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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[/math], whereas the decadic logarithm of this ratio is defined as the absorbance, [math]A[/math]; accordingly

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

and

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

Both these quantities are dimensionless and therefore independent of the units (of which there are many alternatives) in which the light intensity is measured. Alternative names for [math]A[/math] are absorbancy and optical density and one-tenth of this quantity has been called the transmission loss(in decibels).

The primary usage of Beer’s law in chemistry is in measuring concentrations via photometry or spectrophotometry [4]. The “spectro” in the latter name reflects the frequent utility of measuring absorbance as a function of the radiation’s wavelength, thereby generating an absorption spectrum. The medium is most often a liquid solution housed in a cuvette (a rectangular “box” of standard interior length, with transparent window-like walls); but gaseous and solid media are investigated similarly. Figure 1 resembles a schematic diagram of a spectrophotometer. The intensity of the monochromatic light exiting the medium is measured and compared either with the incident intensity at [math]x=0[/math], or with the emergent intensity from the same or a matched cuvette with the absorber absent. The output of the instrument is generally in the form of a transmittance and/or an absorbance measurement. A standardizing experiment is also performed, using a solution of known absorber concentration, whereby [math]\varepsilon[/math] is determined or confirmed. In photometry, the concentration of absorber in the cuvette is generally uniform and unchanging, but a uniform concentration is not mandatory. Beer’s law holds whether or not the concentration is uniform, because the solution to the differential equation

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

is given by aversion of equation (3) in which the [math]“c”[/math] in that equation is replaced by the average concentration along the light path, so that

[math] A=log_{10}\frac{I(0)}{I(L)}=\varepsilon \bar{c} L where \bar{c}=\frac{1}{L}\int\limits_0^Lc(x)dx [/math]

This conclusion is easily rationalized because it is the number of light-absorbing molecules, and not their distribution, that in crucial in abbreviating the light intensity.

3 3. The bivariate Beer laws

A spatial variability of concentration can arise from the illumination itself if the light-absorbing molecules, on capturing the light’s photons, suffer a chemical change. Typically, only a small fraction of photon absorptions lead to chemical transformation, most photons being absorbed harmlessly. Though molecular destruction or reorganization may be infrequent, each chemical change causes the absorber concentration to decrease with time and also to become non uniform. Figure 2 shows the diminution of light intensity along the path length and the concomitant development of a gradient in concentration. Not only does the absorber concentration become a temporal – as well as a spatial – variable but, in consequence, the light intensity is forced to follow suit, becoming perturbed away from the exponential spatial dependence predicted classically. Time, which has no relevance to the classical Beer law, becomes a player. Provided the decomposition product does not itself emit or absorb light, the partial derivative

[math] \frac{\partial }{\partial x} ( I(x,t) )=-\alpha c(x,t)I(x,t) [/math]

replaces equation (1). Because it involves two independent variables –distance and time –equation(8) is named the bivariate Beer intensity law. The rate of light-induced decomposition would be expected to be proportional both to the local light intensity and to the concentration of absorber, and this expectation is generally realized. Accordingly, the equation,

[math] \frac{\partial }{\partial t} ( c(x,t) )=-\beta I(x,t)c(x,t) [/math]

named the bivariate Beer concentration law, describes the local rate of chemical change, [math]\beta[/math] being a proportionality constant. Notice the similarities between equations (9) and (8): in form, the dependence of concentration on time exactly matches the dependence of light intensity on distance. A single function must describe both! The identity of that function was recently established [5]; it is called the bleaching function in recognition of the prototype problem to which it applies being the bleaching, by light, of dye solutions. The bleaching function solves the problem posed by the twin differential equations (9) and (8), subject to the boundary conditions [math]c(x,0)=c_{0}[/math] a constant, and [math]I(0,t)=I_{0}[/math], another constant; thus

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4 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.