Independent stoichiometric equations

When several reactions occur simultaneously a degree of advancement is associated with each stoichiometric equation. Problem P4.01.26 is a application of this point. Some processes, for instance cracking of petroleum fractions, involve many substances. Then a correct number of independent stoichiometric equations must be formulated before equilibrium can be calculated. Another technique is to apply the principle that equilibrium is at a minimum of Gibbs free energy. This problem, however, is beyond the scope of this book. 

At this point, we should mention the difference between independent chemical equations and independent chemical reactions. The former are of mathematical significance, being helpful to carry out consistent material balance. The latter are useful for describing the chemical steps implied in a chemical-reaction network. They may be identical with the independent stoichiometric equations, or derived by linear combination. This approach is useful in formulating consistent kinetic models. 

Let us consider a complex reaction involving c components Cls C2,. .., Cc, none of which is chemically inert under the experimental conditions and which may appear in the formulation of s independent stoichiometric equations. Stoichiometric equations are independent if none of them may be obtained by a linear combination of the others. On the other hand, one particular component is not independent when it may be obtained from the other constituents. Therefore, the number, c, of independent constituents is given by the relationship... 

Let us consider a complex reaction described by s independent stoichiometric equations between c components for which... 

The second class of complex systems is that in which concurrent reactions yield different sets of products. These are again simple to characterize in that at least two or more independent stoichiometric equations will be required to represent the reaction at any time. Thus, for the pyrolysis of toluene we need at least two such equations to account for the products ... 

Cyclohexane is rly easily dehydrogenated into benzene, and even at very low extents of reaction, stoichiometry reaction (6) can be replaced by the secondary stoichiometry reaction (12). For cyclohexane, the constituents are (apart from cyclohexane) hydrogen, methane, ethane, ethylene, acetylene, propene, 1-butene, 1,3-butadiene, cydohexene, and benzene [3]. However, one can check that the equations written are independent, using the Jouguet [IS] criterion, (t.e., if/ = n-o)). In this criterion, the number of the independent constituents, 1//, for a chemical system is equal to the required constituents. n, (i e., H3, Cl, C3, CsHe, Q, C4, Cj. c-Q, CaJ, subtracting the number of independent stoichiometric equations. q>. 

Moreover, by combining the Brinkley s [19] criterion, m rt-if/, one can calculate the number of independent stoichiometric equations. In this criterion the number y/ of independent constituents of a chemical system is equal to the rank of the matrix of the indexes of the elements in the formula of the constituents, hence cd = n-y/ = 9-2 = 7. Kinetic considerations have led to write 11 stoichiometric equations. It must be checked by Jouguet s criterion [18] that these 11 stoichiometric equations are independent. In the case of cycohe.xane the stoichiometric equations are 2, 3, 4, 8, 9, 11, and 12. Thus these seven stoichiometric equations are independent and describe the decomposition of cyclohexane. 

In this work, however, tests were desired to allow screening of hydrocaibon/catalyst combinations through measurenoenis of endotberm of reaction, extent of conversion, and identification of reaction products. However, the atmospheric-pressure bench scale test apparatus has been described elsewhere [11]. Parametric studies of the catalytic decomposition of n-heptane and cyclohexane were conducted to provide their stoichiometric analysis. The propo independent stoichiometric equations describing the catalytic decomposition of cyclohexane are ... 

We shall characterize as a complex heterogeneous catalytic reaction any reaction system which consists of at least of two independent stoichiometric equations. The compounds participating in a heterogeneous catalytic reaction can be divided into two groups, viz. reagents and intermediate surface compounds (ISCs). The regents are the starting compounds and products of the reaction. Their concentrations or partial pressures can be determined at any moment of the reaction and at any point in the reaction space. 

Contemporary chemical kinetics and the theory of reaction mechanisms are characterized not only by increased complexity of the mechanisms (hypotheses of mechanisms) but also by the considerable number of hypotheses (the possible mechanisms describing each reaction). Cases are known where the mechanism of formation of a certain product in a complicated multiroute mechanism incorporates completely different sequences of elementary steps and intermediates" even in the case of reactions that have one linearly independent stoichiometric equation The greater mechanistic complexity and high number of hypotheses raise the issue of the formalization and automation of the procedure adopted for the generation of hypotheses. 

At this point it should be emphasized that one cannot legitimately carry out the above integration unless the concentrations of all species are held constant throughout the deactivation process. This can be accomplished easily for a reaction involving one independent stoichiometric equation by varying the flow rate to the reactor to keep one concentration constant [6]. In the case of more than one independent stoichiometric equation, maintaining constant concentrations is a difficult control problem. 

Sheet and Crowe considered 10 reacting species. Six linear independent stoichiometric equations are needed to describe the variation of the amounts of these species along the reactor. 

Consider a system containing hydrogen, chlorine and hydrogen chloride molecules, together with hydrogen and chlorine atoms in low concentration. Between these species it is seen on inspection that there are three independent stoichiometric equations. These may be... 

For every system containing a greater number of constituents, a set of stoichiometric equations may easily be constructed by means of a suitable combination of row vectors of the matrix of constitution coefficients. Only a certain number of these equations will, however, be mutually linearly independent. The other equations may then be expressed by a linear combination of the preceding reactions. It is essential for the following considerations to determine the maximum number of linearly independent stoichiometric equations, and to select out of all possible combinations those which describe a given system in the simplest possible manner. 

The so-called direct search method, described by Hooke and Jeeves, is used for the minimum of the function F. The above mentioned authors Anthony and Himmelblau also studied different possibilities of selecting equilibrium criteria and thus also of choosing the function f i = 1,2,. ..,iV or F. One of the existing possibilities is to choose the function F equal to the overall enthalpy of the system. However, the state of equilibrium may also be characterized by R linearly independent stoichiometric equations, satisfying the relations (see (3.29))... 

Rank of the matrix is if = 4, thus the system contains four independent constituents. Considering the required condition that the independent constituents should be represented in the highest concentration, it is advisable to select CO2, H2O and N2. The fourth constituent selected is CO. A determinant formed from the constitution coefficients of these four constituents is ) = 4, providing that the constituents are linearly independent. The remaining six constituents shall be considered to be derived using the procedure described in Example 5, we now construct the matrix of stoichiometric coefficients for six linerlay independent stoichiometric equations in the form of synthesis reactions ... 

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