# Boltzmann constant

 related topics {math, energy, light} {math, number, function} {acid, form, water} {theory, work, human} {line, north, south}

The Boltzmann constant (k or kB) is the physical constant relating energy at the individual particle level with temperature observed at the collective or bulk level. It is the gas constant R divided by the Avogadro constant NA:

It has the same units as entropy. It is named after the Austrian physicist Ludwig Boltzmann.

## Contents

### Bridge from macroscopic to microscopic physics

Boltzmann's constant, k, is a bridge between macroscopic and microscopic physics. Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure p and volume V is proportional to the product of amount of substance n (in moles) and absolute temperature T:

where R is the gas constant (8.314 472(15) J K−1 mol−1). Introducing the Boltzmann constant transforms the ideal gas law into an equation about the microscopic properties of molecules,

where N is the number of molecules of gas.

### 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 three 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.