Boltzmann Constant

The Boltzmann constant (symbol k or k_B) is the physical constant that links the temperature of a system to the average energy of its particles. Since the 2019 redefinition of the SI units its value is fixed exactly at 1.380649 × 10⁻²³ joules per kelvin.

Temperature measures how much random thermal motion the particles of a substance have, and the Boltzmann constant converts that temperature into energy per particle. For an ideal gas the average translational kinetic energy of a molecule is (3/2)*kT*, and multiplying k by Avogadro's number gives the molar gas constant R, about 8.314 joules per mole per kelvin. In this way it ties the microscopic world of single molecules to the large-scale laws of thermodynamics.

Ludwig Boltzmann showed that entropy counts the number of microscopic arrangements, W, that are consistent with what is observed of a system. The relation S = k ln W is engraved on his tombstone in Vienna, and it explains why heat flows from hot to cold: there are overwhelmingly more ways to spread energy out than to concentrate it.

At body temperature the thermal energy kT is about 4 × 10⁻²¹ joules, a few piconewton-nanometers. This is similar to the energy of weak bonds such as the hydrogen bonds that hold proteins and DNA together, so thermal jostling constantly makes and breaks them. In biophysics the Boltzmann distribution, which gives the chance of finding a molecule in a state of energy E as proportional to e^−*E*/kT, explains how enzymes, channels and molecular motors work.

• Kelvin — the unit the constant is defined with
• Entropy — the quantity Boltzmann counted
• Thermodynamics — the macroscopic laws it underpins