Stages nonlinear reactions

Figure 5 shows the effect of thickness on the loss of conductivity at 150 As in Figure 1, the conductivity decay, Ao/o , is proportional to in the initial stages of reaction. Attempts to treat the data as for a first order reaction, in the manner of Samuelson and Druy (75), were unsuccessful, particularly at the higher temperatures (120 and ISO " C). As with several other conducting polymers (75,18,19), the first-order kinetic plots, ln(a/o ) vs t, were nonlinear at high temperature (> 120 C). The inherent assumption in the first order plot is that a is directly proportional to the concentration of conducting species. 

The reason for constructing this rather complex model was that even though the mathematical equations may be easily set up using the dispersion model, the numerical solutions are quite involved and time consuming. Deans and Lapidus were actually concerned with the more complicated case of mass and heat dispersion with chemical reactions. For this case, the dispersion model yields a set of coupled nonlinear partial differential equations whose solution is quite formidable. The finite-stage model yields a set of differential-double-difference equations. These are ordinary differential equations, which are easier to solve than the partial differential equations of the dispersion model. The stirred-tank equations are of an initial-value type rather than the boundary-value type given by the dispersion model, and this fact also simplifies the numerical work. 

Chemical processes in condensed media often cannot be reduced to simple mono- and bimolecular reactions simply because chains of reaction take place. Therefore their kinetics is described by a set of ordinary differential equations (2.1.1) which are generally nonlinear due to bimolecular stages. Independent variables n ( ), i = 1,..., s (intermediate reactions products) define a number of equations under study. 

When a reaction occurs in an ideal system (i.e., in ideal gas mixture, ideal solution, or ideal adsorbed layer), then rs and r s in (44) are determined by simple mass action law. We shall call linear the stages whose rate, = rs - r s, depends linearly on the concentrations of intermediates (including free sites of the surface) the stages whose rate depends nonlinearly on the concentrations of intermediates (i.e., includes squares of concentrations of... 

A reaction mechanism, all stages of which are linear, will be called linear. For such a mechanism, (44) produces a linear set that always has one solution. This solution can be obtained algebraically in an explicit form. If the reaction mechanism is nonlinear (i.e., if it includes nonlinear stages along with linear ones), the existence of several solutions of a system of (44) (i.e., of several steady states of the reaction) is possible in some cases (28). Sometimes the steady-state course of a reaction is not reached at all and sustained oscillations of the rate (29) or a continuous acceleration of the reaction of an exponential type (30) occur. In the industrial catalytic processes discussed below, these possibilities are not realized even in the cases when the mechanisms are nonlinear therefore, it is not expedient to discuss these possibilities here in more detail. 

A so-called kinetic assay, in which the reaction rate is followed continuously, is advantageous because it is possible to observe directly the linearity or nonlinearity of the response with respect to time. Many enzyme assays, however, are based on a single measurement at a defined time, a so-called fixed-time assay. It is usually not possible to predict the appropriate amount of enzyme in either kinetic or fixed-time assays to obtain an optimum velocity like that of Assay 2 in Figure 11-14. This may be empirically determined by a dilution experiment in two stages. At first, constant volumes of serial 10-fold dilutions of enzyme are assayed to find the range of dilution in which the calculated activity is maximal and constant (see Figure 11-15). 

Next, a second and third series of reaction mixtures should be prepared, with enzyme added at concentrations of half and twice the value used in the first. These reactions are started and sampled, chromatograms are obtained, and the data are plotted as a function of reaction time. At this early stage in the optimization of the assay, it is advisable to continue sampling one of the incubations until the rate of product formation becomes nonlinear or the amount of substrate present is exhausted. This prolonged incubation provides information about the extent of the primary reaction and also allows any secondary reactions to take place and form enough products to be detectable. 

These equations are applicable for a reaction proceeding under pseudo-first-order conditions, i.e. when the concentration of the solute species is constant right up to the gas/liquid interface. It is thus possible to examine the possibility that reaction may occur in a film for the catalyst reoxidation and reduction reactions separately, if the two-stage redox mechanism is appropriate. The penetration theory leads to a series of coupled nonlinear partial differential equations which have to be solved numerically with appropriate boundary conditions. For example, if y is the distance from the melt surface, the equation governing the concentration of species B in time and space is given in (15). 

In certain reactions nonlinearity stems from the fact that a product feeds back to a particular stage of a reaction and either activates or inhibits the reaction causing a form of nonlinearity in the reaction scheme. Higgins (1964) has formalized these concepts in his paper. 

Further, nonlinear model fitting procedure for PE and PE-n-MMT TGA-curves has led to the following triple-stage model scheme of successive reactions (Figure 6 a, b) ... 

With best fidelity, the undertaken nonlinear model fitting for the stabilized samples of PE and PE-n-MMT has provided a triple-stage model scheme of successive reactions, wherein an nth-order autocatalysis reaction (Cn) was used at the first step, while a general nth-order (Fn) reaction was used for both the second and the third steps of the overall process of thermal oxidative degradation (Table 1) ... 

If a FIA procedure is required to reach the same level of sensitivity as a batch procedure, two obstacles have to be overcome the short reaction time, which is due to the short residence time and may result in a relatively low yield of reaction product, and an excessive dispersion of the sample zone, which results in an unwanted dilution of the species to be measured. Leaving aside at this stage the problem of too short a residence time, it might be helpful to consider an approach by means of which the dispersion can be minimized, yet still remain sufficient for supplying the middle of the sample zone with an adequate amount of reagent. Obviously, lack of reagent in the center of the sample zone will result in the absence of product to be sensed (cf. Chapters 2 and 3). Thus, instead of a sharp smooth peak, one will obtain either a double peak —which may be well pronounced or may merely appear as a noisy peak signal—or a nonlinear... 

If a linear-response instrument is used, the A[S] term in this equation simply becomes AS jv with a nonlinear-response instrument A[S] becomes [/( ) — /( )]. Thus, a linear relationship between [E]o and 1/Ar is independent of the linearity of the instrumental response and measurements need not be made during the initial stages of the reaction. 

Despite this, PPC is inherently suited for nonlinear mixed effect modeling of pharmacokinetic data as it utilizes the posterior distribution of parameter estimates to examine whether the salient features of the original data are observed in the derived (replicated) data. Berin and Rubin applied this approach to mixture models of reaction time based on visual tracking experiments conducted in patients with and without schizophrenia. Their application of PPC was directed in the modelbuilding stage and not implemented to validate their final model per se. This technique has appeal in both settings as it can serve as a metric in the establishment of a credible model. 

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