kelvin

SI base unit
Name	Symbol	Quantity
kelvin	K	thermodynamic temperature
Definition The kelvin, symbol K, is the SI base unit of thermodynamic temperature. The kelvin is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380 649 × 10⁻²³ when expressed in the unit J K⁻¹, which is equal to kg m² s⁻² K⁻¹, where the kilogram, metre and second are defined in terms of h, c and Δν_Cs.

The definition of the kelvin implies the exact relation k = 1.380 649 × 10^–23 kg m² s^–2 K^–1. Inverting this relation gives an exact expression for the kelvin in terms of the defining constants h, Δν_Cs and k :

$1 \mspace{4mu} \text{K} \mspace{10mu} = \dfrac{1.380 \mspace{4mu} 649 \times 10^{-23}}{k} \mspace{6mu} \text{kg} \mspace{4mu} \text{m}{^2} \mspace{4mu} \text{s}{^{-2}}\\ \\ \\ 1 \mspace{4mu} \text{K} \mspace{10mu} = \dfrac{1.380 \mspace{4mu} 649 \times 10^{-23}}{(6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34})(9 \mspace{4mu} 192 \mspace{4mu} 631 \mspace{4mu} 770))} \mspace{6mu} \dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{k}\\ \\ \\ 1 \mspace{4mu} \text{K} \mspace{10mu} = 2.266 \mspace{4mu} 665 \mspace{4mu} 264 \mspace{4mu} 601 \mspace{4mu} 104 \mspace{4mu} 867 \text{...} \mspace{6mu} \dfrac{h \mspace{6mu} \Delta \nu _{Cs}}{k}$

The effect of this definition is that one kelvin is equal to the change of thermodynamic temperature that results in a change of thermal energy k T by 1.380 649 × 10^–23 J.

The kelvin is named after the British physicist and engineer William Thomson, 1st Baron Kelvin (1824 – 1907).

Thermodynamic temperature

Thermodynamic temperature is a measure of the average kinetic energy of the particles in a substance.

The absolute temperature of a gas is directly proportional to the average kinetic energy of its molecules. The Kelvin scale is an absolute thermodynamic temperature scale using as its null point absolute zero, the temperature at which all thermal motion ceases in the classical description of thermodynamics.

Ideal gas law

The ideal gas law, also called 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, is equal to the product of two of the SI defining constants – the Boltzmann constant, k, and the Avogadro constant, N_A. Substituting the Boltzmann constant gives an alternative form of the general gas equation:

$p V = n k N_A T\\ \\ p V = N k T$

where:

N is the number of molecules of gas,
k is the Boltzmann constant, in J K⁻¹,
N_A is the Avogadro constant, in mol^-1.

Kinetic temperature

Molecular kinetic theory relates the pressure and volume of a gas to the average molecular kinetic energy.

Using SI coherent units,

$p V = \dfrac{2}{3}\ N \left( \ \overline{\dfrac{1}{2}m v^2} \ \right)$

where:

p is the pressure in pascals, symbol Pa,
V is the volume in cubic metres, symbol m³,
N is the number of molecules of gas,
¹⁄₂ mv² is the average kinetic energy of the gas molecules, in joules, symbol J.

Combining this with the ideal gas law gives an expression for temperature, sometimes referred to as the kinetic temperature.

$N k T = \dfrac{2}{3}\ N \left( \ \overline{\dfrac{1}{2}m v^2} \ \right) \\ \\ \\ k T \mspace{16mu} = \dfrac{2}{3} \left( \ \overline{\dfrac{1}{2}m v^2} \ \right) \\ \\ \\ T \mspace{26mu} = \dfrac{2}{3}\ \dfrac{1}{k} \left( \ \overline{\dfrac{1}{2}m v^2} \ \right)$

where:

T is the absolute temperature in kelvins, symbol K,
k is the Boltzmann constant, in J K⁻¹.

It can be seen that 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.

Triple point of water

The triple point of water is the unique combination of temperature and pressure at which ice, liquid water and water vapour can all coexist in thermodynamic equilibrium. The previous definition of the kelvin set the temperature of the triple point of water, symbol T_TPW, to be exactly 273.16 K.

Due to the fact that the current definition of the kelvin fixes the numerical value of the Boltzmann constant, k, instead of T_TPW, the latter must now be determined experimentally. At the time of adopting the current definition, T_TPW was equal to 273.16 K with a relative standard uncertainty of 3.7 × 10⁻⁷ based on measurements of k made prior to the redefinition.

The triple point of water occurs at a partial vapour pressure of 611.66 Pa.

Black body radiation

The spectral radiance of a body varies with its temperature. It is a description of the amount of energy that it emits at different electromagnetic radiation frequencies. Spectral radiance is the power emitted per unit area of the body, per unit solid angle of emission, per unit frequency.

A black body is an idealised object which absorbs and emits all frequencies of electromagnetic radiation. Planck’s radiation law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature, when there is no net flow of energy between the body and its surroundings.

Planck’s law shows that for any given temperature, there is a unique wavelength of electromagnetic radiation where the spectral radiance is at a maximum. At higher temperatures the wavelength of the peak radiance is shorter. For example, at a temperature of about 4000 K the peak radiance occurs at the red end of the visible spectrum, and at 7600 K it is at the violet end.

Wien’s displacement law

Wien’s displacement law encapsulates the relationship described in Planck’s law between the wavelength of the peak radiance and the temperature of a black body. Wien’s law states that the black body radiation curve for a given temperature has a peak value at a wavelength that is inversely proportional to the temperature.

Using SI coherent units, the proportionality constant is Wien’s displacement constant, b:

$\lambda_{\text{max}} = \dfrac{b}{T}$

where:

λ_max is the wavelength in metres, symbol m,
T is the absolute temeprature in kelvins, symbol K,
b is Wien’s displacement constant in metre kelvins, symbol m K.

Wien’s displacement constant is 2.897 771 955 … × 10⁻³ m K.

Colour temperature

The kelvin is used as a measure of the colour temperature of light sources. Colour temperature is based upon the principle that a black body radiator emits light whose colour depends on the temperature of the radiator. Black bodies with temperatures below about 4000 K appear reddish, whereas those above about 7500 K appear bluish.

Temperature in K

Colour temperature is important in the fields of image projection and photography, where a colour temperature of approximately 5600 K is required to match “daylight” film emulsions.

Image editing software and digital cameras often use colour temperature in K for colour balancing. The higher the colour temperature, the more white or blue the image will be. A reduction in colour temperature gives an image more dominated by reddish, “warmer” colours.

In astronomy, the stellar classification of stars and their place on the Hertzsprung-Russell diagram are based, in part, upon their surface temperature, known as effective temperature. The photosphere of the Sun has an effective temperature of 5778 K.

Noise temperature

In electronics, the kelvin is used as an indicator of how noisy a circuit is in relation to an ultimate noise floor, i.e. the noise temperature. The Johnson-Nyquist noise of discrete resistors and capacitors is a type of thermal noise derived from the Boltzmann constant and can be used to determine the noise temperature of a circuit using the Friis formulas for noise.

kelvin

SI base unit

Definition

Thermodynamic temperature

Ideal gas law

Kinetic temperature

Triple point of water

Black body radiation

Wien’s displacement law

Colour temperature

Noise temperature