1352
The Journal of Physical Chemistry, Vol. 82, No. 12, 1978
K. Bar-Eli and R. M. Noyes
Detailed Calculations of Multiple Steady States During Oxidation of Cerous Ion by Bromate in a Stirred Flow Reactor K. Bar-Eli and Richard M. Noyes" Department of Chemistry, University of Oregon, Eugene, Oregon 97403 and Depa~mentof Chemistv, University of Tei Avlv, Israel (Received December 29, 1977)
reel Aviv,
Publication costs assisted by the National Science Foundation
Under certain conditions, addition of acidic bromate, bromide, and cerous ion solutions to a stirred flow reactor can generate two different locally stable steady states for the same input conditions. We have used a mechanism developed previously to calculate effects to be expected for various inputs of reactants and total flow rates. The calculations suggest specific experimental tests that would support or invalidate the proposed mechanism and refine rate constants assigned to specific steps.
Introduction The oxidation of cerium(II1) by acidic bromate has been studied in batch reactors by various investigators.' A detailed mechanism for the reaction has been proposed by Noyes, Field, and Thompson2 (NFT), and its predictions have been compared to experimental data by Barkin et alS3 This mechanism was also used by Bar-Eli and Noyes4 to explain the flow reactor results of Geiseler and Fo11ner.5 Unlike previous investigators, the latter authors conducted the reaction in a continuously stirred tank reactor (CSTR).I7 They followed the reaction both potentiometrically with an electrode specific to bromide ion and spectrophotometrically by ceric ion absorbance. They found that given sufficient time a CSTR system would reach one of two possible steady states under identical flow rate and concentrations of reactants in the feed. After a perturbation, the system would either return to its previous steady state or transform to the other one. All of these phenomena are accounted for by the NFT mechanism, and a detailed account of the appropriate calculations has been given by Bar-Eli and no ye^.^ The object of that paper was to reproduce the experimental findings of Geiseler and Follner.5 In the present manuscript, we investigate the theoretical behavior of the system under a variety of experimental conditions (not previously reported) and suggest and predict the results of further feasible experiments. These suggested experiments would serve a double function. First, they would probe further the very interesting and complex reaction between cerium(II1) and bromate. That reaction exhibits a difficultly reproducible induction period,'^^ and experiments under steady state conditions would be less susceptible to slight variations in the initial conditions. Of course our calculations depend upon the particular values we select for various rate constants, and any discrepancy between experiment and calculation can be used to estimate some of the constants that are difficult to measure directly. Secondly, the different behavior of closed and open systems has been under investigation recently both from theoretica16J8 and experimental5 points of view. The present calculations support a theorem due to Gavalas6 that the number of steady states in an open system is odd (2m + 1)of which at least m are unstable. In this system, *Address correspondence to this author at the Department of Chemistry, University of Oregon.
m = 1 and there are three steady states one of which is unstable.
Mechanism The NFT mechanism is given by seven reversible steps Br0,- t Br- + 2H' k , = 2.1 M-3 s-'
F?
HBrO, t HOBr
(1)
k-' = 1 X l o 4 M-'
s-'
HBrO, + Br- t H+e 2HOBr k, = 2 X l o 9 M-, s-' k-, = 5 x 10-5 M-1 s"
(2)
HOBr t Br- + H+ Br, k, -. 8 X l o 9 M-' s-'
+ H,O
(3)
Br0,- + HBrO, + H" k, = 1 X l o 4 M-ls-'
2Br0,. + H,O k-, = 2 x l o 7 M-l s-'
= 110 s-1
Ce3+t BrO,. + H' 3 Ce4++ HBrO, k, = 6.5 x 10' M-' S-' k-$= 2.4 Ce4'
+
BrO,.
t
H,O
k, = 9.6 M-'s-'
(5)
x 10' M-' s-,
Ce3++ Br0,- t 2H"
k-, = 1.3 x lo-,M - 3 sml
2HBr0, e BrO,' + HOBr + H' h , = 4 X lo' M-' S-' k-, = 2
X
(4)
M-, s-'
(6) (7)
to which we have assigned the indicated rate constants at 25 "C. Ratios of forward and reverse rate constants have been made consistent with the free energy assignments of Field, Koros, and Noyes7 for these species.
Method of Computation The seven steps of this mechanism involve nine species (in addition to water whose concentration we assume constant at unit activity). The four species Br03-, Br-, Ce3+, and H+ were assumed added to the reactor a t a constant rate and instantaneously mixed while an equal total volume of solution was simultaneously withdrawn from the uniform contents of the reactor. We wish to determine the composition of a steady state such that dci/dt = 0 simultaneously for all nine species i. For each of these species, a differential equation can be written of the form d[Ce3+]/dt = ko([Ce3+]o- [Ce3'1) k 5 [H+][Br0,.][Ce3'] + h-5 [HBrOz][Ce4'] + h 6 [ B r ~ 2 . ] [ C e 4 +-]h-6 [H+]2[Br03-][Ce"'l In this equation, ko is the ratio between the total flow rate and the volume of the tank reactor; its reciprocal is sometimes called the "retention time".8,9 The quantity [Ce3+Iois the concentration this species would attain for
0022-365417812082-1352$01.0010 0 1978 American Chemical Society
The Journal of Physical Chemistry, Vol. 82, No. 72, 7978 1353
Oxidation of Cerous Ion by Bromate in a Stirred Flow Reactor
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-41
-
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8
7
6
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Figure 2. Changes in steady state bromide concentration with bromide flow for different bromate flows with [Ce3+Io= 1.5 X M, [H+Io = 1.5 M, k,, = 4 X IO-, s-'. Values of [BrO,-], were 2 X M for A, I O x io-, M for B, 20 x io-, M for C, 100 x IO-, M for D, and 200 X lo-, M for E.
-8
-7
-6 Ig[Br-]oin Flow
-5
J
-4
Flgure 1. Changes in steady state concentrations with bromide flow. M, [Ce3+], = 1.5 X Other constraints were [BrO,-l0 = 2 X M. [H+I, = 1.5 M. k, = 4 X IOw38'.
the same flow conditions if no chemical reactions took nlnre in the tank. Of cniirse 1x1, was zero in the eauations ror the rive species that were not being introaucea to tne reactor. The external parameters or constraints8 for this system are [BrO3-Io, [Br-lo, [Ce3+Io,[H+l0,ko, and temperature. We do not know energies of activation for the rate constants and have made no calculations a t temperatures other than 25 "C. The other constraints were varied over wide ranges during the calculations. Each set of constraint values generated either one or three steady states. States designated SSI contained relatively large concentrations of bromide and relatively small concentrations of ceric ion and of oxybromine species other than bromate. States designated SSII reversed the relative importance of these constituents. The composition of an SSI or SSII state could be estimated first by solving these stiff differential equations by the method of Gearlo until rates of change were small. The components of the initial concentration vector do) so obtained were used to calculate matrix elements Pij = (ati/acj) of the Jacobian matrix P. An improved composition vector c(l)was then obtained by solution of the approximation equation C(O) + P(c(l) - do)) = 0. This Newton method for obtaining c = 0 was superior to descent methodsll for solving this problem. Whenever SSI and SSII steady states both satisfied the same set of constraints, it was possible to satisfy the equations with an additional SSIII steady state having concentrations intermediate between those of SSI and
SSII. The compositions of SSIII states were refined from initial estimates based on previously determined SSI and SSII compositions. The stability of a steady state could be established by finding the eigenvalues for the Jacobian P with the aid of routine library programs. States SSI and SSII had only negative eigenvalues of the Jacobian and were stable to infinitesimal perturbation. Each SSIII state had at least one positive eigenvalue and was physically unattainable. Neither the Gearlo program nor an experiment would be expected to move a system toward an SSIII state. Points on SSI (or SSII) very near to SSIII were difficult to calculate exactly because the Jacobian matrix P approached singularity. Of course such states are particularly sensitive to fluctuations that can initiate transitions to SSII (or SSI).
Effects of Varying Constraints Bromide Ion Concentration in Flow. Figure 1 shows calculations of various concentrations as a function of bromide ion concentration in the feed. Values of the other constraints are the same as those used in our previous calculations4 and imitate the experimental conditions of Geiseler and FOllnera5 Note that at very low values of [Br-lo the system goes to SSII with all concentrations virtually independent of bromide flow. As [Br-], increases, [Ce4+]increases slightly, [HOBr] increases somewhat more, and [Br-] becomes almost proportional to [Br-1,. For [Br-1, greater than 4 X loW5 M, SSII can no longer exist for these values of the other constraints. At high values of [Br-lo, the system goes to SSI with [Br-] proportional to [BY], while all the other concentrations are virtually independent of bromide flow. For SSI can no longer exist. When [Br-lo less than 4.17 X [Br-lo is varied slowly, transitions between SSI and SSII will show typical hysteresis behavior. The concentration of bromide is approximately the same for all SSIII systems in Figure 1,while the concentrations
K. Bar-Eli and R. M. Noyes -3
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Figure 4. Changes in steady state ceric ion concentration with bromide flow for different bromate flows with [Ce3+], = 1.5 X M, [H+Io = 1.5 M, k , = 4 X s-l. Values of [Br03-]o were 2 X loT3M for A, 10 X M for B, 20 X M for C, and 100 X M for
D.
Figure 3. Regions of single and multiple steady states as functions of [Br-lo and BrO, for [Ce3+Io = 1.5 X lO-'M, [H+Io = 1.5 M, k o = 4 x 10- s-1 .-I
5
of all other species change by several orders of magnitude during the transition between SSI and SSII. Reasons for these various results will become apparent during the subsequent discussion. Bromate I o n Concentration i n Flow. Figure 2 shows variation of [Br-] in the reactor against [Br-], in the flow for five different values of [BrO,-],. Changing bromate flow has rather little effect on systems in SSII. However, increasing [BrO,-], shifts the composition of SSI and increases the steady state concentration of bromide ion in SSIII. Increasing [BrO,-], also decreases the range of [Br-1, values for which multiple steady states exist, and there is a critical bromate flow concentration between 0.02 and 0.1 M such that only one steady state can exist at higher concentrations. The curves at constant [BrO1O1O M-l s-l are observed with many solutes, it is assumed that the reaction occurs a t every collision. The reaction radii in aqueous solutions, estimated from the Debye-Smoluchowski equation, are usually in agreement with the distances between the reactants a t contact as calculated from crystallographic data. However, it has been noticed that in aqueous solutions,l and also recently in alcoholic solutions,' strong oxidizing agents have reaction radii two or three times greater than radii estimated from crystallographic data. Relatively very little work has been done on solvated electron reaction rates in water-alcohol solution^.^ The addition of alcohol to water has a pronounced effect on the kinetics of slow electron transfer reaction^.^ For fast, diffusion-controlled reactions, such as reactions of the solvated electron with neutral solutes, the effect of similar changes in the composition of the solvent could be expected to depend solely on the values of the diffusion coefficients of the reactants. In our earlier paper5 the diffusion coefficients of the solvated electron in wateralcohol solutions were determined from direct conductance measurements. In this work a study of the solvated electron reactions with several different neutral solutes in water-ethanol and water-methanol solutions was undertaken to shed additional light on the mechanism of solvated electron reactions. Experimental Section
cohols of analytical reagent grade were used without further purification. Air-free samples were prepared by bubbling with argon. Deaerated solutions of different solutes were prepared by injecting, after bubbling, appropriate concentrations of a solute into the solvent. Hydrogen peroxide concentration was determined by permanganate titration, and that of iodine by mixing pulse-irradiated solutions with 0.2 M KI and by measuring the optical density a t 350 nm. The extinction coefficient of the Is- was taken to be 2.5 x lo4 M-l cm-1.6 Different oxygen concentrations were obtained by mixing appropriate volumes of Ar- and O'-saturated solutions from two syringes. Oxygen concentrations for all samples were determined by gas chromatography using a Perkin-Elmer 154 DG apparatus and a molecular sieve column. A pulse radiolysis technique was used to measure the decay rates of solvated electrons at 600 nm in the presence of different solutes. This technique has been described previously7 and it suffices here to outline the method. Concentrations of several micromoles of e; were produced in the samples by irradiation with 20-ns pulses from a Febetron 707 (Field Emission Corp.) electron accelerator. The absorbed doses were in the range 0.5-1 krd/pulse. The variation of e; concentration with time was observed by fast spectrophotometric methodsa8 The rise time of the light measuring system is 10 ns. The measurements were taken a t 19 f 1 "C. Results Solvated electrons react with solutes present in water-alcohol solutions according to eq 1 and also with es-
Solutions were prepared from analytic grade chemicals (BDH or Merck). The water was triply distilled and al-
+
S + product
alcohol according to eq 2.
0022-3654/78/2082-1359$01,00/00 1978 American Chemical Society
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