J. Phys. Chem. 1903, 87.3774-3776
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therefore, misleading. Only a sound mathematical procedure for making the proper adjustments in the rate constants will be meaningful. A further difficulty lies in the fact that, at the bistability limits, the Jacobian is zero; i.e., the matrix is singular and has an infinite slope as described elsewhere.6 This makes all the mathematical procedures in this vicinity more cumbersome. The problem can be slightly reduced by trying to adjust only those rate constants whose values are doubtful or those which are known to be more influential on the position of the surface^.^ A problem of a completely different nature is the generality of this phenomenon. In other words, does every chemical system, which by its nature contains quite a few species and parameters, that shows bistability also oscillate near the critical point? If this is not the case, one should look for the necessary and sufficient conditions to be fulfiied in order for the system to oscillate or to be bistable or both. The existence of oscillations in these situations is also treated by the method of “cross shaped phase diagrams” due to Boissonande and De Kepper.21 One should note that the oregonator,20Field-Koros-Noyes
(FKN),7and present NFT2 mechanisms all show oscillations in a region near the critical point.1sB21 Also the chlorite-iodideZ4 and Briggs-Rauscher systems25 show oscillations near the critical point. In the case of the above-mentioned systems, the oscillation domain extends to large portions of the constraints space, while in the present NFT mechanism this portion is rather small and therefore difficult for experimental detection. Apart from the above disadvantage, this oscillator is the simplest known chemical oscillator, the mechanism of which is completely known in terms of elementary reaction steps. Moreover, unlike the better known BZ oscillator, this one is not influenced by oxygen.1s It is, therefore, a challenge to find a simpler oscillator. Registry No. Cerium, 7440-45-1; bromate, 15541-45-4; bromide, 24959-67-9. (24) C. E. Dateo, M. Orban, P. De Kepper, and I. R. Epstein, J . Am. Chem. SOC.,104, 504 (1982). (25) (a) T. C. Briggs and W. C. Rauscher, J . Chem. Educ., 50, 496 (1973); (b) P. De Kepper, Ph.D. Thesis, L’Universitd de Bordeaux 1, France, 1978; (c) P. De Kepper and I. R. Epstein, J. Am. Chem. SOC.,104, 49 (1982).
Feasibility of Measuring Diffusion Coefficients of Short-Lived Species by Taylor Diffusion Robert M. Marot hpartment of Chemical Physlcs, Welzmann Institute of Science, 76 100 Rehovot, Israel (Received: November 2, 1982: I n Final Form: February 23, 1983)
Taylor diffusion in a fluid where the diffusing species is chemically reactive is considered. The question of whether this phenomenon can be used as a tool for the measurement of diffusion coefficients of short-lived species is investigated. It is concluded that such a technique is feasible if the lifetime of the diffusing species is of the order of 1 min or more, in a condensed phase.
Introduction Almost 30 years ago, Taylor1 discussed the dispersion of soluble matter in solvent flowing slowly through a tube. One would at first expect that a band of solute in such an experiment would be drawn out by convection into a profile determined by the parabolic velocity profile of the flowing solvent. However, Taylor found that, under certain conditions, the band would move with the mean velocity of the fluid, diffusing about this mean position with an effective diffusion coefficient which can be appreciable larger than the molecular diffusion coefficient. To quote Taylor,l “This theoretical conclusion seemed so remarkable that I decided to set up apparatus to see whether the predictions of the analysis could be verified experimentally”. In fact, Taylor did verify the predictions by observing the behavior of potassium permanganate (chosen because its intense color facilitated observations) in water and measuring its diffusion coefficient. The theory of this phenomenon, now called Taylor diffusion, has been further developed by Taylor himself,2 as well as Ark3 and other^.^ Furthermore, the effect of heterogeneous chemical reaction of the solute on the walls of the tube has been incorporated into the t h e ~ r y .The ~ purpose of this paper is to propose that Taylor diffusion ‘Permanent address: Department of Chemistry, University of Oregon, Eugene, OR 97403. 0022-3654/83/2087-3774$0 1.5010
can be used to measure the diffusion coefficient of relatively short-lived species in solution, and to determine the circumstances under which such a measurement is feasible. The measurement of the diffusion coefficient of a short-lived species in a condensed phase is extremely difficult. This is because diffusion in condensed phases is quite slow. Hence, particles cannot diffuse very far during their short lifetime, and no measurable concentration profile characteristic of a diffusion experiment is generated. However, since the Taylor diffusion coefficient can be larger by several powers of 10 than the molecular diffusion coefficient, it is possible to magnify the effects of diffusion, using the Taylor phenomenon, so that diffusion coefficients ought to be measurable in favorable cases. The conditions under which Taylor diffusion manifests itself restrict the lifetimes with which one can work to do the order of several tens of seconds. The very interesting region of fractions of a second is still not accessible. Nevertheless, there may well be problems of interest which (1) Taylor, G. I. Proc. R. SOC.London, Ser. A 1953,219, 186. (2) Taylor, G. I. Proc. R. SOC.London, Ser. A , 1954, 225, 473. (3) Aris, R. R o c . R. SOC.London, Ser. A, 1956, 235, 67. (4) For a review with references, see: Aris, R. In “Dynamics and Modeling of Reactive Systems”; Stewart, W. E., Harmon, W. H., Conley, C. C., Eds.; Academic Press: New York, 1980. (5) De Gance, A. E.; Johns, L. E. Appl. Sci. Res. 1978,34, 189. This is merely the most recent of a number works of the subject.
0 1983 American Chemical Society
The Journal of Physical Chemistry, Voi. 87, No. 79, 7983 3775
Diffusion Coefficients of Short-Lived Species
can be investigated with this technique. Quite recently, Clifford et aL6 have used the technique of Taylor diffusion to measure the diffusion coefficient of hydrogen atoms in the gas phase. In their case, the lifetime-limiting reaction was heterogeneous and took place on the walls of the flow tube. When the recombination reaction is homogeneous, but first order, the effect of chemical reaction can be removed from the equations governing the problem by a simple change of variable. The analysis is very similar to that of Clifford et al. In this paper, we shall therefore concentrate on the case of a bimolecular homogeneous recombination reaction. This paper is organized as follows. We first review briefly the mathematical description of the phenomenon, and the conditions of its validity. Then we discuss the type of experiment that we have in mind, and estimate whether the conditions of validity are fulfilled in practical situations. The paper ends with a brief discussion.
Theory Taylor's result can be succinctly stated as follows. Let a fluid be moving through a cylindrical tube of radius a, in laminar flow with a mean flow velocity (averaged over the tube cross section) u. Let a second component be introduced, at concentration c. Then, in a coordinate system moving with the mean velocity u, c obeys the diffusion equation
a q a t = K(a2C/aX2)
requirement is necessary to ensure that the concentration of the dispersed material is essentially constant over a cross section of the tube on the chemical time scale. An additional limitation is that one is interested in times of the order of f or less. For if the time is too long, the chemistry alone would reduce the amount of dispersed material to an amount difficult to measure. There are two interesting cases to consider. The first case is that of first-order reaction, f ( c ) = c. Equation 1, augmented by the chemical term, is
a q a t = K(a2C/ax2) - kc The substitution
c = ce-kt (6) reduces eq 5 to eq 1 for E , and the problem is reduced to an ordinary diffusion problem. In the second case f ( c ) = c2. Here, the governing equation
a q a t = K(a2C/aX2)- k 2 C 2
where D is the ordinary molecular diffusion coefficient. Since D is very small in condensed systems, it is quite feasible to construct situations in which K >> D. We shall henceforth neglect the additive contribution of D , which is negligible. In the case when a homogeneous chemical reaction is depleting c with a rate law -kf(c), one can follow through Taylor's original analysis, or any of the later variants, and conclude that the right-hand side of eq 1 should be augmented by a term -kf(c), as would, indeed, be intuitively expected. Equation 1 is not exact. Its conditions of validity can be determined from its derivation and are2 4 L / a >> u a / D >> 6.9 (3) where L is a length characteristic of the size of the region where the concentration of dispersed matter is changing. To these conditions, derived by Taylor for the nonreactive case, one must add the condition
(3.8)2fD/a2>> 1 (4) where f is a characteristic time for the chemical reaction. For example, f would be taken as kl, for a first-order reaction, or as (k2co)-' for a second-order reaction, where k and k 2 are the rate constants for the two cases, and co is the initial concentration. Inequality 4 says that the time for radial homogenization of the concentration by diffusion (of the order of a2/ [(3.8)2D]) is much smaller than the chemical time, f. The factor of 3.8 is explained in ref 1. It comes from making a better estimate of the diffusion time than one would guess from purely dimensional arguments, a2/D. This (6) Clifford, A. A.; Gray, P.;Mason, R. S.;Waddicor, J. I. Proc. R. SOC. London,Ser. A 1982, 380,241.
(7)
cannot be reduced to an equation soluble by known analytical methods. It is useful to introduce dimensionless variables. Let us set E = c/co
5
(1)
where the tube axis has been taken as the x axis. x is the distance variable in the moving coordinate system, and K is the so-called Taylor diffusion coefficient, given by K = a2u2/48D+ D (2)
(5)
= (k2co/K)1/2x 7
= k2cot
(8)
Then eq 7 becomes
ac/ar
= a 2 c / a p - 62
(9)
This form is very convenient for the numerical investigation of the equation.
Estimates The type of experiment that we have in mind is the following. One has fluid, solvent plus solute, flowing through a capillary tube. A short-lived species is generated in some region of the tube by flash photolysis, pulsed electrolysis, or some other method. The region over which the species is produced should have some well-defined geometry. One then waits a time to and, downstream in a region displaced by uto from the irradiation region, measures the concentration profile. In order to investigate the validity of expression 3, one needs to choose u and a. a should be as small as possible, but not so small as to violate the rightmost inequality. For the sake of estimation, we will always take in this paper D= cmz/s. This is a typical order of magnitude for a small molecule in a liquid. Let us take a = 0.01 cm, an easily attainable value, and u = 0.1 cmls, also easily attainable. The u a / D = 100, which is sufficiently greater than 6.9 for the rightmost inequality to hold. In order to investigate the leftmost inequality of expression 3, one must know L, which is but vaguely defined. If the initial concentration distribution has a sharp edge, then a typical length scale for the concentration change after a time to is (2Kto)1/2, The numerical results which we shall presently discuss indicate that L should be taken as about 5 times this value. If we then demand that L l a 1000 (since u a / D N loo), this requires that to 60 s (10) or longer. Now, we want toto be not much more than E, in order to have enough material left to measure at the downstream measurement site. If we therefore substitute t ointo ine-
-
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The Journal of Physical Chemistty, Vol. 87, No. 19, 1983
quality 4, we find the left-hand side to be 87, so inequality 4 is well satisfied. We conclude then that, for reactive species with characteristic lifetimes of about 1 min or longer, the measurement of the diffusion coefficient by Taylor diffusion ought to be feasible. In order to extract the diffusion coefficient from a measured downstream concentration profile, one must be able to solve eq 5 or 7. As we have seen, eq 5 is reducible to eq 1,and solutions of eq 1have been so well studied that it would be superfluous to discuss this case further here. Equation 7, on the other hand, has not been solved in closed form; some approximate solutions, of doubtful accuracy, have been ~uggested.~ Therefore, we have resorted to a numerical solution. It is for this reason that the dimensionless form, eq 9, is so useful; one does not have to redo the calculation for every choice of parameters. One merely has to rescale the numerical solution back to the physical distance and time scales. Of course, the solution to eq 9 also depends on the initial distribution E([,O). This may well depend on the parameters. There is no reason why an arbitrary initial condition should be made dimensionless by the same transformation which renders the equation dimensionless. However, there is a convenient initial condition which avoids this problem. Assume that the flow is from left to right and set E([,O) = 1 t
