A More Pedagogically Sound Treatment of Beer's Law: A Derivation

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A More Pedagogically Sound Treatment of Beer’s Law: A Derivation Based on a Corpuscular–Probability Model William D. Bare Department of Chemistry, University of Virginia, Charlottesville, VA 22904-4319; [email protected]

A survey of recent editions of the most popular undergraduate analytical chemistry texts reveals that most current texts provide a derivation or proof of Beer’s law that is based on calculus. The starting point for the derivation in most texts is the assumption of an infinitesimally thin section of solution and the postulate that the absorption of light (dI ) in this film is proportional to the power of the incident light (I ), the concentration of the absorbing species (c), and the thickness of the film (dx), as indicated in eq 1. From this, integration is used to derive eq 2, the familiar form of Beer’s law. dI = ᎑k I c dx

(1)

A = εbc

(2)

This derivation presents several problems. First, what is presented is not actually a derivation of Beer’s law. To the extent that it appears to be so it is a bit of mathematical sleight of hand. In this argument, Beer’s law is implicit in the initial proposition (eq 1). What is presented as a derivation, therefore, is only a mathematical rearrangement of the initial statement to a more useful form. Second, this argument is often presented with an abstruse conceptual model, the initial proposition being far from obvious. The very fact that the argument begins with a proposition concerning the properties of an infinitesimally thin film presents a serious conceptual difficulty. We obviously have no physical experience with such a film. How then, do we postulate—as a starting point—the physical properties expressed in eq 1? A physical model that is based on observable properties of samples with macroscopic dimensions is more intuitively clear and is more easily assimilated by students. Finally, and perhaps most importantly, many students lack the necessary skill in calculus to fully understand the equations employed. Even students who can correctly manipulate derivatives and integrals may not fully comprehend the subtle, but important, difference between finite and infinitesimal changes. This is a particularly important consideration in this case because the initial premise that the absorption of light is proportional to the concentration and the thickness of the solution is potentially misleading. Although this is mathematically true in the limit of infinitesimal thickness, it is not true of finite sections and may present a false impression of this relationship that can lead to confusion. Many students who can correctly answer questions pertaining to the linear relationship of absorbance measurements to concentration and path length also (incorrectly) ascribe a linear relationship to the number of photons absorbed. This may indicate that the commonly seen treatments of Beer’s law overemphasize the linear relationship of absorbance while neglecting the nature of the underlying physical interactions, which are fundamentally nonlinear. Because of the problems associated with the commonly seen derivations of Beer’s law, others have proposed alternate

proofs (1–5). An excellent scientific explanation of Beer’s law based on a corpuscular model and absorption cross sections has been presented by Berberan-Santos (4 ). Unfortunately, this derivation borrows equations from advanced kinetic molecular theory, which are likely to be beyond the grasp of first- and second-year undergraduates who have not yet had a course in physical chemistry. A derivation without calculus has also been published, which includes a mathematical treatment of logarithms (5). This derivation, however, rests on essentially the same initial proposition as the calculus-based derivations and, further, seems to suggest that the absorption of light is proportional to path length in a thin but finite section. Previous authors have also discussed the concept of probability as a basis for understanding Beer’s law. An article by Ricci, Ditzler, and Nestor (6 ) presents a thought experiment involving BB’s rolling down a perforated plane as a macroscopic model for photon absorption and suggests a lab activity to test the model’s validity. More recently, Daniels (7) described a mathematical modeling activity using a computer spreadsheet program and random number generation to demonstrate the effect of atom-shadowing in photon absorption. Bodner and Domin discussed the importance of mental representations in successful problem solving (8). It may also be argued that the pedagogical value of a mathematical derivation lies not in the derivation itself, but rather in the extent to which the derivation serves to form a mental connection between a useful mathematical equation (Beer’s law, in this case) and a conceptual model. With this in mind, I examine the corpuscular–probability model of photon absorption presented previously and use it to generate a derivation of Beer’s law. This simple derivation, which is based on a corpuscular model and the laws of probability, is intuitively logical, is supported by experimental observations of macroscopic absorbing samples, and requires only simple mathematics. An alternate model is also considered and shown to be inconsistent with experimental results. Models To see how Beer’s law may be generated from simple principles and from experimental evidence, we begin by considering a section of a dilute, ideal solution of some absorbing solute. The solution contains an arbitrary number of solute molecules (m), giving a concentration (c), and has an arbitrary but finite path length, which we will call the “standard path length” (d), as in Figure 1. We specify that the standard path length is of macroscopic dimensions (i.e., several orders of magnitude greater than a molecular diameter) and that m is statistically large (i.e., at least several hundred). The reader may verify that these criteria are satisfied for a hypothetical sample with a thickness of 1 mm, cross-sectional area of 1 cm2, and a concentration of 10᎑6 M. As light passes through the solution, some photons are absorbed so that the

JChemEd.chem.wisc.edu • Vol. 77 No. 7 July 2000 • Journal of Chemical Education

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Path length

Figure 1. A thin section of solution showing incident (Io) and transmitted (It) light intensity.

Figure 2. Plot showing the amount of light transmitted (It) and absorbed (Ia) as a function of path length according to the saturation model.

intensity of the transmitted light (It) is less than that of the incident light (Io). There will be some loss of light due to scattering and to reflections at interfaces, as has been discussed in depth by Swinehart (9), as well as some absorption of light by the solvent. For our purposes, however, we will assume that these effects are small and are largely compensated for by solvent blanks and instrument calibration. We neglect any small effects that cause deviations from Beer’s law. We begin a mathematical treatment of the interaction of light with the solution by defining the quantities that will be necessary for our discussion. Although the light intensities involved are properly defined as fluxes (with units of photons per square centimeter per second), we will specify that the solution has a constant cross section and is in a steady state with the photon flux. With this restriction, one may conveniently refer to these intensities as an “amount of light” or a “number of photons”, with the units of photons per square centimeter per second being implicit. The intensities of the incident light (Io) and the transmitted light (It) are shown in Figure 1. Since we are ignoring losses of light due to scattering and other effects, we may assume that all light is either absorbed or transmitted and we may define the amount of light absorbed (Ia) as the difference between the incident and transmitted light, as in eq 3. Ia ≡ Io – It (3)

From the observation that light is absorbed by the sample, we may propose two physical models, which we will call the “saturation model” and the “corpuscular–probability model”. The first has as its fundamental postulate that the sample is capable of absorbing a finite number of photons, after which it becomes saturated. All photons below this limit are absorbed, and those in excess of this limit are transmitted. The second model assumes that the solution contains a multitude of randomly dispersed absorbing bodies, and the absorption of light is related to the probability of a photon encountering an absorbing body while traversing the solution. Clearly, other models are possible, but these two will suffice for the current discussion. With these two intuitively reasonable propositions we may proceed to derive relevant equations and determine if the results thus obtained are consistent with experimental observation. This is a pedagogically useful reminder of the use of models and the scientific method and is particularly relevant given the fact that many students appear to hold the reasonable, but incorrect, beliefs embodied in the saturation model. We can evaluate the two proposed models by considering their effectiveness in predicting the result of increasing the thickness of the absorbing sample. We will consider the effect of passing light through two identical layers of solution, so that the new path length, is 2d and the total number of absorbing molecules is 2m. If the saturation model is correct, doubling the amount of sample will double the saturation limit and double the number of photons absorbed. Thus, increasing the sample length from d to 2d increases the amount of light absorbed from Ia to 2Ia. In general, increasing the sample path length to nd increases the absorbed intensity to nIa while the transmitted intensity decreases to (Io – nIa). In short, this model predicts that the dependence of transmitted light on total path length (b) is linear and that a sample of absolute opacity (Ia = Io, It = 0) is achieved with a sufficiently long path length. The relationship, which is plotted in Figure 2, can be expressed in an algebraic form (eq 7) in which the constant k depends upon the concentration of the sample (c), the saturation limit, and the size of the standard path length.

We define the fractional absorbance (FrA) and fractional transmission (FrT) as the ratio of the amount of light absorbed or transmitted to the total incident light, as expressed in eqs 4 and 5. Of course, the fractional transmission is the same as that which has traditionally been given the name “transmittance” and the symbol “T ”.

Ia Io

(4)

It ≡T Io

(5)

FrA ≡ FrT ≡

Since all light is either transmitted or absorbed, the sum of the fraction of light absorbed and that transmitted must be equal to unity. FrA + FrT = 1 (6) 930

It = Io – kb

(7)

A different mathematical relationship between transmitted

Journal of Chemical Education • Vol. 77 No. 7 July 2000 • JChemEd.chem.wisc.edu


intensity and path length is obtained from the corpuscular– probability model. We note that the transmittance is the fraction of photons that pass through the sample without being absorbed and is by definition a number between 0 and 1. This may be regarded as the probability of a photon passing through the solution without encountering an absorbing center in transit. We may therefore equate the fractional transmission (FrT) and fractional absorption (FrA) with the probability of transmission (PrT) and the probability of absorption (PrA), respectively, as given in eq 8. FrT = PrT

and

FrA = PrA

(8)

Furthermore, since all light is either transmitted or absorbed, the sum of the probabilities for absorption and transmission must be equal to unity. PrT + PrA = 1

(9)

Recall that the probability (Pr) of several events all occurring is equal to the product of the probabilities for each event, as in eq 10. n

Pr e 1 and e 2 and…and e n , all occurring = Π Pr e

i

i=1

(10)

The probability of any one of several events occurring is equal to the sum of the probabilities of the individual events, as shown in eq 11. n

Pr e 1 or e 2 or…or e n , any occurring = Σ Pr e i=1

i

(11)

In this probabilistic model, the motion of a photon through a layer of solution with a thickness 2d is best understood as the sequential transmission through two equal sections with a thickness of d. The probability of a photon being transmitted through both layers then is PrTlayer 1 × PrTlayer 2. If we assume that the solution is ideal and that there are enough absorbing centers to obey laws of probability, then the probability of transmission through each layer is the same, and the probability of transmission through both layers is (PrT )2. In general, the net fractional transmission through a layer of thickness nd is the same as the probability of a photon being transmitted sequentially through n layers of thickness d and is given by eq 12.1 FrTnd = PrTnd = (PrTd )n

(12)

The mathematical expression for the probability of absorption is slightly more complicated. The probability of a photon being absorbed in a layer of thickness 2d is the probability of a photon being absorbed in the first layer or being absorbed in the second layer, which is PrAlayer 1 + PrAlayer 2, and appears at first glance to be equal to 2 × PrA. This result, however, is incorrect because it does not account for the fact that a photon can be absorbed only once. In order for a photon to be absorbed in the second layer of solution it must first be transmitted by the first layer. Therefore, the probability of a photon being absorbed in the second layer is the probability of being both transmitted through layer 1 and absorbed in layer 2, which is PrTlayer 1 × PrAlayer 2. The total probability of a photon being absorbed in either layer 1 or layer 2 is given by eq 13. PrAlayer 1 + (PrTlayer 1 × PrAlayer 2) = PrA + (PrT × PrA) (13)

Path length

Figure 3. Plot showing the amount of light transmitted (I t) and absorbed ( I a ) as a function of path length according to the corpuscular–probability model.

In general, the probability of a photon being absorbed in the nth layer is the probability of being transmitted through (n – 1) layers times the probability of being absorbed in a single layer, which is (PrT)n-1 × PrA. The net probability of a photon being absorbed in any one of the layers 1 through n is, therefore, given by the sum in eq 14. n

PrA = Σ nd

i=1

PrT

n–1 d

× PrA

d

= 1 – PrT

n d

(14)

Recalling that PrA is equal to 1 – PrT (eq 9), it may be shown that the above sum reduces to 1 – (PrT)n, as it must. This proof is left as an exercise for the reader. We have previously linked PrT and PrA to the fractional transmission and absorption, respectively (eqs 7 and 8). With these definitions, we see that the corpuscular–probability model predicts that the amount of light transmitted and absorbed as a function of total path length b may be described by eqs 15 and 16. In these equations the probability of transmission in the standard path length is represented by a constant (PrT) between 0 and 1; b is expressed in terms of the standard path length (b = path length/d ) These relationships are shown in Figure 3. It = Io × (PrT)b It = Io – Io × (PrT)

(15) b

(16)

Experimental Evidence After determining the expected results of each model, we may turn to experimentally obtained results to see which (if either) model is consistent with observation. If time and logistics permit, the presentation of Beer’s law may be integrated with a lab activity allowing students to test the models themselves, as suggested by Ricci et al. (7). In the absence of an integrated lab activity, one may simply state that the predictions of the corpuscular–probability model are supported by experiment, present tabulated (“dry lab”) data to support the correct model, or refer to a historical account. The formulation of what is typically called “Beer’s law” is actually the work of many scientists, and the story of their contributions has been related in this Journal (10–12). One citation from the work of the 18th century French scientist


931


Bouguer is particularly appropriate here (12): At first we might think that if we imagined the transparent body to be divided into parallel layers of the same thickness, all these layers would intercept the same number of rays; thus the light, diminishing by the same amount in its passage through each layer, would decrease according to an arithmetic progression. To examine the truth or falsity of this idea, I passed a light of 32-candlepower perpendicularly through two sheets of glass, after which I found its intensity to have been halved, for it was then only 16-candlepower. Now, if another thickness of two sheets of glass had caused an equal diminution, all the rays would have been intercepted. Nevertheless, the addition of two more sheets certainly did not form an absolutely opaque body. The light was still very bright, and when I passed it through ten sheets, it was still as intense as that from one candle. … it is obvious that the light will always be diminished in geometric [exponential] progression.

Path length

Figure 4. Plot showing the linear relationship between the log of transmittance and path length, as predicted by the corpuscular– probability model.

The work of Bouguer is particularly relevant because one sees that he began with a model that is conceptually similar to the saturation model discussed here and discovered that it is clearly not correct. Although the corpuscular–probability model that we are using may have seemed strange to him, some 70 years before Dalton’s atomic theory, his results clearly show that it is compatible with experiment. In his experiment, we may assign the incident intensity Io as 32-candlepower, the standard path length d as two sheets of glass, and the probability of transmission in the standard path length PrT as 0.5. In this case, we see that his results match precisely those predicted by the corpuscular–probability model and eq 15. 32 candlepower × 0.510/2 = 1 candlepower

Figure 5. Schematic separation of a solution containing 2m absorbing bodies and having a concentration equal to 2c into two equal sections, each containing m absorbing bodies and having concentration c.

The Beer’s Law Equation Having seen that the corpuscular–probability model is consistent with experiment, we may look at the derivation of the familiar form of Beer’s law from the relationships shown in Figure 3. It is most convenient to begin the derivation by considering the fractional transmission, or “transmittance”, as a function of path length. The combination of eqs 5 and 15 provide the following equation: T = (PrT )b

(16)

Taking the base ten log of both sides of eq 16 yields a linear relationship, which is plotted in Figure 4 and is described by eq 17. log T = b log(PrT)

or

log T = ᎑kb b

(17)

In eq 17, the log of the transmission probability (which is a constant less than zero for any given concentration and standard path length) has been replaced by a proportionality constant, kb, and has been given a negative sign to indicate the downward slope of the line in Figure 4. Multiplying through by ᎑1 gives -log T, which is equal to log(1/T ), on the left-hand side, as in eq 18.

log 1 = k b b T

932

or

log

Io = kb b It

(18)

If one now defines the new unitless quantity “absorbance” (A) as the log term in eq 18, a quantity that shows a linear dependence on path length is obtained.

A ≡ log

Io = kb b It

(19)

Although we have not yet addressed the subject of concentration, we have laid the necessary groundwork to do so rather simply. We may easily predict the effect of doubling the concentration by adding m molecules to a solution with the thickness of the standard path length, bringing the total number of absorbing molecules to 2m and the concentration to 2c. With our restriction that the solution is dilute and ideal, we may assume that there is no substantial interaction between the analyte molecules. The first set of m molecules and the second set of m molecules are identical and may be treated separately. According to our model, the transmittance has been equated with the probability of a photon passing through the solution without encountering an absorbing center. For a sample with m absorbing centers and concentration c we may designate this probability as PrTc. In this model, the probability of transmission through a solution with concentration 2c (containing 2m molecules), PrT2c, is equal to the probability

Journal of Chemical Education • Vol. 77 No. 7 July 2000 • JChemEd.chem.wisc.edu


of a photon not encountering a member of the first set of m molecules and not encountering a member of the second set of m molecules, which is the product of these probabilities, as in eq 20. PrT2c = PrTset 1 × PrTset 2 = (PrTc )2

(20)

In general, increasing the concentration of a sample to nc, results in a probability of transmission equal to (PrTc )n. This argument parallels that employed previously for the variation of path length. In fact, since we are ignoring the effects of solvent, one may imagine separating the two sets of m molecules into two sections with the standard path length and with concentration c, as in Figure 5. At this point, the discussions of the variation of path length and concentration become identical. It is clear that the relationship of transmittance to concentration with a constant path length is the same as that of transmittance to path length with a constant concentration. The plots shown in Figures 3 and 4 could apply equally well to concentration as path length. As in the path length studies, this relationship is confirmed by observation. Given that the mathematical relationship is identical, the same arguments may be followed to show that absorbance has a linear dependence on concentration, with a proportionality constant kc, as shown in eq 21. The explicit steps are omitted here and left as an exercise. A = kc c

(21)

We see that absorbance is directly proportional to both path length and concentration, with proportionality constants kb and kc. The constant kb is independent of path length but depends upon concentration, whereas the constant kc is dependent on path length but independent of concentration. These two proportionality constants may be combined into a single constant, which is independent of both path length and concentration. The new constant is called the molar extinction coefficient, designated ε, and thus we arrive at the familiar form of Beer’s law shown in eq 22. The mathematical justification for the combination of proportionality constants has been presented previously (13, 14). A = εbc

order kinetics to determine the values of important parameters, students generally understand that the concentration of a reacting species actually decays exponentially. This is in contrast to students’ perception of Beer’s law in that most students do not understand that, at the most fundamental level, the absorption of light also obeys an exponential, rather than linear, decay and that the apparent linear relationship expressed in Beer’s law arises only from a convenient definition of absorbance. While most students correctly answer numerical questions pertaining to absorption spectroscopy, it seems that many do so while embracing a conceptual model (the saturation model) that is actually inconsistent with Beer’s law. The commonly seen discussions of this topic do little to disabuse students of this false conception, and in fact may contribute to it. To help students reject this erroneous model, it is necessary to expose its inadequacy and also to present a suitable substitute that correctly accounts for the phenomena and that is easily assimilated. The derivation of Beer’s law from the corpuscular–probability model accomplishes these goals. Acknowledgments I am thankful for helpful discussions with James N. Demas and Kristi A. Kneas and for support from the National Science Foundation (CHE 97-26999). Note 1. Additional subscripts are occasionally added to identify parameters, such as path length or concentration, which have two different values in the same equation. When such subscripts are omitted, the standard values (c and d ) are to be assumed.

Literature Cited 1. 2. 3. 4. 5. 6.

(22) 7. 8.

Conclusions The corpuscular–probability model is a simple and easily understood concept that allows for a derivation of Beer’s law. This derivation is mathematically much simpler than many others that have been proposed and is believed to be more pedagogically sound. The evidence that serious pedagogical problems exist with the common treatments of the Beer’s law derivation is clearly seen by a comparison of students’ understanding of light absorption to their understanding of the mathematically similar phenomenon of first-order reaction kinetics. Although linear relationships derived by logarithms are often used in first-

9. 10. 11. 12. 13. 14.

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A More Pedagogically Sound Treatment of Beer's Law: A Derivation

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