# Boltzmann constant

The Boltzmann constant (symbol or ) is a constant of nature, which plays a central role in the fundamental equations of statistical mechanics. It was introduced by Max Planck and named after the Austrian physicist Ludwig Boltzmann, one of the founders of statistical mechanics.

## Value

The Boltzmann constant is universally valid and has the dimension of energy / temperature.

Its value is:

With:

- - Universal gas constant [J / (mol K) ]
- - Avogadro constant [ 1/mol ].

From the Boltzmann constant, the universal gas constant is calculated:

## Definition and context of the entropy

The ideas of Ludwig Boltzmann präzisierend, is found from the Max Planck fundamental relationship:

That is, the entropy S of a macro state in a closed system in thermal equilibrium is proportional to the natural logarithm of the number of Ω ( sample space ) corresponding to the possible states of micro- (or in other words, the measure of the "disorder" of the macro- state ). Ω is the statistical weight of a measure of the probability of a particular micro- state.

This equation linked - via the Boltzmann constant proportionality factor - the micro-states of the closed system with the macroscopic size of the entropy and forms the central basis of statistical physics. It is engraved in a slightly different nomenclature on the grave stone of Ludwig Boltzmann at Vienna's Central Cemetery.

The entropy is defined in the classical thermodynamics:

With the amount of heat Q.

An increase in entropy corresponds to a transition into a new macro-state with a larger number of possible microstates. This is in a closed ( isolated ) system is always the case ( Second Law of Thermodynamics ).

With respect to the microscopic partition entropy can be defined as a dimensionless quantity:

In the "natural" form, the entropy corresponds to the definition of entropy in information theory, and there is a key measure. The term kBT thereby represents that energy to a nit to raise the entropy.

## Ideal gas law

The Boltzmann constant allows the calculation of the average thermal energy of a free particle from the temperature

And occurs, for example in the gas law for ideal gases as one of the possible proportionality on:

Meaning of symbols:

- P - pressure
- V - volume
- N - number of particles
- T - absolute temperature

Based on standard conditions (temperature T0 and pressure p0 ) and the Avogadro constant, the gas equation can be reformulated as:

## Related to the kinetic energy of

General results for the average kinetic energy of a classical point-like particle in thermal equilibrium with f degrees of freedom that enter quadratically in the Hamiltonian ( equipartition theorem ):

For example, a point-like particle has three translational degrees of freedom:

A diatomic molecule has

- Without symmetry three additional rotational degrees of freedom, ie a total of six
- With an axis of symmetry, two additional rotational degrees of freedom, ie a total of five ( by rotation along the symmetry axis no energy can be saved because the moment of inertia here is relatively small).

Then, for sufficiently high temperatures or vibrations of the bonds. Thus, water has an extremely high heat capacity by a large number of such vibrational degrees of freedom.

## Role in statistical physics

General takes the Boltzmann constant on the thermal probability density of arbitrary systems of statistical mechanics in thermal equilibrium, this is:

With

- The Boltzmann factor
- Of the partition function as a normalization constant.

## Example from solid state physics

In semiconductors, the dependence of the voltage across a pn junction of the temperature, can be described with the aid of the temperature or voltage of:

It is

- Is the absolute temperature in Kelvin
- The elementary charge.

At room temperature ( T = 300 K) of the value of the thermal voltage is approximately 25 mV and 1/40 V.

## Applications

- Planck's radiation law
- Maxwell - Boltzmann distribution
- Nernst equation
- Gibbs -Thomson effect
- Johnson noise
- Noise figure
- Desorption
- Noise temperature
- Thermistor
- Fugacity
- Doppler temperature
- Curie constant
- Debye temperature
- Debye length
- Jeans criterion
- Hawking radiation
- Breakdown voltage
- Lawson criterion