Stefan-Boltzmann constant

What is the Stefan-Boltzmann constant?

The Stefan-Boltzmann constant, symbolized by the lowercase Greek letter sigma (σ), is a physical constant expressing the relationship between the heat radiation emitted by a black body and its absolute temperature. According to the Stefan-Boltzmann law, σ has a value of 5.670374419 × 10⁻⁸ watt per square meter per kelvin to the fourth (W / (m² x K⁴).

σ represents the constant of proportionality between these two variables. The law itself only applies to black bodies, which are theoretical surfaces that absorb all incident heat radiation. In practice, such bodies do not exist.

However, the Stefan-Boltzmann law and constant are still useful to study radiation phenomena, such as in Planck's radiation law, and to establish the relationship between an object's temperature and the heat radiated by it.

Black bodies and Stefan-Boltzmann constant

The Stefan-Boltzmann constant is named after its two formulators, Austrian physicists Josef Stefan and Ludwig Boltzmann, who each formulated it in 1879 and 1884, respectively. σ is a constant value and involves black body radiation.

A black body -- sometimes spelled as blackbody -- is also called an ideal radiator. It is an object that radiates or absorbs all radiant energy falling on it with perfect efficiency. The word black refers to the fact that incident visible light gets absorbed into the body instead of being reflected, so the surface appears to be black. Although black bodies are purely theoretical, a box with a small hole and a blackened interior provides a good -- in fact, the best possible -- practical approximation of a black body.

The Stefan-Boltzmann constant defines the power per unit area emitted by a black body as a function of its thermodynamic temperature.

Energy radiated by an ideal radiator

The constant is based on Stefan-Boltzmann's law, which states that the radiant heat energy emitted from a unit area of the black body in one second (E) is directly proportional to the fourth power of its absolute temperature, or E = σ x T⁴.

E is distributed over a range of wavelengths of radiation. Further, as T increases, the wavelength for maximum energy emission shifts to shorter values, so high energy means short wavelengths and high frequency.

Deriving the Stefan-Boltzmann constant

The value of the Stefan-Boltzmann constant can be derived or determined experimentally, like Stefan and Boltzmann did. Using the Boltzmann constant, the formula to derive the constant is expressed as the following:

σ = (2 x π⁵ x k_B⁴) / (15 x h³ x c²) = 5.670367 x 10^-8 W / (m² x K⁴)

Here:

• k_B = Boltzmann constant = 1.380649 x 10^-23 joules per K (J x K^-1);
• h = Planck's constant = 6.62607015 x 10^-34 joule-seconds (J x s); and
• c = speed of light = 3 x 10⁸ m / s.

According to the Committee on Data of the International Science Council, σ can also be derived from the gas constant as the following:

σ = (2 x π⁵ x R⁴) / (15 x h³ x c² x N_A⁴) = (32 x π⁵ x h x R⁴ x R_∞⁴) / (15 x A_r(e)⁴ x M_u⁴ x c⁶ x α⁸)

Here:

• R = universal gas constant = 8.3144598 J per mole per K (J x mol^-1 x K^-1)
• N_A = Avogadro constant = 6.02214076 x 10²³ mol^-1
• R_∞ = Rydberg constant = 10,973,731.6 m^-1
• A_r(e) = relative atomic mass of the electron = 1/1,840 mass of the proton or neutron
• M_u = molar mass constant = 1 gram per mol (g / mol)
• α = fine structure constant = 1 / 137 = 0.0072973525

Expressing Stefan-Boltzmann constant in various unit systems

The dimensional formula for the Stefan-Boltzmann constant is expressed as M¹ x T^-3 x K^-4, where M is mass.

σ can also be expressed in other systems as follows:

Stefan-Boltzmann constant applications

The Stefan-Boltzmann constant is used in many practical applications in physics. In particular, the constant helps with the derivation of many physical quantities, such as the amount of heat a black body radiates. It is also used to calculate the temperature required to generate a particular amount of radiation by a black body over a specific area.

Other Stefan-Boltzmann constant applications are the following:

• convert K to the units for intensity, or W / m²;
• calculate how hot the sun is based on how much power strikes the Earth in a square meter;
• predict how much heat the Earth radiates into space;
• calculate the radii of stars; and
• calculate the emissivity of an object, which is the ratio of its total emissive power to the total emissive power of a black body.

See also: table of physical constants.