Difference between revisions of "Bouguer-Lambert-Beer Absorption Law"

Revision as of 23:13, 1 March 2017

Leif Gerward

Department of Physics, Technical University of Denmark, DK-2800 Lyngby/Denmark.

1 Introduction

It is well known that the intensity [math]I[/math] of light (or other electromagnetic radiation) should decrease exponentially with the distance d that it enters an absorbing medium, i.e. [math]I=I_{0}exp(-\mu d)\tag {1}[/math] where [math]I_{0}[/math] is the intensity of the incident radiation, and μ is the linear absorption (or attenuation) coefficient. Equation (1) is usually known as Lambert’s law of absorption (e.g. Ballentyne and Lovett, 1970). However, there is some confusion about the origin of that law, since other names are Lambert–Beer’s law and Bouguer’s law. Therefore, it should be interesting to know a little more about the scientists associated with the Absorption Law, and about their contributions to the subject. As it turns out, the story has already been told. Fred Perrin (1948) has published a note with the somewhat provocative title “Whose Absorption Law?” Perrin’s paper, however, appeared almost sixty years ago, and few may be aware of it today. Thus, for the benefit of the reader of the Bulletin, it should be worthwhile once again to discuss the origin of the Absorption Law. I have also extended the scope of the present paper by including biographical sketches of Bouguer, Lambert and Beer.

2 Pierre Bouguer (1698–1758)

Pierre Bouguer was a child prodigy trained by his father, Jean Bouguer, who was a professor of hydrography at Croisic in Lower Brittany, France. The father died when Pierre was 15 years old, but by then the son had already acquired enough knowledge in mathematics and natural sciences to succeed in the chair of his father. In spite of his young age, Pierre lectured with great authority on topics such as mathematics, physics and astronomy. The French Académie des Sciences awarded him several prizes for his work, and in 1731 he became himself a member of the Academy. In 1735 he participated in an expedition to Peru to measure the length of a degree of meridian at the equator (this expedition was part of a larger project aimed at determining the shape of the Earth). Bouguer also measured gravity at different altitudes and was the first to estimate the gravitational pull of mountains, the so-called Bouguer correction.

Pierre Bouguer (Fig. 1) devoted most of his professional life to the solving of nautical problems. However, he is also known as the father of photometry. In 1729 he published an extensive work, Essai d’optique sur la gradation de la lumière, where he laid the foundation of this branch of science. In the present context it is interesting to note that Bouguer, among other things, investigated the absorption of light in the atmosphere and other transparent media. Bouguer’s optical work was posthumously published in 1760 as Traité d’optique sur la gradation de la lumière.

Chapter two of the Essai d’optique deals with transparency and opacity of matter. Bouguer takes the trouble to explain in detail how the intensity of light decreases upon traversing a transparent body. In his model, the body is divided into a number of layers of equal thickness, and the light rays are incident perpendicular to the layers. At first sight, one might think that the intensity of light should decrease in arithmetical progression, but Bouguer argues that this cannot be true. Instead, as he correctly points out, the intensity will always decrease in geometrical progression, since each layer absorbs a constant fraction of the intensity incident upon it. Therefore,

the intensity of light, after traversing different thicknesses, can be represented by the ordinates of a logarithmic [curve], which has the thickness of the medium as its axis [of abscissae] (Bouguer, 1729).

\; As noticed by Bouguer, the logarithmic curve, which is well known to mathematicians and surveyors, has the properties that the subtangent is constant and, most importantly in the present case, the ordinates decrease in geometrical progression.

In order to illustrate his point, Bouguer considers a perfectly homogenous body ABCD, divided into layers of equal thickness (Fig. 2). The light is incident perpendicular to the face AB. The length QB represents the incident intensity. After having traversed the depth BF or AE, the light enters the second layer through the face EF. At this depth the intensity is RF. In this way, the intensity as a function of depth can be described by the logarithmic curve QRXY, where BC is the axis of abscissae (the x-axis). For a less transparent body the intensity will decrease more rapidly. In fact, the shape of the transmission curve is determined solely by its subtangent and by the intensity of the incident radiation. Bouguer then goes on, giving several examples of how transmission curves can be constructed and used in various problems.

It follows from above that Bouguer has given a correct description and interpretation of the transmission of light in a transparent body, although he has not expressed the Absorption Law in the concise form of Equation (1), later to be derived by Lambert.