Boltzmann constant

Given a thermodynamic system at an absolute temperature T, the average thermal energy carried by each microscopic degree of freedom in the system is 1/2kT (i.e., about 2.07×10⁻²¹ J, or 0.013 eV, 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 3/2kT 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 1370 m/s for helium, down to 240 m/s for xenon.

Kinetic theory gives the average pressure p for an ideal gas as

${\displaystyle p={\frac {1}{3}}{\frac {N}{V}}m{\overline {v^{2}}}.}$

Combination with the ideal gas law

${\displaystyle pV=NkT}$

shows that the average translational kinetic energy is

${\displaystyle {\tfrac {1}{2}}m{\overline {v^{2}}}={\tfrac {3}{2}}kT.}$

Considering that the translational motion velocity vector v 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. 1/2kT.

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.

More generally, systems in equilibrium at temperature T have probability P_i of occupying a state i with energy E weighted by the corresponding Boltzmann factor:

${\displaystyle P_{i}\propto {\frac {\exp \left(-{\frac {E}{kT}}\right)}{Z}},}$

where Z is the partition function. Again, it is the energy-like quantity kT that takes central importance.

Consequences of this include (in addition to the results for ideal gases above) the Arrhenius equation in chemical kinetics.

Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula.

In statistical mechanics, the entropy S of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of W, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E):

${\displaystyle S=k\,\ln W.}$

This equation, which relates the microscopic details, or microstates, of the system (via W) to its macroscopic state (via the entropy S), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone.

The constant of proportionality k serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:

${\displaystyle \Delta S=\int {\frac {{\rm {d}}Q}{T}}.}$

One could choose instead a rescaled dimensionless entropy in microscopic terms such that

${\displaystyle {S'=\ln W},\quad \Delta S'=\int {\frac {\mathrm {d} Q}{kT}}.}$

This is a more natural form and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy.

The characteristic energy kT is thus the energy required to increase the rescaled entropy by one nat.

In semiconductors, the Shockley diode equation—the relationship between the flow of electric current and the electrostatic potential across a p–n junction—depends on a characteristic voltage called the thermal voltage, denoted by V_T. The thermal voltage depends on absolute temperature T as

$${\displaystyle V_{\mathrm {T} }={kT \over q},}$$

where q is the magnitude of the electrical charge on the electron with a value 1.602176634×10⁻¹⁹ C.[5] Equivalently,

$${\displaystyle {V_{\mathrm {T} } \over T}={k \over q}\approx 8.61733034\times 10^{-5}\ \mathrm {V/K} .}$$

At room temperature 300 K (27 °C; 80 °F), V_T is approximately 25.85 mV[6][7] which can be derived by plugging in the values as follows:

$${\displaystyle V_{\mathrm {T} }={kT \over q}={\frac {1.38\times 10^{-23}\ \mathrm {J{\cdot }K^{-1}} \times 300\ \mathrm {K} }{1.6\times 10^{-19}\ \mathrm {C} }}\simeq 25.85\ \mathrm {mV} }$$

At the standard state temperature of 298.15 K (25.00 °C; 77.00 °F), it is approximately 25.69 mV. The thermal voltage is also important in plasmas and electrolyte solutions (e.g. the Nernst equation); in both cases it provides a measure of how much the spatial distribution of electrons or ions is affected by a boundary held at a fixed voltage.[8][9]

The Boltzmann constant provides a mapping from this characteristic microscopic energy E to the macroscopic temperature scale T = E/k. In physics research another definition is often encountered in setting k to unity, resulting in temperature and energy quantities of the same type. In this context temperature is measured effectively in units of energy and the Boltzmann constant is not explicitly needed.[17]

The equipartition formula for the energy associated with each classical degree of freedom then becomes

${\displaystyle E_{\mathrm {dof} }={\tfrac {1}{2}}T}$

The use of natural units simplifies many physical relationships; in this form the definition of thermodynamic entropy coincides with the form of information entropy:

${\displaystyle S=-\sum _{i}P_{i}\ln P_{i}.}$

where P_i is the probability of each microstate.