Question

Define the spectral emissivity of a medium of thickness \(L\) in terms of the spectral absorption coefficient.

Short answer

Question: Define the spectral emissivity of a medium with a given thickness L in terms of its spectral absorption coefficient. Answer: The spectral emissivity of a medium with thickness L can be defined in terms of its spectral absorption coefficient using the equation: \(E(\lambda) = 1 - \exp{(-\alpha (\lambda) \cdot L)}\).

Step-by-step solution

Recalling the Beer-Lambert Law

The Beer-Lambert law describes the process of absorption of radiation as it passes through a medium. The law states that the absorbed fraction of radiation is proportional to the medium's thickness and the spectral absorption coefficient. The law can be represented by the equation: \(A(\lambda) = 1 - \exp{(-\alpha (\lambda) \cdot L)}\) Here: - \(A(\lambda)\) represents the spectral absorptance - \(\alpha (\lambda)\) represents the spectral absorption coefficient - \(L\) represents the thickness of the medium

Kirchhoff's Law of Radiation

Kirchhoff's law states that the emissivity of a material equals its absorptivity at a specific frequency or wavelength. In mathematical terms: \(E(\lambda) = A(\lambda)\) Here: - \(E(\lambda)\) represents the spectral emissivity - \(A(\lambda)\) represents the spectral absorptance

Defining Spectral Emissivity in Terms of the Spectral Absorption Coefficient

Using Kirchhoff's law and the Beer-Lambert law, we can determine the spectral emissivity \(E(\lambda)\) in terms of the spectral absorption coefficient \(\alpha(\lambda)\) and the thickness \(L\) of the medium as follows: \(E(\lambda) = 1 - \exp{(-\alpha (\lambda) \cdot L)}\) This equation defines the spectral emissivity of a medium with thickness \(L\) in terms of its spectral absorption coefficient.