THE BEER-LAMBERT LAW AND THE COMBINATION OF PROPORTIONAL DEPENDENCES
0
EUGENE M. HOLLERAN St. John's University, BrooMyn, New York
IN
A recent article in THIS JOURNAL^ Lohmau pointed out that Lambert's law and Beer's lam are often derived separately and theu combined without mathematical justification into the Beer-Lambert law. He then suggested a procedure for effecting the combination. A method described by Slifkinz for combining proportional dependences can also he applied to this case. However, since various similar situations (Boyle's aud Charles' laws; law of mass action) are usually encountered by chemistry students before they have studied calculus, which is required by both these methods, it is well to realize that combinatio~lsof this sort can be accomplished in a much simpler manner. Oue such method, for example, can be applied to the radiatiou absorption laws as follows. Lamhert's law states that A = kJ where A is the absorbance, 1 is the length of the light path in the solution, and k , is independeut of 1 but varies IT-ith the concentration c. For constant 1, lc, is proportional to A. But for constant 1, Beer's law states that A is proportional to c. Therefore k , is proportional t,o c for a given 1. This is true with the same proportionality constant for every 1, since kl cannot. depend on 1. Thus the combiued law, A = lccl, follows. .knother elemeutary method, similar to one attributed to A. T. Lonseth hy Willard and Dieh13can be stated i n a more general way as follo~vs. I t is assumed that :
y = .i(x,,
x2,
. . . x,,)
(1)
aud that, for each independent variable, x.: 11 =
k.r,
(2)
where k, is a funrtiou of all the independent variables except x,. Divisiou of equation (2) by the product of all the independent variahles, Wx,, gives:
ill which the right member mallifestb does not depend ou the one independent variable, xi. But since this equation can he writteu for every xi, the left member must be independent. of all t,he independent variables; that is, it mnst be constant. Therefore: 'IJOHMAN. F. H.. J. CHEM. EUUC.. . 32.. 155 (19.55). . .
' SLIFKIX,'I,., ibid., 25,
This demonstrates the general rule that a functiou of several independent variahles which is proportional to each one separately when all the others are constant is proportional to their product. The main virtue of the combination procedures which require calculus is their not inconsiderable value as mathematical exercises for more advauced chemistry students. Probably the most useful for this purpose is the one employed, for example, by Daniels4 and I < l o t ~ , ~ which may be generalized as follows, beginning again with equations (1) and (2). From equation (1) t,hc complete differential of y is:
From equation (2) :
Substitutiou of these into equation ( 5 ) and divisiou by y gives:
Iutegratiou and removal of logarithms leads again to equation (4). The detailed application of this method to the rombination of the gas lams, VT, p = kln (an extension of Avogadro's law), Tin, T = liJP (Royles' laxT), Vm, = ksT (Charles' lam), serves as an excellent refresher for t.he mathemat,ics of thermodynamirs and fits in easily with the usual preliminary discussiou of properties of mat,ter and equations of state. In the preseutation of the Beer-Lambert law, h o w ever, it seems desirable to derive the combiued law directly. The above mle for the rombination of proportional depelldellces can rollvenieutly he employed in the following manner. Consider a monochromatic light beam of intensity I passiug through an infinitesimal thickness dl of a solution with a concentration c of ahsorhing centers. I t is obvious, or at, least reasonahlr., os~eciallv from the view~ointof col.
346 (1948). 4 DANIELS, F., "Outlines oi I'hysicnl Chrmintt.y," John Wiley ~ r r ~ H.~ H., m AND ~ ,H. D ~ H L"Advanced , Qu:mtit,:~tive York, 1948, p. Oil. Bnslyeis," D. Van SostmndCo., Inc., New York, 1943, p. 112. 8.Sons, Inc., 8 KLOTZ, I. \I.. "Chemied Thennodynamies," Prentioe Hall, The author is indebted to Professor Harold Harm of St. John's Ine., S e w Ymk, 1950, p. 60. Univemity for directing hia att,entioo to this reference.
DECEMBER, 1955
637
lisions bct.weeu photons and absorbing centers as that the decrease in described by Goldstein and intensity, -dl, should obey the following three separate laws:
1
- d l = k,Z
(k, constant for given c and d l )
- d l = kge
(kz constant for given I and d l )
~
417 (1054).
~
J. H., .
~
AN,, ~
11,. , A.~ nAu, ~ jR., ~ J~ . cHEM. ~ ,I , : ~
-dZ = k3dl (k. constant for given I and c )
(10)
The reason for considering an infinitesimal thickness, dl, is to limit thedecrease of intensity to an infinitesimal value so that I may be considered constant as required by equations (9) a i d (10). A finite 1 and an infinites(8) imal c could just as well have been considered. The (9) combined form of these three statements, - d I = ~ 31, ~ . , IcIedl, integrates to the combined law which includes the separate laws as sperial cases.
