# Boltzmann constant

(Redirected from Boltzmann's constant)

The Boltzmann constant (k or kB) 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 is (in SI units, 2002 CODATA value)

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. In principle, the Boltzmann constant could be a derived physical constant, as its value is determined by other physical constants and in the definition of unit of absolute temperature. However, calculating the Boltzmann constant from first principles is far too complex to be done with current knowledge. In Planck's system of natural units, the natural unit temperature is such that the Boltzmann constant is one.

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. Room temperature, 300 K (27 °C or 80 °F), corresponds to a kT/2 of 2.07 × 10−21 J, or 13 meV.

In classical statistical mechanics, homogeneous ideal gases possess kT/2 per degree of freedom per atom. 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 definition of entropy

In statistical mechanics, the entropy S of a system 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):

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

The constant of proportionality k is the Boltzmann constant. 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.

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