The Beer-Lambert law in optics is an empirical relationship that relates the absorption of light by a material, to the properties of the material through which the light is traveling. This is the basis of absorption spectrometry, which allows to measure the concentration of different gases in a chamber.

	The principle of an absorption spectrometer, illustrated on the left, is as follows. Consider the following two experiments. First, we do a control experiment, where we illuminate from one side a container containing some reference gas with light at a certain frequency. We measure the light intensity (say, [latex]I_0[/latex]) at the other side of the container. Then, we add some other gas to the container, repeat the experiment, and measure the light intensity again (say, [latex]I[/latex]). Depending on the absorption properties, as well as the concentration [latex]x[/latex], of the added gas, the light will be more or less absorbed with respect to the reference situation.
	The Beer-Lambert law postulates that the log-ratio [latex]\log(I/I_0)[/latex] is linear in the concentration [latex]x[/latex]. In other words, [latex]y = \log(I/I_0) = ax[/latex], where the constant [latex]a[/latex] depends on the light frequency and on the gas.

If the container has a mixture of [latex]n[/latex] ‘‘pure’’ gases in it, the law postulates that the logarithm of the ratio of the light intensities is a linear function of the concentrations of each gas in the mix. The log-ratio of intensities is thus of the form [latex]y=a^Tx[/latex] for some vector [latex]a \in \mathbb{R}^n[/latex], where [latex]x[/latex] is the vector of concentrations. The coefficients [latex]a_j[/latex], [latex]j= 1, \cdots, n[/latex] correspond to the log-ratio of light intensities when [latex]x= e_j[/latex] (the [latex]j[/latex]-th vector of the standard basis, which correspond to the [latex]j[/latex]-th pure gas). The quantity [latex]a_j[/latex] is called the coefficient of absorption of the [latex]j[/latex]-th gas and can be measured in the laboratory.

See also: Absorption spectrometry: using measurements at different light frequencies.