Boltzmann constant

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The Boltzmann constant (k or k_B) is the physical constant relating temperature to energy.

It is named after the Austrian physicist Ludwig Boltzmann, who made important contributions to the theory of statistical mechanics, in which this constant plays a crucial role. Its experimentally determined value (in SI units, 2002 CODATA value) is:

  [k = \ ] 1.380 6505(24)×10⁻²³ joules/kelvin

:[ = \ ] 8.617 339×10⁻⁵ electron-volts/kelvin.

The digits in parentheses are the uncertainty (standard deviation) in the last two digits of the measured value. The conversion factor between the values of the constant in the two different units of measure is the magnitude of the electron's charge:

q = 1.602 177×10⁻¹⁹ coulomb per electron.

Physical significance

Boltzmann's constant k is a bridge between macroscopic and microscopic physics. Macroscopically, one can define a (gas scale) absolute temperature as changing in proportion to the product of the pressure P and the volume V that a sample of an ideal gas would occupy at the temperature:

[P V \propto T]

Introducing Boltzmann's constant transforms this into an equation about the microscopic properties of molecules,

[P V = N k T \,]

where N is the number of molecules of gas, and k is Boltzmann's constant. This reveals kT as a characteristic quantity of the microscopic physics, having the dimensions of energy, and signifying the volume × pressure per molecule.

The numerical value of k has no particular fundamental significance in itself - it merely reflects a preference for measuring temperature in units of familiar kelvins, based on the macroscopic physical properties of water. What is physically fundamental is the characteristic energy kT at a particular temperature. The numerical value of k measures the conversion factor for mapping from this characteristic microscopic energy E to the macroscopically-derived temperature scale T = E/k . If, instead of talking of room temperature as 300 K (27 °C or 80 °F), it were conventional to speak of the corresponding energy kT of 4.14×10⁻²¹ J, or 0.026 eV, then Boltzmann's constant would simply be the dimensionless number 1.

In principle, the joules per kelvin value of the Boltzmann proportionality constant could be calculated from scratch, rather than measured, using the definition of the kelvin in terms of the physical properties of water. However this computation is too complex to be done accurately with current knowledge.

(Note: the ideal gas equation can also be written

[P V = n R T \, ]

where n = N / N_A, the number of molecules divided by Avogadro's number, is the number of molecules measured in moles, and R = N_A × k, the Boltzmann constant multiplied by Avogadro's number, is called the universal gas constant. Working in terms of moles can be more handy when dealing with everyday chemical quantities of substances).

Role in the equipartition of energy

Given a thermodynamic system at an absolute temperature T, the 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 3 degrees of freedom per atom, corresponding to the three spatial directions, which means a thermal energy of 1.5kT per atom. As indicated in the article on heat capacity, this corresponds very well with experimental data. The thermal energy can be used to calculate the root mean square speed of the atoms, which is 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.

From kinetic theory one can show that for an ideal gas the average pressure P is given by:

[ P = \frac\frac m }]

Substituting that the average translational kinetic energy is:

[ \fracm \overline = \frac k T ]

gives:

[ P = \frac k T ]

and so the ideal gas equation is regained.

The ideal gas equation is also followed quite well for molecular gases; but the form for the heat capacity is more complicated, because the molecules possess new 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 in total approximately 5 degrees of freedom per molecule.

Role in Boltzmann factors

More generally, systems in equilibrium with a reservoir of heat at temperature T have probabilities of occupying states with energy E weighted by the corresponding Boltzmann factor:

[p \propto \exp} ]

Again, it is the energy-like quantity kT which takes central importance.

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

Role in definition of entropy

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

[S = k \, \ln \Omega.]

This equation, which relates the microscopic details of the system (via Ω) 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 appears in order to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:

[\Delta S = \int \frac]

In hindsight however, it is perhaps a pity that it was not chosen to introduce a rescaled entropy such that:

[ = \ln \Omega} \; ; \; \; \; \Delta S^ = \int \frac]

These are rather more natural forms; and this (dimensionless) rescaled entropy exactly corresponds to Shannon's subsequent information entropy, and could thereby have avoided much unnecessary subsequent confusion between the two.

Role in semiconductor physics

In semiconductors, the relationship between the flow of electrical current and the electrostatic potential across a p-n junction depends on a characteristic voltage called the thermal voltage, denoted V_T. The thermal voltage depends on absolute temperature T (in kelvins) as:

[ V_T = ]

where q is the magnitude of the electrical charge (in coulombs) on the electron. At room temperature (T ≈ 300 K), the value of the thermal voltage is approximately 26 millivolts. See also semiconductor diodes.

Boltzmann's constant in Planck units

Planck's system of natural units is one system constructed such that the Boltzmann constant is 1. This gives:

[ T } \ ]

as the average kinetic energy of a gas molecule per degree of freedom; and makes the definition of thermodynamic entropy coincide with that of information entropy:

[ S = - \sum p_i \ln p_i ]

The value chosen for the Planck unit of temperature is that corresponding to the energy of the Planck mass —a staggering 1.41679{{e.

Historical Note

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 an accurate value for it, in his derivation of the law of black body radiation in December 1900. The iconic terse form of the equation S = k log W on Boltzmann's tombstone is in fact due to Planck, not Boltzmann.

As Planck wrote in his 1918 Nobel Prize lecture,

"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. 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." [link]

Before 1900, equations involving Boltzmann factors were not written using the energies per molecule and Boltzmann's constant, but rather using the gas constant R, and macroscopic energies for macroscopic quantities of the substance; as for convenience is still generally the case in Chemistry to this day.

References

• [Boltzmann's constant CODATA value] at NIST
• Peter J. Mohr, and Barry N. Taylor, "CODATA recommended values of the fundamental physical constants: 1998", Rev. Mod. Phys., Vol 72, No. 2, April 2000

 

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