be too large to be affected by the collisions of molecules for some temperature range. Thus the effective degrees of freedom may depend upon the temperature under consideration.

With these considerations taken into account in assessing the degrees of freedom, in a gas of molecules having n effective degrees of freedom the total energy of the gas is divided equally among those n degrees of free of freedom. This is the Equipartition of Energy Theorem that the name is based upon. It can be extended. In a gas in equilibrium at a temperature T made up of a mixture of molecules the energy of the gas would be divided among the different components of the gas such that the average translational energies would be the same for all components. It is in this form that the Equipartition of Energy Theorem is often stated.

A Statement of the Equipartition of Energy Theorem

A generalized momentum coordinate which occurs in the Hamiltonian function only as a squared term contributes an energy ½kT to the mean kinetic energy of the system.

(where T is the absolute temperature and k is the Boltzmann constant.)

Proof:

Let {q1, … , qn} be the generalized position coordinates of a molecule of the gas and let {p1, … , pn} be their canonical conjugate momentum coordinates. Let ζ be a momentum coordinate such that the Hamiltonian is of the following form

H = f(Z)ζ² + H0(Z)

where Z is the set of position and momentum coordinates other than ζ.

For a fixed value for Z the probability distribution for ζ is given by

P(ζ:Z)dζ = exp(−βf(Z)ζ²dζ)/[∫exp(−f(Z)α²dα]

This is the conditional probability distribution; i.e., conditional on Z.

The conditional expected value of f(Z)ζ² is then given by

E{f(Z)ζ²: Z)} = [ ∫f(Z)ζ² exp(−βf(Z)ζ²dζ)/[∫exp(−f(Z)α²dα]

where β is equal to 1/(kT).

The expression on the right can be put into the form −(∂ /∂β)[ln(∫exp(−βf(Z)α²)dα]. Thus

E{f(Z)ζ²: Z)} = −(∂ /∂β)[ln(∫exp(−βf(Z)α²)dα]

If a new variable of integration z=ζ√β is introduced then

E{f(Z)ζ²: Z)} = −(∂ /∂β)[ln(β∫exp(−f(Z)z²)dz]

But ∫exp(−f(Z)z²)dz is equal to 1 so the above equation reduces to

E{f(Z)ζ²: Z)} = −(∂ /∂β)[−½ln(β)]
and still further to
E{f(Z)ζ²: Z)} = 1/(2β)

Since 1/β is equal to kT

E{f(Z)ζ²: Z)} = ½kT

Thus the contribution of the ζ variable to mean translational energy (kinetic energy) is ½kT.

The conditionality of the expected value was made on the basis that f(Z) might depend upon the other variables Z. But the conditional expectation is independent of Z so the unconditional expected value is

E{f(Z)ζ²)} = ½kT

If each of the momenta for the three position coordinates can serve as ζ then the average kinetic energy of the molecules is given by

E{½mv²} = (3//2)kT

where m is the mass of a molecule and v is its velocity.

If a gas is made up of more than one component then each component is at temperature T then for each component j the average translational kinetic energy is equal to (3/2)kT; i.e.,

E{½mjvj²} = (3/2)kT

In other words, the average translational kinetic energies of the component gases of a mixture are all equal to each other. Where there are rotational degrees of freedom, such as two more for diatomic molecules being elongated with an axis of symmetry, the average translational and rotational kinetic energy is (5/2)kT.

Heat Capacities

Heat capacities measure the total amount of energy that must be provided to a system to raise its temperature by one degree. It makes a difference if this is measured at constant volume or constant pressure. The heat capacity at constant volume better fits the concept dealt with here.

Cv = (∂U/∂T)v

where U is energy.

Temperature is a measure of the average translational kinetic. Thus the heat capacities of all monatomic molecule gases like helium and neon should be equal and likewise for diatomic molecule gases like hydrogen, oxygen and nitrogen. But the heat capacities of diatomic molecule gases should be larger than for monatomic molecule ones. The heat capacities should be proportional to the degrees of freedom. As noted previously the effective number of degrees of freedom may depend upon the temperature because any internal degrees of freedom may subject to the quantization of the energy in the internal structure.

Empirical Measurements

Molar Heat Capacities at Constant Volume of Various Gases
GasCv
(J/mol)/K°
cv
Monatomic
Helium12.47171.5R
Neon12.47171.5R
Argon12.47171.5R
Diatomic
Hydrogen20.472.46R
Oxygen21.02.52R
Nitrogen20.82.50R
Triatomic
Carbon
Dioxide
CO2
28.463.42R
Water
Vapor<
H2O/td>
28.033.37R
Hydrogen
Sulfide
H2S
26.23.15R
Complex Molecule
Ammonia
NH3
26.73.21R

The third column expresses the molar heat capacity in terms of the gas constant R which is equal to 8.3144621 (J/mol)/K°.

The confirmation is perfect for the monatomic gases and indicates that their molecules have 3 degrees of freedom. The confirmation is good for the diatomic gases and indicates that their degree of freedom is 5. The data for the triatomic gases indicate that the degrees of freedom for carbon dioxide is 7, indicating 3 translational, 3 rotational and 1 internal mode. For water vapor and hydrogen sulfide the data indicate at the temperature of measurement there is some internal energy mode involved but it is not complete. This is also the case for ammonia.

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