Reaction, consecutive

The simple consecutive reaction A — B — C yields the Jacobi matrix 

Both concentrations can be graphed in a reduced form depending on the ratio of the rate constants of the individual steps. The functions are given in Figs. 2.5 and 2.6. 

89) gives X- and K-diagrams in parametric form. The elimination of t using 

The eq. (2,90) above represents the solutions for the case The case X = 1 yields dependencies discussed in Example 2.17 in Section 2.2.1.1. The corresponding -diagram is given in Fig. 2.7 using 

Another example is a mechanism containing two linear independent steps, the latter including a back reaction A — B C 

In this section consecntive reactions will be investigated. In the previous two examples, maximization of the yield of component B could be achieved by maximizing the conversion of component A, which is equivalent to the minimization of the outlet concentration of component A. 

It is assumed that the reaction takes place according to the following scheme  

The overall transfer function between cc and c, can easily be obtained through linearization and Laplace transformation of Eqn. (12.27)  

Third-order systems are usually not difficult to control using standard-type controllers. 

If B is the preferred component, one should realize that the concentration of B has a maximum. To see this, first the responses of Ca and Cb to c, have to be derived. The response of ca to changes in c, was already given in Eqn. (12.4) and is the same in this case  

The rates of the forward and reverse reactions for the dimerization of proflavin (1), an antibacterial agent that inhibits the biosynthesis of DNA by intercalating between adjacent base pairs, were found to be 8.1 x 10 dm mok s (second order) and 2.0 X 10 s (first order), respectively. The equilibrium constant for the dimerization is therefore 

In general, biological prooesses have complex mechanisms and we need to know how to build an overall rate law from the rate law of each step of a proposed mechanism. 

A reactant commonly produces an intermediate, a species that does not appear in the overall chemical equation for the reaction but which has been invoked in the mechanism. Biochemical processes are often elaborate versions of this simple model. For instance, the restriction enzyme EcoRI catalyzes the cleavage of DNA at a specific sequence of nucleotides (at GAATTC, making the cut between G and A on both strands). The reaction sequence it brings about is 

We can discover the characteristics of this type of reaction by setting up the rate laws for the net rate of change of the concentration of each substance. 

In any sequence involving two or more reactions, in order to derive the rate law, separate kinetic equations must be written for each step in the mechanism  

The mechanistic steps of reaction (2.31) have three kinetic equations that serve to define the reaction (Mahler Cordes, 1966 Plowman, 1972)  

If ah the first-order and pseudo-first-order rate constants, k A, k,B, ka, and k are of the same order of magnitude, the explicit solution is not possible and the computer is the simplest means to arrive at a solution (Fig. 3). 

Although the explicit solution is not possible, several limiting cases provide solutions that are capable of simple interpretation. If 3 is much smaller than k, the rate of the product formation, dD/dt, is governed by the unimolecular breakdown of C, which in turn is formed rapidly and reversibly from A plus B kmeticahy the reaction is first order in either reactant  

If 3 ki s kn, the intermediate C is converted to D as rapidly as it is formed and the reaction is again first order in either reactant, despite the difference in the rate-fimiting step  

A characteristic kinetic feature of many complex reactions is the occurence of two or more steps following one another. Such reactions are known as consecutive. The first step of such a reaction yields a compound capable of further chemical conversions in the next step. The compounds formed in the subsequent reaction step are known as intermediates. 

An example of a consecutive reaction is that between sodium vapor and mercuric chloride vapor observed in highly rarefied flames (see Section II.6), i.e. 

The mechanism of this reaction involves two consecutive steps Na -b HgCla NaCl -f HgCl, 

The two-step nature of such a reaction is recognized by the existence of twa reaction zones that with maximal yield of reaction products and that with maximal yield of light. The intermediate in this reaction is the univalent radical HgCl. 

To understand the kinetic characteristics of consecutive reactions consider the general case of a two-step consecutive reaction that involves first the conversion of species A to B, and second of B to C, 

In addition to the situation of parallel reactions, it is also possible for the product of the reaction to react further to additional (usually undesired) products. This is termed reactions in series or consecutive reactions. Continuing with the combustion reaction as an example, we note that complete oxidation of a fuel may actually occur by partial oxidation of C to CO, followed by oxidation of CO to CO. In this case, the overall reaction must be written as 

Although this is described as fundamentally different from the situation with parallel reactions, both types of problems can be solved by defining two extents of reaction and 

TABLE 6.3. Example Stoichiometric Table for General Reactions in Series 

Species Inlet Stoichiometric Coefficient Reaction 1 Stoichiometric Coefficient Reaction 2 Outlet 

Assuming that A is the only component in the feed to the reactor, we obtain a stoichiometric table very similar to the one developed for the case of parallel reactions (Table 6.3). 

The equilibrium constant at 333 K in the numerical example considered above is 

Reactor design becomes more challenging when yield as well as conversion must be considered. One common situation in which this arises is when there are consecutive irreversible reactions such as the following  

The desired product is C. The undesired product is D. There are many important industrial examples of this type chlorination, oxidation, and nitration of a variety of hydrocarbons. The specific reaction rates for the first and second reactions are k and k2, respectively. 

There are several ways to define yield (or selectivity ) of the desired product C, as shown in Eq. (2.48). One is on the basis of the amount of A fed. The other is on the 

The desirable product C is produced by the first reaction whose rate depends on the concentrations of A and B in the reactor. But C is consumed by the second reaction whose rate depends on the concentrations of C and B in the reactor. 

This implies that they measured kt and not an integral multiple of it and that 

The reaction of hydrogen atoms with hydrazine has only been studied at temperatures between 25 and 200 °C53 54. At these temperatures the reaction proceeds via (5). This is confirmed by the observation that in the reaction D+N2H4 no NH2D could be detected this rules out reaction (4) at low temperatures. For the rate coefficient of reaction (5) Schiavello and Volpi54 quoted the Arrhenius expression 

Birse and Melville55 measured the disappearance of N2H4 in the presence of hydrogen atoms generated by mercury-sensitized photolysis of H255. They could not discriminate between reactions (4) and (5) and assumed that the H atoms were removed by reaction (4). In view of the more recent experiments with D atoms it is, however, more likely that they determined the rate of reaction (5). They report a rate coefficient 

If experimental data on Ca, Cq and Cs versus time are available, the assumed orders can be checked and the values of ki and kz can be obtained by fitting the experimental curves. 

The maximum in the Cq versus t curve is found by differentiating the equation for Q with respect to fand setting the result equal to zero  

Consecutive first order reactions. Concentration versus time profiles for various ratios of 

Dividing the rate equations by one another to find the point selectivity gives  

Integrating after switching to the conversion of A gives the integral selectivity  

Till now we have used only symbolic resources of Mathcad system to solve differential equations. We again accentuate, that all kinetic equations from above are the equations with separatable variables. User need to separate those variables without assistance to obtain a solution. And getting final solution is in fact an integration of both parts of the obtained equalities. 

We now will separately discuss the possibilities of Mathcad s symbolic transformations. Symbolic transformations in Mathcad became possible after authors had instilled the core of Maple V R4 package symbolic operations in the program. But, symbolic core was instilled in a slightly topped version, apparently in order not to overload the package - only the simplest symbolic constructions could be calculated. Unfortunately, functions present in Maple to solve differential equations were not included in the list of functions, available for work in MathCAD. 

Therefore we need to accept, that many kinetic problems could not be solved analytically in this package. And it is often very important to know analytic expressions for the time-dependences of current concentrations in particular. 

In this case Maple system opens wide facilities for the analysis of mathematic models of complex reactions, primarily owing to the built-in function dsolve for solution of ordinary differential equations and sets. 

We need to apply dsolve in following format in order to find partial solution of differential equation  

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The EROS (Elaboration of Reactions for Organic Synthesis) system [26] is a knowledge-based system which was created for the simulation of organic reactions. Given a certain set of starting materials, EROS investigates the potential reaction pathways. It produces sequences of simultaneous and consecutive reactions and attempts to predict the products that will be obtained in those reactions. 

Equation 1 is referred to as the selective reaction, equation 2 is called the nonselective reaction, and equation 3 is termed the consecutive reaction and is considered to proceed via isomerization of ethylene oxide to acetaldehyde, which undergoes rapid total combustion under the conditions present in the reactor. Only silver has been found to effect the selective partial oxidation of ethylene to ethylene oxide. The maximum selectivity for this reaction is considered to be 85.7%, based on mechanistic considerations. The best catalysts used in ethylene oxide production achieve 80—84% selectivity at commercially useful ethylene—oxygen conversion levels (68,69). 

Dehydration of ethanol has been effected over a variety of catalysts, among them synthetic and naturally occurring aluminas, siUca-aluminas, and activated alumina (315—322), hafnium and 2irconium oxides (321), and phosphoric acid on coke (323). Operating space velocity is chosen to ensure that the two consecutive reactions. 

Experiments with cyclic thioethers (80JCS(P1)1693), thiourea, thiocyanate and ethyl xan-thate always led to destruction of oxaziridines (73AJC2159). Products of complicated consecutive reactions could be isolated but only with some difficulty, e.g. (92) from a reaction with carbon disulfide (74JOC957), and (93), obtained by trapping with butadiene a product of a reaction between an oxaziridine and a thiirane (80JOC1691). 

Selectivity A significant respect in which CSTRs may differ from batch (or PFR) reaclors is in the product distribution of complex reactions. However, each particular set of reactions must be treated individually to find the superiority. For the consecutive reactions A B C, Fig. 7-5b shows that a higher peak value of B is reached in batch reactors than in CSTRs as the number of stages increases the batch performance is approached. 

For the consecutive reactions A B C, a higher yield of intermediate B is obtained in batch reac tors or PFRs than in CSTRs. 

Tubular flow reactors—minimum volume for second-order reversible reactions, maximum yield of consecutive reactions, minimum cost with and without recycle, and maximum profit with recycle... 

Selectivity of parallel and consecutive reactions and of reac tions in a porous catalyst... 

Maximum eld of first-order consecutive reactions in CSTB by apphcation of Lagrange multiphers... 

For the consecutive reactions 2A B and 2B C, concentrations were measured as functions of residence time in a CSTR. In all experiments, C o = 1 lb moPfF. Volumetric flow rate was constant. The data are tabulated in the first three columns. Check the proposed rate equations,... 

Present research is devoted to investigation of application of luminol reactions in heterogeneous systems. Systems of rapid consecutive reactions usable for the determination of biologically active, toxic anions have been studied. Anions were quantitatively converted into chemiluminescing solid or gaseous products detectable on solid / liquid or gas / liquid interface. Methodology developed made it possible to combine concentration of microcomponents with chemiluminescence detection and to achieve high sensitivity of determination. 

The perspective of using consecutive reactions is grounded on the example of the analysis of isomeric mono-nitrophenols and anion surface-active substances. The variants of systematic analysis of mixtures of tri-, di- and mono-nitrophenols, anion surface-active substances, based on the combination of measurements of consecutively received extracts at different pH values are discussed. 

On Figure 6.1.1, the four consecutive reaction steps are indicated on a vertical scale with the forward reaction above the corresponding reverse reaction. The lengths of the horizontal lines give the value of the rate of reaction in mol/m s on a logarithmic scale. In steady-state the net rates of all four steps must be equal. This is given on the left side with 4 mol/m s rate difference, which is 11 mm long. The forward rate of the first step is 4.35 molW s and the reverse of the first reaction is only 0.35 mol/m s, a small fraction of the forward rate. 

The formation of resins, tarry matter by consecutive reaction, is prevalent in organic reactions. Figure 3-13a shows the time variations in the concentrations of A, B, and C as given by these equations. The concentration of A falls exponentially, while B goes through a maximum. Since the formation rate of C is proportional to the concentration of B, this rate is initially zero and is a maximum when B reaches its maximum value. 

The computer results from Table 5-13 show the calculated compositions of benzene, diphenyl, triphenyl, and hydrogen. At a fixed feedrate, increasing V/F values correspond to movement through the plug flow reactor (i.e., increasing reactor volume). Thus, these results illustrate how the composition varies with position in the reactor. Here, the mole fraction of benzene decreases steadily as the reaction mixture progresses in the reactor, while the composition of diphenyl increases and reaches a maximum between 1,684 and 1,723 hr and thereafter decreases. This is often typical of an intermediate in consecutive reactions. 

Consecutive reactions are those in which the product of one reaction is the reactant in the next reaction. These are also called series reactions. Reversible (opposing) reactions, autocatalytic reactions, and chain reactions can be viewed as special types of consecutive reactions. 

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

Applications have been made to consecutive reactions,with several methods being developed to extract the rate constants. Consider Scheme XIV. 

A kinetic scheme that is fully consistent with experimental observations may yet be ambiguous in the sense that it may not be unique. An example was discussed earlier (Section 3.1, Consecutive Reactions), when it was shown that ki and 2 in Scheme IX may be interchanged without altering some of the rate equations this is the slow-fast ambiguity. Additional examples of kinetically indistinguishable kinetic schemes have been discussed.The following subsection treats one aspect of this problem. 

These statements refer to an elementary reaction, which from this point of view may be defined as a reaction possessing a single transition state. A complex reaction is then a set of elementary reactions, the potential energy surface of the whole being continuous. Thus, for two consecutive reactions the product of the first reaction is the reactant of the second. Each reaction has its own transition state. 

This argument can be extended to consecutive reactions having a rate-determining step. - P The composition of the transition state of the rds is given by the rate equation. This composition includes reactants prior to the rds, but nothing following the rds. Thus, the rate equation may not correspond to the stoichiometric equation. We will consider several examples. In Scheme IV a fast acid-base equilibrium precedes the slow rds. 

An alkene activated by an electron-withdrawing group—often an acrylic ester 2 is used—can react with an aldehyde or ketone 1 in the presence of catalytic amounts of a tertiary amine, to yield an a-hydroxyalkylated product. This reaction, known as the Baylis-Hillman reaction, leads to the formation of useful multifunctional products, e.g. o -methylene-/3-hydroxy carbonyl compounds 3 with a chiral carbon center and various options for consecutive reactions. 

Virtually any aldehyde or ketone and any CH-acidic methylene compound can be employed in the Knoevenagel reaction however the reactivity may be limited due to steric effects. Some reactions may lead to unexpected products from side-reactions or from consecutive reactions of the initially formed Knoevenagel product. 

The reaction of a cyclic ketone—e.g. cyclohexanone 1—with methyl vinyl ketone 2 resulting in a ring closure to yield a bicyclic a ,/3-unsaturated ketone 4, is called the Robinson annulation This reaction has found wide application in the synthesis of terpenes, and especially of steroids. Mechanistically the Robinson annulation consists of two consecutive reactions, a Michael addition followed by an Aldol reaction. Initially, upon treatment with a base, the cyclic ketone 1 is deprotonated to give an enolate, which undergoes a conjugate addition to the methyl vinyl ketone, i.e. a Michael addition, to give a 1,5-diketone 3 ... 

Compared with uncatalyzed reactions, catalysts introduce alternative pathways that, in nearly all cases, involve two nr more consecutive reaction steps. Each of these steps has a lower activation energy than does the uncatalyzed reaction. We can nse as an example the gas phase reaction of ozone and oxygen atoms. In the homogeneons uncatalyzed case, the reaction is represented to occur in a single irreversible step that has a high activation energy ... 

Electron transfer to a suitable monomer leads to the formation of an ion pair like M, NaL The negative monomer" ion may react again with sodium forming a Na—M—Na unit. The latter arises, therefore, as a product of two consecutive reactions, e.g.,... 

The procedure for solving the relations between concentrations has been used in kinetic studies of complex catalytic reactions by many authors, among the first of them being Jungers and his co-workers 17-20), Weiss 21, 22), and others [see, e.g. 23-25a). In many papers this approach has been combined with the solution of time dependencies, at least for some of the single reactions. Also solved were some complicated cases [e.g. six-step consecutive reaction 26,26a) 3 and some improvements of this time-elimination procedure were set forth 27). The elimination of time is... 

Concerning consecutive reactions, a typical example is the hydrogenation of alkynes through alkenes to alkanes. Alkenes are more reactive alkynes, however, are much more strongly adsorbed, particularly on some group VIII noble metal catalysts. This situation is illustrated in Fig. 2 for a platinum catalyst, which was taken from the studies by Bond and Wells (45, 46) on hydrogenation of acetylene. The figure shows the decrease of... 

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