Blackbody Radiation and Wein’s Law – Learn


Blackbody Radiation

The atoms in any object which is at a temperature above 0K will have some energy which causes them to vibrate back and forth. As the particles are vibrating, they must be accelerating, hence so are the charges that make up the atoms. As a result of these charges accelerating, electromagnetic radiation will be emitted from any object above 0K – all objects continuously radiate energy in the form of electromagnetic radiation.

Any body at a temperature above 0K will emit radiation at all wavelengths of the electromagnetic spectrum. The intensity of each wavelength varies according to the temperature of that body. It is important to note here that the intensity of any particular radiation is dependent on the temperature of the body only and not the type of material itself. You can observe the radiation emitted from lava yourself – very hot lava will be yellow and cooler lava will be red. Even cooler lava will be black, it is however still emitting radiation from the infrared part of the spectrum.

 

As an object becomes hotter there are two main features of the emitted radiation that change:

  • the total amount of radiation emitted increases
  • the peak intensity of the radiation shifts towards shorter wavelengths

These ideas will be explored further below.


What is a Blackbody?

A blackbody is an ideal surface that completely absorbs all wavelengths of electromagnetic radiation falling on it. A black body will also be a perfect emitter of electromagnetic radiation at all wavelengths. The radiation is characteristic of the temperature of the black body but not of the atoms it is made from. When a body is at room temperature, the strongest emissions are in the infrared part of the electromagnetic spectrum.

A blackbody is a theoretical concept and in reality perfect black bodies do not exist. However, we can visualise a blackbody by imagining a material with a hollow space at its core. If a pin sized hole was made to allow radiation to pass into the cavity this radiation will ‘bounce’ around inside and reflect off the surface. If the material was a perfect black body then all of the radiation would eventually be absorbed. If we assume that the body is in thermal equilibrium with its surroundings, the radiation that was absorbed would be emitted at the same rate at which it was absorbed. The features of the radiation that is emitted will only be dependent on the temperature of the body and not the type of material it is made of.

 

The image below illustrates a blackbody with radiation falling on its surface at a single, particular frequency. The radiation that is emitted is dependent on its temperature only. Whilst it emits a continuous spectrum of radiation (radiation of all wavelengths) the intensity of each wavelength varies according to the temperature of the black body:


Blackbody Spectrum

The radiation emitted from a blackbody forms a continuous spectrum. The spectrum of observed radiation and the relative intensities of each wavelength is displayed on a black body curve. The black body model and curve is useful as it allows us to determine the temperature of distant objects.

The ‘classical’ wave-theory of light predicted that, as the wavelength of radiation emitted becomes shorter, the radiation intensity would increase. Classical theory predicted that as the energy that was emitted decreased in wavelength, the intensity of the radiation emitted would approach infinity. This would violate the principle of conservation of energy and could not be explained by existing theories. This effect was called the ‘ultraviolet catastrophe’.

 

Data collected from blackbody experiments clearly showed that the black body curve had a definite peak. As the temperature of a blackbody increased this peak shifted toward shorter wavelengths. Further to this, the area under the curve also increased, indicating that the total amount of radiation emitted also increased.

 

Max Planck proposed an idea to explain the experimental data observed for blackbodies. He suggested that the energy radiated from a blackbody was emitted in packets. Each ‘packet’ represented a particular frequency or wavelength of radiation and the size of that packet varied depending on the temperature of the black body. Essentially, Planck was explaining that the atoms in a black body could only oscillate with energies of specific sizes. He called these packets of energy quanta. This was the beginning of quantum physics.

These quanta of energy are quantitatively explained by the equation:

E = hf

where:

E = energy, measured in J

h = Planck’s constant = 6.626 × 10−34 Js

f = frequency in Hz.


Wein’s Law

Wilhelm Wein extended on the theory that as an object became hotter the peak intensity of this radiation shifted toward shorter wavelengths, to provide a quantitative model for determining the peak wavelength. He summarised his theory using the equation:

{ \lambda }_{ max }\;=\;\cfrac { b }{ T }

where:

{ \lambda }_{ max } is the wavelength where the peak occurs (in m)

b is Wein’s constant = 2.898 × 10−3 mK

T is the temperature in K (0°C = 273.15K)


Example 1:

How much energy does a photon from the visible spectrum have if it has a frequency of 7.2 × 1014 Hz?

Answer: using E = hf

E=(6.626\times { 10 }^{ -34 })\times (7.2\times { 10 }^{ 14 })

E=4.8\times { 10 }^{ -19 }\:J

Example 2:

Lava from a volcanic eruption glows a dull red colour and emits light with a wavelength of 1.2 μm. What is the surface temperature of the lava?

Answer: using { \lambda }_{ max }\;=\;\cfrac { b }{ T } and rearranging to give: T=\cfrac { b }{ { \lambda }_{ max } }

T=\cfrac { 2.898\times { 10 }^{ -3 } }{ 1.2\times { 10 }^{ -6 } }

T=2415K

T=2415-273.15={ 2141.85 }^{ \circ }C

Close Menu