Richard C. Pinkelton North Carolina Stote of the Universitv of North Carolina at Raleigh
II
Beer's Low wi+hou+C U M U S
Presentations of the Beer-Lambert or Bouguer-Beer law in textbooks of analytical or physical chemistry vary between two extremes. I n simpler, more utilitarian texts, the law is taken as given, stated in either exponential or logarithmic form. This is probably quite satisfactory for ordinary purposes. The curious student is bound to be left a little perplexed (just why, a t first glance, isn't the absorption of light a linear function of distance?). I n more complete texts, the presentation based on calculus is usually given. The manipulations themselves are not difficult. Nevertheless, many students who have been exposed to calculus cannot use it in a very meaningful way, as we all know. The calculus-based presentation has in it some conceptual difficulties, and several articles have been written dealing with these.'e2 For example, there is the problem of combining the dependencies on concentration and length, when postulated separately. Beyond this, there is the idea of absorption occuring in an unimaginably thim layer of material, according to
' HOLLERAN, E. M., J. CHEM. EDUC.,32.636 (1955).
SWINEHART, D. F., J. CHEM.EDUC.,39,333 (1962).
366
/
Journol of Chernicol Education
rules which do not follow immediately from any observable macroscopic behavior. These difficulties are inherent in the calculus, which contributes nothing whatsoever to our basic understanding of the law. Some years ago, Liebhafsky and Pfeiffer wrote an article in which they examined the history of Beer's law.3 They also took the opportunity to critize some of our present concepts. In a later paper, they advocated a presentation more closely related to the historical origins of the law.4 One statement is worth recalling in particular. After setting down the usual differential relations, they comment: Some of the textbooks have given the impression that the student is to regard the foregoing mathematical exercise as s derivalia of Beer's law. It is, of course, no such thing; the basio postulates are implicit in equations (1) and (2) [their equations].
This is true of the derivation of any physical relationship, but their point is well taken. Very few textbooks have followed their suggestion, which was to return to PFEIFER,H. G.,
AND
LIEBHAFSKY, H. A., J. CHEM.EDUC., R.,AND YOE, J. H.,J.
D. 28, 123 (1951). See also MALININ, CHEM. EDUC..38.129 (1961).
a formulation in terms of absorbing centers. Liebhafsky and Pfeiffer still resorted to the calculus in their treatment. So do most textbooks. In some of these it is implied, intentionally or unintentionally, that it is necessary for "proof" of the law to consider an infinitesimal increment in length, or in concentration, as if nature could not function starting on the human scale. The following treatment is offered because it is simpler. No apology is made for it on that account. Many will find it intellectually sounder. Consider first a beam of light having power POpassing through a cell of some standard unit length, b,. Upon leaving this cell of finite length, the power will be diminished to a value PI,which will be some definite fraction of the ~ r i g i n a lso , ~that PI = fPo
Next, it is imagined that a second cell, also of unit length, is placed in the path of the beam, so that the total length traversed is bz = 2b1. The logical assumption is then made that the power of the beam is further reduced by the same fraction, or Pz = fP,. Therefore, Pe = This is the axiomatic basis of the present treatment. By merely repeating the operation, one can see that for a length nb,,
m.
P, = f"Po
4 t this point it will be helpful to consider a problem of a sort which must have been used many times as a student exercise. The example is: In passing through a certain material, 2.5y0 of the incident light is absorbed. What percentage is absorbed in the same material if the thickness is doubled? The solution is: P9 = (1
- '/dZP0 = (*/,dPo
The percentage absorbed is (1 - P1/Po) X 100, or about 44%. Such simple illustrations can be solved without even using logarithms, but they are perfectly valid. The above example can be solved in reverse, to find either the absorbance or transmittance through onehalf the length b,. In fact, more generally, suppose we start out with some standard length b and wish to find the amount of light which would pass some fraction of that length, b,/m. We would have that.
How is n to be determined? Ignoring the concentration variable for the moment, it is clear that for a general path length b, n = b/bl. Suppose, for example, that b = 1.5 b,, or 3b/2. We could readily solve the problem first by finding the transmittance after three standard path lengths 3bl, and then taking the square root to find that transmitted in half the length 3bl. Ultimately, of course, logarithms will be used. It is now easy to see that, in any real situation, n need not be an integer or even a rational number, but may take on any value. I n considering the concentration dependence, it is convenient to use an idea related to absorbing center^.^ Suppose that we have a cell which is two standard units in length so that again P2 = ?PO. This cell is filled with a solution having a concentration el of absorbing material. Now suppose that the walls of the cell can be moved a t will, so that the length of the light path may be varied and the volume contained by the cell changed proportionately. The path length is decreased until it is again b,. At the same time, solvent is evaporated from the absorbing solution until the volume is also reduced to one-half its previous value. I n this event, the new concentration will he c2 = 2cl. Now, it is logical to expect that, by the end of this process, the fraction of the incident light transmitted will be no different from what it was originally. This is because, in passing through the solution, the probability of a photon being absorbed will be exactly what it was previously, the number of absorbing centers not having changed a t all. I n similar fashion, doubling the concentration while maintaining the length constant will have just the same effect as doubling the length and holding the concentration the same. No mathematical slight-of-hand is needed to combine these ideas. We may immediately take an expanded definition of n: n = bc/b,c,
in which el is some agreed upon standard unit of coucentration. Incidently, this simple statement expresses the principle upon which the Duboscq colorimeter is based. It is usually derived from Beer's law, whereas the reverse might better be the case. The usual form of Beer's law can be obtained directly. Since in general loga PIPo = n log. f
where f' is some new fraction (closer to unity). Then in m such lengths, totaling bl, PI
=
(fYP0
or bc(loglof)/blc,
where
But so that
The fraction of light absorbed in length b,/m would be 1 - f1Im. 5 In a. more conventional treatment one would consider that fraction which is adsorbed: (Po - P1)/P = (1 - j). The latter would he set equal to k dl and the interval (Po - P ) taken as very small. This is purposely avoided here.
All quantities making up a are constant. These are conveniently lumped together to make up the molar absorptivity. Of course, the unit of length is cornmonly chosen so that b, = 1 cni, and CI has the dimensions of 1mole per liter. Although they are basically equivalent, the assumptions made in the foregoing treatment are more readily assimilated than those ordinarily used. There is no very good reason for using calculus in the presentation of Beer's law, even in more sophisticated texts. Such practice may lend a misleading aura of authority, hut little else. Volume 41, Number
7,July 1964 / 367
