Test4

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.

2 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.