The total radiant intensity B(T ) of a blackbody at the absolute temperature T can be derived by integrating the Planck function over the entire wavelength domain from 0 to∞. Since blackbody radiation is isotropic, the flux density emitted by a blackbody is therefore F = π B(T ) which is proportional to the fourth power of the absolute temperature T through the Stefan-Boltzmann constant σ = 5.67 × 10−8 J m−2 sec−1 deg−4.
Kirchoff's law establishes that for a medium at the thermodynamic equilibrium, the spectral emissivity ε(λ) at a given wavelength λ, is equal to the its spectral absorbance, A(λ) at the same wavelength λ. Hence ε(λ)=A(λ) at each fixed λ, for a blackbody ε(λ)=A(λ)=1 at whatever λ. Kirchoff's law is valid also in Local Thermodynamic Equilibrium (LTE) conditions as the ones usually occurring in (small volumes of) the Earth's atmosphere even in the most turbulent conditions.
Kirchoff's law has important applications also for the study of spectral signatures of mineral and rocks and, in general, of opaque - i.e. with spectral transmittance T(λ)=0 - bodies. In that case, the relation which relate the spectral reflectance R(λ), absorbance A(λ) and transmittance T(λ) of a body: R(λ)+A(λ)+T(λ) =1
reduce to R(λ)+A(λ)=1 and in LTE conditions, thanks to the Kirchoff's law:
R(λ)+ε(λ)=1 which allows to obtain measurements of spectral emissivity indirectly through (more simple and stable) measurements of spectral reflectance:
ε(λ)=1-R(λ)
Rocks and mineral exhibit important (diagnostic/discriminating) signatures in their spectral emissivity in the thermal infrared (TIR) region. Measuring spectral emissivity in a laboratory (particularly if samples have to be characterized for their properties in natural conditions) is a quite difficult task due to the difficulty to insolate the sample from the lab environment (and instruments themselves) all emitting approximately at the same (environmental) temperature. Kirchoff's law allows to obtain, for opaque bodies, spectral emissivities from spectral reflectances measurements which are much easy to realize in normal remote sensing labs.