Spatial Inhomogeneity Effects in Photochemical Kinetics S. R. Logan University of Ulster at Coleraine, N. Ireland BT52 1SA In a ohotochemical reaction, if a substantial proportion of the iniident light is absorbed within the reacGon>ell, then the rate of lieht absor~tion(in quanta per unit volume per unit time) m i s t vary considerably with position inside the cell, leading to appreciable spatial inhomogeneity of the initial light-induced process. Having regard to the increasing use of light absorption to initiate chemical reactions, it is the nurDose of this . Daoer . to look a t the fundamental factors involved in the kinetics of such reactions, recognizing the effects of the nonuniformitv that will almost invariablv be present. t by the ahsorpWhereachemical reaction is b r o u ~ habout tion of light,any consideration of t h i kinetics Gill require an exoression for the rate of the photochemical initiation step. his can he simply expressedas the product of +, the nu& her of molecules dissociated, or otherwise caused to react, per quantum of light absorhed, and l a b , the number of Einsteins of light absorhed per unit volume per unit time. Since + is also the number of moles dissociated per Einstein of light absorbed, the product +Iabs has the dimensions of moles oer unit volume oer unit time, which are those of a reaction rate as i t is usually expressed. The ouantitv I.&. which denotes the rate of absorption of light, may weli n i t h e constant throughout the solution being illuminated. Referring to Figure 1, which depicts a beam of monochromatic light traversing an aperture of area 1 cm2 and impinging a t normal incidence on the face of a cell of path length 1 cm, we will assume that the flux of light a t the front face of the cell is Q Einsteins per second. In addition, we shall assume that within the cell the light is absorbed hy one species only, of concentration c and decadic absorption coefficient r At any distance x 5 1 from the front face of the cell, the rate of light absorption will be given by:
Photochemical Reaction Rate Let us suppose that the rate of the photochemically initiated reaction occurring in the cell, to give a product that is totally transparent a t the wavelength of irradiation, depends on the rate of initiation t o the power n.
. .
= 2.303ccQ. 10-"" Einstein cm-3 s-' = 2.303 X 10beQ. lo-'" Einstein L-' S-I
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light Area
= 1 c r,J2n
-Lcm-
Figure 1. Dlagrarn of s parallel beamof light passing througha defined opening of area 1 cm2 and entering a cell of path length I. .
(1)
The variation of Isbs with position and with the concentration of c is illustrated in Figure 2. If rcl > 1, then Iabs varies enormously across the cell, and virtually all the light is absorbed within it. This makes the point that, if most of the light is absorbed, the rate of light absorption will necessarily show considerable variation with position; if 90% is absorbed, then it varies by a factor of 10. Figure 2 also demonstrates that for a finite path length, Isba shows interesting variations with concentration. By differentiatine ea 1 with respect to c, one can show that for a specified 6 t h length 1 the maximum value of lab, occurs when 2.303rcl= 1,or when the ahsorhance A = rcl = 0.4343. The foregoing assumes monochromatic light, implying a single value of r. Where the incident light is polychromatic, the situation becomes more complex, but the variations in the rate of light ahsorption with position become even greater. We- -shall our attentions to~monochromatic lieht - - - ~ confine ~ ~ ~ purely because of its relative simplicity. ~
monochromatic
Figure 2. Plots 01 lab against x f w indicated values of the absorbance A = eel. illustratlno " how I.*. varies wrth concentration and with oositian in the cell. Thssecuwes relale to 0 = 10-8Elnste n cm ' s 'and I = 8 cm. tor valasof me absoroance A = m a s shown.
In most steady-illumination experiments, because of the techniques such as dilatometw used to monitor the reaction, the relevant parameter is thk total amount of product P formed during a certain period of illumination, rather than the instantaneous rate a t any particular point within the cell. Summing over the whole cell we obtain, for the number of moles of P formed in unit time,
where the Iabs figure relates t o the number of Einsteins per liter per second and the factor of arises because the cell volume is 1cm3 = 10-3 1L. Thus we have:
specification are those in which a free radical chain reaction is initiated photochemically, and this group includes polymerization Drocesses. A reneral characteristic of chain reactions in which the chainlends by mutual termination is that a lthe rate of the initiation the reaction rate is ~ r o ~ o r t i o nto stage to the power lj2. If we assume that, in solutions containing the monomer M, the initiator A is theonly species that absorbs themonochromatic incident light, and that @, assumed independent of the concentrations of A and M, represents the quantum efficiency for the formation of m,., the growingpolymer chain incorporating one monomer unit, then the rate of initiationat any point will he given by 6Iab. The subsequent reactions of the growing polymer chain, mi., are the well-known (1) propagation and termination steps, which may be written as:
my + m i k=dead polymer For the simple case where the index n and the proportionality constant x are both unity, eq 3 simplifies to: which shows that the amount of product P formed per unit time is simply @ times the number of Einsteins of light absorbed within the cell. In this instance, regardless of whether rcl >> or = or > 1, H W S is then independent of c. ~t low values of c, where rcl
