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(9)R. J. Richardson, H. T. Powell, and J. D. Kelley, J. Phys. Chem., 77, 2601 (1973). (IO) A. A. Westenberg and N. de Haas, J. Chem. Phys., 46,490(1967). (11) A. G. Gaydon and H. G. Wolfhard. "Flames", Chapman and Hall, London, 1970,p 55. (12)W. H. Breckenridge and T. A. Miller, Chem. Phys. Lett., 12,437 (1972). (13)G. Hancock, C. Morley, and I. W. M. Smith, Chem. Phys. Lett., 12, 193
(1971). (14)H. T. Powell and J. D. Kelley, J. Chem. Phys., 60, 2191 (1974). (15)A. Tewarson and H. E. Palmer, Symp. (lntl.) Combust., [Proc.], 13th, 1970, 99 (1971). (16)G. W. Taylor, J. Phys. Chem., 77, 124 (1973). (17)See, for example, ref 3. (18)H. S.Pillof, S. K. Searles, and N. Djeu, Appl. Phys. Lett., 19,9 (1971).
An Equation Describing the Rate of the Photochemical Reaction of a Bulk Powdered Sample E. L. Slmmons* Department of Chemistry, University of Natal, Durban, South Africa
and W. W. Wendlandt Department of Chemistry, University of Houston, Houston, Texas 77004 (Received May 3 1, 1974: Revised Manuscript Received October IO, 1974) Publication costs assisted by the University of Natal
An equation is derived which describes the rate of the photochemical reaction of an infinitely thick powdered sample. The equation is expressed in terms of the sample diffuse reflectance which is the most conveniently measured parameter for following the extent of the photochemical reaction of a powder. The equation is applied to experimental results for the photochemical reaction of powdered K3[Mn(C204)3] 3H20.
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Introduction A large number of solid-state photochemical reactions have been studied and the studies have been reviewed.l Most of the studies, however, have been concerned with the determination of reaction stoichiometries. Few quantum yield determinations have been made. The major reason for this is that the study of solid-state photochemicd reactions is complicated by the fact that the diffusion of solid photoproducts is restricted and hence a concentration gradient is created in the sample by the photochemical reaction. Equations have been derived which describe the rate of the photochemical reaction of a sample with slab geomet r ~ . These ~ - ~ equations appear to be applicable for the determination of quantum yield values of solid-state photochemical reactions. Equations have also been derived which describe the rate of the photochemical reaction of a thin powdered l a ~ e r . ~Most - ~ solid samples, however, are most conveniently handled in the form of bulk powdered samples. As yet, the most promising method of studying photochemical reactions of bulk powdered samples appears to be reflectance spectroscopy. The technique has been applied to study a number of photochemical reactions in a qualitative manner over the past few years.8-12 The theory of reflectance spectroscopy, however, has not been sufficiently developed to allow calculations of quantum yield values from reflectance measurements of powdered samples undergoing photochemical reactions. Recently, a model representing a powdered sample as a collection of uniformly sized rough-surfaced spherical particles has been used to relate the reflectance of a bulk powThe Journal of Physical Chemistry, Vol. 79, No. 12, 1975
dered sample to its fundamental optical p a r a m e t e r ~ . l 3 In -~~ this investigation, the same model is used to derive an equation describing the rate of the photochemical reaction of a bulk powdered sample in terms of the sample reflectance as a function of time. Although several approximations are made in the derivation, the equation appears to have promise as a means of estimating quantum yield values of solid-state photochemical reactions from reflectance measurements. The equation was applied to experimental results for the photochemical reaction of K3[Mn(Cz04)31.3Hz0.
Phenomenological Description Consider a powdered sample which can be considered as infinitely thick made up of substance, C, which reacts photochemically to give product, P. The sample is illuminated with monochromatic radiation of intensity, Io, at the sample surface. The wavelength of the radiation is considered small compared with the diameters of the sample particles so that Rayleigh scattering can be ignored. Consider the ith particle located below the sample surface. The radiation intensity impinging on the ith particle from the upward direction is I , while that impinging on it from the downward direction is J L + l .Hence, the rate of change of the number of moles, Ni, of C in the ith particle due to photochemical reaction is given by
where A,i is the fraction of the impinging radiation absorbed by C in the ith particle, 4 the quantum yield, and ( I ;
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Photochemical Reaction of a Bulk Powdered Sample
+
J,+1) r d 1 2 / 4the total number of einsteins of radiation impinging on the ith particle in unit time. Since the concentration, Ci, of C in the ith particle is given by
C, = [ C O / ( A c- A,)](l - A, - 2Ri/(l
(3)
It has previously been shown7J4 that A,, is given by Aci = A,Ci/Co = (2/3)n2kdiCi/C,
( 5)
dCi/dt = Q
