First-order reactions INDEX

The formation of the major UV degradation peak at about 287 nm in the weathered PC appears to correlate well with the formation of the yellow color in the weathered sample. In Figure 8 the formation of both the peak at 287 nm and the yellow color have been assumed to be products of a first order reaction. This figure shows a plot of the log of the percentage of a scaling constant minus the yellowness index divided by the constant, versus a measurement of the exposure. In this case, the exposure is expressed as cal/cm2, obtained from ENC0N data. 

Here c(x, t)dx is the concentration of material with index in the slice (x, x + dx) whose rate constant is k(x) K(x, z) describes the interaction of the species. The authors obtain some striking results for uniform systems, as they call those for which K is independent of x (Astarita and Ocone, 1988 Astarita, 1989). Their second-order reaction would imply that each slice reacted with every other, K being a stoichiometric coefficient function. Only if K = S(z -x) would we have a continuum of independent parallel second-order reactions. In spite of the physical objections, the mathematical challenge of setting this up properly remains. Ho and Aris (1987) have shown how not to do it. Astarita and Ocone have shown how to do something a little different and probably more sensible physically. We shall see that it can be done quite generally by having a double-indexed mixture with parallel first-order reactions. The first-order kinetics ensures the individuality of the reactions and the distribution... 

Since the dimensionless time for a first-order reaction is the product of the reaction time t and a first-order rate constant k, there is no reason why k(x)t should not be interpreted as k(x)t(x), that is, the reaction time may be distributed over the index space as well as the rate constant. Alternatively, with two indices k might be distributed over one and t over the other as k x)t(y). We can thus consider a continuum of reactions in a reactor with specified residence time distribution and this is entirely equivalent to the single reaction with the apparent kinetics of the continuum under the segregation hypothesis of residence time distribution theory, a topic that is in the elementary texts. Three indices would be required to distribute the reaction time with a doubly-distributed continuous mixture. 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

When the Dirac delta distribution is placed closer to the permeate side (i.e., a subsurface step distribution) of an CMR, the total conversion is actually lower than that with a uniform catalyst distribution (Figure 9.7). For a performance index other than the total conversion (such as product purity or product molar flow rate), the optimal distribution of the catalyst concentration can be rather complex even for reversible first-order reactions as displayed in Figure 9.8. 

Figure 6 (7). The catalyst activity index, a, is the isomerization rate calculated from the reaction time and the octane numbers of the product, a first-order reaction being assumed.

Total height of the holding vessel including 40% excess volume (m) Height of tapered portion of the primary fluid nozzle (m) Consistency index for pseudo-plastic non-Newtonian liquid (Pa s") First-order reaction rate constant (m /kg catalyst s)... 

The gaseous and solid reactants are denoted as A and B respectively it is assumed that they react in a stoichiometric ratio of one. A first order reaction at the massive core surface (index s) is assumed, its rate constant is k. ... 

Those experimentalists who use spectrophotometry or spectrofluorimetry to measure rates of biochemical reactions should always be mindful that bubble clearance frequently displays first-order kinetics. This applies to bubbles adhering to the inside wall of the cuvette as well as bubbles released from solution itself. The presence of bubbles within a cuvette may introduce artifactual kinetic behavior resulting (a) from refractive index differences between the gas trapped in the bubbles and that of the test solution, and (b) from the high reflectance of the air/water interface surrounding some bubbles. 

As can be seen from the table, the values of gRP do indeed remedy the insufficiency of the first order index r and correctly predict the reaction preferred... 

This approach is based on the introduction of molecular effective polarizabilities, i.e. molecular properties which have been modified by the combination of the two different environment effects represented in terms of cavity and reaction fields. In terms of these properties the outcome of quantum mechanical calculations can be directly compared with the outcome of the experimental measurements of the various NLO processes. The explicit expressions reported here refer to the first-order refractometric measurements and to the third-order EFISH processes, but the PCM methodology maps all the other NLO processes such as the electro-optical Kerr effect (OKE), intensity-dependent refractive index (IDRI), and others. More recently, the approach has been extended to the case of linear birefringences such as the Cotton-Mouton [21] and the Kerr effects [22] (see also the contribution to this book specifically devoted to birefringences). 

The rate equation 5.5 and the first-order plot remain valid in terms of any physical variable f that is a linear function of the concentrations. This is usually true for properties such as refractive index, electric conductivity, specific gravity, and rotation of the plane of polarized light. In all such cases, x is defined as before by eqn 5.3, but with f instead of Ct. Equation 5.5 remains valid even if the variable changes its sign in the course of the reaction, as it might with rotation of the plane of polarized light. 

When the decay of intermediate B is faster than its formation, kBC > kAB, the absolute value of the pre-exponential term will become small and thus the transient concentration cB will be small at all times. For kBC 3> kAB, the appearance of product C will approach the first-order rate law, Equation 3.6 (replace the index B by C), because the transient concentration of intermediate B becomes negligible. This shows that observation of a first-order rate law for the reaction A —> C does not guarantee that there is no intermediate involved, that is, that the observed reaction A —> C is an elementary reaction. 

The reaction rate increases with substrate concentration to the point where the available enzyme is saturated and v = Emax the reaction becoming zero order (Fig. 9.2). When [S] Km the reaction becomes first order, v = En,ax/ m[S] assuming [ , is constant. When [S] = Km, v = Emax/2. The parameter Km is also an index of the slope of the v/[S] relation. 

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