BOUGUER-BEER LAW J. H. GOLDSTEIN and R. A. DAY, JR. Emory University, Emory University, Georgia
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usual presentation of the Bouguer-Beer (or Lambert-Beer) law in elementary texts devotes little attention to the details of the process of light absorption. Likewise no convincing physical interpretation is normally given of the important parameter of the process, the so-called absorbancy index (or extinction coefficient). Liebhafsky and Pfeifferl have recently given an excellent derivation of the law in more general terms than normal. For the beginning students in analytical or physical chemistry courses, however, a simple kinetic picture remedies the above-mentioned defects. An incident beam of monochromatic light of intensity I impinges on a length b of solution containing c molecules per cubic centimeter. I t is desired to derive an expression for the intensity I of the emergent beam as a function of c and b. For monochromatic light I is proportional to N, the number of photons striking a one square centimeter surface per second. We may ask then for the decrease, - d N , in the latter quantity after the light passes through a layer of solution db in thickness. The absorption of one photon by a molecule requires a L'collision" or encounter of the two. Accordingly, the number of photons absorbed per second is proportional to the total number of collisions: -dN
=
keN db
This approach avoids separate derivations of the lams of Bouguer and Beer which are so frequently encountered in textbook treatments of the subject. Methods used to combine the two laws are frequently confusing to the student. An interpretation of the constant k is best approached from the standpoint of dimensionality. Since the quantity kcb must be dimensionless, it is necessary that k have the dimensions of l l c b , i. e., of area per molecule. This suggests then that k be identified as the effective "collision cross section" presented by one molecule to a photon of the given frequency. (The term "cross section" is quite commonly used in nuclear physics to describe the absorption by nuclei of various bombarding particles.) The dimensionless exponent, kcb (absorbancy or optical density), may be identified as the fraction of the unit cross-sectional area presented to the incident light beam which is occupied by photonabsorbing surfaces in the segment of solution. Thus it is easy to see that the fractional decrease in light intensity produced by passage through a layer db thick is equal to kc db, which may be recognized as the usual starting point in "deriving" the Bouguer-Beer law. Equation (4) is commonly written as
(1)
Here c is the concentration in moles per liter and a, is the molar absorbancy index (or molar extinction coefficient, 6 ) . The spectral properties of a solution are normally Here k , the constant of proportionality, can be in- presented graphically as a plot of a, (usually log a,) terpreted as the probability that a collision will lead to against the frequency or wave length of the radiation. absorption. The quantity c db gives the number of Ideally this curve is sharply peaked at the absorption molecules in a segment of solution of unit cross section frequencies and falls avay rapidly above and below the and db centimeters thick. The product Nc db meas- maximum. The finite width of the peak reflects the ures the number of collisions per second. In kinetic fact that molecules absorb radiation over a wide range terminology the process is said to be first order in pho- around the absorption maximum. According to the tons and molecules. uncertainty principle this arises from the fact that the Since I is proportional to N, equation (2) is equiva- two energy levels involved in the transition induced by lent to photon absorption are themselves correspondingly broad rather than sharp. Maximum probability for transition occurs a t or near the center of the hand. This characteristic dependence of a, upon wave Integration yields the form of the Bouguer-Beer length can be easily explained to beginners, however, by law an analogy drawn from baseball. A shorts to^'^ efficiency in-converting groundballs into ~ u t o u t s a first t ,,, I - "-kb \=I I0 " base may be definedin terms of a "cro& section1'-the LIEBUMSKY, H. A,, AND H. G.PFEIFFER, J. CHEM.EDZTC., 30, segment of the illfield ~r-ithinwhich a given shortstop 450 (1953). can convert hits into putouts. This cross section de-
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pends upon the speed of the ball, however. If the speed is too great the ball is through the infield before the shortstop reaches it, and if too slow, he cannot reach it in time to make the play a t first base. Obviously, a t some intermediate speed maximum efficiency is attained. There are, of course, many situations in which de-
JOURNAL OF CHEMICAL EDUCATION
viations from the Bouguer-Beer law occur. An extension of the present picture might be useful in explaining such deviations. However, our purpose here is merely to present a simple derivation of the law under ideal conditions in an attempt to give beginners a more meaningful picture than usual of the process of light absorption.
