Irreversible first-order kinetics

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

Example 4.12 used N stirred tanks in series to achieve a 1000-fold reduction in the concentration of a reactant that decomposes by first-order kinetics. Show how much worse the CSTRs would be if the 1000-fold reduction had to be achieved by dimerization i.e., by a second order of the single reactant type. The reaction is irreversible and density is constant. 

Figure 2. Inhibition of eel AChE by ANTX-A(S) - the secondary plot. P, the first-order rate constant which was the rate of inhibition at that ANTX-A(S) concentration obtained from the primary plot (insert). The intercept on the 1/P axis is 1/k and the intercept on the 1/[I] axis is -1/K. Figure insert Progressive irreversible inhibition of eel AChE by ANTX-A(S). The inactivation followed first-order kinetics. ANTX-A(S) concentrations, xg/mL (A) 0.083 ( ) 0.166 (o) 0.331 ( ) 0.497 (V) 0.599 ( ) control. Each point represents the mean of 3 or 4 determinations.

As Levenspiel points out, the optimum size ratio is generally dependent on the form of the reaction rate expression and on the conversion task specified. For first-order kinetics (either irreversible or reversible with first-order kinetics in both directions) equal-sized reactors should be used. For orders above unity the smaller reactor should precede the larger for orders between zero and unity the larger reactor should precede the smaller. Szepe and Levenspiel (14) have presented charts showing the optimum size ratio for a cascade of two reactors as a function of the conversion level for various reaction orders. Their results indicate that the minimum in the total volume requirement is an extremely shallow one. For example, for a simple... 

Reagent A undergoes an essentially irreversible isomerization reaction that obeys first-order kinetics. 

For the case where all of the series reactions obey first-order irreversible kinetics, equations 5.3.4, 5.3.6, 5.3.9, and 5.3.10 describe the variations of the species concentrations with time in an isothermal well-mixed batch reactor. For series reactions where the kinetics do not obey simple first-order or pseudo first-order kinetics, the rate expressions can seldom be solved in closed form, and it is necessary to resort to numerical methods to determine the time dependence of various species concentrations. Irrespective of the particular reaction rate expressions involved, there will be a specific time... 

Let us now consider how the external surface concentrations can be eliminated when our reaction follows simple irreversible first-order kinetics. In this instance equation 12.4.20 becomes... 

For this situation an irreversible chemical reaction with first-order kinetics with respect to A and B has been used, where a very high value of the reaction rate constant has been taken to simulate an instantaneous... 

A,A-Dialkylformamide acetals (7) react with primary amines to give the corresponding amidines (8). Kinetics of the reaction of a range of such acetals with ring-substituted anilines—previously measured in neutral solvents such as methanol or benzene —have been extended to pyridine solution. In pyridine, the reactions are irreversible, with first-order kinetics in each reactant, and mechanistically different from those in non-basic solvents. Two mechanisms are proposed to explain Hammett plots for a range of anilines, in which the p value switches from negative to positive at a cr value of ca 0.5. The pyridine solvent substantially enhances the rate in the case of very weakly basic anilines. 

If the reaction is irreversible and follows first-order kinetics... 

Consider the simple unimolecular reaction of Eq. (15.3), where the objective is to compute the forward rate constant. Transition-state theory supposes that the nature of the activated complex. A, is such that it represents a population of molecules in equilibrium with one another, and also in equilibrium with the reactant, A. That population partitions between an irreversible forward reaction to produce B, with an associated rate constant k, and deactivation back to A, with a (reverse) rate constant of kdeact- The rate at which molecules of A are activated to A is kact- This situation is illustrated schematically in Figure 15.1. Using the usual first-order kinetic equations for the rate at which B is produced, we see that... 

The interaction between the kinetics, the initial composition and the measurements would be taken into account in a full analysis. This is important for a complete aliasing of kinetics can arise under special circumstances. For example, in a continuum of reactions [22] all of which were first order and irreversible, we might denote the fraction of material with rate constant in the interval (/c, k + dk) at time t by x(t, k) Xo(k) = Jt(0, k). Then for parallel first order reaction we would have... 

The NH acidities of some sterically hindered ureas, namely the ureido esters (93), have been reported.81 The kinetics and mechanism of the alkaline hydrolysis of urea and sodium cyanate, NaCNO, have been studied at a number of temperatures.82 Urea hydrolysis follows an irreversible first-order consecutive reaction path. Tetrahedral intermediates are not involved and an elimination-addition mechanism operates. Sodium cyanate follows irreversible pseudo-first-order kinetics. The decomposition of the carcinogen /V-mcthyl-/V-nitrosourca (19) was dealt with earlier.19 The pyrolysis of /V-acctylurca goes by a unimolecular first-order elimination reaction.83... 

Further chlorination reactions can also take place, but since they involve insignificant amounts of reactants they have been considered to be negligible. The kinetics of the process were studied by McMullin (1948), who showed that the chlorination of benzene (A), monochlorobenzene (B) and dichlorobenzene (C) is in all cases first-order and irreversible. 

This model assumes irreversible first-order kinetics for nitrification, denitrification, mineralization, immobilization, and plant uptake. It takes the following form, in which sinks and sources are aggregated ... 

The approach is developed for a fairly wide class of processes, i.e., the class of irreversible nonchain reactions characterized by first-order kinetics. Although this is not the most general case, it encompasses several real reactive processes. 

Mechanism-based inhibition should be irreversible. Dialysis, ultrafiltration, or washing the protein (e.g., by isolating microsomes by centrifugation and resuspending them in drug-free buffer) will not restore enzyme activity, and the inhibition is highly resistant to sample dilution. Mechanism-based inhibition should be saturable. The rate of inactivation is proportional to the concentration of the inactivator until all enzyme molecules are saturated, in accordance with Michaelis-Menten kinetics. Additionally, the decrease in enzymatic activity over time should follow pseudo-first-order kinetics. 

Consider an exothermic irreversible reaction with first order kinetics in an adiabatic continuous flow stirred tank reactor. It is possible to determine the stable operating temperatures and conversions by combining both the mass and energy balance equations. For the mass balance equation at constant density and steady state condition,... 

First order kinetics with respect to monomer conversion (no irreversible... 

Figure 5.20 shows concentration-time profiles for the decomposition of hydrocortisone butyrate at 60°C in a buffered aqueous propylene glycol (50 w/w%, pH 7.6). Consecutive, irreversible, first-order kinetic models [i.e., Equation (5.119a), Equation (5.119b), and Equation (5.119c)] fit reasonably well with the experimental... 

It is worthwhile to compare the conversion obtained in an isothermal plug flow reactor with that obtained in a CSTR for given reaction kinetics. A fair comparison is given in Fig. 7.3 for irreversible first-order kinetics by showing the conversion obtained in both reactors as a function of To- The conversion of A obtained in a plug flow reactor is higher than that obtained in a CSTR. This holds for every positive partial reaction order with respect to A. For multiple reactions selectivities and yield enter into the picture. 

The above analytical relations have been derived for irreversible first-order kinetics. Arbitrary kinetics do not generally allow an analytical solution. The continuity equation for A, written again for a slab geometry ... 

It is easily shown that, for irreversible first-order kinetics, Eqn. 7.118 reduces to 7.111. 

It follows from Eqn. 7.134 that for irreversible first-order kinetics the Hatta number, y, can be calculated from ... 

Fig. 7.14. Concentration profiles in a liquid phase with a homogeneous catalyst for irreversible first-order kinetics at different values of the Hatta number. Reactant A has to be transferred from...

Two prototype reaction examples (reversible first-order and irreversible second-order kinetics) were discussed to address issues of rounding when switching from deterministic variables to stochastic (i.e., conversion of real numbers to integers), as well as the thresholds of population sizes and transition probabilities to control accuracy in the first two moments of the population (mean and variance). Other more complex examples were also mentioned. The... 

Based on this premise, y-acetylenic GABA (IV) waB synthesized (11) and found to be an irreversible inhibitor of GABA-T, in vitro and in vivo (12). Thus, when GABA-T, partially purified from pig brain, is incubated for varying time periods with y-acetylenic GABA, a time-dependent inactivation process is observed which follows pseudo first-order kinetics. Enzyme half lives range from 28 minutes to 9 minutes with concentrations of inhibitor between 0.029 mM and 0.29 mM. Time dependent inactivation is... 

The multireaction model accounts for irreversible retention in two ways. First, as a sink term, Q, which represents a direct reaction between the solution phase C and Skr (e.g., precipitation/dissolution or immobilization) as a first-order kinetic process, where kin is the associated rate coefficient (h ). 

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