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Theoretical Basis o f the Bouguer-Beer law o f RadiationAbsorptionFREDERICK C. STRONG', VilZar+ovaCollege, Villanova, Pa.

Presentations of the mathemathl law governingthe absorption of radiation as a function of thethickness and concentration of absorbing materialare customarily based upon experimental observa-tion. A complete derivation is presented, basedonly upon modern theories of matter and energy.Considering the radiant power of a beam as thenumber of photons passing per unit time and theprocess of absorption as the capture of photons bymolecules or ions, the probability of capture isrelated to a ratio of areas in an element of material.This is then integrated over three dimensions for

HE author has observed that the relationship of absorptionT f radiation to thickness of absorbing mate rial and concen-tra tion of an absorbing component is customarily presented asthe result of experiment.As the initial radiant power varies, the fraction absorbed by agiven sample of material remains constant.The rate of decrease of radiant power with thickness and withconcentration is proportional to th e initial power:

For example, one reads th at :

dP , dP,dx dchere ~ and- re the rates of change of power of mono-chromatic radiation, P,, with thickness and concentration, re-spectively, b is a fixed thickness, and a: and a; are positive con-

stants, The usual presentation then continues with the integra-tion of these equations, yielding( 3 )= e -a 'bcmPm,o

or related forms, P,,, being the initial radiant power. [Rad iantpower P , rather than intensity, I, has been recommended by theCommittee on Letter Symbols of the American Physical Society(6) and by the Joint Committee on Komenclature in AppliedSpectroscopy, established by ASTM Committee E-2 on Emis-sion Spectroscopy and the Society fo r Applied Spectroscopy.]The author has often wished to see a derivation from theoryonly. Such a derivation can be carried out much more readilytoday, with the improved modern theories of ma tter and energy,tha n was possible in the times of Bouguer and Beer.The exponential nature of the law suggests that probability

plays a part. This has been recognized, but apparently neverfollowed up by a published derivation. For example, Lange (6 )stated in 1932 th at the exponential dependence of radiation ab-sorption by sols could be obtained by a consideration of prob-ability . Fundamentally the process of absorption by colloidalparticles and by molecules and ions would be identical.DERIVATION FOR MONOCHROMATIC RADIATION

The quantum theory of radiation accounts for t,he phenom-enon of absorption more readily than does the wave theory.According to the former, radiation consists of particles (quantaor photons) of energy hv = hc/X , where h is Planck's constant,Y is the frequency of the radiation (when considered as a wave),

1 Present address, Stevens Insti tute of Technology, Hoboken, N. J.

identical photons (monochromatic radiation) toyield the familiar absorption law. The general caseof a beam with a continuous distribution of wavelengths is obtained by further integration and theresult is given in several forms. In the belief thatsome of the equations are difficult to comprehend,graphical interpretations are presented, followed byanalysis of possible methods of employing the lawin practice. Finally, cauges of apparent deviation areconsidered. Both the derivation and the graphicalinterpretations of the absorption law should makeit more intelligible and fruitful for analysts.

ci s its velocity, and Xis its wave length. ilbsorption is then to beinterpreted as the capture of a photon by a molecule or portionof a molecule, thereby increasing its energy. Likewise, theradiant power is interpreted as the number of photons passing perunit time.Consider a parallel radiation beam (Figure 1) n which the pho-tons are identical ( radiation is monochromatic) and the cross sec-tion of the beam is S.Let the beam pass through a homogeneous absorbing mediumwith: (1 ) parallel faces perpendicular t o the beam; ( 2 ) thicknessbi (3) initial power of t he beam entering t he medium after reflec-tion equal to P,,,; and (4) ower after traveling distance x equalto Pm.Consider an element of absorbing material with cross section Sand thickness dx (Figure 1) . Passage of P , through thickness dxwill change it by dP,. Because P, decreases by absorption, dP,is negative. Let dS, be the projected area of regions in the ele-ment which will capture photons. I t will be infinitesimal be-

cause the thickness is infinitesimal.If a photon is passed perpendicularly through the element,the probability of its capture &-illbe the ratio of capture area tototal area, dS,/S . If this is done at random a great many times(as would be the case for the photons in a beam), the count of thefraction captured would be -- nd would be statisticallyPmP,equal to the probability of capture:

dP, dS,- -P, = s (4)

0 dx X bFigure 1. Diagram for Derivation of Absorption Law

Although it is true that any given beam will have a fixed den-sity of radiation determined by the optical cystem and will,therefore, not correspond t o random passage of photons, theprobability of photon capture will be unaffected, as the materialis assumed to be macroscopically homogeneous,338

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V O L U M E 2 4 , NO. 2, F E B R U A R Y 1 9 5 2The material will in general contain more than one absorbing

( 5 )substance, each contributing to th e area, dS,:

dS, = a:dn , + aidns + . . . . .where a ; is the capture cross section of a unit quant ity (molecule,mole, gram, et c.) of species I, d n l the number of units of species1 in the volume element Sdx, tc. Substituting and changingsigns,

339

( 6 )1_ - -P,p , ( a l d n , + aLdn, + . . . . .[As the quest ion of whether a photon can be absorbed by amolecule depends upon whether the energy of the photon (hc/A)

is capable of raising the energy of the molecule to a permi ttedstate, the as will be functions of wave length .] Summing overthe interval 0 2 2 b,

or

Substituting the spectrophotometric quantities, transmittance( T = , and absorbance ( A = -log T ) , and retaining thesubscript m to denote the monochromatic case,

P

where ai = a,! log e= 0.4343 a

Equations 8, 9, and 10 may be considered as equivalent state-ments of t he absorption law in its most fundamental form-namely, in terms of quant ity of substance in the pat h of the beam.[According to a recent intensive study ( 7 ) f Beers original paper(f), his concept of the basis of absorption was also th at of Beer.However, he chose to express it in terms of length and com-pensated for the use of different concentrations by consideringthat he had in effect changed the length of path. In other words,he did not use concentration as an explicit variable, contrary toprevailing opinion. ]In the interests of practicality, both concentration and pathlength are introduced as follo w:

The volume, T of th e path through the medium is bS.Therefore:

1 bs = vandsince n l / V = c l, the concentration of component 1.

(15)(16)

P ,In- - a:bcl- aibc2 - . . . , .Pm.0ubstituting,P,log ~ = - a h l - a h . . .P W8 , = alba + azbcp + . . . . . (17)

(18)~ -- - a l b a - albcl - . . . .EXTENSION TO FINITE BAND WIDTH

The above equivalent expressions for the absorption la w in-volve monochromatic radiation, a condition that may be ap-

proached in practice but probably never quite attained. Filterphotometers pass a wide wave-length band and spectrophotom-eters a band whose narrowness is limited by the intensity of thesource of radiation, the sensitivity of th e detector, and theresolution of the optical system (Figure 2) ( 4 ) . Even an isobtedline has been shown to have width. Hence, it is desirable toobtain a form of t he law which applies to a band of finite width.In Figure 2, it is apparent th at the rad iant power of the bandis the area under the distribution curve-that is, the definiteintegral:

P = L $ A

where y is the vertical axis in the figure. (I t is of course neces-sary to avoid integration over a path containing the points of dis-continuity of curves a and b. However, this can be done bydividing the area vertically at these points and integrating overeach portion separately .) The meaning of y can be readilyascertained, as the integral is the sum of an infinite number ofinfinitesimal areas, d P :dP = yclX (20)

An equivalent mathematical notation for this derivative wouldbe P i and i t is frequently so represented, though usually with theprime omitted.

Figure 2. Wave-Length Distribution of Bandsa.6.C. Filtered band

Exit slit of approximately same w idth as entrance slitExit slit wider or narrower than entra nce slit

In order to apply the absorption law for monochromatic radia-tion to a finite band, a n infinitesimal interval of the band can beconsidered. For band width dX , the radiation is ( a ) essentiallymonochromatic and ( b ) of infinitesimal power, dP,. If dP , isreduced exponentially to dP b y passage through a given solution,

which provides an improved definition of T , and a new statementof the absorption law fo r the monochromatic case:p , = P , , ~ - :bct - ib cz - .. (23)

To obtain the expression for finite band width,

ande - l b a - nbcz . A (25)

This is the correct form of t he absorpt ion law for finite bandIn order to evaluate the integral mathematically, theidth.variation of PA.^, a ; , a; , etc., with ?.must be known.

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340 A N A L Y T I C A L C H E M I S T R YFor application to analysis, the absorption law takes theform:

However, it is more common t o measure rclative transmittance,Tr = P / P B ,where P B is the radiant power transmitted by ablank containing all the substances present in the sample solutionexcept t he substance or substances being determined and all ofthe same thickness. If only one substance (component 1 ) is))ringdetermined, and if the solution is sufficiently dilute, or theabsorpt ivities of the other components are sufficiently smallthat their absorbance in the sample solution is equivalent to theabsorbance of the blank,

( 2 7 )p B = p x , ,& - :h s - . dX

PA

0 CCurve of Equation Px = PxJWakigure 3.

b mustant

Dividing Equation 24 by Equat'ion 27,

which is another form of the absorption equa tion for mono-chromatic radiation.d P = EB - a:ba dX (29)a h

( 3 1 )

Substituting,

with

GRAPHICAL INTERPRETATION OF ABSORPTION EQUATIONSThe meaning of the foregoing equations may not be readily

Forrasped. Figures 3 to 6 will perhaps make them clearer.

simplicity, it is assumed that az = a3 = . . . . .. = 0, b is con-stant, and the subscript 1 is dropped from al and cl .The exponential decrease of PA with concentration of on ecomponent for constant thickness is shown in Figure 3 . Forboth concentration and wave-length variable, Figures 2 , c , and3 can be combined perpendicularly to give the three-dimensionalgraph of Figure 4.

P A

0 C X CFigure 4 . Surface of Equation Px = P x . ~ ~ O - ~ ~

a = f(k) an d b constant

The equation for PA= f(c, A ) is tha t of a surface which inter-sects the P A ,X plane in the distribution curve for with wavelength and the parallel plane through c = c z in the curve for P i =j ( c , , A) . The cross-hatched areas are, from left to right ,

and P ( l > x dX)and their ratio is T . The portion of the surface marked L repre-sents a region of t he band spectrum not absorbed by the solution.The form of a graph for comparison with a blank-i.e., P i =P ~ , BO - abc-will be the same.The three-dimensional graph of T , = is shown inFigure 5 . The intersection of t he surface with the T,,&, planewill be a straight line at T , = 1 because here dP = dP,. Theregion L of Figure 4 becomes a horizontal rectangle. The areascorresponding to those evaluated in the previous figures no longerhave significance, as T , cannot be integrated directly to give 7'.Figure 5 is transformed into Figure 6 by taking the negativelogarithm of the vertical axis: -log T, = abc. The surfacepasses through the wave-length axis because for c = 0, A , = 0.The plane in which c = 1intersects the surface in a curve which isthe variation of a with X or the familiar absorption spectrumcurve. Every plane perpendicular to the wave-length axis atbonie value X4 will intersect the surface in a straight line mithequation A , = a(X,)bc = kc .

APPLICATION OF THE ABSORPTION LAWBecause the practical use of the absorption law in cheniicalanalysis involves integrat ion of unknown functions, a ny one of anumber of procedures can be followed.-1. An experimental curve for the variation of absorbancewith concentration can be determined, using a given band of fixedwidth and solutions of known concentrations. This curve canthen be used for analyses under these conditions and with thesame instrument. Even installation of a new radiation sourcemight require revision of the curve.

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V O L U M E 2 4 , N O , 2, F E B R U A R Y 1 9 5 213. The approximation that the absorptivities are constant inthe wavelen gth interval can be made:

341grands of the expression for the relative transmittance and thevalue of the absorptiv ity at 8 given wave length calculated forknown concentrations. Thi s can then be used with any instru-ment for determining unknown concentrations, provided that theconstants in the approximate functions are also determined forthe instrument.

Eberhardt (5) as proposed appi oxiriiate functions for spectro-photometry with the Beckman DU and similar instruments.Replacing a portion of his notation by th at used in this paper, hisfunction for P i is:

N , e - 5; ci - 5; cz - . . ( 3 6 )( 3 7 )

Whether this approximation is sufficiently accurate can bedetermined experimentally by the procedure desoribed in A, astraight line indicating that it is. It will be observed that theapproximation consists of applying the equation for monochro-matic radiation,

or^=- ~ , - - n ; b c i - a a ; b c 2 - . . .Po -

to finite band width.approximate equation becomesWhen the sample solution is compared with a blank, the

( 38 )- e - - o b c = 1 O - a b c7 -and

A, = ab c (39)where the subscript r denotes measurelnent relative to the blank.

, ,I f i t/ ,0 CA C

Figure 5 . Surface of Equation T, =a = f(i) an d b constant

The monochromatic equation will give the best results when( a ) he pass band is as narrow as possible, ( b ) he band is centeredon a wave length for which the absorptivity is a maximum or aminimum, and ( c ) the concentration range is kept small. Condi-tion ( b ) is desirable fo r two reasons. Variation of absorptivitywith wave length is small and the distribution of radiant powerwith wave length (when triangular) is at a maximum in thecenter of the band where variation of absorptivi ty with wavelength is least.Provided that the resulting curve is approximately linearand the instrument is suitable, transmittance or absorbance canbe measured at varying band widths and extrapolated to aeroband width.Approximate functions can be substituted in the inte-

C.

D.

The constants OL and p correct for slight deviation from theXo =riangular shape of PA,^ caused by nonlinear dispersion.

l / p ( A 1 + XZ ) and AA = l / a ( X 2 - XI).

A m

C 1 CFigure 6. Surface of Equation A , = abc

a f(X) an d b constant

For the exponential factor i n the function for the blank hee - a:bc, - a i % - . . . 5X r(X) = r ( A o j / l + p ( X - A,)] (42)

This is satislactory if absorption is notFor the absorptivity function

suggests

where r(A,) is a constant.great and the band not too wide.he substitutes one of t,wo alternative functions:a ; ( h ) = a;(x,, + y(X - A,) + 6 (A - A,)*

a:@) = a:(x,, + -&- A,) for Xo 5 X 5 Xo + PAX(43(44)(45)

These approximate functions are substituted in the equation forY7 Equation 3 2 ) and integrated. Eberh ardt gives equationsfor the two values of T , correspondiiig to the function chosen fora ; and also the simpler equations for cases where a = 0 = 1and p = 0.Philpotts, Thain, and Smith (8 )have proposed an error functionas the approximation for n ; ( x ) when x single absorption band int,he infrared can be isolated:

or a:(A) = a: (A , ) - yl(X - A,) for ho - aAX 5 h 5 A,

where 1 is a slightly different band width measurement (distancebetween points of inflection). The exponential is expanded ande-a;(X)bc is again expanded. The integrals are evaluated bysubstituting five terms of the second expansion and assuming thatFrom the resulting equation,a( h o ) s determined.Hardy and Young ( 3 )have inverted the integrals by means of aFourier transformation and a Taylor expansion, obtaining an

is constant over the interval.

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342 A N A L Y T I C A L C H E M I S T R Yexpression made up of a sum of derivatives. The trea tmen t isvery general, though mathematically complex, and deservesattention.

An attachment could perhaps be devised fo r a spectropho-tometer which would automatically correct transmittance meas-urements to zero slit width. Hardy and Young have indicatedhow this ideal operation might be carried out electronically.E.

EXPERIMENTAL DEVIATIONS FROM THE ABSORPTION LAWUpon examination, it is realized that few deviations from theabsorption law observed experimentally are true deviations, asthe y result from nonfulfillment of the condit ions of application ofthe law. A number of possible situations are enumerated below.The phrase I . . .does not obey Beers law, , , commonly refersto the nonlinearity of the variat ion of absorbance with concen-tration. If the deviation is large, it usually indicates a mobileequilibrium that is shifted by dilution. Obviously, the deviationis not a failure of the law bu t of the esperinienter in not includingth e absorption and correct concentrations of all components.Nonlinearity in absorbance is to be expected for a pass bandfinite width. Here the deviation is present not in the measure-ment but in the approximate equation employed.Scattering of a portion of the radiation represents a decreasein rad iant power not caused by absorption, since some of it willnot reach the detector. Fortunately] the relative error is usuallysmall if absorption is significant.Fluorescence and the Raman effect represent emission thatcan cause a positive error in the measurement of transmittance.These effects are also usually small in absorptiometry.Absorptivities may change in concentrated solutions or at

different temperatures] owing to changes in solvation. As thisinvolves a change in the structure of the substance being deter-mined, deviation is to be expected.Change in absorptivity with temperature can be expected foranother reason: The distribution of the energy states of themolecules would be changed, altering their average capture crosssection for photons of given energy.Photochemical decomposition of th e sample may occur duringirradiation, causing a change in concentration. The lo w intensityof pass bands of photoelectric ins truments usually avoids thisdifficulty whenever it is theoretically possible..4t high values of r adian t power, a saturation effect might beobserved. With a relatively large number of molecules in ahigher energy state due to absorption, the absorptivity wouldbe altered. The obvious remedy for such a hypothet ical situa-tion would be to revert to lo w power radiation.

LITERATURE CITED(1 ) Beer, A ., Ann . Physik (Poggendorf), 86, 78 (1852).(2 ) Eberhardt, W . H., J . Optical SOC. m. , 40, 172 (1950).(3 ) Hardy. A . C ., and Young, F. M., b i d . , 39,265 (1949).(4 ) Hogness, T. R.. Tscheile,F. P., and Sidwell,A . E., J . Phys. Chem.,(5 ) Hughes, H. K., et al., Am . J . Ph ys . , 16, 176 (1948).(6 ) Lange, B.. Z . physik. Chem., A159,288 (1932).(7 ) Pfeiffer, H. G. , and Liebhafsky, H. A . , J . Chem. Education,28,123 (1951).(8) Philpotts, A. R., Thain, W., and Smith, P. G., ANAL.CHEM. , 3.268 (1951).R E C E IV E D pril 17, 1981. Presented in par t at Pittsburgh Conference onBnalytical Chemistry and Applied Spectroscopy, Pittsburgh, Pa., March7 , 1951.

41, 394 (1937).

PRECISION COLORIMETRYGeneral Method of Improving Precision in Multicomponent Spectrophotometryand Some Practical Consequences Thereof

C . F. H I SK EY AND DAVID FIRESTONEl, Polytechnic Insti tute of Brooklyn, Brooklyn 2 , N. Y.

The principles of relative absorbance colorimetry have been extended t o m ulti -component systems. In order to achieve the h ighest possible precision in th econcentration measurement, th e analyst mu st work with absorptivity ratioswhich are in turn relatively determined. The principles of the method forobtaining these ratios and for conducting such analyses are developed herein.Application is made to a two-component system consisting of the permanganateand dichromate ions. A tenfold gain is obtained in the precision of the analyti-cal results obtained by the relative method as compared with the results forthis same system analyzed by the absolute method. Extension of this relativeapproach to multicomponent systems is outlined, as well as a general methodfor chemical analysis without sample weighing.

HIS paper presents a number of considerat ions dealing withT he problem of analyzing multicomponent systems usingspectrophotometr ic techniques, By a combination of a relativeabsorbance technique coupled with the use of absorptiv ity ratiosprecisely determined it is possible to reduce the error of measure-ment in cases such as these so as to make their spectrophotometricdetermination comparable in precision to those of volumetric orgravimetric techniques.The principles of relative absorpt iometry have been extensivelytreated in a group of papers dealing with the fundamental theory1 Present address, Food and Drug Administration, 201 Varick St., New

York. N. .

(7-9) as well as with a number of practical applications thereof(8 , 3, 5, I S ) . In all of these papers the emphasis has been on thedetermination of the unknown concentration of an absorber rela-tive to that of a reference standard. Since such relative con-centrations may be determined precisely, it follows that relativeabsorptivities may be determined with similar precision and , con-sequently, multicomponent systems possessing considerable over-lap of their absorption bands may be analyzed with the same in-crease in precision available to systems already described in theliterature.Although some other rrorkers have utilized relative absorptivity ratios in multicomponent analysis, the essential character-