Boltzmann constant (k_B)

The Boltzmann constant k_B relates the thermal energy scale of a system to its temperature. It appears in S = k_B ln Ω, in the ideal-gas law written per molecule PV = N k_B T, and in the Boltzmann factor exp(−E/k_B T). Its value, fixed exactly by the 2019 redefinition of the SI, is 1.380649 × 10⁻²³ joules per kelvin.

Physically, k_B is best read as an exchange rate between two units that history defined separately: the kelvin, invented for thermometers, and the joule, invented for mechanics. The product k_B T is the natural energy of thermal agitation — about 4 × 10⁻²¹ J at room temperature, the typical kinetic jiggle of one molecule. Had temperature been measured in energy units from the start, k_B would equal one and disappear from the equations.

k_B is the gas constant R divided by Avogadro's number: R = N_A k_B. It is the per-molecule counterpart of the per-mole constant R.

Though it bears Boltzmann's name and underlies his 1877 entropy formula, the constant was first evaluated numerically by Max Planck in 1900, in the same blackbody-radiation work that introduced Planck's constant; Planck also gave the first accurate value of Avogadro's number from it.