Supplement 4.1: Lambert's Law and the Lambert-Beer Law

Lambert's law in chemistry

In the section about absorption of light, Lambert's law, which shows the decline in the intensity of light on its way $x$ through an absorbing medium having an absorption coefficient $a$, has been presented like this:

In chemistry, the law is mostly written as an exponential function to the base 10:

The variable $\chi$ is known as decadic extinction coefficient, which is obviously not of the same number as the absorption coefficient $a$. How can one quantity be converted into another? The algebraic relation

is very useful in order to transform exponential functions from one base to another. In this case we write:

For Lambert's law we conclude:

Taking into account Euler's number $\text{e}=2.\mathrm{718...}$ and with ${log}_{10}\text{e}=0.\mathrm{434...}$ it can be obtained:

By the same procedure it can be shown that:

When using data from literature, it is always important to pay attention to what is actually presented: the absorption coefficient or the extinction coefficient respectively, i.e., which base of the exponential term is underlying.

In experiments, the intensity loss of light by absorbing matter is measured with an absorption photometer. The functionality is presented on the following pages. Because photometers are used especially in chemist's laboratories, the data format orientates at the typical notion of Lambert's law in chemistry: As an exponential function to the base 10. In consequence they do not present data of the extinction coefficient $\chi$, but the absorbance A, which is derived from the decadic logarithm of Lambert's law:

The absorbance is also named optical density.

The term absorption is a rather colloquial expression for the weakening of light. It is not further defined in physics.

The Lambert-Beer law

In nature, absorbing matter nearly always is a composite of different materials. A photometric measurement provides values of the absorbance, defining the concentrations is not always possible. However, when experimenting with pure substances one usually knows about their specific absorbing features. This is especially convenient for the analysis of liquids in which absorbing pure substances are dissolved. That way it is possible to determine the concentration from the knowledge about the extinction coefficient. For the absorbance one can write:

This is the Lambert-Beer law, where $c$ is the concentration of the dissolved substance which is given unit of mol/L. $\epsilon$ is the molar decadic extinction coefficient in L/(mol·cm), if the optical path length $x$ is given in cm. The writing of L/(mol·cm) is often neglected so that the extinction coefficient is commonly given without a unit. Sometimes the concentration is indicated in g/L instead of mol/L. In consequence L/(g·cm) applies to the extinction coefficient as well.

As solvents, liquids of low absorbance are favourable, for instance purified water, methanol, ethanol and other organic substances. Their absorbances are small, nontheless they have to be taken into account in precise measurements. Photometers correct the absorbance of the solvent through a comparative measurement of the solution and the solvent. This is done by measuring data from the pure solvent first. Then a second measurement is done with the sample. Both series of data have to be set off against each other in the way it is described in the section about absorption photometry.