Linearly independent stoichiometric equation

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. 

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. 

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))... 

For a complex system, determination of the stoichiometry of a reacting system in the form of the maximum number (R) of linearly independent chemical equations is described in Examples 1-3 and 14. This can be a useful preliminary step in a kinetics study once all the reactants and products are known. It tells us the minimum number (usually) of species to be analyzed for, and enables us to obtain corresponding information about the remaining species. We can thus use it to construct a stoichiometric table corresponding to that for a simple system in Example 2-4. Since the set of equations is not unique, the individual chemical equations do not necessarily represent reactions, and the stoichiometric model does not provide a reaction network without further information obtained from kinetics. 

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... 

The relationship between the reaction rates for the reagents and the rates along the linearly independent stoichiometric routes may be expressed by the following equations ... 

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 ... 

The number of independent rate equations is the same as the number of independent stoichiometric relations. In the present example. Reactions (1) and (2) are reversible reactions and are not independent. Accordingly, C,. and C, for example, can be eliminated from the equations for and which then become an integrable system. Usually only systems of linear differential equations with constant coefficients are solvable analytically. 

Due to the fulfilment of this law of conservation, the number of linearly independent intermediates is not three but one fewer, i.e. it amounts to two. To the right of mechanism (1) we gave a column of numerals. Steps of the detailed mechanism must be multiplied by these numerals so that, after the subsequent addition of the equations, a stoichiometric equation for a complex reaction (a brutto equation) is obtained that contains no intermediates. The Japanese physical chemist Horiuti suggested that these numerals should be called "stoichiometric numerals. We believe this term is not too suitable, since it is often confused with stoichiometric coefficients, indicating the number of reactant molecules taking part in the reaction. In our opinion it would be more correct to call them Horiuti numerals. For our simplest mechanism, eqn. (1), these numerals amount to unity. 

Horiuti stoichiometric rule. This rule is applied to find the number of linearly independent routes. Stoichiometric numbers must satisfy the equation... 

This is a homogeneous system of c linear equations in the s unknowns Xj. Let s be the rank of the matrix of stoichiometric coefficients. If the stoichiometric equations are independent, the X, s are all zero, and thus s = s. 

The expression independent has essentially the same meaning as in mathematical terminology. A series of reactions is described as independent if no one of the stoichiometric equations can be derived from the others by linear combination. For example, in an aqueous solution of ammonia the following reactions can occur ... 

The reactions 1. .. / are said to be independent if none of the stoichiometric equations can be derived by linear combination of any of the others. 

Even if the actual chemical reaction pathways are not known, a set of linear independent reactions can be generated from some given set of components, thereby allowing Equation (4.541). This approach is known as the stoichiometric formnlation and is considered in some detail in Section 4.5.5. We mnst emphasize that the independent reactions generated by this method do not necessarily describe the actnal chemical reactions that occur in the system. These independent reactions relate the mole changes of each species in each reaction to changes in the extent of reaction via a stoichiometric coefficient, i.e., Eqnation (4.540). Since the final equilibrinm state will be independent of the particnlar pathway taken, the final eqnilibrinm compositions of the given species will stiU be correctly described by Eqnation (4.541). 

Eq. (2.33) allows a direct transformation to the differential equations for the linear independent degrees of advancement. Multiplying by the inverse of the reduced matrix of the stoichiometric coefficients v , one obtains... 

By independeni is meant that no one of the stoichiometric equations can be derived from the others by a linear combination. Discussions of this are giver, by Denbigh [1], Prigogme and Defay [2], and Aris [3]. Actually, some of the definitions and manipulations are true for any set of reactions, but it is convenient to work with the minimum, independent set. 

Each linear combination of stoichiometric equations is itself a stoichiometric equation. This is the reason why the number of stoichiometric equations is fixed, but not their nature. The writing of a complex stoichiometric system is therefore likely to lead to equations which are not independent. One must make certain that this is not the case. JOUGUET s criterion indicates that the equations are independent if the rank of the matrix of the stoichiometric coefficients is equal to I. 

The number of stoichiometrically independent reactions is given by the rank of the matrix Rp, which can be determined with e.g. the aid of the Gaussian method of elimination. As a result, the stoichiometrical coefficients of linearly independent equations for the reaction system are necessary and sufficient for, for example, calculation of the conversion of the key variables and therefore also for all other components. Thus... 

First system of a linear homogeneous equations in an expression (1.22) has m-rk(0 ) fundamental solutions to which m-rk(0 ) of linear independent vectors V/ correspond. Each of these vectors determines a stoichiometric note of the / of the final reaction. That is why, in accordance with the criterion of Brinkley, we have... 

The second system of linear homogeneous equations in expression (1.22) has m + n-rk(P) fundamental solutions determining a linearly independent vector cfe. Each of these vectors assign a stoichiometric note of k to the elementary reaction and that is why... 

This equation represents the Horiuti rule [13], but it is specified by us as a number of a stoichiometric independent routes. Expression (2.23) means that, knowing the basis from the linearly independent vectors /, the rest can be founded from the solution of equation (2.22) as a linear combination of the base ones. 

On substitution of N — H concrete linearly independent vectors r,H+2> r,Ar) = 1, 2,..iV —into equations (2.18) we evidently obtain N — H linearly independent vectors (v i, v 2 ) r = 1, 2,JV — if. Such a set of linearly independent vectors of stoichiometric coefficients is called the particular solution of the set (2.17) or (2.10). For practical calculations, it is useful to find a particular solution of such a type that the matrix of stoichiometric coefficients should contain a maximum number of zeros in every row. From linear algebra theory follows, that two cases must be distinguished ... 

If consideration is directed to processes rather than to species, the multiplicity (j) is the number of independent stoichiometric processes necessary to describe any arbitrary overall process. Then any possible combination of coefficients for which equation (T) is balanced can be generated by a linear combination of

independent component processes each of the form of equation (C). 

The most significant stoichiometric matrix tool, which plays an important role in solving kinetic problems, is its rank. As it is known, a matrix rank defines the number of its linearly independent rows or columns. Using of the notion of a matrix rank allows to reduce the number of differential equations in a reaction mathematical model and, thereby, to make solving the direct and inverse kinetic problems easier. For example, let us consider a reaction scheme ... 

It turns out the number of independent equations can also be found from the rank of the stoichiometric matrix, V/. Recall from linear algebra that the rank of a matrix is defined by the number of linearly independent rows in the matrix. It can be found using Gaussian elimination with partial pivoting or simply by using the rank(...) function in MATLAB. Once the rank is determined, we need to specify that number of independent... 

Introduction of stoichiometric number concept and linear transformation of the "conventional" QSSA equations (16) to the equivalent system (20) was essentially the major (and, possibly, only) result of theory of steady reactions developed independently by J. Horiuti in 1950s and M. I. Temkin in 1960s. 

To explain the first approach let us consider S chemical involved in r chemical reactions, from which only R are independent. The stoichiometric relations give a set of linear algebraic equations ... 

This is an homogeneous system of s linear equations in the c unknowns 7i 72, . 7c- The rank of the matrix of stoichiometric coefficients being equal to s > s, it is possible to choose arbitrarily c — s coefficients 7) and to calculate the others from the Cramer system of the remaining equations. Thus, there exist c —s sets of independent solutions for the coefficients jj, i.e. the number of invariants is equal to c —s. As for stoichiometries, any linear combination of invariants is an invariant. 

To obtain the stoichiometric number experimentally, it is necessary to measure i by two independent methods from extrapolation of the linear Tafel region to q = 0 and from micropolarization measurements. Equation 51F can then be used to calculate v, since the value of n is determined independently. Unless one considers a very complex system, both n and v are integers, and it is easy to distinguish experimentally among a small number of possible values of v. 

The fact that chemical reactions are expressed as linear homogeneous equations allows us to exploit the properties of such equations and to use the associated algebraic tools. Specifically, we use elementary row operations to reduce the stoichiometric matrix to a reduced form, using Gaussian elimination. A reduced matrix is defined as a matrix where all the elements below the diagonal (elements 1,1 2,2 3,3 etc.) are zero. The number of nonzero rows in the reduced matrix indicates the number of independent chemical reactions. (A zero row is defined as a row in which all elements are zero.) The nonzero rows in the reduced matrix represent one set of independent chemical reactions (i.e., stoichiometric relations) for the system. 

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