Beer–Lambert law

Remarks

The idea of the proof: a thin layer of thickness $dz$ removes a fixed fraction $\mu(z) \, dz$ of the light that enters it. This gives the linear differential equation $I' = -\mu I$, whose solution is an exponential of the integral of $-\mu$. With $\mu = \ln(10) \, \epsilon c$ constant, taking $\log_{10}$ gives the absorbance form.

Example: if a layer of $1$ cm transmits half of the light, then $\mu = \ln 2$ per cm, and $3$ cm of the same medium transmit $(1/2)^3 = 1/8$ of the light.

The hypotheses are needed. At very high intensity the absorption becomes nonlinear (saturation), and $\mu$ depends on $I$. If the medium scatters light into the beam or emits light, the intensity obeys the full radiative transfer equation, of which this law is the case with no source terms. At high concentration, interactions between molecules make the absorbance depend nonlinearly on $c$.

Going deeper

For a student

The law is the simplest solution of the radiative transfer equation: a beam along one line, with only extinction. The quantity $\int_0^\ell \mu \, dz$ is the optical depth of the path, and the law says the transmission is its negative exponential. In spectroscopy it is used to measure concentrations; in astrophysics and atmospheric physics it describes the extinction of starlight and sunlight.