Boltzmann constant

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 J/K.

The digits in parentheses are the uncertainty (standard deviation) in the last two digits of the measured value.

Physical significance

The numerical value of k in itself has no particular fundamental physical significance. What is physically significant is the direct relationship between a temperature T and a corresponding characteristic energy kT (as discussed below).

The constant k merely reflects a preference for communicating this temperature in units of familiar kelvins, rather than in terms of the corresponding characteristic energy. 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 26 meV, 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 first principles, rather than measured, as the kelvin is defined in terms of the physical properties of water. However this computation is far too complex to be done accurately with current knowledge.

The universal gas constant R is simply the Boltzmann constant multiplied by Avogadro's number. The gas constant is more useful when calculating numbers of particles in moles.

Role in relating temperature to 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 (ie about 2.07 × 10⁻²¹ J, or 13 meV at room temperature).

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 speed at room temperature ranges from 1370 m/s for helium, down to 240 m/s for xenon. The situation is more complicated for molecular gases; diatomic gases, for example, possess 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:

$p\propto \exp {\frac {-E}{kT}}$

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 equilibirum 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:

$\Delta S=\int {\frac {dQ}{T}}$

With 20:20 hindsight however, it is perhaps a pity that Boltzmann did not choose to introduce a rescaled entropy such that

${S^{\,'}=\ln \Omega }\;;\;\;\;\Delta S^{\,'}=\int {\frac {dQ}{kT}}$

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.

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

${E={\frac {1}{2}}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 × 10³² K

Reference

• 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