I
R. C. Michaelson and L. F. Loucks' ~
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Expressions for the Light Absorbed by a Single Component of a Multi-Absorber Solution
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University of Prince Edward island Charlottetown, P.E.I. Canada CIA 4P3
1
In the course of preparing for publication a study of the photolysis of mixtures of Con and N20,2 one of us became interested in the justification of the assumption that the ratio of light intensity absorbed by the components of the mixture is given by the ratio of the products of extinction coefficient and concentration, i.e. -Ink.& =-
I
(a%
.
fhCI?
This is indeed an exact relationship and although a more eeneral form is stated in a t least one monograph? there ap;ears to be no derivation in the accessible iiterature. In view of the increasing use of photochemical procedures, many of which involve the photolysis of mixtures, an elaboration on the absorption characteristics of multi-absorber mixtures may help to avoid errors that easily could be made. It is well known that the total absorhance of a mixture of absorbing species, a,h, . . . , i s given by A ,,,,:,I = A * ,
+
A ,>...
a relationship that derives directly from I
I,,exp (-[w,
=
+
t,,c, 11) I...
(1)
where 6 is the absorption coefficient of the various absorbers, e is the concentration, and 1 is the path length. Also, from this latter relationship the total light absorbed is found t o be given by the relationship =
I,ll - exp
(-[(,A
+
t,,c,, ... 1l)l
(2)
In this paper two alternative derivations for the intensity absorbed bv a sinele comoonent of a multi-absorber system are presented. Although the case of a two component system is considered. the derivations are ceneralizable to any number. The app;oaches are rather difierent and it may be interestine to comoare the usefulness of these approaches .. in teaching this concept. The first approach is a rather straightforward extension of the usual Beer's Law derivation. The differential expression for the loss of intensity due'to a single component a, is dl., = -I(l)e.c,dl (3)
Integration of both sides of eqn. (4) yields
The quantity I , represents the intensity of the radiation that has not been absorbed bv component a. Thus. a t anv distance 1, I , will be the sumvof the light intensit; 10) at distance 1, plus the intensity that has been absorbed by component b. T o obtain the intensity ahsorhed by component a, the integral given in eqn. ( 5 ) must be evaluated over the limits of I . = Io a t 1 = 0 to I . = I.' at apoint where 1 = 1'. I t follows that the general expression for the intensity of light that has been absorbed by component a a t any distance 1, is given by
Observe that under the condition that cb = 0, eqn. (6)reduces to the usual expression for a single component system. In exactly the same manner the intensity absorbed by component h is found to be
I t follows, therefore, that the ratio of the intensities absorbed by components a and b is given exactly by
-
Notice that the intensity is a function of the position 1, along the optical path and 10) represents the intensity a t any position 1 after considering absorption by all absorbing components of the system. The expression for I a t any distance 1 in a mnlticomponent system has already been given in eqn. (1).Insertion of this expression for I(1) for the case of a two component system leads to
'To whom enquiries should he addressed. Loucks, L. F., and Cvetanovie, R. J., J Chem. Phys., 57,1682
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11972). -, Noyes, W. A., Jr., and Leighton, P. A,, "The Photochemistry of Gases," Dover, New York, 1966,p. 152. Steinfeld,J., Aects. Chem. Res., 3,313 (1970). 652 / Journal of Chemical Education
The second derivation, perhaps more modern in apnroach. treats absorntion as a random orocess and the ireatment is inspiredby Steinfeld's derivaiion of the SternVolmer eauation bv stochastic methods4 The oresent treatment 'is in f a c t formally identical to ~teinfeid'shut with the difference that the present case is not a truly stochastic process. Current usage limits the word stochastic to processes that are random in time and in the present case the process is random in space. Conceotuallv the method mav be stated quite simply: As passes through an absorbing mkdinm, a beam bf the number of photons absorbed hy component a is the result of absorpcon over the complete distribution of path distances, reduced by the probability that the photons have been absorbed by component b. First, consider only absorption by component a: The intensity of the radiation not absorbed by component a is given for any distance along the absorption path by the expression
At this stage only the absorption by component a is being considered because the probability of absorption by component b will be considered later in the derivation. Per unit length the change in the intensity resulting from absorption by component a, of photons that have reached a par-
ticular position 1 = L along the absorption path, is given by Therefore Now, the probability that over the distance 1 = 0 t o 1 = L a photon will not have been absorbed by component b may be expressed as 1 - P*,b (I < L). The probability that a photon will be absorbed by component b over this distance can be derived from absorption expressions for a system containing only b, and thus P&h
=
I& d l
Thus, P s b . b ( 1 < L ) = I. ( I (-tbcd.) Hence,
