J. Phys. Ckem. 1985,89, 1054-1058
1054
Criteria for the Absence of Thermal Convection in Photochemical Systems Arne J. Pearlstein Aerospace and Mechanical Engineering Department, University of Arizona, Tucson, Arizona 85721 (Received: June 15, 1984)
The problem of avoiding thermal convection in a photochemically reacting fluid is considered. On the basis of an understanding of the light absorption, heat generation and transfer, and fluid mechanical processes occurring in a light-absorbing fluid, design criteria are developed that are sufficient to ensure the absence of thermal convection in photochemical experiments. The reaction cell should have a horizontal top and bottom and vertical, insulating side walls. The light entering the cell should be directed either vertically upward or downward and should fill the entire cross section of the cell. A simple equation is developed which gives the minimum temperature difference that must be imposed between the top and bottom boundaries in order to ensure a gravitationally stable density distribution. It is shown that the required temperature difference will be smaller if the illumination is from the bottom rather than the top. Finally, the results are applied to a recent experiment, showing that the criteria can easily be satisfied in practice.
I. Introduction Recently, considerable interest'-5 has been directed to the role of thermal convection in the temporal oscillations observed experimentally in various photochemically reacting and other light-absorbing fluids. In some of these systems, the temporal oscillations are thought to result from the fluid motion',* rather than directly from oscillatory processes intrinsic to the chemical kinetics or light absorption and emission events. Furthermore, it has been pointed o ~ tthat ~ *very ~ small temperature differences are sufficient to drive the convective motion. The purpose of the present paper is to present definite criteria that will be useful in the design of photochemical experiments in which the absence of thermal convection can be assured, thus allowing the underlying kinetic rate processes (and their possible interaction with molecular diffusion) to be. studied in the absence of fluid motion. 11. Preliminary Considerations The basic driving force for thermal convection in photochemical systems is the existence of a temperature gradient, and hence a density distribution, which is not compatible with a quiescent fluid. The physical processes involved are well-known and will be discussed only briefly here in connection with the development of our physical and mathematical model. We consider a light-absorbing Newtonian fluid in a container. Light enters from the outside through one or more transmitting boundaries and is attenuated as it propagates through the fluid. We will suppose that all of the attenuation results from absorption and that scattering is unimportant. We also neglect any contribution to the intensity that arises from emission within the fluid. The radiant energy that is absorbed may be converted or degraded in various ways. In the event that emission (fluorescence, phosphorescence) occurs, the emitted quanta will be at a longer wavelength (lower energy) than the absorbed quanta, with the remaining energy being transferred into vibrational, rotational, and translational modes. In the case of internal conversion or collisional deactivation, all of the absorbed energy is converted to heat. If photochemical reaction occurs, the thermal energy absorbed by the fluid will be the energy of the photon minus the enthalpy change (per molecule) resulting from reaction. Thus, for an endothermic reaction, some (or all) of the photon's energy will be applied to an enthalpy change, and the remainder will be dissipated via a vibrationally (or otherwise) excited product molecule. We suppose that, regardless of the mechanism by which the absorbed quanta are in part or in whole degraded to heat, the (1) Laplante, J. P.; Pottier, R. J. J . Phys. Chem. 1982, 86, 4759-66. (2) Epstein, I. R.; Morgan, M.; Steel, C.; Valdes-Aguilera, 0. J . Phys. Chem. 1983, 87, 3955-8. (3) Epstein, I. R. J . Phys. Chem. 1984, 88, 187-98. (4) Epstein, I. R. Physica D (Amsterdam) 1983, 7 0 , 47-56. (5) Zimmermann, E. C.; Ross, J. J . Chem. Phys. 1984, 80, 720-9
0022-3654/85/208~-1054$01 .50/0
degradation occurs at the point of absorption. This is equivalent where L is a characteristic length to assuming that L >> r ( c 2 )'Iz, (e.g., cell size), 7 is a characteristic time associated with the thermal dissipation of an absorbed photon's energy, and (c,2)'I2 is a root-mean-square velocity of an excited species. Thus, the distance traveled by an excited molecule must be small in comparison to the dimensions of the container. W e shall further assume that the fluid density depends only on the local temperature, and that this dependence is monotonic. The first assumption is made to eliminate "doubly diffusive" effects6 that may arise in binary (or multicomponent) fluids when the density also depends on composition. This does not appear to be a serious restriction in the systems studied e~perimentally.'~~~~ The monotonicity assumption has as its only significant consequence the elimination from consideration of aqueous solutions when the temperature at which the density is a maximum (3.98 O C for pure water) is included within the temperature range of interest. Finally, we will limit consideration to steady and quasi-steady systems, in which the rates of light absorption and heat generation do not vary rapidly in time.
111. Direction of Illumination A necessary condition for the fluid to be at rest is7 0 = -vp + pg where p is the (hydrostatic) pressure,
p
is the local density, and
g is the gravitational acceleration. Taking the curl, we obtain
o=gxvp (1) so that the density gradient must be parallel or antiparallel to the (vertical) gravitational field. Thus, in a fluid for which p depends only on T , (1) requires that there be no horizontal temperature variation. But the temperature distributions in a motionless medium with constant thermal conductivity k must under steady conditions satisfys
+
0 = k(d2T/dx2 dzT/dyz
+ d 2 T / d z z )+ Q(x,y,z)
(2)
where Q(x,y,z)is the rate at which heat is produced in the fluid by the processes following light absorption. Thus, if we take the vertical direction to coincide with the z axis, the existence of a motionless state and the resulting requirement of no horizontal temperature variation lead to the conclusion that Q cannot depend on the horizontal coordinates x and y . The heat source term can be expressed as9 o ( x , y , z ) = - V . l S a ( h ) l Z(x,y,z,h$)Q dQ dh 0
4*
(3)
(6) Turner, J. S. "Buoyancy Effects in Fluids"; Cambridge University Press, 1973; Chapter 8. (7) Batchelor, G. K. "An Introduction to Fluid Dynamics"; Cambridge University Press, 1970; p 19. (8) Benson, S. W. 'The Foundations of Chemical Kinetics"; McGraw-Hill: New York, 1960; pp 427-8.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 6,1985 1055
Thermal Convection in Photochemical Systems where X is the wavelength, n is the direction of propagation, Z(x,y,z,X$) is the radiation intensity (quanta per unit area per unit time), and M
Here, aj and S,(X) are, respectively, the quantum yield of thejth overall process and the amount of thermal energy produced (per quantum absorbed) by the j t h process. If the three processes considered are luminescence, reaction, and nonradiative decay (including internal conversion and collisional deactivation), then S l u m = hcd 1/ h , b - 1/h,,,,,), ,S = hcI/X,ps - AHIN, and S n r d = hcI/Xab,where cIis the speed of light, h is Planck‘s constant, N is Avogadro’s number, Xabs and Alum are the wavelengths of the absorbed and emitted quanta, and AH is the molar enthalpy change associated with reaction. If more than one reaction occurs, then a(X) will contain contributions from each. In any case, a(X) can be interpreted as the average thermal energy dissipated per quantum of wavelength X absorbed. Equation 3 can then be rewritten as
In the absence of scattering, we have
v m , Y , z , x , Q ) = -n €(A)
C(X,Y,Z)
I(x,y,z,X,Q)
(5)
where €(A) is the molar extinction coefficient and c(x,y,z) is the concentration of the single absorbing species. If there are several absorbing species, the right-hand side of ( 5 ) will require a summation. Equation 5 is simply a multidimensional version of the Beer-Lambert law and is subject to the same limitations (no multiphoton absorption, no saturation of the absorbing species, etc.). Substitution of ( 5 ) into (4) yields
From this it follows that if Q depends only on z, then I and c must not depend on x and y . If I and c are independent of x and y , then the left-hand side of ( 5 ) has only a z component. Thus, it follows from ( 5 ) that must have only a z component for any wavelength that is absorbed, from which we conclude that the light must enter the container from above or below, be parallel or antiparallel to the gravity vector (Le., vertical), and be horizontally uniform across the cell. If these conditions are not satisfied, there can be no motionless steady state, regardless of the viscosity of the fluid or the weakness of the light absorption. The magnitudes of the induced convective velocities will, however, depend on the details of the situation, such as the angle of propagation and intensity of the light, optical density, cell size, and the thermophysical properties of the fluid (viscosity, thermal conductivity, thermal expansivity, and heat capacity). The dependence of the light intensity (and hence Q) on only the vertical coordinate is not, however, a sufficient condition to ensure the existence of a motionless steady-state temperature distribution T(x,y,z). In addition, we require that (a) the side walls be vertical and insulating, i.e. aT/ax = aT/ay = o at the lateral boundaries; (b) the top and bottom of the cell be horizontal; and (c) there be no lateral temperature variations on the top and bottom of the cell. Conditions b and c are easily satisfied, and a can be approximated to an excellent degree by constructing the side walls of a material with a low thermal conductivity and making them very thick. If the light beam is vertical and the bounding surfaces satisfy a-c above, there will exist a motionless steady solution of the full equations. In that case, the presence or absence of thermal convection in the fluid will depend on the hydrodynamic stability (9) Bandini, E.; Stramigioli, C.; Santarelli, F. Chem. Eng. Sei. 1977, 32, 89-96.
of the density distribution associated with the purely vertical temperature gradient that results from light absorption.
IV. Stability of the Motionless Solution The motionless rest state is one solution of the full equations of motion but, as the full equations are nonlinear, the rest solution is not necessarily unique. Under some conditions, the equations may also admit one or more “convecting” solutions as well. The stability of the rest state with respect to other possible solutions can be examined in several different ways. One such way is by use of linear stability theory,’O which provides, as does eq 1, necessary conditions for the physical realizability of the rest state. The linear stability of a light-absorbing fluid layer illuminated from aboue has been previously considered by Yucel and Bayazitoglu,” who considered a horizontally infinite fluid layer (no side walls) illuminated from above. They computed, in a dimensionless form, the smallest adverse temperature difference that would be dynamically unstable with respect to one class of infinitesimally small disturbances. Unfortunately, this work is not suitable for design purposes for three reasons. First, the analysis in ref 11 is for a layer of infinite horizontal extent (with no side walls), whereas the onset of thermal convection in finite containers for various temperature distributions is known to differ, both qualitatively and quantitatively, from the horizontally unbounded case.’* Second, Yucel and Bayazitoglu only considered stability with respect to infinitesimally small disturbances and, furthermore, restricted the onset of motion to be via steady convection, rather than via time-dependent motion. Thus, the conditions obtained are necessary, but not sufficient, for the absence of convection in an infinite horizontal layer. The restriction to infinitesimally small disturbances is an important limitation, in light of the theoretical work of Tveitereid,” who has shown that, in a horizontally infinite layer with uniform (no vertical variation) internal heating, subcritical instability occurs. That is, although the layer may be stable with respect to infinitesimally small perturbations, it may be unstable with respect to larger disturbances. Third, the analysis in ref 11 is restricted to a layer illuminated from above, and, as we shall show, this orientation of the incident beam is less advantageous than illumination from below. A second way to examine the stability of the rest state is by use of energy theory,14 which provides sufficient conditions for stability. Unfortunately, the predictions of energy theory are almost always overly con~ervative.’~When the criteria of linear theory and energy theory coincide, then that criterion is both necessary and sufficient for stability. This occurs only rarely, a notable example being the classical Rayleigh-BBnard problem for a fluid layer heated from below by its lower boundary. In general, however, such agreement does not occur, and the energy theory can give results that differ substantially from experiment. For these reasons, we will develop and present, in addition to the necessary conditions for the existence of a motionless steady state developed in section 111, criteria that are sufficient to guarantee the stability, and hence the physical realizability, of this motionless state. These sufficient conditions are obtained much more simply than the sufficient conditions of energy theory, with which they probably share the disadvantage of not being necessary and sufficient. Typically, our conditions will be more conservative than those of the energy theory, but given their simplicity and attainability in real experiments (see the example in section V), it is thought that they will be of considerable utility ’ to those involved in the planning of experimental work. We consider a reaction cell with interior height Lf and transparent top and bottom horizontal walls of thickness L, and &, respectively, a side view of which is shown in Figure 1. The (10) Chandrasekhar, S. “Hydrodynamic and Hydromagnetic Stability”; Oxford University Press, 1961. (11) Yucel, A.; Bayazitoglu, Y. J. Heat Transfer 1979, 101, 666-71. (12) Catton, I. Proc. 6th Intl. Heat Transfer Con$ 1978, 6, 13-31. (13) Tveitereid, M. Int. J. Heat Mass Trans. 1978, 21, 335-9. (14) Joseph, D. D. “Stability of Fluid Motions I”; Springer-Verlag: New York, 1976.
1056 The Journal of Physical Chemistry, Vol. 89, No. 6, 1985
- Z" -L,
1 Tf(Lf) = Tb(Lf) dTf(Lf) k
f
T
=k
dTb(Lf) b 7
(7e) (7f)
Equations 7a-b account for heat transfer between the environment and the extemal horizontal boundaries of the cell. The coefficients h, and hb are the (convective) heat transfer coeflcients that obtain for the loss of heat from the cell to the surroundings. In the event that there is a vigorous air flow (forced or free convection) at these horizontal ends, then the wall temperature at z = -4 (or z = Lf 4 ) will very nearly be To(or TI). The boundary conditions (7c-f) simply express the continuity of temperature and heat flux at the fluid-solid boundaries. W e nondimensionalize (6a-2, 7a-f) by introducing 9 = Z/Lf Ti = To + 6&dZt + Ib)a/kf i t, f, b
fluid
+
bottom
7 =
I- F L f tL, Figure 1. Schematic drawing of the photolysis cell showing the orientation with respect to gravity.
horizontal cross section may be rectangular, circular, or have any other shape subject to the conditions that the cross section is constant (vertical side walls) and that the incident light beam completely fills the cross section. Satisfaction of the latter is required in order that there be no horizontal variation in the absorbed radiation. We suppose that the cell is filled with a fluid containing a light-absorbing species with concentration co and molar extinction coefficient e, and that the fluid has a constant thermal conductivity kEwhich is independent of temperature. We further suppose that the incident radiation is monochromatic or, equivalently, that t and u are constant over the range of wavelengths included in the incident beam. In the usual case when the density of the fluid decreases monotonically with increasing temperature, a sufficient condition for hydrodynamic stability is that the temperature increase monotonically as one moves upward in the layer. Such a condition is not necessary for stability, but the fact that it is sufficient, as well as relatively easy to implement, makes it a desirable design criterion. In order to ensure that the temperature gradient is always upward, we begin with the equations governing the steady temperature distribution in the fluid and the top and bottom walls
0 = k, dZT,/dz2
+
(6a)
0 = kf(d2Tf/dz2) utco[Zte-fc@+ Zbe-cco(Lfz)]
(6b)
0 = kb d2Tb/dz2
(6c)
which follow from (2). Here, z is the vertical coordinate measured downward, as shown in Figure 1. To avoid duplication, we simultaneously consider irradiation from above and below. At the conclusion of the analysis, we will specialize to the cases of upward (Z, = 0) and downward (1, = 0) beams. We have included thermal conduction in the top and bottom walls for the sake of generality; as we shall see, the results are relatively insensitive to the thickness and thermal properties of the horizontal walls for typical situations of interest. In addition to (6a-c), we will need boundary conditions at the top and bottom
and at the fluid-solid boundaries
ecOLf
fit = Lt/Lf fib = Lb/Lf = kf/k, b b = kf/kb
Bit = h,Lf/k,
Bib = h&/kb
Here, y is the optical density, Bit and Bib are the Biot numbers at the top and bottom ends, and 6 is the fraction of the incident light that is directed downward. We will be primarily interested in the cases 6 = 0 and 6 = 1. The dimensionless versions of (6a-2,
The Journal of Physical Chemistry, Vol. 89, No. 6, 1985 1057
Thermal Convection in Photochemical Systems
I .o
1- - - ’ °1 c 100
0.8
10-4
o’2
IO-^
to-2
IO-’
IO0
IO‘
IO2
Figure 2. Minimum required dimensionless temperature difference R,, as a function of optical density y for upward illumination (6 = 0).
In order that the temperature be a monotonically decreasing function of 7,we require that dur(q)/dq = a 2 / u t
+ (1 - 6)eT - 6 + 6e7n - (1 - 6)e7(V1) (11)
be less than zero for all q between 0 and 1. Setting r = e’? and s = (1 - 6)eT, we obtain from (1 1) the condition
O
< s?
+ (6 - s - a2/ut)r - 6
(12)
which is to be satisfied for 1 Ir e?‘ 5 e?. Inequality (12) will be satisfied for all r in the indicated range if a2/ut< 0, from which it follows [using (lo)] that
Thus, for a given cell and absorbing fluid, (13) is an inequality involving the overall temperature difference To - TI across the cell, and the intensity and direction of the incident radiation. Any A greater than R will guarantee stability, but we would like to ensure the monotonicity of the temperature distribution in the fluid with the smallest topto-bottom temperature difference consistent with (13). The desirability of operating with the smallest temperature difference consistent with monotonicity is due to the fact that large vertical temperature variations may give rise to a number of effects which can complicate the interpretation of experimental data. Among the undesirable consequences are the thermal variation of molecular diffusion coefficient^,'^*'^ as well as kinetic rate constants for the primary photoprocesses and subsequent reaction^,"*'^ which in turn may lead to the falsification of kinetic parameters and misinterpretation of the reaction mechanism,lg or variation in the product distributionn20 It can easily be shown that the right-hand side of (13) is, for 0 I6 I1, a positive and monotonically increasing function of 6, from which it follows that the choice 6 = 0 will minimize the dimensionless temperature difference A. (1 5) Christie, M. I.; Roy, R. S.; Thrush, B. A. Tram.Furuduy Soc. 1959, 55, 1139-48. (16) Cooper, S.R.; Calvin, M. Science 1974, 185, 376. (17) Malkin, S.; Fischer, E. J. Phys. Chem. 1964, 68, 1153-63. (18) Williamson, D. G.; Bayes, K. D. J. Am. Chem. SOC.1968, 90, 1957-65. (19) Goldfinger, P.; Hybrechts, G.; Mahieu-Van der Auwera, A. M.; Van der Auwera, D. J. Phys. Chem. 1960,64,468-70. (20) Bailey, R. T.; Cruicbhank, F.R. J. Phys. Chem. 1976,80, 1596601.
t
L
0.0
IO-^
IO-^
IO-’
o 10’
loo
1 IO2
Figure 3. Ratio of R,, to R,(minimum required temperature differences for upward and downward illumination)as a function of optical density Y.
Figure 2 shows R [the right-hand side of (13)] as a function of y for 6 = 0 (upward illumination) and several values of the parameter r = (1 /Bib + &,)ab characterizing the bottom wall. We note here that the thickness, external heat transfer coefficient, and thermal conductivity of the top wall do not appear in (1 3) because the critical condition (corresponding to A = R) gives dOr(O)/dq = 0, so that from (9c) there is no heat flux through the upper wall, and its thermal behavior is irrelevant. Although illumination from below will always be preferable in terms of minimizing the end-to-end temperature difference, it may prove more convenient in some situations to illuminate the cell from above, especially if the bottom of the cell is being cooled by a liquid (e.g., water) bath. In that case, the minimum dimensionless temperature gradient is obtained by setting 6 = 1 in (13). Figure 3 shows the ratio of minimum dimensionlesstemperature differences for upward and downward illumination for three values of r, and it is clear that the temperature difference required to ensure stability is considerably smaller for upward illumination, especially for larger optical densities y. At large optical densities, the minimum required dimensionless temperature difference R asymptotically approaches r for upward illumination, whereas for downward illumination, R approaches 1 r. Thus, at large y, R will be significantly smaller for upward illumination, especially at the small values of r likely to be encountered in practice, as we will see in the next section.
+
V. Discussion and Example The recognition of convection in photochemical systems goes back to at least 1912.21 The necessity of eliminating convection in quantitative photochemical experiments was pointed out in 195122and 195823by Noyes, who, in discussing the feasibility of a method for measuring the molecular diffusion coefficients of photochemically generated iodine atoms in solution, realized that measurements of a diffusive transport process “would be vitiated by any convection currents that carried iodine atoms from illuminated into dark areasneZ3In several later papers, Noyes’ “photochemical space intermittency” method was actually used, ~~~
~
(21) Plotnikow, J. 2.Phys. Chem. 1912, 78, 293-8. (22) Noyes, R. M. J. Am. Chem. SOC.1951, 73, 3039-43. (23) Noyes, R. M. J. Am. Chem. SOC.1959, 81, 566-70.
Pearlstein
1058 The Journal of Physical Chemistry, Vol. 89, No. 6,1985
and it was pointed out that vertical illumination would minimize convective t r a n ~ p o r t . ~The ~ - ~authors ~ originally illuminated the cell from below, but later redesigned the apparatus to provide illumination from the top, in the belief that this would reduce convective transport still further at high optical densities.zs They also considered the possibility of operating with an overall, endto-end temperature difference in order to completely eliminate convection. This was not attempted because the authors believed that elimination of the residual "vertical convective flow is obviously of very little importance in a cell with such short light path". They apparently neglected the fact that the existence of vertical flow of an incompressible fluid in a closed cavity implies the existence of a horizontal flow according to the continuity equation
where z and w are the vertical coordinate and velocity component, respectively, and (x,y) and ( u p ) are the corresponding horizontal quantities. The conclusion follows from the fact that, if there is a vertical velocity component, it must vanish at the impermeable horizontal walls, so that awl& is nonzero somewhere in the fluid. Thus, the horizontal derivatives of the horizontal velocity components are also nonzero, which implies that the horizontal velocity itself is nonzero. The authors were interested only in suppressing horizontal convective transport and overlooked the connection between horizontal and vertical fluid motion implied by (14). We now show, by means of an example, how the results of the previous section can be used to design an experiment in which the temperature increases monotonically with height and the layer is guaranteed to be stable. For illustrative purposes, we consider system b of Eptein et a1.* The cell height Lf is 5.5 an,the thermal J s-l cm-' K-' , and the conductivity kf of the liquid is 2.17 X concentration of the absorbing species (acetone) in acetonitrile is 0.3 M. The molar extinction coefficient 6 of acetone at the exciting wavelength (296 nm) is taken as 6.5 M-l cm-' based on a spectrum in methanol2' and the fact that the spectra of acetone = 274 nm, E,,, = 15.5 M-' cm-I) and in in acetonitrile (A, = 270 nm, emax = 17 M-' cm-') are very similar.28 methanol (hX The gas-phase value at 296 nm is e = 9 M-' cm-1.29 We will (24) Salmon, G. A.; Noyes, R. M. J . Am. Chem. SOC.1962,84, 672-3. (25) Levison, S.A.; Noyes, R. M. J . Am. Chem. SOC.1964,86,4525-9. (26) Burkhart, R. D.; Boynton, R. F.;Merrill, J. C. J . Am. Chem. SOC. 1971, 93, 5013-7. (27) 'Sadtler Handbook of Ultraviolet Spectra";Sadtler Research Laboratories: Philadelphia, 1979; p 556. (28) Hayes, W. P.; Timmons, C. J. Spectrochim. Acta 1965, 21, 529-41.
suppose that the same incident light intensity that was used to illuminate the cell from the side2 is used to uniformly illuminate the cell from below, which gives 6 = 0 and Zb = (2 X 1015quanta s-')/37 mm2 = 5.4 x ioi9 quanta s-l m-2. As for the average thermal energy dissipated per absorbed quantum, a, the literature2s30provides no quantitative data reSo), phosgarding the quantum yields of fluorescence (Sl phoresence (TI So), or biacetyl, CO, CH4, and C2H6formation from acetone in acetonitrile solution. The quantum yields of the < 0.01) for luminescent processes appear to be very low (alum solutions of acetone in various other liquid ~olvents.~'This fact, and the low quantum yields for production of biacetyl, CO, CH4, and C2H6 in solution, have been i n t e r ~ r e t e das~ ~evidence in support of the conclusion that most of the excited singlet molecules (or the triplets produced by intersystem crossing) are either physically quenched or, in the event of an a-cleavage to form methyl and acetyl radicals, a very large fraction of the radicals recombine in the solvent cage. Thus, most of the absorbed energy will be dissipated as heat, and we will take a = hcl/habs. In the absence of any information about the top and bottom ends of the cell used in ref 2, we will assume that there is no thermal resistance at the ends (Ob = ut = O), which is very reasonable for the high conductivity brass holder employed. Then, (1 - e T ) / y , or (13) reduces to A L -e-Y
-
-
+
The optical density is y = 6.5(0.3)5.5 = 10.7, so that the required minimum temperature difference is To - T I = 0.86 K. Thus, if the cell had been illuminated uniformly from below, and if the side walls were sufficiently insulating, it would only have been necessary to maintain the top of the cell 0.86 K (or more) warmer than the bottom in order to ensure the absence of thermal convection. Such a temperature difference is of the same order of magnitude as that produced by the use of heaters in ref 2 and should be easily attainable. Acknowledgment. The author thanks N. R. Armstrong, C. F. Chen, L. S. Forster, S. Lichter, and the anonymous referees for their comments on the manuscript. Registry No. Acetone, 67-64-1; acetonitrile, 75-05-8. (29) Calvert, J. G.; Pitts, J. N. "Photochemistry";Wiley: New York, 1966; p 371. (30) Nemzek, T. L.; Guillet, J. E. J . Am. Chem. SOC.1976,98, 1032-4. (31) Borkman, R. F.; Kearns, D. R. J . Chem. Phys. 1965, 44, 945-9. (32) Berces, T. In "Comprehensive Chemical Kinetics"; Bamford, C. H., Tipper, C. F. H., Eds.; Elsevier: Amsterdam, 1972; Vol. 5, pp 234-380, especially p 335.
