Reactant time-dependent concentrations

Multichannel time-resolved spectral data are best analysed in a global fashion using nonlinear least squares algoritlims, e.g., a simplex search, to fit multiple first order processes to all wavelengtli data simultaneously. The goal in tliis case is to find tire time-dependent spectral contributions of all reactant, intennediate and final product species present. In matrix fonn tliis is A(X, t) = BC, where A is tire data matrix, rows indexed by wavelengtli and columns by time, B contains spectra as columns and C contains time-dependent concentrations of all species arranged in rows. 

Once the stoichiometric coefficients Ny and rate functions v, for all reactions and transport processes are assembled, the time-dependent concentration of a metabolic reactant 5,(f) is described by the dynamic mass balance equation... 

The rates of liquid-phase reactions can generally be obtained by measuring the time-dependent concentrations of reactants and/or products in a constant-volume batch reactor. From experimental data, the reaction kinetics can be analyzed either by the integration method or by the differential method ... 

The Well-Stirred Mixture. A key assumption of most kinetic measurements is that of a w ell-mixed solution of reactants. Then any component can be characterized by a single time-dependent concentration, applicahlc lo the entire sy stem. 

As was the case for techniques based on potential excitation, current-excitation methods are best understood by studying the time-dependent concentration changes in solution caused by the excitation signal applied to the electrode. Concentration-distance profiles for the case of species O being reduced to R by a current-step excitation signal (application of constant current to the cell) are shown in Figure 4.2. Consider first the profiles in Figure 4.2A for the reactant, O. An important concept is the relationship between the applied current and the slope of the profile at the electrode surface as expressed by... 

The shape of the transient shown in Fig. 13K depends on electrode kinetics, althougli the transition time T is independent of it. For the reversible case, this can be obtained by introducing the time-dependent concentrations of reactants and products at the electrode surface, C (0,t) and Cj (0,t), respectively, into the Nemst equation. The result is... 

Determination of reaction kinetics alone is not enough to get a reasonably complete picture of surface processes during deposition. Often the gaseous reactants and products are identified by mass spectroscopy and time-dependent concentrations are measured. Adsorbed species are identified by in situ spectroscopy (infrared, Raman, or laser-induced fluorescence). The morphology of the product as studied by electron microscopy also contributes to an understanding of CVD reaction mechanisms. 

In real situations diffusion and reaction will take place in both solid phases, giving rise to time dependent concentration gradients of both reactants in each phase. The calculation of the concentration changes with place and time can be based on equations (5.28) and (5.29), which have to be solved by numerical means. In many cases the diffusion coefficients cannot be considered as constants, and have to be expressed as functions of the concentrations. For such processes a lot of specific data have to be gathered in order to arrive at an adequate description. 

Thus, for multi-step reactions which involve a large number of reactants and products, corresponding mathematical models can be quite awkward. In most cases, it is not necessary to have a system of N equations for modeling N kinetic curves. Fulfillment of a condition r ank (M) < N means that some substances entering a general kinetic scheme can be not excluded from an initial model at all, since their kinetic curves can be calculated on the basis of information about a time-dependent concentration behaviour of other key components which have not been excluded from an overall equation system. 

Next, the elements related to the stable reactants should be extracted from the vector of the right parts of the differential equations and the intermediate concentrations, which enter the equations, should be replaced with the obtained expressions. This stage of the s5mibolic evaluation is shown in Fig. 2.20. Thereby, by applying the steady-state concentration method, we obtained the reduced system of the differential equations describing the time-dependent concentration decrease or increase of the stable reactants. 

Figure B2.5.1 schematically illustrates a typical flow-tube set-up. In gas-phase studies, it serves mainly two purposes. On the one hand it allows highly reactive shortlived reactant species, such as radicals or atoms, to be prepared at well-defined concentrations in an inert buffer gas. On the other hand, the flow replaces the time dependence, t, of a reaction by the dependence on the distance v from the point where the reactants are mixed by the simple transfomiation with the flow velocity vy...

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6-9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is N-n(t), and the concentration of product states is N-pU). We assume that we can define the reactants and products as distinct macrostates that are separated by a transition state dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1). 

In the genuine low-temperature chemical conversion, which implies the incoherent tunneling regime, the time dependence of the reactant and product concentrations is detected in one way or another. From these kinetic data the rate constant is inferred. An example of such a case is the important in biology tautomerization of free-base porphyrines (H2P) and phtalocyanins (H2PC), involving transfer of two hydrogen atoms between equivalent positions in the square formed by four N atoms inside a planar 16-member heterocycle (fig. 42). 

Sections 3.1 and 3.2 considered this problem Given a complex kinetic scheme, write the differential rate equations find the integrated rate equations or the concentration-time dependence of reactants, intermediates, and products and obtain estimates of the rate constants from experimental data. Little was said, however, about how the kinetic scheme is to be selected. This subject might be dismissed by stating that one makes use of experimental observations combined with chemical intuition to postulate a reasonable kinetic scheme but this is not veiy helpful, so some amplification is provided here. 

The mass spectrometer sampling capillary or the dispersive infra-red analyzers used for continuous analysis and monitoring of the gas phase composition are situated between the reactor and the sampling valve, as close to the reactor as possible, in order to avoid any delay in the recording of changes in the composition of reactants or products. This delay should be taken into account when plotting simultaneously the time dependence of catalyst potential or current and gas phase concentration of the reactants or products. 

FIGURE 13.14 The characteristic shapes of the time dependence of the concentration of a reactant during a second-order reaction. The larger the rate constant, k, the greater is the dependence of the rate on the concentration of the reactant. The lower gray lines are the curves for first-order reactions with the same initial rates as for the corresponding second-order reactions. Note how the concentrations for second-order reactions fall away much less rapidly at longer times than those for first-order reactions do. 

Influence of the mode of operation on process performance. The mode of operation of stirred-tank reactors can also significantly affect reactor performance. The history of concentrations will be changed by the time policy of reactant(s) addition to the reaction mixture. In view of our very limited possibility of controlling of temperature in stirred-tank reactors, the temperature-time dependencies for different policies of dosing will also be different. For example, the result of nitration depends upon the method of addition of nitric acid to aromatics, and the choice which phase is dispersed and which is continuous. Consequently, if the reaction is concentration- or temperature-sensitive the result will be dependent on the mode of operation (see Example 5.3.1.5). 

Equation (3) in essence states that the rate of change of the concentration of A at time t is equal to that of B and that each of these changes at time t is proportional to the product of the concentrations of the reactants raised to the respective prowers. Note that CA[t) and CB( ) are time-dependent variables. As the reaction proceeds, both CA and CB( ) will decrease in magnitude. For simplicity, these concentrations can be denoted simply by Ca and CB, respectively. 

Expressions for the time dependence of reactant and product concentrations may be obtained in the usual fashion... 

Here fcd is the infinite-time specific rate, and D is the mutual diffusion coefficient of the reactants. Using the time-dependent specific rates, Schwarz reports an increase of molecular yields that is 2% at low concentration, and 5% at high concentration of solutes. 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

We consider an ensemble of reactants in the reduced state situated at the interface. Their concentration is kept constant by an efficient means of transport. We denote the perturbation describing the interaction between one reactant and the electrode by M(r,R). According to time-dependent first-order perturbation theory, the probability per unit time that a reactant will pass from the initial to the final state is ... 

More sophisticated calculations (14,20), using either stochastic Monte Carlo or deterministic methods, are able to consider not only different Irradiating particles but also reactant diffusion and variations In the concentration of dissolved solutes, giving the evolution of both transient and stable products as a function of time. The distribution of species within the tracks necessitates the use of nonhomogeneous kinetics (21,22) or of time dependent kinetics (23). The results agree quite well with experimental data. 

We seek to describe the time-dependent behavior of a metabolic network that consists of m metabolic reactants (metabolites) interacting via a set of r biochemical reactions or interconversions. Each metabolite S, is characterized by its concentration 5,(f) > 0, usually measured in moles/volume. We distinguish between internal metabolites, whose concentrations are affected by interconversions and may change as a function of time, and external metabolites, whose concentrations are assumed to be constant. The latter are usually omitted from the m-dimensional time-dependent vector of concentrations S(t) and are treated as additional parameters. If multiple compartments are considered, metabolites that occur in more than one compartments are assigned to different subscripts within each compartment. 

The time dependence of accumulation of product P ([P]) depends on the initial concentration of substrate (reactant), S ([So]) ... 

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