Boltzmann constant

Boltzmann constant
Ludwig Boltzmann, the constant's namesake
Symbol:	kB, k
Value in joules per kelvin:	1.380649×10−23 J⋅K−1[1]

The Boltzmann constant (k_B or k) is the

Macroscopically, the

$${\displaystyle pV=nRT,}$$

where R is the

) transforms the ideal gas law into an alternative form:

$${\displaystyle pV=NkT,}$$

where N is the

number of molecules

of gas.

Role in the equipartition of energy

Main article:

Equipartition of energy

Given a thermodynamic system at an absolute temperature T, the average thermal energy carried by each microscopic degree of freedom in the system is 1/2 kT (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

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. 1/2 kT.

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 T have probability P_i of occupying a state i with energy E weighted by the corresponding

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

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.

Role in the statistical definition of entropy

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

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

One could choose instead a rescaled

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

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

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

The thermal voltage

In

—depends on a characteristic voltage called the thermal voltage, denoted by V_T. The thermal voltage depends on absolute temperature T as

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

where q is the magnitude of the electrical charge on the electron with a value 1.602176634×10⁻¹⁹ C.^[6] Equivalently,

$${\displaystyle {V_{\mathrm {T} } \over T}={k \over q}\approx 8.617333262\times 10^{-5}\ \mathrm {V/K} .}$$

At room temperature 300 K (27 °C; 80 °F), V_T is approximately 25.85 mV^[7][8] 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.^[9][10]

History

The Boltzmann constant is named after its 19th century Austrian discoverer, Ludwig Boltzmann. Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k, and gave a more precise value for it (1.346×10⁻²³ J/K, about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901.^[11] Before 1900, equations involving Boltzmann factors were not written using the energies per molecule and the Boltzmann constant, but rather using a form of the gas constant R, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation S = k ln W on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his eponymous h.^[12]

In 1920, Planck wrote in his Nobel Prize lecture:^[13]

This constant is often referred to as Boltzmann's constant, although, to my knowledge, Boltzmann himself never introduced it – a peculiar state of affairs, which can be explained by the fact that Boltzmann, as appears from his occasional utterances, never gave thought to the possibility of carrying out an exact measurement of the constant.

This "peculiar state of affairs" is illustrated by reference to one of the great scientific debates of the time. There was considerable disagreement in the second half of the nineteenth century as to whether atoms and molecules were real or whether they were simply a

In versions of SI prior to the 2019 redefinition of the SI base units, the Boltzmann constant was a measured quantity rather than a fixed value. Its exact definition also varied over the years due to redefinitions of the kelvin (see Kelvin § History) and other SI base units (see Joule § History).

In 2017, the most accurate measures of the Boltzmann constant were obtained by acoustic gas thermometry, which determines the speed of sound of a monatomic gas in a triaxial ellipsoid chamber using microwave and acoustic resonances.

This convention simplifies many physical relationships and formulas. For example, the equipartition formula for the energy associated with each classical degree of freedom (${\displaystyle {\tfrac {1}{2}}kT}$ above) becomes

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

As another example, 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.

See also

• Committee on Data of the International Science Council
• Thermodynamic beta
• List of scientists whose names are used in physical constants

Notes

References