Irreversible First-Order Consecutive Reactions

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... 

Example 2.1 is the following irreversible first order consecutive reaction... 

The concentrations of reactant and products at the outlet of a packed bed reactor can be easily calculated with the mass balances for each compound supposing ideal plug flow behavior. For irreversible first-order consecutive reactions (Eq. (11.5)), the concentrations at the reactor outlet depend on the inlet concentration, Cj g, the rate constant, and the residence time, r. For reaction systems with constant fluid density, the residence time corresponds to the space-time defined as, r = V/Vg, with V the reactor volume and Vq the volumetric inlet flow. The space time... 

It is also noteworthy that the valne of 10 at 60% v/v CH3CN and, consequently, one might tend to believe that the rate of reaction should follow a simple first-order rate law at 60% v/v CH3CN because k3 is 10-fold larger than kj. Bnt the practical reality is that althongh the observed data do follow an apparent first-order rate law, the calculated kinetic parameters are very unreliable, and the observed data fit equally weU to a kinetic equation derived for a two-step irreversible first-order consecutive reaction. Thus, it appears that the reliable fit of observed data to a kinetic equation is a necessary but not sufficient requirement for the correctness of the kinetic equation. The reliability of the calculated kinetic parameters from the kinetic equation must also be tested if possible. 

To illustrate, a chemical example in which the rate of a reaction follows an irreversible two-step first-order consecutive reaction path is the spectrophotomet-ric kinetic study on the aqueous cleavage of Al-methylphthalamic acid (R) in mixed H2O-CH3CN solvent. The rate of hydrolysis of R at 0.03-M HCl has been carried out spectrophotometrically by monitoring the change in absorbance (A bs) 310 nm with a change in the reaction time (t). The hydrolysis in mixed acidic water-acetonitrile solvent follows the reaction scheme as expressed by Equation 7.42... 

This consists of two consecutive irreversible first-order (or pseudo-first-order) reactions. The differential rate equations are... 

Generalization of Scheme X to any number of consecutive irreversible first-order reactions is obviously possible, although the equations quickly become very cumbersome. However, Eqs. (3-42) and (3-44) reveal patterns in their form, and West-man and DeLury have developed a systematic symbolism that allows the equations to be written down without integration. 

Consecutive reactions figure prominently in Part II. Since complex ions have a number of reactive centers, the product of one reaction may very well take part in a subsequent one. The simplest and very common, but still surprisingly involved, sequence is that of two irreversible first-order (1.66) or pseudo first-order (with X and Y in large excess) (1.67) reactions. 

Figure 8 shows a plot of the concentration of methanol produced by the hydrolysis of SiQAC at pH 4.07 in water and the nonlinear regression curve of equation (18) assuming three consecutive, irreversible first-order reactions. A summary of the observed rate constants at each pH studied is shown in Table 4. Regression fits produced R2 values of better than 0.99 for all the pH values investigated. Plots of the observed values of k, k2, and k3 vs. pH are linear in all cases, with R2 values greater than 0.99, and with slopes of -0.997, -0.992 and -0.999, respectively. The ratio of kt k2 k3 is approximately 20 3 1. 

Figure 16. Product distribution for a consecutive irreversible first order reaction A -> R -> S as a function of the space time.

Let us now apply the above birth model to a well-known process, i.e., a consecutive-irreversible z-stage first order chemical reaction, with a single initial substance, the "first member of the family". The various states are Sj s Aj (i = 0,... 

The simple form of Eq. (2-91) shows that selectivity and yield calculations can advantageously be carried out by dividing the rate for one reaction by that for another, eliminating time in the process. Since yield and selectivity are usually more important than total conversion for complex-reaction systems, this procedure will be emphasized in the following section. The possible combinations of simultaneous, parallel, and consecutive reactiom are very large. A few irreversible first-order cases will be analyzed in Sec 2-10 to illustrate the method of approach. Then in Sec. 2-11 a different type of complex system, chain reactions, will be discussed. 

Fig. 7.2 Plots of the concentration of A, B, and C against time for a consecutive reaction scheme with irreversible first-order processes. The solid lines show the results based on the exact solution assuming ki = s and 2 = 100 s the points were calculated on the basis of the steady-state approximation (equations (7.2.31) and (7.2.32)).

If the intrinsic reaction rate is fast compared to the internal and/or external mass transfer processes, the reactant concentration within the porous catalyst and on its outer surface is smaller compared to the bulk concentration, whereas the concentration of the intermediate will be higher. Consequently, the consecutive reaction is promoted and the yield diminishes. The degree of yield losses depends on the ratio between transfer time and the intrinsic rate of the consecutive reaction, which is characterized by the corresponding Thiele moduli and Damkohler numbers referred to the consecutive reaction. For irreversible first-order reactions, the equations are as follows ... 

Gas (or gas with homogeneous catalyst) heat of reaction endothermic reaction rate, fast capacity 0.001-200 L/s good selectivity for consecutive reactions and irreversible first order volume of reactor 1-10000 L OK for high pressures or vacuum. For temperatures < 500 °C. For temperatures > 500 °C use fire tube. For example, used for such homogeneous reactions as acetic acid cracked to ketene. Liquid (or liquid with homogeneous catalyst) heat of reaction endothermic reaction rate, fast or slow capacity 0.001-200 L/s good selectivity for consecutive reactions volume of reactor 1-10000 L OK for high pressures. For temperatures... 

Gas with fixed bed of solid catalyst heat of reaction endothermic or slightly exothermic reaction rate, fast good selectivity and activity for consecutive reactions and for irreversible first order reactions volume of reactor 1-10000 L OK for high pressures. High conversion efficiency, simple, flexible, gives high ratio of catalyst to reactants. 

To discuss the influence of internal transport processes on consecutive reactions, we assume simple irreversible first order reactions. With and k2 being the intrinsic rate constants, the production rate of A2 and the disappearance of Aj are given by ... 

A simple consecutive reaction (the case of two consecutive irreversible first-order reactions is selected as the first demonstration) ... 

Consecutive reactions (two irreversible first order steps)... 

It has been shown by Wei (1962) that any complex network of first-order reactions consisting of combinations of consecutive and parallel paths including reversible paths can be decomposed into a system involving only independent parallel reaction paths of irreversible first-order reactions. Let rc be a vector of rate expressions corresponding to a vector of concentrations C for all species. Then ... 

In 89.8% HaS04, diacetylmesitylene undergoes two successive protodeacetylations and a final sulphonation to give mesitylenesulphonic acid. A kinetic analysis of the component reactions has been reported as an example of consecutive, irreversible, first-order reactions in which each of the three steps is much slower than the subsequent one. ... 

The schemes considered are only a few of the variety of combinations of consecutive first-order and second-order reactions possible including reversible and irreversible steps. Exact integrated rate expressions for systems of linked equilibria may be solved with computer programs. Examples other than those we have considered are rarely encountered however except in specific areas such as oscillating reactions or enzyme chemistry, and such complexity is to be avoided if at all possible. 

Irreversible Reactions in Series. We first consider consecutive unimolecular-type first-order reactions such as... 

To introduce the appropriate features and concepts of reaction in flow systems, we start by considering the simplest irreversible examples—a single first-order step and then two consecutive first-order steps. 

For the general case of trialkoxysilanes, although the hydrolysis reaction is reversible, under the conditions employed it may be considered as three consecutive, irreversible pseudo-first-order reactions shown in the following equations ... 

The enhancement factors are either obtained by fitting experimental results or are derived theoretically on the grounds of simplified model assumptions. They depend on reaction character (reversible or irreversible) and order, as well as on the assumptions of the particular mass transfer model chosen [19, 26]. For very simple cases, analytical solutions are obtained, for example, for a reaction of the first or pseudo-first order or for an instantaneous reaction of the first and second order. Frequently, the enhancement factors are expressed via Hatta-numbers [26, 28]. They can be used in combination with the HTU/NTU-method or with a more advanced mass transfer description method. However, it is generally not possible to derive the enhancement factors properly from binary experiments, and a theoretical description of reversible, parallel or consecutive reactions is based on rough simplifications. Thus, for many reactive absorption processes, this approach appears questionable. 

Let us apply these general first-order solutions to two specific reactions irreversible consecutive and simple reversible reactions. Consecutive reactions (A —> B —> C) do not have reverse reactions. This leads to ... 

In order to avoid the restrictions to complicated adsorptive reactions in the MOC3D, Selim et al. (1990) developed a simulation system based on the multireaction model (MRM) and multireaction transport model (MRTM). The MRM model includes concurrent and concurrent-consecutive retention processes of the nonlinear kinetic type. It accounts for equilibrium (Freundlich) sorption and irreversible reactions. The processes considered are based on linear (first order) and nonlinear kinetic reactions. The MRM model assumes that the solute in the soil environment is present in the soil solution and in several phases representing retention by various soil... 

Type of Reaction and Application. An increased emphasis on gas-solid reactions has been evident for about a decade. Three of the papers in this symposium treat gas-solid reactions, two (13,18) dealing with coal combustion and the other (11) with catalyst regeneration. Of the four papers which consider solid-catalysed gas-phase reactions, one (15) deals with a specific application (production of maleic anhydride), and one (12) treats an unspecified consecutive reaction of the type A B C the other two (14,16) are concerned with unspecified first order irreversible reactions. The final paper (17) considers a relatively recent application, fluidized bed aerosol filtration. Principles of fluid bed reactor modeling are directly applicable to such a case Aerosol particles disappear by adsorption on the collector (fluidized) particles much as a gaseous component disappears by reaction in the case of a solid-catalysed reaction. 

For the situation in which each of the series reactions is irreversible and obeys a first-order rate law, eqnations (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 consecutive reactions in which all of the reactions 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 at which the concentration of a particular intermediate passes through a maximum. If interested in designing a continuous-flow process for producing this species, the chemical engineer must make appropriate allowance for the flow conditions that will prevail within the reactor. That disparities in reactor configurations can bring about wide variations in desired product yields for series reactions is evident from the examples considered in Illustrations 9.2 and 9.3. 

Figure 9.6 Dimensionless representation of product distributions for consecutive first-order irreversible reactions A -> V -> W (stirred-tank reactor).

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