Coupling between continuous-flow, stirred tank reactors (CSTR’s), each having multiple steady states, can produce new steady states with different concentrations of the chemical species in each of the coupled tanks. In this work, we identify a kinetic potential ψ that governs the deterministic time evolution of coupled tank reactors, when the reaction mechanism permits a single-variable description of the states of the individual tanks; examples include the iodate-arsenous acid reaction, a cubic model suggested by Noyes, and two quintic models. Stable steady states correspond to minima of ψ, and unstable steady states to maxima or saddle points; marginally stable states typically correspond to saddle-node points. We illustrate the variation in ψ due to changes in the rate constant for external material intake (k0) and for exchange between tanks (kx). For fixed k0 values, we analyze the changes in numbers and types of steady states as kx increases from zero. We show that steady states disappear by pairwise coalescence; we also show that new steady states may appear with increasing kx, when the reaction mechanism is sufficiently complex. For fixed initial conditions, the steady state ultimately reached in a mixing experiment may depend on the exchange rate constant as a function of time, kx(t) : Adiabatic mixing is obtained in the limit of slow changes in kx(t) and instantaneous mixing in the limit as kx(t)→∞ while t remains small. Analyses based on the potential ψ predict the outcome of mixing experiments for arbitrary kx(t). We show by explicit counterexamples that a prior theory developed by Noyes does not correctly predict the instability points or the transitions between steady states of coupled tanks, to be expected in mixing experiments. We further show that the outcome of such experiments is not connected to the relative stability of steady states in individual tank reactors. We find that coupling may effectively stabilize the tanks. We provide examples in which coupled CSTR’s can be operated stably with one of the tanks at or beyond the single-tank marginal stability point.
Photoinduced reactions when confined to the interior (or perhaps the immediate vicinity) of a micellar assembly are known to exhibit remarkable kinetic effects (e.g., drastically modified reaction rates, transition to pseudo-first-order behavior for reactions of any molecularity) which have been well documented over the last decade. In earlier papers in this series we have shown how a stochastic approach can be mobilized to describe the dynamics of reactions in compartmentalized, distributed systems. A master equation was derived and solved both for the case of irreversible and reversible photoinduced reactions. In this contribution we study the asymptotic properties of the stochastic master equation for (irreversible and reversible) intramicellar reactions and show how the correct homogeneous-system (macroscopic) kinetic description is reached in the limit of large micellar volume and attendant molecular occupancy. We display numerically the kinetic behavior at intermediate stages in the taking of this limit and, for reversible reactions, show how the ’’apparent’’ equilibrium constant Q for a reaction carried out in a compartmentalized, distributed system relaxes to the ’’canonical’’ equilibrium constant K when the homogeneous system limit is reached. The simulations reported have considerable relevance to recent work on microemulsions and in the concluding section we discuss, from the vantage point of the theory presented in this paper, the possible kinetic effects observable experimentally in these systems.
In this paper, we present a combined experimental and theoretical study of monomer–eximer dynamics in spread monolayers. The system studied here is a spread monolayer of 12-(1-pyrene) dodecanoic acid (PDA) and oleic acid at the air–water interface of a neutral aqueous solution. The microscopic kinetic behavior of the PDA probe in the monolayer was monitored through the eximer–monomer, steady-state photoexcitation of the pyrene moiety. The steady-state ratio of eximer to monomer intensities, I2/I1, was found to depend linearly on the mole fraction of PDA in oleic acid over the concentration range studied. These data were interpreted theoretically by constructing a reaction-dynamic model to describe the lateral diffusion and subsequent interaction of the pyrene species. Using analytical results from the theory of random walks on lattices with traps, and from data from Monte Carlo simulations, an estimate is obtained for the two-dimensional diffusion coefficient of the pyrene probe in the monolayer; this estimate is 1.7×10−6 cm2/sec.
We develop in this paper a stochastic master equation to describe the dynamics of a certain class of micellar kinetic processes—those in which the reactants are confined to the interior (or, perhaps, the immediate vicinity) of the micellar assembly. At the outset, the problem is formulated in some detail in order to display the theoretical factors involved in accounting both for chemical reaction and physical diffusion (exit/re-entry of solubilized reactants). Then, for definiteness, we confine our attention to the study of irreversible intramicellar reactions where complete compartmentalization of reactants is assumed. For this case, chemical processes of molecularity 1, 2, and 3 are considered, and a solution to the resulting master equation for each process is obtained. Our theoretical predictions are compared with experimental results reported by the groups of Grätzel and Singer and nearly exact, quantitative agreement with the data is found.
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