Discovering the Beer-Lambert Law Robert W. Ricci, Mauri A. Ditzler, and Lisa P. Nestor College of the Holy Cross, Worcester, MA01610 The Chemistry Department a t Holy Cross College has been developing a series of discovery experiments for general chemistry (I, 2). In a discovery experiment the instructors guide the students toward the re-discovery of a law or concept throueh a shared laboratorv - experience. The laboratok period-opens with a prelab meeting where the instructor introduces the topic through a series of questions. This is followed by a l~boratorysessionwhere the students eather their data. At the postlab meeting, the students share their data and the in&ructor guides them toward the re-discovery of a chemically relevant concept. In this article we will describe our laboratory experiment for re-discovering the Beer-Lambert Law. The readers of this Journal have shown an enduring interest in the BeerLambert Law (3).At least 22 accounts have appeared on these pages over the past 40 years. Many of the articles deal with the "derivation" of the law. This contribution is uniaue in its use of a simple ~hvsicalmodel to help studenis taking general chemistry gain a more me-ngful picture of the process of light absomtion. The approach is .empirical and 'free of c a l c u h . An understanding of the Beer-Larnbert Law by firstyear college students is becoming increasingly important now thatspectroscopic instrumintation i s being introduced early in the chemistry curriculum. The relationship between light absorption and path length and concentration is simple enough to present from a utilitarian point of view. Our aim, however, is to present the law so that the students are led to an intuitive understanding of its reasonableness. Also, in each of our discovery exercises we try to introduce or reinforce one or more of the fundamental aspects of the chemist's approach to investigation. Several of these investigative tools are employed in the Reer-Lambert Law experiment. The primary focus is on the utility of "thought" experiments. Students are asked to consider BB's rolline down a ~crforatedplane as a model for liaht passing through a solution containing absorbing molecules. Students perform a series of mental experiments with the model; their conclusions are used to derive the Beer-Lambert Law. In another process-related feature, students use graphical methods arrive at a mathematical description of their data. This is accomplished for both path length and concentration effects. Finally, when students combine path length, concentration and absorptivity effects into a single equation, they have participated in the process of generalizing the specific, an activity central to the scientific process. A
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Prelab Meeting At the orelab meeting. -. our students are told that the goal of thls laboratory experiment is to investigate the phenomena of licht absomtion in solution. We beein by showing the stud& a simple diagram illustrating aeuvette together with the usual 10 and symbols and arrows to represent the initial and transmitted beam intensities. They are then asked to predict what variables might effect 10. The students have no problem in predicting that the path length of the cuvette and concentration of absorbing species will be important variables to study. The instructor then draws a blank plot of I versus path length on the board and asks the students to predict the shape of the
Figure 1. Absorption of light by molecules in a cuvene modelled by 66's rolling down an inclined plane. versus path length curve. As you might guess, most students will suggest a linear decrease of I with increasing path length. The remaining students normally leave this incorrect prediction unchallenged, though some will express an uneasiness with the possibility that at some path leueth I must stast throueh 0 and become newtive. The ploFis left on t6e blackboard and the students &e asked to participate in a thought exoeriment. A drawine of an in;lined plane that hasmany holes randomly drilGd into the surface is sketched on the blackboard. The edee of the plane is marked off in 10 equal intervals. The instructor tells the students that 1,000 BB's are to be placed at the top of the incline. The for the students is to predict how mauv BB's will arrive at the bottom of the inclined plane given the probability that any one BB has a one in 10 chance of falling through a hole during its passage along any one interval of the inclined plane. The BB's are a model for the photons of light and the number of BB's are analogous to the light intensity, I. The holes on the inclined plane represent the absorbing molecules in the cuvette. The BB's are lost in a gravity hole while the photons fall through an electromagnetic hole (are absorbed). The inclined plane model as a counterpart to the light beam and cuvette is illustrated in Figure 1. To solve the problem we mentally release the 1,000 RHS and allow thcm to roll down throueh one internal. The students are then asked how many BB's will have fallen through the holes and they have no problem in predicting that 100 BB's, or one in 10. Nine hundred BB's remain to roll down the next interval, and they are asked again how many BB's will he lost as they roll down the second interval. The students realize they have two choices. Either another 100 BB's will be lost or one in 10 of the remaining 900 BB's, i.e., 90 BB's. Most of our students will correctly predict that 90 BB's will fall into the holes during their passage down the second interval, leaving 810 BB's to roll down throueh the third interval. The instructor can quickly c o n s t k t a table on the board for both options; one where the decrease in the number of BB's is ~ r o ~ o r t i o n a l to the number of BB's remaining and a second representinga constant decrease in the number of BB's. (See the table.) Recognizing that the light intensity is analogous to the number of remaining BB's, the students begin to realize their original prediction that the light intensity will decrease linearly with path length corresponds to the incorrect constant decrease option illustrated in the table. It is more reasonable to expect that the light beam, as it travVolume 71 Number 11 November 1994
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Loss in the Number of BB's erses the cuvette, will decrease in intensity more gradually a s illusDecrease Proportional to Number of BB's Remaining t r a t e d by the proportional decrease option. 0 1 2 3 4 5 6 7 8 9 1 0 Path Length At this point the students have #ofBB's 1000 900 810 729 656 590 531 478 430 387 348 both a qualitative a n d pheno- Remaining menological understanding of the BB'sLost 100 90 81 73 66 59 53 48 43 39 35 path length dependence. I t is inConstant Decrease in Number of BB's structive for the student to express t h i s relationship mathe- Path Length 0 1 matically. We find a graphing #of BB's 1000 900 program with a limited selection Remaining (library) of functions useful i n BB's Lost 100 100 this process. Students can iterate through the list to find the function of their variable(s) that eives or the best fit to a linear relationship. With practice they develop the ability to select several most probable functions ln[III,]=KxCxL by qualitatively evaluating the raw data. The graphing to avoid a nonzero intercept. program is run on a computer equipped with a n overhead display. For this exercise they find that their "mental" data Laboratory Assignment and Postlab Meeting is linear when in(# BB's left) is plotted versus length segThe laboratory assignment is to test the model. We ment. Using the equation for a straight line they deterdivide the students into small groupings. Each group is mine that assiened a uniaue nath leneth and solute for testine the In (#BB'sleft) = K(1ength segment) + In (initial #BB8s) concentration dependence of percent transmittance. Students run a n absorotion soectrum of their solute. seFor the case examined, the students find a slope of lecr an appropriate wavelength and verif,v the concen0.10 that can be thought of a s representing the arbitrarily tration drorndrnce oredicted b\. their model bv showine selected one in 10 probability of the BB falling in a hole in a plot of -joglo T v e k s concentration to be &ear. ~ i & any one length segment. If the students have had calculus, ure 3 illustrates typical pooled student data for the ammoit is appropriate a t this point to allow them to state their nia complex of cupric ion. When the slopes of the individphenomenological understanding of a n equal fraction of ual lines are plotted against the pathlength a t the cells, a the remaining BB's falling in any segment a s -dlIdl = kI; linear relationship is obtained, thereby confirming the preintegration between limits leads to the same equation dedicted dependence of the light intensity, I, on the pathtermined graphically. length. Having verified their predicted equation, students expand the equation by showing that the absorptivity conOnce the relationship between light intensity and path stant varies with solute type. length has been established, the effect that changing the We also point out that the dimensions of the extinction concentration has on the amount of lieht absorbed can be coefficient can be expressed a s area/molecule, which can be approached by simply nsking the studcnts what would bc interpreted a s a type of "cross-sectional" area of the molethe chnnw in on)bahihtv of a BB fallme throueh a hole if cule for capturing photons. This fact provides a n addiyou halve the number i f holes. The p&babilib also will tional link to the size of the holes on the inclined plane. halve to 1in 20 and the slope of the ln(#BB's) versus path The model also can be used to illustrate other exoonential length curve would be-0.05 (Fig. 2). Clearly, the slope conrclalionship such us a first-order rate proress but seems tains the concentration term, and, by analogy, we would particularl.v apt in rnodellmg the Beer-Larnbcrt Law. predict that the intensity of the light emerging from the cuvette would have the general form:
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BEER-LRNBERT LRW GRAPH 1.00
MODELLING THE BEER-LRMBERT LRW
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Figure 2. Modelling the Beer-Lambelt Law by BB's rolling down a perforated inclined plane.
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Figure 3. Pooled student data on the absorbance of an aqueous solution of the copper ammonium complex. Four differentpath length cells were used: I mm, 2 mm, 5 mm. and 10 mm.
Acknowledgment
Literature Cited
the National Science Foundation (I)U14:9254016~and the PEW Charitable Trust is griluhlly acknowledged.
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