Frederick C. Strong Ill1 [nstitut i i r Spektrochemie
und angewandte Spektroskopie
Dortmund, Germany
Use of Polychromatic Radiation in Absorption Photometry
In these days of recording double-beam spectrophotometers, Duboscq calorimeters would seem to be out of date, yet they are still available in equipment catalogs and presumably some people have them and use them They are always described in introductions to spectrophotometry and their principleradiant power matching by thickness variation-is fundamental in spectrophotometric theory. This principle is emoloved in the variable-thickness cell for removal of solvent absorption bands in infrared spectrophotometry. Perhaps someone will build a double-beam instrument based upon automatic thickness variation and thickness recording. The Duboscq colorinieter is sometimes provided with a light filter but is often used with white (ie., polychromatic) light. Some presentations of the equation for matching with white light, brf, = b.c.
(1)
use Beer's Law,
wavelength range XI-X2. In mathematical language, this can be truly monochromatic only if Xi-Xs is infinitesimal (dX) because the source is continuous. The distribution of the radiant power reachmg the sample with respect to wavelength is the partial derivative 5Po/5\, also denoted as Px,o. Radiant powers for infinitesimal and finite bandwidths are therefore dPo = PA,&
P
and
= ~
Ã
ˆ
,
After the beam passes through thickness 6 of the sample a t concentrations with absorbing components 1,2, CI, c.>, . ., the radiant power is reduced to
where PÃ (5P/SX) is the new distribution resulting from the absorption spectra of the solutes. From a consideration of photon capture probability, the equation
-log T = A = ate
(applicable only t o monochromatic radiation) to prove its validity! Others do not emphasize that white light will give a valid result; some even state that it will not do so The author proposes to prove below that the Duboscq equation is valid for white light. A recent paper in THIS JOURNAL^ entitled "The Necessity of Using Monochromatic Radiation in Spectrometry" is not inconsistent with the above if the use of white light is not called spectrometry. However this author feels that the thesis of this paper was adequately covered in an earlier paper not cited by Loudon
can be derived, where the a's (absorptivities) are wavelength functions characteristic of each substance. Monochromatic versus Polychromatic Radiation
If the bandwidth is sufficiently narrow for the absorptivities to be substantially independent of wavelength, the exponential in the above equation can be factored out, yielding Beer's law: P = 10-aibci - osbca -
Mathematical Formulation
=
In an instrument providing a continuous source of radiation, there will be a certam radiant power, Po,4 supplied to the sample that will extend over some
and
* Heinrictl Hertz Research Fellow. Present address (to which reprint requests should be sent): Dept. of Chemistry, University of Bridgeport, Bridgeport, Connecticut. LOUDON, G. M., J. CHEM.EDCC.41, 391 (1964). STRONG, F. C., "The Theoretical Basis of the Bouguer-Beer Law of Radiation Absorption," Anal. Chew. 24,338, 2013 (1952). The probability basis of absorption of photons has been recognized for some time, e.g., LANOE,B., Z, Physik. Chem. A159, 277 (1932); C.A. 26, 3187. However, this author felt that a clear presentation of a derivation (not just the usual integration) was needed for chemists, especially students, to whom Beer's Law has always been a mvsterions relationshin. Re~rintsof this paper wereexhausted years ago, but a condensed version will be sent on request. Ignoring reflection and window absorption. This complication can be avoided by defining P y as the radiant power after passing through a blank.
Taking
342
/ Journal of Chemical Education
P = Po
10-aibci - aihci -
T
=
10-ah
fi
pn
- ~ b -o
(4)
+ ate, +
(5)
- log of both sides, -log T = uibci
or A=&+&+
(6)
where T is transmittance and A is absorbance. For polychromatic radiation, equation (3) must be used. It is a perfectly valid equation but not the one conventionally called Beer's law. There are various mathematical and instrumental ways of handling it.% Matching with Polychromatic Radiation
For matching of similar solutions, the situation is quite simple and polychromatic radiation may be used.
In two solutions of one solute, unknown (x) and standard (s),
But
V,
and
If matched by varying the thickness until P,
=
P.
then
Obviously this must be true if But this tt ill be true if equal cross sections (8) of the two cells contain the same number of n~olecules(n) between the cell windows. Let V be the volume of this region: =
b.S
va = b&S Multiplying by n,
6,
or
31
=
or
31
= b.
bl - = -b. vi V. b xnE =
n
and
:.
bvcX = btc,
n -=
v.
cs
when n is the same m both volumes. Thus monochromatic radiation is not necessary for matching by thickness variation. Even solutions of more than one absorbing component can be matched with polychromatic radiation if the ratio of component concentrations is constant. The mathematical proof is analogous to the above Of course, the sensitivity of matching will be greater (as can easily be demonstrated on a Duboscq calorimeter) if most of the radiation not affected significantly by the sample is filtered out. Conclusion
bicX = b.c.
V,
= c,
Proportionality between concentration and absorbance (Beer's law) requires approximately monochromatic radiation to the extent that the variation of absorptivities with wavelength results in less error than other instrumental and chemical errors. However polychromatic radiation is acceptable when similar solutions are matched by thickness variation. Acknowledgment
The author wishes to thank Professor H. Kaiser for his comments during the preparation of this paper
Volume 42, Number
6, June 7 965
/
343
