### 6.8 Kirchhoff’s Law explains why nobody is perfect.

Remember that when radiation encounters matter it may be absorbed or transmitted or scattered (including reflected). For an object acting as a perfect Planck distribution function, it must absorb all radiation completely with no scattering and no transmission. Some objects absorb very well at some wavelengths but not at others. For instance, water vapor absorbs little visible radiation but absorbs infrared radiation at some wavelengths very well.

At the same time, the sun, like other objects, does not radiate perfectly according to the Planck distribution function spectral irradiance, but instead radiates at a fraction of it at some wavelengths. This fraction, which goes from 0 to 1, is called the emissivity and is denoted by *ε*. How is an object’s emissivity related to its absorptivity?

**Kirchhoff’s Law** states that at any given wavelength, an object’s emissivity *ε* is equal to its absorptivity, that is:

$$\epsilon \left(\lambda \right)=\alpha \left(\lambda \right)$$

Thus, if an object has some wavelengths at which radiation is scattered or reflected, then the object will have an emissivity less than 1 at the wavelength, and the fraction that is absorbed will be equal to the emissivity at each wavelength.

Thus, when we integrate the Planck distribution function spectral irradiance over wavelength to obtain the irradiance emitted by the object, it first has to be multiplied by the wavelength-dependent emissivity, thus leading to the modified form of the Stefan–Boltzmann law:

$$F=\epsilon \text{\hspace{0.17em}}\sigma \text{\hspace{0.05em}}{T}^{4}$$

where we understand that *ε* is some form of averaged emissivity.

Watch the following video (1:07), where the Stefan–Boltzmann Law is described in greater detail:

#### Note

Some typical average emissivities are listed in the table below. These are emissivities averaged over all wavelengths. At any particular wavelength, the emissivity may be greater or less than the average.

Material | Emissivity, ε |
---|---|

ice | 0.97 |

pure water | 0.96 |

snow | 0.8–0.9 |

trees (oak, beech, maple, pine) | 0.97–0.98 |

grass | 0.98 |

soil | 0.93 |

aluminum foil | 0.03 |

asphalt | 0.88–0.94 |

What about gases? Gases absorb and thus emit like all other matter. To know more about the emissivity of all objects, we need to know more about the absorption of objects.

#### Quiz 6-2: Thank You, Planck

- Find
**Practice Quiz 6-2**in Canvas. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz. - When you feel you are ready, take
**Quiz 6-2**. You will be allowed to take this quiz only**once**. Good luck!