Boltzmann constant

Values of k[1] Units
1.38064852(79)×10−23 JK−1
8.6173324(78)×10−5 eV⋅K−1
1.38064852(79)×10−16 erg⋅K−1
For details, see § Value in different units below.

The Boltzmann constant (kB or k), named after Ludwig Boltzmann, is a physical constant relating energy at the individual particle level with temperature. It is the gas constant R divided by the Avogadro constant NA:

$k = \frac{R}{N_\text{A}}.\,$

The Boltzmann constant has the dimension energy divided by temperature, the same as entropy. The accepted value in SI units is 1.38064852(79)×10−23 J/K.

Bridge from macroscopic to microscopic physics

Boltzmann's constant, k, is a bridge between macroscopic and microscopic physics. Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure p and volume V is proportional to the product of amount of substance n (in moles) and absolute temperature T:

$pV = nRT \,$

where R is the gas constant (8.3144621(75) J⋅K−1⋅mol−1[1]). Introducing the Boltzmann constant transforms the ideal gas law into an alternative form:

$p V = N k T ,$

where N is the number of molecules of gas. For n = 1 mol, N is equal to the number of particles in one mole (Avogadro's number).

Role in the 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 on the order of magnitude of kT/2 (i.e., about 2.07×10−21 J, or 0.013 eV, at room temperature).

Application to simple gas thermodynamics

In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. Monatomic ideal gases possess three degrees of freedom per atom, corresponding to the three spatial directions, which means a thermal energy of 1.5kT 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

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

Combination with the ideal gas law

$p V = N k T$

shows that the average translational kinetic energy is

$\tfrac{1}{2}m \overline{v^2} = \tfrac{3}{2} k T.$

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. kT/2.

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 Pi of occupying a state i with energy E weighted by the corresponding Boltzmann factor:

$P_i \propto \frac{\exp\left(-\frac{E}{k T}\right)}{Z},$

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

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):

$S = k \,\ln W.$

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:

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

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

${S' = \ln W}, \quad \Delta S' = \int \frac{\mathrm{d}Q}{k T}.$

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.

Role in semiconductor physics: the thermal voltage

In semiconductors, 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 VT. The thermal voltage depends on absolute temperature T as

$V_\mathrm{T} = { k T \over q },$

where q is the magnitude of the electrical charge on the electron with a value 1.602176565(35)×10−19 C[1] and k is the Boltzmann's constant, 1.38064852(79)×10−23 J/K. In electronvolts, the Boltzmann constant is 8.6173324(78)×10−5 eV/K,[1] making it easy to calculate that at room temperature (≈ 300 K), the value of the thermal voltage is approximately 25.85 millivolts ≈ 26 mV.[2] The thermal voltage is also important in plasmas and electrolyte solutions; 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.[3][4]

History

Although Boltzmann first linked entropy and probability in 1877, it seems the relation was never expressed with a specific constant until Max Planck first introduced k, and gave a 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.[5] 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.[6]

In 1920, Planck wrote in his Nobel Prize lecture:[7]

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 heuristic tool for solving problems. There was no agreement whether chemical molecules, as measured by atomic weights, were the same as physical molecules, as measured by kinetic theory. Planck's 1920 lecture continued:[7]

Nothing can better illustrate the positive and hectic pace of progress which the art of experimenters has made over the past twenty years, than the fact that since that time, not only one, but a great number of methods have been discovered for measuring the mass of a molecule with practically the same accuracy as that attained for a planet.

In 2013 the UK National Physical Laboratory used microwave and acoustic resonance measurements to determine the speed of sound of a monatomic gas in a triaxial ellipsoid chamber to determine a more accurate value for the constant as a part of the revision of the International System of Units. The new value was calculated as 1.38065156(98)×10−23 J⋅K−1 and is expected to be accepted by the International System of Units following a review.[8]

Value in different units

Values of k Units Comments
1.38064852(79)×10−23 J/K SI units, 2010 CODATA value, J/K = m2⋅kg/(s2⋅K) in SI base units[1]
8.6173324(78)×10−5 eV/K 2010 CODATA value[1]
electronvolt = 1.602176565(35)×10−19 J[1]

1/k = 11604.519(11) K/eV

2.0836618(19)×1010 Hz/K 2010 CODATA value[1]
1 Hzh = 6.62606957(29)×10−34 J[1]
3.1668114(29)×10−6 EH/K EH = 2Rhc = 4.35974434(19)×10−18 J[1]
= 6.579683920729(33) Hzh[1]
1.0 Atomic units by definition
1.38064852(79)×10−16 erg/K CGS system, 1 erg = 1×10−7 J
3.2976230(30)×10−24 cal/K steam table calorie = 4.1868 J
1.8320128(17)×10−24 cal/°R degree Rankine = 5/9 K
5.6573016(51)×10−24 ft lb/°R foot-pound force = 1.3558179483314004 J
0.69503476(63) cm−1/K 2010 CODATA value[1]
1 cm−1 hc = 1.986445683(87)×10−23 J
0.0019872041(18) kcal/(mol⋅K) per mole form often used in statistical mechanics—using thermochemical calorie = 4.184 joule
0.0083144621(75) kJ/(mol⋅K) per mole form often used in statistical mechanics
4.10 pN⋅nm kT in piconewton nanometer at 24 °C, used in biophysics
−228.5991678(40) dBW/(K⋅Hz) in decibel watts, used in telecommunications (see Johnson–Nyquist noise)
1.442 695 041... Sh in shannons (logarithm base 2), used in information entropy (exact value 1/ln(2))
1 nat in nats (logarithm base e), used in information entropy (see Planck units, below)

Since k is a physical constant of proportionality between temperature and energy, its numerical value depends on the choice of units for energy and temperature. The small numerical value of the Boltzmann constant in SI units means a change in temperature by 1 K only changes a particle's energy by a small amount. A change of °C is defined to be the same as a change of 1 K. The characteristic energy kT is a term encountered in many physical relationships.

Planck units

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 the Planck units or natural units for temperature and energy. In this context temperature is measured effectively in units of energy and the Boltzmann constant is not explicitly needed.[9]

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

$E_{\mathrm{dof}} = \frac{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:

$S = - \sum P_i \ln P_i.$

where Pi is the probability of each microstate.

The value chosen for a unit of the Planck temperature is that corresponding to the energy of the Planck mass or 1.416833(85)×1032 K.[1]

References

1. P.J. Mohr, B.N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 6.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http://physics.nist.gov/constants [Thursday, 02-Jun-2011 21:00:12 EDT]. National Institute of Standards and Technology, Gaithersburg, MD 20899.
2. 300 kelvin * k / elementary charge in millivolts – Google Search
3. Kirby BJ., Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
4. Tabeling (2006), Introduction to Microfluidics<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
5. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).. English translation: "On the Law of Distribution of Energy in the Normal Spectrum". Archived from the original on 2008-12-17.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
6. Duplantier, Bertrand (2005). "Le mouvement brownien, 'divers et ondoyant'" (PDF). Séminaire Poincaré 1 (in French): 155–212. Unknown parameter |trans_title= ignored (help) <templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
7. Planck, Max (2 June 1920), The Genesis and Present State of Development of the Quantum Theory (Nobel Lecture)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
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