Consecutive reactions numerical example

Domino transformations combining two consecutive anionic steps exist in several variants, but the majority of these reactions is initiated by a Michael addition [1]. Due to the attack of a nucleophile at the 4-position of usually an enone, a reactive enolate is formed which can easily be trapped in a second anionic reaction by, for example, another n,(5-urisalurated carbonyl compound, an aldehyde, a ketone, an inline, an ester, or an alkyl halide (Scheme 2.1). Accordingly, numerous examples of Michael/Michael, Michael/aldol, Michael/Dieckmann, as well as Michael/SN-type sequences have been found in the literature. These reactions can be considered as very reliable domino processes, and are undoubtedly of great value to today s synthetic chemist... 

For multiple reactions, material balances must be made for each stoichiometry. An example is the consecutive reactions, A = B = C, for which problem P4.04.52 develops a closed form solution. Other cases of sets of first order reactions are solvable by Laplace Transform, and of course numerically. 

Equations 2.22 and 2.23 become indeterminate if kg = kA. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. Identical rate constants do exist. Examples include multifunctional molecules where reactions at physicaliy different but chemically similar sites can have the same rate constant. Polymerizations are an important example of consecutive reactions where it is common to assume that reaction rates are independent of the length of the polymer chain. Unlike analytical solutions, numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

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. 

There are numerous examples of consecutive reactions such as radioactive decay series, hydrolyses of dicarboxylic acid esters or tertiary alkyl halides, as well as nitrations of aromatics. Conversions of gases on catalyst surfaces are also examples of this. 

The next phase of the problem is to find those values for T and V that will give the lowest product cost. This is a problem in optimization rather than root-finding. Numerical methods for optimization are described in Appendix 6. The present example of consecutive, mildly endothermic reactions provides exercises for these optimization methods, but the example reaction sequence is... 

A. (a) Each ion below has two consecutive acid-base reactions (called stepwise acid-base reactions) when placed in water. Write the reactions and the correct symbol (for example, K2 or for the equilibrium constant for each. Use Appendix B to find the numerical value of each equilibrium constant. 

The quantity of the array x elements as well as the quantity of the right-hand values is equal to the quantity of equations in the system. Let s clarify how rkfixed works by an example of a numerical solution of the direct problem for a three-step consecutive chemical reaction (Fig. 3.7). 

For processes involving consecutive protonation and redox steps, even more complex reaction schemes such as fences, meshes, or ladders can be constructed. An example for the analysis of this kind of more complex reaction scheme by a stepwise dissection based on numerical simulation is given in the introduction... 

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