The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, and in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann.
As part of the 2019 redefinition of SI base units, the Boltzmann constant is one of the seven "defining constants" that have been given exact definitions. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly .
Roles of the Boltzmann constant
Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure and volume is proportional to the product of amount of substance and absolute temperature :
where is the molar gas constant (). Introducing the Boltzmann constant as the gas constant per molecule transforms the ideal gas law into an alternative form:
where is the number of molecules of gas.
Role in the equipartition of energy
Given a thermodynamic system at an absolute temperature , the average thermal energy carried by each microscopic degree of freedom in the system is (i.e., about , or , at room temperature). This is generally true only for classical systems with a large number of particles, and in which quantum effects are negligible.
In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. Monatomic ideal gases (the six noble gases) possess three degrees of freedom per atom, corresponding to the three spatial directions. According to the equipartition of energy this means that there is a thermal energy of per atom. This corresponds very well with experimental data. The thermal energy can be used to calculate the root-mean-square speed of the atoms, which turns out to be inversely proportional to the square root of the atomic mass. The root mean square speeds found at room temperature accurately reflect this, ranging from for helium, down to for xenon.
Kinetic theory gives the average pressure for an ideal gas as
Combination with the ideal gas law
shows that the average translational kinetic energy is
Considering that the translational motion velocity vector has three degrees of freedom (one for each dimension) gives the average energy per degree of freedom equal to one third of that, i.e. .
The ideal gas equation is also obeyed closely by molecular gases; but the form for the heat capacity is more complicated, because the molecules possess additional internal degrees of freedom, as well as the three degrees of freedom for movement of the molecule as a whole. Diatomic gases, for example, possess a total of six degrees of simple freedom per molecule that are related to atomic motion (three translational, two rotational, and one vibrational). At lower temperatures, not all these degrees of freedom may fully participate in the gas heat capacity, due to quantum mechanical limits on the availability of excited states at the relevant thermal energy per molecule.
Role in Boltzmann factors
More generally, systems in equilibrium at temperature have probability of occupying a state with energy weighted by the corresponding Boltzmann factor:
where is the partition function. Again, it is the energy-like quantity that takes central importance.
Consequences of this include (in addition to the results for ideal gases above) the Arrhenius equation in chemical kinetics.
Role in the statistical definition of entropy
In statistical mechanics, the entropy of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of , the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy ):
This equation, which relates the microscopic details, or microstates, of the system (via ) to its macroscopic state (via the entropy ), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone.
The constant of proportionality serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:
One could choose instead a rescaled dimensionless entropy in microscopic terms such that
This is a more natural form and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy.