Equipartition theorem

The equipartition theorem (also known as equipartition ) is that in thermal equilibrium in the middle of each degree of freedom, has the same average energy at a temperature T:

It is the Boltzmann constant.

So true for particles with degrees of freedom:

The equipartition theorem applies only to degrees of freedom, the variable in the expression for the energy, that is, in the Hamiltonian, occur as a square. Furthermore, these degrees of freedom may not be "frozen", i.e., this degree of freedom to be actually excited. For example, molecular vibrations " small molecules " such as H2 O2, or not stimulated at room temperature because the time required for the transition to the lowest excited state energy is not attained.

Degrees of freedom, the variable does not appear in the Hamiltonian, also not lead to a contribution to the energy; for degrees of freedom as found anywhere in purely quadratic form, can the average energy not so easy to calculate.

• 3.1 monatomic ideal gas
• 3.2 Diatomic ideal gas
• 3.3 Thermal equation of state

Examples

Specific heat of gases

From the equipartition theorem, the heat capacity (specific heat) of an ideal gas can for example be calculated at constant volume. We first consider a monatomic gas ( inert gas ):

The energy of the gas is given by the kinetic energy of the atoms; applies for each atom

Wherein the mass of the atom, and the components of the speed vector. Thus, there are three degrees of freedom per atom present as square, so the average energy per atom

It follows by differentiating with respect to temperature, the heat capacity of each atom, ie

For a monatomic gas with N atoms.

Diatomic gas molecules as they are formed, for example, hydrogen, oxygen and nitrogen, two rotational degrees of freedom are also taken into account ( the rotation around the molecular axis, that the third spatial direction, is not relevant, and molecular vibrations are "frozen "). This results in five degrees of freedom per molecule, and thus

For a gas of N molecules. Vibrational degrees of freedom are only produced at temperatures above 1000 K.

Heat capacity of solids

In solids, the vibration of the atoms can be approximated to its rest position by the potential of a harmonic oscillator. The spatial direction is the corresponding energy by

Optionally, wherein the angular frequency of the oscillator, and the deflection means of the atom from its rest position in the direction. The first term is the kinetic energy of the second potential energy. So there are two degrees of freedom per atom and space dimension as the square before, in three dimensions, ie six degrees of freedom per atom. Therefore, the average energy per atom

Atoms therefore degrees of freedom are taken into account. From this immediately follows the heat capacity of

This equation is known as the Dulong - Petit law. Again, that the degrees of freedom may not be "frozen" ( the temperature must be well above the Debye temperature ); otherwise the heat capacity can be calculated only with the Debye model.

Derivation

In the following, the equipartition theorem for classical systems is derived. The starting point is a closed system, which is energetically coupled to a heat bath. It therefore lends itself to the consideration of the canonical ensemble.

We consider the average size, which can stand for local () or momentum coordinates (). denotes the Hamiltonian function of the system.

The integration is carried out over the accessible phase space. denotes the inverse temperature, the canonical partition function. Partial integration leads to:

Where we assume that sufficiently fast for large drops, so that the boundary terms can be neglected:

The most general formulation of the equipartition theorem for classical systems in thermal equilibrium is

The derivation was performed here using the canonical ensemble, it is possible by means of microcanonical ensembles.

Applications

From the general equipartition theorem can be deduced: Any variable that is squarely in the Hamilton function, contributes to the mean energy at:

This is also true for the general quadratic form:

Monatomic ideal gas

For non-interacting particles ( monatomic ideal gas) in three spatial dimensions is the Hamilton function only of the kinetic component:

The application of the above result

Provides

That is, each degree of translational freedom (here) is the average kinetic energy.

Diatomic ideal gas

A diatomic ideal gas, which means that the individual molecules do not interact with each other, the Hamiltonian is the rotation-vibration neglecting coupling (constant moment of inertia)

The total mass, the reduced mass and the moment of inertia of a molecule. describes the deflection from the equilibrium ratio. Overall, therefore, go seven sizes square in the Hamilton function:. It follows:

This average energy is only valid at high temperatures, even if rotations and vibrations are excited thermally.

Thermal equation of state

Consider a real gas in a container, so is the Hamiltonian

Wherein the potential between the wall and the particles, and the potential between the particles. For a cubic container with side length of the wall potential for example, writes as follows

The Heaviside function was used. The application of the virial theorem (in the sense of quantum mechanics ) provides:

Consider now the first term on the right side

It has been exploited that the distributive derivation of the Heaviside function is the Delta function. In the penultimate step, the volume could be introduced. Now one introduces an ensemble averaging by and used that the pressure is defined by ( see, eg, here: Canonical ensemble).

Thus we obtain the thermal equation of state:

This corresponds to the ideal gas equation to an additional term - is expanded - the virial. The virial can be developed in powers of the particle density (see: virial expansion ).