Algebra of Stoichiometry

It is useful to divide Wald s work into three main historical periods The first is devoted to thermodynamics, particularly entropy (approximately 1889-1893), the second to an algebraic reconstruction of stoichiometry on the basis of Gibbs phase rule (approximately 1893-1907), and the third to a polydimensional deduction of chemical variety on the basis of the notion of phase (approximately 1907-1930). 

The numerical methods in this book can be applied to all components in the system, even inerts. When the reaction rates are formulated using Equation (2.8), the solutions automatically account for the stoichiometry of the reaction. We have not always followed this approach. For example, several of the examples have ignored product concentrations when they do not affect reaction rates and when they are easily found from the amount of reactants consumed. Also, some of the analytical solutions have used stoichiometry directly to ease the algebra. This section formalizes the use of stoichiometric constraints. 

The stoichiometry and heats of reaction in Equations (7.21) and (7.22) are algebraically summed to give... 

For mechanisms that are more complex than the above, the task of showing that the net effect of the elementary reactions is the stoichiometric equation may be a difficult problem in algebra whose solution will not contribute to an understanding of the reaction mechanism. Even though it may be a fruitless task to find the exact linear combination of elementary reactions that gives quantitative agreement with the observed product distribution, it is nonetheless imperative that the mechanism qualitatively imply the reaction stoichiometry. Let us now consider a number of examples that illustrate the techniques used in deriving an overall rate expression from a set of mechanistic equations. 

The algebra required to express the stoichiometry of chemically reacting systems is well established. The species present in the reacting mixture are A, i = 1, N. The reactions are denoted by... 

While linear algebraic methods are present in almost every problem, they also have a number of direct applications. One of them is formulating and solving balance equations for extensive quantities such as mass and energy. A particularly nice application is stoichiometry of chemical systems, where you will discover most of the the basic concepts of linear algebra under different names. 

An early objective in a mechanistic investigation is to establish the rate law (see Chapter 3) which is an algebraic equation describing the instantaneous dependence of the rate on concentrations of compounds or other properties proportional to concentrations (e.g. partial pressures). Rate laws cannot be rehably deduced from the stoichiometry of the overall balanced chemical equation-they have to be determined experimentally. The functional dependence of rates on concentrations maybe simple or complicated, and concentrations may be of reactants, products or even materials not appearing in the overall chemical equation, as in the case of catalysis (see Chapters 11 and 12) [3-7]. 

A natural language accounting for the stoichiometry of chemcial reactions is that of linear algebra. Let us remind ourselves of its basic concepts. 

In the usual case, t and ain will be known. Equation (1.49) is an algebraic equation that can be solved for aout. If the reaction rate depends on the concentration of more than one component, versions of Equation (1.49) are written for each component and the resulting set of equations is solved simultaneously for the various outlet concentrations. Concentrations of components that do not alfect the reaction rate can be found by writing versions of Equation (1.49) for them. As for batch and piston flow reactors, stoichiometry is used to relate the rate of formation of a component, say Sl-c, to the rate of the reaction SI, using the stoichiometric coefficient vc, and Equation (1.13). After doing this, the stoichiometry takes care of itself. 

The purpose of stoichiometric analysis is to insure that element balance is maintained. In the present case the stoichiometry is fairly straightforward. In more complex cases linear algebra can be used to perform stoichiometric analysis in a generalized manner (1 ). 

The final visualization of the reduced B matrix allows finding the basic set of independent chemical equations. Note that C = rank (B) gives the number of component species that may form all the other noncomponent species by a minimum of independent chemical reactions. The procedure can be applied by hand calculations for simple cases, or by using computer algebra tools for a larger number of species. More details can be found in the book of Missen et al. [7], or at www.chemical-stoichiometry.net. 

In this fashion, the set of rate equations of any simple pathway (unless it is part of a network) can be reduced to a single rate equation and the algebraic equations expressing the stoichiometry. To illustrate how much work can be saved in this way, let us return to the Gillespie-Ingold mechanism of nitration of aromatics, for which a repeated application of the Bodenstein approximation provided a rate equation in Example 4.4 in Section 4.3. 

Table 7.3. Algebraic forms of segment coefficient Ay, of segment with stoichiometry Xk + B — P and different configurations (from Helfferich and Savage [7]).

Chemical kinetics is an enormous subject just a few basic principles will be treated in this chapter. Section 2.1 deals with reaction stoichiometry, the algebraic link between rates of reaction and of species production. Section 2.2 considers the computability of reaction rates from measurements of species production the stoichiometric constraints on production rates are also treated there. The equilibrium and rate of a single reaction step are analyzed in Sections 2.3 and 2.4 then simple systems of reactions are considered in Section 2.5. Various kinds of evidence for reaction steps are discussed in Section 2.6 some of these will be analyzed statistically in Chapters 6 and 7. 

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