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1 1. The classical law and its history

When electromagnetic radiation, such as ultraviolet or visible light, passes through a transparent medium that contains an absorber of that illumination, the radiation’s intensity diminishes steadily with passage through the medium. Commonly the radiation is in the form of a collimated beam that impinges perpendicularly on a slab of width [math]L[/math] of the medium, as suggested diagrammatically in Figure 1. One may conjecture that, at any illuminated plane [math]x[/math] within the medium, the decrease in the intensity [math]I[/math] of the radiation with distance would be proportional to the uniform concentration [math]c[/math] of the absorber and to the local intensity of the light at that point; that is

[math] \frac{\mathrm d}{\mathrm d x} ( I(x) )=-\alpha cI(x) (1) [/math]

where [math]\alpha[/math] is a proportionality constant. Integration of this equation leads to

[math] \ln\frac{I(0)}{I(x)}=\alpha c x [/math]

whence, on choosing [math]x[/math] to be the exit plane for the radiation, and with [math]\varepsilon=\alpha ln(10)=2.303\alpha[/math]:

[math] \log_{10}\frac{I(0)}{I(L)}=\varepsilon c L [/math]

Citing the earlier findings [1] of Pierre Bouguer, the 1760 treatise [2] of Johann Lambert publicized the linear dependence of the logarithm of the [math]I(0)/I(L)[/math] ratio on [math]L[/math], whereas its analogous dependence on c remained unrecognized until the work [3] of August Beer almost a century later. Equation (3) is the form in which Beer’s law (also known as the Beer-Lambert or Beer-Lambert-Bouguer law) is most commonly encountered.

The constant [math]\varepsilon[/math] goes by a variety of names, of which absorptivity, extinction coefficient, absorbancy index and attenuation coefficientare four. Frequently and confusingly, those names, or similar ones, are also given to [math]\alpha[/math] moreover, the symbol [math]\varepsilon[/math] often replaces in equation (2). Further misperception can arise because [math]\varepsilon[/math] has the inverse units of (concentration) [math]\times[/math] (distance) and its numerical value therefore reflects the unit chosen for concentration, which variously may be moles per litre, molecules per cubic metre or one of several kinds of percentage. Absorptivity is generally a function of wavelength and therefore Beer’s law is strictly valid only if the radiation is monochromatic. Furthermore, the law fails at high concentrations or if there is significant light scattering from turbidity.

Any of equations (1), (2) or (3), as well as equations (4) and (5) below, may be said to be mathematical expressions of Beer’s law. A verbal statement of the law would be:

When radiation traverses a medium containing an absorbing but photostable component, the decrease of radiant intensity with distance, at any point in the medium, is proportional to the concentration of the absorber and to the local light intensity, and therefore the radiant intensity varies exponentially with distance.

2 2. Photometry

The reciprocal of the [math]I(0)/I(L)[/math] ratio is known as the transmittance [math]T\lt/math?, whereas the decadic logarithm of this ratio is defined as the absorbance, \ltmath\gtA[/math]; accordingly

3 References

[1] Bojarski, Czesław, and Joachim Domsta. "Theory of the Influence of Concentration on the Luminescence of Solid Solutions." Acta Physica Academiae Scientiarum Hungaricae 30, no. 2 (1971): 145. [1]

[2] Bojarski, C. "Influence of the Reversible Energy Transfer on the Donor Fluorescence Quantum Yield in Donor-Acceptor Systems." Zeitschrift für Naturforschung A 39, no. 10 (1984): 948-951 [2]

[3] Sienicki, K., and M. A. Winnik. "Donor-acceptor kinetics in the presence of energy migration. Forward and reverse energy transfer." Chemical physics 121, no. 2 (1988): 163-174.

[4] Twardowski, R., and J. Kuśba. "Reversible energy transfer and fluorescence decay in solid solutions." Zeitschrift für Naturforschung A 43, no. 7 (1988): 627-632.

[5] Sienicki, K., and G. Durocher. "Time‐dependent chemical reactions: A revision of monomer–excimer kinetics?." The Journal of chemical physics 94, no. 10 (1991): 6590-6597

[6] Kułak, L., and C. Bojarski. "Forward and reverse electronic energy transport and trapping in solution. I. Theory and II. Numerical results and Monte Carlo simulations." Chemical physics 191, no. 1-3 (1995): 43-66 and 67-86.