Abstract
The Beer–Lambert law is inadequate to describe the absorption of radiation by a medium if the absorbing component is being simultaneously destroyed by the radiation. A replacement law is derived and solved in terms of a family of polynomials. The solution is confirmed numerically and by simulation.
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Notes
Alternatively one might prefer to express \(I\) in Einsteins per square metre per second and \(C\) in moles per cubic metre.
Only minor changes in the theory are needed if the product also absorbs light, but is otherwise inert.
The cell width \(L\) does not appear in the ensuing mathematics. Though necessarily finite in experimental practice, there is no theoretical limit to the cell width. In the simulation \(L\) is accorded the value \(2.5/[\upvarepsilon C(X,0)]\).
As is their sum \(f(y,z)+g(y,z)\)
Provided that F remains finite for all positive values of \(x\) and \(t\).
Applicable, after augmentation by the italicized entries in Table 2, to all positive integer values of \(k\) and \(h\).
Not to be confused with the Bernoulli polynomials for which the same notation is in use.
Other than by retracing the derivation backwards through the \(b\) integers and the \(a\) coefficients.
By truncating the summations after \(k\) = 12 and adding one-half of the \(k=13\) term.
In the examples used to create the figures, the values \(M=N\) = 499 were employed.
References
P. Klàn, J. Wirz, Photochemistry of Organic Compounds; From Concepts to Practice (Wiley, Chichester, 2009)
W.J. Youngblood, S.-H.A. Lee, K. Maeda, T.E. Mallouk, Visible light water splitting using dye-sensitized oxide semiconductor. Acc. Chem. Res. 42, 1966–73 (2009)
K. Rajeshwar, J. Ibanez, Environmental Electrochemistry: Fundamentals and Applications in Pollution Abatement, chap 6 (Academic Press, San Diego, 1997)
P. Bouguer, Essai d’optique sur la graduation de la lumiere (Jombert, Paris, 1729)
J.H. Lambert, Photometria sive de mensure et gradibus luminis (Klett, colorum et umbrae, Augsberg, 1760)
A. Beer, Bestimmung der Adsorption des rothen Licht in farbien Flüssikeiten. Annalen der physikalischer Chemie 86, 74–88 (1852)
J.M. Parnis, K.B. Oldham, Beyond the Beer–Lambert law: the dependence of absorbance on time in photochemistry. J. Photochem. Photobiol. A Chem. 267, 6–10 (2013)
Y. Pinchover, J. Rubinstein, Introduction to Partial Differential Equations (Cambridge University Press, New York, 2005)
L. Lapidus, G.F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering (Wiley, Chichester, 1982)
M.L. Boas, Mathematical Methods in the Physical Sciences, 2nd edn. (Wiley, New York, 1983), p. 365
S. Wolfram, The Mathematica Book, 4th edn. (Cambridge University Press, New York, 1999)
Microsoft. Excel spreadsheet software. (The Microsoft Corporation, Redmond, 2010)
Acknowledgments
This unfunded research was kindly assisted by Jan Myland.
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Oldham, K.B. The propagation of radiation through a medium containing a component that absorbs the radiation and is steadily destroyed by it. J Math Chem 52, 1007–1019 (2014). https://doi.org/10.1007/s10910-014-0309-1
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DOI: https://doi.org/10.1007/s10910-014-0309-1