# Boltzmann constant

##### SI defining constant
Name Symbol Value Unit Expressed in terms of SI base units
Boltzmann constant k 1.380 649 × 10−23 J K-1 kg m2 s−2 K−1

### Definition

The numerical value of the Boltzmann constant, symbol k, is defined to be exactly 1.380 649 × 10−23 when expressed in the unit joule per kelvin, symbol J K-1, or kg m2 s−2 K−1.

The Boltzmann constant is a proportionality constant which relates the average relative kinetic energy of particles in a gas to the thermodynamic temperature of the gas.

The Boltzmann constant, k, together with the Planck constant, h, and the hyperfine transition frequency of the caesium 133 atom, ΔνCs, forms the basis for the definition of the kelvin.

### Nature

The Boltzmann constant k is a proportionality constant between the quantities temperature (with unit kelvin) and energy (with unit joule), whereby the numerical value is obtained from historical specifications of the temperature scale. The temperature of a system scales with the thermal energy, but not necessarily with the internal energy of a system. In statistical physics the Boltzmann constant connects the entropy S with the number Ω of quantum-mechanically accessible states, S = k ln Ω.

### Ideal gas constant

The ideal gas law, also known as the general gas equation, is the equation of state of a hypothetical ideal gas. It approximates the behaviour of gases under many conditions.

Using SI coherent units,

$p V = n R T$

where:

• p is the pressure in pascals, symbol Pa,
• V is the volume in cubic metres, symbol m3,
• T is the absolute temperature in kelvins, symbol K,
• n is the amount of gas in moles, symbol mol,
• R is the ideal gas constant, in J K−1 mol−1,

The ideal gas constant, R, also known as the molar gas constant, is the molar equivalent of the Boltzmann constant. It is equal to the product of two of the SI defining constants –
the Boltzmann constant, k, and the Avogadro constant, NA.

$R = k N_A$

The exact value of the ideal gas constant can be calculated as follows:

$R = 1.380\ 649 \times 10^{-23} \mspace{6mu} \text{J K}^{-1} \ \times \ 6.022\ 140\ 76 \times 10^{23} \mspace{6mu} \text{mol}^{-1}\\ \\ R=8.314\ 462\ 618\ 153\ 24 \mspace{6mu} \text{J K}^{-1} \text{mol}^{-1}$