Number of independent reactions

The discussion so far in this chapter has been based on the supposition that there is only a single stoichiometric process in the system of interest. As noted in 4 2, this means simply that the changes in the amounts of all substances in the system can be expressed as small multiples or submultiples of l he change in the amount of any one of them. For example, if the only stoichiometric process is 

When only a small number of substances are present it is usually quite easy to write down by inspection the minimum number of chemical equations which will represent the complete stoichiometry. In more complex systems it is desirable to use a more systematic procedure. For example, it would be time-consuming to determine, by trial and error, what are the minimum number of independent chemical equations involving, say, a dozen hydrocarbons, as in a cracking process. 

A simple rule for determining this minimum number is as follows. Chemical equations are first written down for the formation from their component atoms of all compounds which are regarded as being present in the system.f These equations are then combined in such a way as to eliminate ifrom them any free atoms which are not actually present. The result is the minimum number, B, of chemical reactions which are sufficient to represent the stoichiometry (although not necessarily the kinetics)] of the system. 

X Free radicals or other intermediates may be important in regard to the kinetic mechanism, but these may be insignificant in regard to the overall stoichiometry. 

by hypothesis, the system is not regarded as containing free hydrogen atoms in significant amount, the first equation may be used to eliminate H from the other three. This gives 

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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). 

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 ... 

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. 

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. 

A method for calculating the number of independent reactions discussed... 

This is easily extended to an arbitrary number of independent reactions r and produces the expected generalization of equation 244 where... 

The ammonium chloride acts as a reducing agent, reducing the chromic anhydride of the dichromate to the green sesquioxide. The reaction is not a simple one but consists of a number of independent reactions. 

Remark 7. The number of independent reactions in a given set lr is the rank of the stoicheiometric matrix and may be determined by elementary row operations. 

When the equilibrium constants for the reactions (A) and (B) are expressed in terms of the partial pressure of the various species (in atm), the equilibrium constants for these reactions have the values KpA = 0.046 and KpB = 0.034. Determine the number of independent reactions, and then determine the equilibrium composition of the mixture, making use of a simple MATLAB program that you develop for this purpose. 

A graph circle is a final sequence of the edges in which no node except the starting point occurs twice. A graph for the isomerization reaction has one circle, whereas that for vinyl chloride synthesis contains two circles. Every route of a chemical reaction corresponds to a graph s circle and vice versa. The number of independent reaction routes is equal to the number of elements in the basis of circles. It permits us to determine independent reaction routes from the graph type. 

In matrix 5.1-15 the three components are CO, H2, and CH4. However, if the order of the columns were changed, other components would be chosen. Thus the conservation matrix is not unique. A set of components must contain all the elements that are not redundant. The rank of the stoichiometric number matrix is equal to the number of independent reactions. 

R number of independent reactions in a system specified concentrations of coenzymes... 

The rank of the stoichiometric number matrix is equal to the number of independent reactions, which is 2. 

Notice that the number of equations, i.e., the number of independent reactions (r) is related with the number of components (c) and the number of elements (e) by the equation... 

Chemical reactions reduce the number of degrees of freedom of a system. For each independent chemical reaction that can occur in a system, AnnG = 0 is required. If r is the number of independent reactions at equilibrium in the system, this reduces the degrees of freedom by r. 

In this formulation, N is the number of independent components, Nr the number of independent reactions, z and t are dimensionless space and time according to... 

Number of independent reactions Universal gas constant Defined in Eqn. (14)... 

The number of independent reactions R can be found simply as the rank of the matrix of stoichiometric coefficients %J with dimension Sx r such that R < r. Different methods can be applied, such as reduction to triangular matrix by Gaussian elimination for small-size matrices, or computer methods for larger problems. 

In the structure CS1 the flow rates of both fresh reactants are fixed. As shown in Chapter 4, this strategy can be used for complex reactions, when the number of independent reactions matches the number of products for which the self-regu-lahon principle is applied. Figure 5.23 indicates the response at a step change of... 

The lacking special description of the Gibbs phase rule in MEIS that should be met automatically in case of its validity is very important for solution of many problems on the analysis of multiphase, multicomponent systems. Indeed, without information (at least complete enough) on the process mechanism (for coal combustion, for example, it may consist of thousands of stages), it is impossible to specify the number of independent reactions and the number of phases. Prior to calculations it is difficult to evaluate, concentrations of what substances will turn out to be negligibly low, i.e., the dimensionality of the studied system. Besides, note that the MEIS application leads to departure from the Gibbs classical definition of the notion of a system component and its interpretation not as an individual substance, but only as part of this substance that is contained in any one phase. For example, if water in the reactive mixture is in gas and liquid phases, its corresponding phase contents represent different parameters of the considered system. Such an expansion of the space of variables in the problem solved facilitates its reduction to the CP problems. 

The obvious reason is that in this system a relation apart from the equations expressing the conservation of the elements exists, viz., that the number of atoms of two of the elements are the same in all compounds occurring in the mixture. This causes two of the equations of conservation to merge into one, so that the number of independent reactions must be increased by one. If generally there are r such relations, of which the... 

P + 2 — r, where F is degrees of freedom, C is number of components, P is number of phases, and r is the number of independent reactions. In this case, C is 2, P is 2, and r is 1 (namely, W-C4H10 —v A0-C4H10) therefore, F = 1. This means that we can choose either temperature or pressure alone to specify the system when the temperature is given (311 K in this case), the system pressure is thereby established. 

The number of independent reactions is just the rank of the matrix S, that is, the rank of the largest nonsingular square submatrix of S. In the example above, the second row is the sum of the other two, so that the rank cannot be 3. On the other hand, the four elements in the upper left-hand corner obviously form a nonsingular submatrix, so that the rank is 2. A nonsingular square submatrix s having the rank of S, is selected... 

Another situation in which the number of independent reactions is less than the rank of the formal stoichiometric matrix arises when the rates of two reactions are always in the same ratio. In this situation, the two reactions should be combined into one, and instead of the two corresponding rows of the matrix, the appropriate linear combination of them should be used. 

For A = 4, / is obviously 1 for A = 5, / = 2 and for N = 6, /f = 4, because the reaction 2A3 = Ag cannot be written as a linear combination of reactions of the type of Eq. (165). However, after that adding a new olefin only increases the number of independent reactions by one, so for N> 5 one has R = N — 2. For every component A/ / > 3, there is one linkage class all pairs which, as a complex, have the same carbon number (this still leaves out a large number of compatible complexes, because compatible triplets, etc., are excluded). However, the number of complexes and the deficiency now grow very rapidly with N, since they are delivered by a modified Fibonacci series ... 

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