GUEST AUTHORS Henry Shaw and Sidney Toby Rutgers University N e w Brunswick, N e w Jersey
Textbook Errors, 70
Light Absorption in Photochemistry
N o t many years ago photochemistry was a small suhdivision of physical chemistry. Within the last decade, however, the importance of organic photochemistry has increased immensely and it is now very difficult (and probably meaningless) to decide whether some aspects of photochemistry are "organic" or "physical." As a result of many researchers' efforts, there now exists an imposing body of knowledge on the chemistry of the triplet and singlet excited states. 'k impnnanre of ~pevtro.w)~y and qurinrum r~l(~(:hnni in forminu thr fmmeu.olk oi ~~hotochemiwv is unquestionable. Many of the elementary texts, however, ignore some of the basic physics involved in the prncess we call light absorption. The purpose of this article is to review some of the basic concepts and to point out an inconsistency common to many textbooks. '
-
Manuscripts for this column, or suggwtions of material suitable for it, are rarely solicited. These should be sent with as many details as possible, and particdarly with references to modern textbooks, to W. H. Eherhardt, School of Chemistry, Georgia. Institute of Technology, Atlanta, Georgia 30332. Since the purpose of this column is to prevent the spread and continuetion of errors and not the evaluation of individual texts, the source of errors discu.sed will not be cited. To be presented, the error must occur in a t least two independent standard books.
408
/
Journal of Chernicol Education
The Electromagnetic Wave
Photochemistry is the study of the effects of light on chemical systems. The fundamental principles of light propagation and absorption are determined from electromagnetic theory. James Clerk Maxwell (183179) formulated the basic four equations which completely determine the electromagnetic field. Maxwell's equations lead directly to the equation for an electromagnetic wave (Appendix A). The flow of energy associated with the electromagnetic field of a monochromatic plane wave can be obtained from Poynting's theorem (Appendix A)
where S represents the energy flux along the xdirection for an electric field E and a magnetic field H and c is the velocity of the wave. We define n as a unit vector in the direction of light propagation so that Substituting eqn. ( 2 ) into eqn. (1) and noting that E.n = 0 we obtain
Defining p as energy density we can write?
from x = 0 where the intensity is I'owe obtain the well known Beer-Lambert relation
Since the field is propagated with the velocity of light
The only dimensional restriction for eqn. (9) is that I' (or I)must have the same units as IJo (or Io). The primary photochemical process
S = cpn
(5)
Eq. (3) leads to the important result that energy flux = energy density,velocity. Let us define light intensity, 1', as the quantity of electromagnetic energy impinging on unit area in unit time. Therefore, light inten-itg is simply an energy flux and can be equated to the time average of Foynting's vector
-
S = I' = epn
(6)
We ran drop the vector notation since the intensity is understood to be in the direction of light propagation. We should also recall that we are dealing with monochromatic light of frequency v and write I' = INh"
(7)
where X is Avogadro's number. The units of I' are the same as those of S, ergs/cm2-sec. Most photochemists prefer to use I as defined by (7) in einsteins/em2-sec because these units provide a direct correlation vith the Einstein law of photochemical equivalence: one einstein of energy Nhv absorbed will activate one n1ole.s The Absorption of Ligh!
I n n photochen~icalexperiment one generally uses a collinmted light beam of intensity 1'. The energy passing through surface Z (see figure) in the frequency
Figure 1.
+
range v to v dv is I'Zdv or in terms of the radiation density pcZdu. When an atom or molecule A absorbs light it will be excited from its ground state to an excited state A*. This excitation refers to the change in internal energy AU. The frequency of light absorbed is v = AU/h. The nbsorption coefficient per mole of A for light frequency v is denoted as a("). Let I,' be the inten. sity a t n plane at x for a beam of light moving in the x direction. The number of moles in the ground state dx is [ A ] Z dx (note: units of [A] between .c and x are moles cnlr3). The light absorbed per second in Zdx is proportional to the absorption coefficient per mole, the number of moles, and the incident energy.
+
-dl'Zdu
=
I'dv[A]n(u)Zdz
(8)
The frequency interval dv can be omitted if we assume the radiation is monochromatic. Integrating eqn. (8) ' L a r ~ a u ,L., A N D L I F S ~ I TE., Z , "The Classicd Theory of Fields," -4ddison-Wesley Pnhlishing Ca., Reading, Maw., 1951, Chap. 6 . Exceptions have appeared. See OGRYZLO,E. A,, J. C H E M . Eorr., 42, 647 (196.5).
A
+ h"-.A*
can be treated as a bimolecular reaction of a photon with species A. -d[A]/dt = @(All
(10)
where p represents an absorption cross-section for an einstein of radiation. Solving eqns. (8) and (10) simultaneously where 4 represents the primary quantum yield. Differentiating eqn. (9) and substit,uting into eqn. (11) one obtains for the local rate of reaction d [ A ] / d t = -,,+I,[A]n(v)e-lAI=(4z
(12)
If we assume that we have a uniform concentration in the photolysis cell then we can average the reaction rate over the length of the cell, L:
LL LL
(d[A],dt)dz
average rate of reaction -
-
h:
-LAI=b)z ,Aln(v)dz
dz
=
+Io(l - e-=(u)lAlL)/L =
S:
dz
,,+is
(13)
Equation (13) holds for a photolysis cell of slab geometry. More complicated expressions are obtained when the light intensity is averaged over cells of other geometries.' The assumption of uniform reactant concentration is satisfactory for most photochemical systems. Experiments are generally designed so as to have sufficient diffusion or convection t,o assure uniform concentration throughout the photolysis cell. Noyes and Leight o n v u o t e experimental work where 50yQof the incident energy was absorbed. Kevertheless, the local and average rates still did not differ appreciably. It should be emphasized that the units of fa are einsteins sec-'. The quantity usually defined by I. = lo- I and called "absorbed intensity" is in flux units, energy or einsteins sec-' and is clearly not immediately relatable to the kinetics of the reaction. The quantity I. involves the cell geometry and is not in the appropriate dimensions. In photochemical kinetic expressions one should always use fa,the intensity averaged ouer the photolysis cell and defined by equation (18). The rate of a Primary Process
A typical photoinitiated mechanism begins with the primary step A
+ hv-2R
(a)
'HUFF,J. E., A N D WALKER, C . A,, J . Am. Inst. Chem. E m. .. , 8.. 193 (1962). NOYES, W . A., JR.,A N D LEIGUTON, P . A,, "The Photochemist r y of Gases," Reinhold Pnblishing Corp., New York, 1941. Volume 43, Number 8, August 1966
/
409
and for simplicity a primary quantum yield of unity (units moles einstein-') will be a ~ s u m e d . ~The rate of disappearance of species A is commonly expressed in three different ways: -d[A]/dt
=
klI.
(1h)
=
&[A1
(14b)
=
I.
(14~)
Equation (14a) is unsatisfactory for the following reasons. If I., like incident intensity 10,has dimensions of einsteins per area-time, then k, would have dimensions of length-' which makes little physical sense. If, on the other hand, I, is taken to mean then the kl is redundant. The weakness of equation (14b) becomes apparent as soon as we introduce a secondary process such as R
+A
-
products
(b)
Then -d[A]/dt = kz[A] f k8[R][A]; ka is a rate constant, dependent on temperature but independent of concentration. However, kz corresponds to no recognizable property: it is virtually independent of temperature but dependent on concentration (except for very low optical densities). We conclude that the best way of describing the rate of disappearance of species A in step (a) is simply -d[A]/dt = Te where it is understood that Ta is an averaged absorbed intensity with dimensions of einsteins per volume-time. I n the general case Ta should be multiplied by the quantum yield.
in the appropriate units. Equation (6) is equally true if we use H instead oi E. Let us consider the special case of plane waves. These are electromagnetic waves in which the field depends only on one coordinate. say x
This equation can be solved exactly when suitable boundary conditions are imposed. We will simplify eqn. (7) further by imposing the conditions for monochromatic plane waves of frequency v and obtain the easily verifiable solution Equations (7) and (8) are equally correct for the magnetic field, H, but eqn. (I) or (3) requires the direction of the two vectors to be diierent. They are mutually perpendicular and are both perpendicular to the direction of propagation as implied by the equation The Poynting Vector
Poynting's Vector is defined as
and, as shown in the text,, is just the energy flux in the direction of propagation, n. For the solutions t o R'laxwell's Equations represented by eqn. (8) above,
Appendix A MaxweN's equations and the electromagnetic wave
and the time-average of S is just
Maxwell's equations for a vacuum are:
div H = 0
(2)
1 aE eurl H = - e a1
(3)
div E = 0
Clcluical Electromagnetic Theory
(4)
for an electric field E and a magnetic field H. Taking the curl of equation (1) and substituting equation (3) into it: curl curl E
=
- e21 ats -
Substituting equations (4) and (5) in the vector identity for curl curl E we obtain the (D'Alembert) wave equation
The magnetic field H a t any point in the wave is equal to the electric field E provided they are expressed
'Distinotion should he made csseiully between primary . and over-dl quantum yield a. For s. full quantum yield $ J . G., discussion see NOYES,W. A,, JR., PORTER,G., AND JOLLEY, Chem. Rm., 56, 49 (1956).
410 / journal of Chemical Educafion
Bibliography
SEARS, FRANCIS W., "Electricity and Magnetism," AddisonWesley Puhlishing Co., Reading, Mass., 1951, Chap 17. J. C., AND FRANK, N. H., c'Electr~m&gnet~$m," jMcGrawSLATER, Hill Book Company, Inc., New York, 1949, Chap. 1JIII. LANDAU, L., AND LIPSHITZ,E., "The Classical Theory of Fields," Addison-Wesley Publishing Co., Reading, Mass., 19:il. Chau.
.
ti.
MITCHELL. A,. AND ZEMANSRT. 31. W.. "ResonanceRadiation and ~ x e i t e d~' & m s , " cambridge ~ n k e r s i t yPress, New York, 1934, pp. 06101. The Absorption of Light
DAVIDSON,N., 'iStatisti~sl Nechanic..," MeCraw-Hill Book Company, Inc., New York, 1962, Chap. 12. KAUZM.ANN, W., "Quantum Chemistry," Academic Press, Inc., New York, 1957, Chap. 15. E. A,, "Physical Chemistry," 2nd ed., MOELWYN-HUGHES, Pergaman Press, London, 1961, pp. 1216-25. Photochemistry
NOYES, W. A,, JR., AND LEIGHTON, P. A,, "The Photochemistry of Gases." Reinhold Puhlishine CO~D.. .. New York. 1941. DOKEN,E. J., "Chemicd Aspects of Light," Oxford University Press, New York, 1946. C.
