Gordon M a r Loudonl
Louisiana State Universitv Baton Rouge
II
The Necessity of Using Monochromatic Radiation in Spectrometry
The necessity of utilizing radiation which is as nearly monochromatic as possible in spectrometry is universally recognized, hut it seems that a formal proof of this requirement has not been given. Charlot and Gauguin2 give an intuitive argument for the case of two specific wavelengths of incident radiation, and suggest the possibility of a generalization. We shall prove formally that only by the use of monochromatic radiation is it possible to maintain a constant absorbancy for a given system. We assume the validity of Beer's law: Ai = a b C = log Pai/Pi
(1)
where A, = absorbancy of the sample a t some wavelength Xi; b = sample thickness in the radiation path; C = sample concentration; Psi = intensity of radiation incident on the sample a t some wavelength A,; Pt = intensity of radiation transmitted from the sample a t some wavelength Xi. The extinction coefficient, a,, is a constant characteristic of the system under consideration and the wavelength of the incident radiation. Let us consider a spectrophotometric experiment with incident radiation having a continuous distribution of wavelength from XI to X1. The resulting intensity distribution function is given by k'f(P.i)h,-~~
(2)
where k' is an arbitrary constant. The intensity of the transmitted radiation also varies continuously according to kg(Pi)h-hl
we require that A %= log
=
aibC = log p"' Pi
(5)
that is,
Equation (6) has the form of a proportion, and thus we may write
and
where ar is a proportionality constant. Differentiating equations (7a)and (7b),
or f(P.i) = alk'
(8a)
(3)
where k is also an arbitrary constant. If we assume the validity of Beer's law, we have, by analogy with equation (I), Thus, f(P,Jmust be constant; since we know its endpoint value [f(PA) = PX,],we know its value everywhere: j(P.,)
Now A , must remain constant over the entire intensity distribution. But, by Beer's law, A, = a,bC, and the ahsorbancy is constant only if A, = log ( P d / P J . Thus 'Senior chemistry mjor, Louisiana State University. The author thanks Professor Paul Delahay, Louisiana. State University, who motivated the author's interest in this problem. C n a n ~ o G., ~ , AND GAUGUIN,R., "Dosages ColarimBtriques, Principes et MBthodes," 1st ed., Masson and Co., Paris, 1952, pp. a 1 0 .
=
a/k' = P.I
(9a)
Similarly, we deduce that g(Pi) = a l k
=
PI
(gb)
But it is necessary for equations (9a) and (9b) to hold that the wavelength be constant. Thus, equations (Qa) and (9b) assert that the assumed continuous distribution of wavelength is indeed a constant; i.e., only through the use of monochromatic radiation is it possible to maintain a constant ahsorbancy for a given system.
Volume 41, Number 7, July 1964
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391
