Reaction independent

The independent reaction network is similar to the parallel network, with one important difference. The reactants are not the same. Reactants A and B above are different compounds. There can be more than one reactant in each reaction. However, there are no common reactants in the two reactions. In general, the products of the two reactions will be different. However, this is not a necessary condition. 

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In section 11.3 vie showed that the difficult problem of solving the flux relations can be circumvented rather simply when the stoichiometric relations are satisfied by the flux vectors, but the treatment given there was limited to the case of a single Independent chemical reaction, when the stoichiometric relations permit all the flux vectors to be expressed in terms of any one of them. The question then arises whether any comparable simplification is possible v en the reactants participate in more than one independent reaction. 

For ease of exposition, let us limit attention to. two independent reactions--the generalization to more reactions is straightforward. Then the material balance equations take the form... 

The procedure in the case of two independent reactions should be compared with that described earlier for a single reaction. When there is a single reaction, p, X2 ...,x are related to by solution of an... 

Matrix methods, in particiilar finding the rank of the matrix, can be used to find the number of independent reactions in a reaction set. If the stoichiometric numbers for the reactions and molecules are put in the form of a matrix, the rank of the matrix gives the number or independent reactions (see Ref. 13). 

The resulting set of / equations is a complete set of independent reactions. More than one such set is often possible, but all sets number / and are equivalent. 

In this procedure, the question of what chemical reactions are involved never enters directly into any of the equations. However, the choice of a set of species is entirely eqmvalent to the choice of a set of independent reactions among the species. In any event, a set of species or an equivalent set of independent reactions must always be assumed, and different assumptions produce different results. 

FIGURE 22.4 The light-dependent and light-independent reactions of photosynthesis. Light reactions are associated with the thylakoid membranes, and light-independent reactions are associated with the stroma. 

More realistically, the catalyst is generated in an independent reaction. Consider a reaction with two paths—a direct reaction between A and B and a product-catalyzed one, as shown ... 

The Diels Alder reactions of maleic anhydride with 1,3-cyclohexadiene, as well the parallel reaction network in which maleic anhydride competes to react simultaneously with isoprene and 1,3-cyclohexadiene [84], were also investigated in subcritical propane under the above reaction conditions (80 °C and 90-152 bar). The reaction selectivities of the parallel Diels-Alder reaction network diverged from those of the independent reactions as the reaction pressure decreased. In contrast, the same selectivities were obtained in both parallel and independent reactions carried out in conventional solvents (hexane, ethyl acetate, chloroform) [84]. 

Lastly, we isolated a tetrabromo compound 21 (ref. 10) which is derived from solvent. IH and NMR spectra of 21 indicates the formation of a highly symmetrical compound whose configuration is not known. Possible configurations are given on scheme 10. On an independent reaction we treated decalin with bromine at high temperature and obtained 21 in high yield. 

Therefore, such sequential in situ reactions are always carried out either in order to prepare a substance for a color reaction that is to follow or to increase the amount of information that is obtained by exploiting a combination of different independent reactions. This provides information that could not be obtained using one single reagent. 

Equilibrium Compositions for Multiple Reactions. When there are two or more independent reactions. Equation (7.29) is written for each reaction ... 

Solution Two independent reactions are needed that involve all four components. A systematic way of doing this begins with the formation reactions but, for the present, fairly simple case. Figure 7.5 includes two reactions that can be used directly ... 

Independent Reactions. In this section, we consider the number of independent reactions that are necessary to develop equilibrium relationships between N chemical species. A systematic approach is the following ... 

Example 7.19 Find a set of independent reactions to represent the equilibrium products for a reaction between Imol of methane and 0.5 mol of oxygen. 

The maximum set will consist of Equations (14.1) and (14.3) and N versions of Equation (14.2), where N is the number of components in the system. The maximum dimensionality is thus 2- -A. It can always be reduced to 2 plus the number of independent reactions by using the reaction coordinate method of Section 2.8. However, such reductions are unnecessary from a computational viewpoint and they disguise the physics of the problem. 

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.

Evidence for (56) includes the almost quantitative formation of chlorocyclo-hexanone and the production of 50 % of the Ir(III) in the form of [IrCl50H2] . At pH 1 an acid-independent reaction predominates with the rate parameters, E = 16.4 kcal.mole and AS = —12.6 eu. [IrCl50H2] also oxidises cyclohexanone by an acid-independent path, with E = 16.6 kcal.mole and AS = — 7.3 eu. 

Methods discussed for a one-reaction system can easily be extended to multireaction systems. For all independent reactions, a separate equilibrium constant is defined as ... 

Zn Natural hydrolysis rate constant for the pH-independent reactions of a chemical with water, L/s... 

This equation is extremely useful for calculating the equilibrium composition of the reaction mixture. The mole numbers of the various species at equilibrium may be related to their values at time zero using the extent of reaction. When these values are substituted into equation 2.6.9, one has a single equation in a single unknown, the equilibrium extent of reaction. This technique is utilized in Illustration 2.1. If more than one independent reaction is occurring in a given system, one needs as many equations of the form of equation 2.6.9 as there are independent reactions. These equations are then written in terms of the various extents of reaction to obtain a set of independent equations equal to the number of unknowns. Such a system is considered in Illustration 2.2. 

This procedure may be repeated as often as necessary until one has l s down the diagonal as far as possible and zeros beneath them. In the present case we have reached this point. If this had not been the case, the next step would have been to ignore the first two rows and columns and to repeat the above operations on the resultant array. The number of independent reactions is then equal to the number of l s on the diagonal. 

Once the number of independent reactions has been determined, an independent subset can be chosen for subsequent calculations. 

The various energy transfer constraints enter into the analysis primarily as boundary conditions on the difference equations, and we now turn to the generation of the differential equations on which the difference equations are based. Since the equations for the one-dimensional model are readily obtained by omitting or modifying terms in the expressions for the two-dimensional model, we begin by deriving the material balance equations for the latter. For purposes of simplification, it is assumed that only one independent reaction occurs within the system of interest. In cases where multiple reactions are present, one merely adds an appropriate term for each additional independent reaction. 

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