The development of Beer-Lambert's law is done from a phenomenological perspective (see Figure below). First, students are asked to reason about the form of the linear relationship they observed. For example, students are asked to explain why the line must pass through the origin. If there is no light, then there can be no change in light. Thus, the observation that the change in the light intensity is linear with respect to the light intensity is equivalent to the difference equation:
I (d+1) I (d) = k·I (d)
I (d+1) = (k + 1)·I (d) = r·I (d)
where I (d) is the intensity of light at depth d, k is a constant between -1 and 0, and r = k+1. This statement is the basis of Lamberts law.
A discrete model of light passes through layers of
water is used to develop Beer-Lamberts law.
Students can verify the reasonableness of the equation by observing that with each new depth, a fraction of the light is absorbed and that this fraction remains constant. Both statements resonate well with the Plexiglas experiment. Furthermore, students are asked to reason on the values of the constants. The constant k is negative for the fraction of light lost, while the constant r is positive representing the fraction of light retained.
Using these equations, students generate the pattern that:
I (1) = r· I (0)
I (2) = r·I (1) = r ^2· I (0)
I (3) = r ^3· I (0)
From which the general solution of I = r ^ d· I (0) can be conjectured and tested against the data. As already noted, this model of light absorbance is known historically as Lamberts law or more recently as Beer-Lamberts law:
I = I (0)·e ^(m·d) =I (0)· r^d
where I (0) is the intensity of the incoming light and m = ln(r ) is the absorption coefficient of the substance (Iavorskii, 1980). As a result of this activity, students are able to make sense of Beer-Lamberts law using their previous experiences, their experiences in conducting the experiments and symbolic reasoning.
Beer-Lamberts law can also be developed from a calculus perspective by considering increasingly small changes in depth. This results in a differential equation rather than a difference equation. An alternative definition of the absorption coefficient is as the constant of proportionality between the rate of change in the light intensity and the light intensity, I' = m·I. Separation of variables or Eulers method can be used to solve the differential equation. In doing so, the definition of the natural logarithm and exponential functions as well as their derivatives and integrals arise naturally. Thus, this activity is useful in a wide-variety of classes including algebra, pre-calculus, calculus and differential equations.
The value of r in the discrete development is equal to the constant e^m. Sometimes the value of r is more meaningful since it more directly expresses the fraction of light absorbed. Discussion of how constants are defined can provide useful insight into the nature of science and mathematics and in this case, provide practice with exponents. Finally it should be noted that Beer-Lamberts law can also be developed from a statistical many-body perspective, but that requires much more mathematical sophistication.
By completing this investigation on Beer-Lamberts law, students have an opportunity to generate, collect, and analyze data. Students also have the opportunity to develop and test conjectures, recognize patterns, fit curves to make predictions established by the data, and represent the situation using recurrence relations. Beer-Lamberts law serves as the basis of spectroscopic instruments which are increasingly being used in the science curriculum. Moreover, research is still in progress to understand and find appropriate models for the light absorbance (e.g. Gordon, 1989; Perovich, 1995). The activity also leads to numerous deeper science and mathematics questions such as why the ocean appears blue and how does the light intensity change if only certain wavelengths are absorbed. For these reasons, the Shedding Light on the Subject has become one of our students favorite activities.
Other activities developing Lamberts law without the use of data driven experiments may be found in the Journal of Chemical Education articles "Discovering the Beer-Lambert Law" by Robert Ricci, Mauri A. Ditzler, and Lisa P. Nestor (1994), and "The Beer-Lambert Law Revisited: A Development without Calculus" by Peter Lykos (1992). These articles also provide a further discussion of the underlying physics as well as indicating numerous other teaching resources on Beer-Lamberts law.