HEINZ 0. PFEIFFER and HERMAN A. LIEBHAFSKY General Electric Company, Schenectady, New York
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examination of many books and articles dealing with photometric analysis had given us the impression that Beer discovered and formulated the exponential absorption law in which concentration and length appear as symmetrical variables. We were led to carry out this examination because of our belief1 that an absorption law so formulated is too restricted for the needs of modem analytical chemistry. When we read Beer's a r t i ~ l e ,we ~ were surprised to find this impression wrong in several important respects, and we have consequently prepared the following discussion of his clarsical investigation. Of the more recent publications we examined, an article by Hague3 comes closest in our estimation to Beer's point of view. We shall begin by giving a free (but, we hope, scientifically accurate) translation of part of Beer's paper. Interpolated parenthetical material and italics are our own. Equations have been numbered in sequence and some of Beer's symbols are explained later. On page 83, Beer says:
the two beams before and after the absorber was inserted in the unknown beam. Let the former value of the angle be $, and the latter a ; then, in Beer's symbols a/A = tang ll. (3) and
A = pD
log [10/11 = a-cb
ua/A = tang u
(4)
where a and A are the amplitudes of the unknown and of the reference beam before the absorber is inserted; and v is a coefficient that gives the diminution in amplitude produced by the absorber-in Beer's case, an aqueous solution of a colored salt contained in a glass cell. From the refractive indices of glass and solution, Beer calculates that about five per cent of this diminution is due to losses a t the cell windows; hence he writes h =
Y/T
= tang m/0.95 tang
+
(5)
where r = 0.95 corrects for these losses and v takes its value from equations (1) and (2). The derivation of We shall take the absorption meficient4 to he the coefficient equation (2) is now clear. giving the diminution in amplitude suffered by a light ray as it To discuss equation (1)in modern terms, we should passes through unit length of su absorbing material. The unit deal with the internal transmittance of the solution in of length will he one decimeter. We then have, according to Beer's cell. Hence, we write theory, and as I have found verified by experiment, (1)
where g is the absorption coefficientand D the length of the absorbing material traversed in the experiment. Finally, we have for the absorption coefficient expressed in t e r m of quantities actually measured p = [tang u/0.95 tang J.]'ID
(2)
Beer's description of the theory and operation of his photometer agrees in every important detail with a modem descriptionGf the method in which polarization is used to measure the relative amplitudes of an unknown and a reference beam. Beer used two oil lamps, carefully adjusted, and red glass as filter. Nicol prisms served as polarizers. In his apparatus, measurements of an angle gave the relative amplitudes of
(6)
where lois the homogeneous radiant energy entering the sample, and I that incident upon its second surface6; c is the concentration in moles/liter; and b the length of solution in cm. Inasmuch as only ratios of I and of a are involved, we may write Io/l= ( a / b ) a
= l/Al =
l/pZD
(7)
hence a,eb = lag [1/p2DI = - 2 0 log p
(8)
Also, owing to the diierent units of length, D = O.lb
or,
LIEBHAFSKY, HERMAN A., paper presented a t the Pittsburgh a,c = -0.2 log p (9) Conference on Analytical Chemistry and Applied Spectroscopy, Equations 7 to 9 show conclusively that Beer did not Pittsburgh, Pennsylvania, February 15 to 17,1950. %BEER, A,, Ann. P h y ~ i k(Poggendorf), 86,78 (1852). treat concentration and length as symmetrical varia H a o w , JOHN L., PTOC. Am. Sac. Testing Materials, 44, 712 ables. Nowhere in his paper does he give any indica(1944). tion of being familiar with equation (6) in this or an ' Here Beer interpolates the Greek phrase mr eIomv which alternative form; all his thinking seems to have tenwe have been told is roughly equivalent to "par ezcellenee." We take this to mean that Beer is thinking of a true absorption tered around only one absorption law, equation (I), coefficient; i. e., with all corrections (such as cell corrections) and (as will aDDear from the followine translated exapplied. National Bureau of Standards, Washington, Letter Circular 6 KO-usc~, F., "Praktische Physik," 18th ed., B. G. LC857. Teubner, hipzig, 1943, Vol. 1, pp. 503-504. 123 1
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JOURNAL OF CHEMICAL EDUCATION
cerpts) all his results were calculated on the basis of this law. Sow let us see how Beer describes his experiments. In most oi my experiments, solutions diluted to various extents were employed. From the values of A, which these experiments yielded, the absorption ooefficient of 1 decimeter of the most concentrated solution ( I ) was calculated on the supposition that the watw added during the dilution did not alter the specific
is taught to i s by observation. tzonU(1)through which it So, for example, a cell of 1 decimeter length was first filled with a solution of copper sulfate that contained 9 volumes of water for each volume of solution concentrated s t 13.5%. (11)-in other words a solution of dilution 1/9. Further, a cell was filled with a solution of dilution 1/19. The latter cell contsins just so much concentrated solution (I or 11) as the former; it has the same tint as the former in white light and gives sensibly the same value of ar in red. The value of this angle was 3"28'1OUfor the short cell and 3'23'0'' for the loneer. - , and these yield the almost identical values 0.065 and 0.063 for the absorption coefficient of the concentrated solution (I).
In contrast r i t h the rest of Beer's paper, this paragraph could confuse the unwary reader because "concentrated" is used in connection with both Beer's stock solution (11) and the strongest solution he actually
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The absorption of light during the irradiation of a colored substance has often been the object of experiment; but attention has always been directed to the relative diminution of the various colors or, in the case of crystdine bodies, the relation between the absorption and the direction of polarization. Concerning the absolute magnitude of the absorption that a particular ray of light suffers during its propagation through an absorbing medium, there i s s o far as I know-no information itvaihhle.
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Finally, why did Beer carry out his investigation? He introduces his paper with these two sentences:
=u
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With copper sulfate as an example, it m y further be shown how observations on a dilute solution lead to the absorption coefficient of one more concentrated. For a thickness of 1 decimeter and for the dilution 1/19, a = 13'48'10" was found for this salt. If A is the corresponding ooefficient of diminution (i. e., the corresponding transmissivity), then this coefficient will have the value AZ for double this thickoess. But, a t double this thichess, just so much cancentrrtted solution will he irradiated as a t a thickness of 1 decimeter and the dilution 1/9. Accordin&. Ax is the ahsorntion coefficient for the dilution 119. One
do -
.u
measured (I). (We have taken the stock solution of the foregoing paragraph to be a solution of copper sulfate saturated a t 13.5'C. although Beer's density data indicate it was only 98 per cent saturated.) Comparison of the values for the tangents of 3"28'10N (0.061) and 3'23'0" (0.059) with those just given for the absorption coefficients leads a t once to the identification accordiig to the Roman numerals given. In view of the fact that Beer tabulated values only for the most concentrated solution measured, it is probable that he was thinking of I where "I or 11" amears in the preceding paragraph. At any rate, it is clear that Beer was thinking primarily of an amount (or thickness) of concentrated solution; in other words, of a mass of absorbing material, not of a concentration. To be logical, anyone who regards this as hair-splitting must also maintain that it is pointless to distinguish between extensive and intensive properties. If there is no evidence of Beer's being familiar with an absorption law containing the concentration as an explicit variable, then how did he correlate his measurements, which were done a t diierent concentrations? Fortunately, he answers this question in the following paragraph.
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I
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80
I
I
I
110XK120
I
I
I
I
I
40 60 8 0 6 C 0 2 0 WAVELENGTH IN YlLLlMlCRONS
I
40
=', I
80
I
The words "relative" and "absolute" deserve attention. Their use here confirms our belief that Beer did not know that concentration and length are symmetrical variables in equation (6). We believe he considered the "relative diminution" as expressed in equation (1) to be a function of the length of absorber, and that he consequently placed the length (D) in the exponent. We believe he may have thought that the change in the amount (or mass) of absorber produces an "absolute diminution" such that
10
Re..tition of Tra. Eme.im.nt. by Bee* The agreement of b with hl measures the internal consistency of Beer'r
The oalculation cannot diatingulsh between k and h,but it aeerm . M e to ignore the latter. (T's for curves I. 11, znd IY are transmittances. "c's for curves I11 and V are tranamittanoies. See reference 6.)
Al = -k(m)
(10)
To summarize: (1)Beer's paper does not contain an absorption law in which concentration appears as an explicit variable; (2) Beer thought primarily of the
MARCH, 1951
amount or mass of absorbing material; (3) Beer may or may not have understood the functional relationship between absorbance and mass (or concentration) of absorbing material; a t any rate, he used length as an intermediate variable in correlating results obtained a t different dilutions. It is accordingly not surprising t o find in Beer's paper no evidence that he believed himself to have discovered a new absorption law. I n a thorough (but necessarily incomplete) investigation of the older literature, we found the phrase "Beer's Law" or "Beer's Absorption Law" first used by Walter7 nearly forty years later. Kayser; in his monumental "Handbuch der Spectroscopie" says: "If we let a represent the absorption coefficient for unit concentration, there follows the absorption law
to be incorrect. There is thus good precedent for any twentieth-century inaccuracies in describing what Beer thought and did. We have made an attempt to evaluate the internal consistency of Beer's work. The figure gives transmittance curves measured by a General Electric Recording Spectrophotometerg on water and on two of Beer's solutions, all measurements being made in the same cell. Taking the diierences in cell length into account, we have compared Beer's absorption coefficients with our measured transmittamies to locate the "effective wave length" (an X-ray term)10 of Beer's red light. The figure shows these wave lengths t o differ very little for the two solutions, which speaks highly for the quality of Beer's work. The principal aim of this article has been to show that Beer, in th'mking primarily of the amount of absorbing material, laid the foundation of an absorption where d is the thickness of the layer and c the concen- law broader than equation (6). That he did not write tration. This law was set up by Beer and is called this amount as an explicit variable in the absorption L' Beer's Law.'' Comparison of equation (11) with law seems to us no reason for not calling that law by equation (1) shows the first half of the last sentence his name, and his alone-after all, didn't Columbus discover America? 'WALTER,B., Ann. Physik (Wiedemmn), 36, 502, 518 (1889). See especially pp. 505, 512, 520, 521. 8 KAYSER, H., "Handbuch der Spectroscopie," 9. Hirzel, L e i p zig, 1905, Vol. 3, pp. 10-25. We used the references there cited as a guide to the older literature.
'
MICHAELSON, J. L., AND H. A. LIEBHAFSKY, Gen. Elec. Rev., 39. 445 (1936). . . io See, for example, LIEBHAFSKY, HERMANA,, Analytical Chemistry, 21, 19 (1949).
