Dependence of the Beer-Lambert Wayne E. Wentworfh
University of Houston Houston, Texas
I
I
Absorption Law on ~ono;hromatic Radiation A n e x p e r i m e n t in rpectrophotometry
A
discussion of the well-known absorption law can be found in most texts of analytical and instrumental analysis as well as in texts of physical chemistry. Examples of how well the law is obeyed by a given substance are frequently given along with a discussion of the application to kinetic and equilibrium studies and multicomponent analysis. The restriction requiring monochromatic radiation is generally stated. However, examples showing the effect using radiation with a finite band width are not given. The purpose of this paper is to present an experiment which clearly emphasizes the necessity of using monochromatic radiation. As will be demonstrated later, an easier yet more critical examination of this restriction is a graph of absorbance versus slit width rather than the conventional absorbance versus concentration. This does not mean to imply that graphs of absorbance versus concentration should be eliminated. They still serve a most useful function by displaying chemical interactions which result in so-called deviations from the absorption law. The absorption law is applicable to the entire electromagnetic spectrum from the X-ray and far ultraviolet through the visible region into the infrared. The law has received a variety of names in attempts to give proper credit to the formulators.' I n mathematical terms it can be stated simply as log Po - = -log T
= A = rbc P where Po = radiant power incident on the sample; P = radiant power transmitted by the sample; T = transmittance; A = absorbance; e = molar absorptivity; b = path length in cm; and c = molar concentration. This nomenclature has been adopted by the Joint Committee on Nomenclature in Applied Spectroscopy whereas an alternative nomenclature is commonly found in physical chemistry journal^.^ A derivation of the absorption law can be found in a paper by S t r ~ n g . ~ The law is valid only when so-called monochromatic radiation is used. The formal proof for the necessity of using monochromatic radiation has been given;4 however, it should be emphasized that the stringency of this
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MALININ, D. R.,
AND
YOE,J. H., J. CAEM.EDUC.,38,
(1061).
129
HUGAES, H. K., (Chairman for Joint Committee on Nomenclature in Applied Spectroscopy), Anal. Chem., 24, 1340-1354
(1952).
STRONG, F . C.,Anal. Chem., 24, 338, 2013 (1952).
'LOUDEN, G. M., J. CAEM.EDUC., 41,391 262 / Journal of Chemical Education
(1964).
requirement is dependent upon the variation of e with wavelength in the region in which the absorption measurement is being made. I n practice monochromatic radiation is never realized; dispersing systems always have a finite resolving power, and any light source, even if it were line-like from an atomic emission, has a finite line width. Therefore, intensity measurements actually consist of integrals over the frequency region of the band pass, and the actual absorbance, A', is a ratio of such integrals of the incident intensity to the transmitted intensity. Assuming the absorption law is obeyed in the limit of monochromatic radiationa
where Io(v) and e(u) are defined as given previously except that they now represent distributions over fr* quency. If r ( u ) is a constant, r , or essentially so in the frequency region, the exponential term can be taken out of the integral and the absorption law results.
The question to be answered then is simply, "When is e considered essentially constant?" At the maximum of an absorption band there is a point of zero slope and a region of essentially constant e. The width of this region generally depends upon the half width of the absorption band. Many compounds with broad peaks of 20-50 mp half widths will obey the absorption law when filters are used to isolate the radiation. Filters are frequently employed in calorimeters. I n other cases prism or grating monochromators are required to obtain a narrow band pass as required in spectrophotometers. Even these are sometimes unsatisfactory when the absorption spectra are extremely sharp, e.g., atomic absorption and absorption of some gases at low pressures. The visible absorption spectrum of Pr+% was run on a Beckman DB Spectrophotometer and is reproduced in Figure 1. One will note the four absorption bands with a variety of '/*-bands widths. The extremes are the sharp band at approximately 482 mp and a broad band at 586 mp. Measurement of absorbance as a function of slit width at these two wavelengths obviously reveals to the student the restrictions of monochromatic light (or more correctly, the constancy of e within the band pass) on the absorption law. Such graphs of
absorbance versus hand pass are given in Figure 2 a t wavelengths of 482 and 586 mp, respectively. It is convenient to plot log slit width in order that a wide range of values can he obtained on a single graph. Absorbance values a t the relatively sharp absorption band, 482 mp, are changing dramatically with band pass in
Figure 1 .
Absorption Spectrum for
at 10.2M concentration.
contrast to similar measurements of the broader absorption band a t 586 mp. Eventually if ahsorbance measurements at 482 mp with smaller slit widths are carried out, the absorbance approaches a constant value a t approximately 0.2 mp band width. On the other hand only a band pass less than approximately 1.0 mp a t 586 mp is required to obtain constant absorbance measure-
ments and agreement with the absorption law. If absorbance measurements are continued on down to the extreme of maximum sensitivity of the photomultiplier and minimum (but not readable) slit width, a decrease in ahsorbance is again observed. This we attribute to stray light which finally becomes significant a t extremely low light levels. Under "normal" operations the stray light is not of consequence. I n order to make such absorbance measurements it is necessary to have a spectrophotometer with adjustable sensitivity of the detector over a wide range along with variable bilateral slits. A Beckman DU with the photomultiplier attachment accessory has this flexibility and was employed in these experiments. The absorbance measurements are made of the three different solutions a t a single wavelength without altering the wavelength dial a t anytime. The wavelength had previously been set at the maximum of the absorption band. The cell holder in the Beckman DU has a capacity of four absorption cells containing the three solutions and the solvent. Therefore, the solutions do not have to be disturbed at any time during this set of measurements. The absorbances of the cells, which are obtained by filling all cells with solvent, can be measured later. I n making these measurements, the technique of pipetting out the solutions and flushing a number of times with solvent from a pipet without removing the cells from the holder is re~ommended.~A syringe has been used in this laboratory for this purpose. Precise relative absorbance measurements, which are necessary for this experiment, can be obtained when these variables, wavelength resetting, and replacement of cuvettes, are eliminated. I n carrying out any study of the absorbtion law it is generally recommended that graphs of ahsorbance versus concentration be made. The criterion for conformity with or deviation from the absorption law is the observed linearity or non-linearity. These graphs, Figures 3 and 4, were made with the same absorbance readings as found in Figure 2. Although the deviations from linearity are ob+,ous it is somewhat surprising, to the author also, that the slope or apparent molar absorptivity is changing so dramatically for a corresponding decrease in band pass. From this it can be concluded that for measurements a t absorption maxima a more critical test of the absorption law can be made from an observation of absorbance versus slit width in preference to the conventional absorbance versus concentration. The latter study not only yields a less critical evaluation of the law, hut also is slower and more time consuming. If the students are asked to make the conventional absorbance versus concentration plots a t the different band passes, extreme care should be exercised in making up the original solutions. Careful volumetric techniques should be employed. One may even ask the students to weigh the solution pipetted to ensure accuracy further (this is common practice in precise studies). Since only four points (including the zero point for solvent) are used in the experiment to test the linearity, one cannot afford to have a single concentration in great error. If more solutions are required of the student, the experiment becomes quise time consuming.
LOG BAND WIDTH (mp) Figure 2. Absorbance of PC+' at 482 m p of slit width at various consentrotionr.
rn and 586 mp 0 0 %Q
function
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Ross, S. O., AND WILBON, D. W.,SpectmVisim, NO.10, 4
1961. Volume 43, Number 5, May 1966
/ 263
I n conclusion, the experiment reveals to the student the nature of deviations from the absorption law which can arise from the use of non-monochromatic radiation. Furthermore, if the absorbance measurements are carried to extremely small slits, the effectof stray radiation can also be shown. The necessity of using essentially monochromatic radiation is extremely critical in atomic absorption spectroscopy. After having done this experiment, the student may better appreciate the neces-
MOLAR CONCENTRATION PR + 3 Figure 3. Abrorbonce of PriJat 482 m p as o function of concentration at various bond widths.
264 / Journal of Chemicol Education
sity of going to various hollow cathode light sources to overcome this problem in atomic absorption spectroscopy. Finally a more critical test of this type of deviation from the absorption law is revealed in absorbance as a function of band pass or slit width rather than the conventional ahsorbance versus concentration graphs.
MOLAR CONCENTRATION P R + ~ Figure 4. Absorbance of Prt' at 586 m p or o function of concentrotion a t mriour bandwidths.
