$$\require{cancel}$$

# 10.17 Kirchoff's Laws

Emission line spectrum of hydrogen in the visible part of the spectrum. Click here for original source URL

In the 1850s, German physicist Gustav Kirchhoff discovered in the laboratory the conditions that produce the three different kinds of spectra: the continuum, absorption lines, and emission lines. When Kirchhoff looked through his spectroscope toward a sodium flame against a dark background, he saw a strong yellow emission line. But when he changed the background to a brilliant beam of sunlight passing through the same flame, he saw a strong absorption line at the same yellow wavelength. In each case, the lines came from the gaseous sodium atoms in the flame.

Gustav Kirchhoff. Click here for original source URL

From laboratory observations, Gustav Kirchhoff derived three physical laws, called Kirchhoff's laws of radiation:

• A gas, or a sufficiently heated solid, will glow with a smooth, continuous spectrum, or continuum.

• A hot gas will produce only certain bright and narrow wavelengths, called emission lines. Each element emits a characteristic set of emission lines.

• A cool gas, if placed between the observer and a hot continuous-spectrum source, absorbs certain wavelengths, causing narrow absorption lines in the observed spectrum.

Hydrogen energy diagram. Click here for original source URL.

The Bohr model for the atom, showing the nucleus and different levels that an electron can occupy. Click here for original source URL.

What is happening at the subatomic level in a hot gas? You can think of electron energy levels like a kind of ladder, with energy increasing upwards. The rungs of the ladder indicate the discrete, or quantized, energy levels. The lowest rung of the ladder is the ground state; all higher levels are called excited states and the spacing of the rungs decreases as the energy increases. Below the highest rung (or energy level) the electron can only have quantized energy levels — they are bound to the atom. Above the highest rung the electron energy level can vary continuously — such electrons are unbound and called free electrons.

Solar spectrum showing the dark absorption lines. Click here for original source URL.

Constant, microscopic motions of atoms and molecules lead to a smooth spectrum of thermal radiation. The peak wavelength of the radiation is an indicator of the temperature, according to Wein's law. If a material is hot enough that electrons are not bound to the atomic nucleus, the electrons can have a continuous range of energies. These free electrons will emit a smooth featureless spectrum. The filament of a light bulb is an example. The electrons in a gas can be liberated from the atomic nucleus either by energy from photons or by collisions with other energetic particles. The smooth continuum of the Sun's radiation tells us that the edge of the Sun is a layer of hot gas. Remember that Wein's law allows us to determine a body's temperature by its dominant radiation. The wavelength of the strongest solar radiation gives the temperature of the solar surface as about 5700 Kelvin.

Emission lines arise when a gas is hot enough that electrons are in excited states, but not hot enough that all the electrons are liberated. Using the ladder analogy, each electron that drops from a higher rung to a lower rung emits a photon of a particular wavelength. Each different downward jump corresponds to a particular energy difference. The energy lost by the electron appears as a photon. The array of possible ways the electron can lose energy maps into a set of sharp emission lines. A neon sign is a familiar example. The neon gas inside the glass tube is excited (or has energy added to it) by an electrical current. As the excited electrons drop back into lower energy states, they emit photons with particular energies. The purity of the color reflects the fact that the radiation is concentrated in a few red emission lines. Other gases have their own distinctive set of emission lines — think of the bluish color of mercury vapor street lights or the yellowish color of sodium vapor street lights or the intense pink glow of a neon sign.

Absorption lines arise in a cooler gas, when electrons are in or near the ground state. If the cool gas is illuminated by a hot source of radiation, the gas will see photons with a smooth and continuous range of energies. However, photons can only be absorbed if their energy corresponds to the difference between two electron energy levels. Returning to the ladder analogy, photons are absorbed if they can raise an electron to a higher rung. Each possible upward jump corresponds to a different wavelength of photon that can be absorbed. Since this energy is removed from the incoming radiation, it leaves a deficit of photons at a particular set of wavelengths. Photons of these wavelengths are emitted again as the excited electrons drop back down into lower energy orbits, but in this case the emission takes place in all directions including back towards the source. The result is that radiation at these specific wavelengths is subtracted from the background source, resulting in dark lines. The cool surface layer of the Sun is an example.

Kirchhoff himself found that the absorption lines and emission lines of a given gas have identical wavelengths. This makes sense since the same electron energy levels are involved in the two processes. What we see depends on the temperature and density of the gas relative to the radiation coming from behind it, as indicated in the third law. Later an important modification was made to Kirchhoff's laws: an absorption spectrum need not originate in front of, or in a cooler gas than, the continuous spectrum. It can arise within the same gas as the continuous spectrum. This is because within a single gas, electrons may be jumping upward in some atoms (making absorption lines) and downward from the free state into other atoms (forming the continuum spectrum). These light-emitting layers of gas form the well-defined visible surface of the Sun.

Astronomers usually make a spectrum by dispersing the light from the entire object. However, it is possible to get more detailed spectral information in a different way. We can form an image by filtering the light so that only a small range of wavelengths is detected. If the narrow wavelength range is centered on a particular spectral feature, we can make a map of the physical conditions that lead to the spectral transition. The Sun contains hydrogen over its whole surface. However, if it is viewed through a filter that transmits the appropriate narrow range of wavelengths, we can get a map where the temperature and density are just right to cause atoms to emit the Hα emission line.