Boltzmann constant

SI defining constant

Name	Symbol	Base units
Boltzmann constant	k	J K^-1
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.
$k = 1.380 \mspace{4mu} 649 \times 10^{-23} \mspace{6mu} \text{J} \mspace{4mu} \text{K}^{-1}$

Heat capacity

The fixed numerical value of the Boltzmann constant, k, is defined exactly:

$k \mspace{6mu} = 1.380 \mspace{4mu} 649 \times 10^{-23} \mspace{4mu} \text{J} \mspace{4mu} \text{K}^{-1}$

Inverting this relation gives an exact expression for the unit of heat capacity, the joule per kelvin, in terms of the Boltzmann constant, k :

$1 \mspace{6mu} \text{J} \mspace{4mu} \text{K}^{-1} \mspace{6mu} = \dfrac{1}{1.380 \mspace{4mu} 649 \times 10^{-23}} \mspace{4mu} k$

Thermodynamic temperature

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, Δν_Cs, forms the basis for the definition of the unit of thermodynamic temperature, the kelvin.

The definition of the kelvin implies the exact relation:

$\dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{k} \mspace{6mu} = \dfrac{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)} {1.380 \mspace{4mu} 649 \times 10^{-23}} \mspace{6mu} \text{K}$

Inverting this relation gives an exact expression for the kelvin in terms of the Boltzmann constant, k, the Planck constant, h, and the caesium frequency, Δν_Cs :

$1 \mspace{4mu} \text{K} \mspace{4mu} = \dfrac{1.380 \mspace{4mu} 649 \times 10^{-23}}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770)} \mspace{6mu} \dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{k}$

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 joules per mole kelvin, symbol 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, N_A.

$R \ = k N_A \\ \\ R \ = 1.380 \mspace{4mu} 649 \times 10^{-23} \mspace{6mu} \text{J K}^{-1} \ \times \ 6.022 \mspace{4mu} 140 \mspace{4mu} 76 \mspace{4mu} \times 10^{23} \mspace{6mu} \text{mol}^{-1}\\ \\ R \ = 8.314 \mspace{4mu} 462 \mspace{4mu} 618 \mspace{4mu} 153 \mspace{4mu} 24 \mspace{6mu} \text{J K}^{-1} \text{mol}^{-1}$

The general gas equation can therefore be written in terms of the Boltzmann constant, k:

$p V = N k T$

where:

N is the number of molecules of gas (which is equal to the amount of gas in moles multiplied by the Avogadro constant),
k is the Boltzmann constant, in J K⁻¹.

Rearranging gives an expression for the Boltzmann constant, k, in terms of the parameters of the general gas equation:

$k = \dfrac{p V}{N T}$

Molar heat capacity

Inverting the definition for the ideal gas constant gives an exact expression for the unit of molar heat capacity, the joule per mole kelvin, in terms of the Boltzmann constant, k, and the Avogadro constant, N_A :

$1 \mspace{4mu} \text{J} / (\text{mol} \mspace{4mu} \text{K}) \mspace{6mu} = \dfrac{1}{8.314 \mspace{4mu} 462 \mspace{4mu} 618 \mspace{4mu} 153 \mspace{4mu} 24} \mspace{6mu} k \mspace{4mu} N_A$

Nature

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.