Independent chemical equations

In more complex cases when several reactions are occurring simultaneously in the system under observation, calculations of the composition of the system as a function of time will require the knowledge of a number of independent composition variables equal to the number of independent chemical equations used to characterize the reactions involved. 

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. 

Following the second approach, only the knowledge of the chemical species involved in the chemistry is sufficient. The problem can be formulated as follows given a list of chemical species S find a proper set of independent chemical equations R that ensures the conservation of atomic species N in terms of the molecular formulas of the system species. A proper set allows that any other chemical equation can be obtained from this by algebraic operations. A chemical species is characterized by formula, isomeric form and phase, and it may be neutral molecules, ionics and radicals. The atomic species can be atoms and charge (+ or -). 

The number of independent chemical equations R can be found simply by the relation ... 

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. 

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. 

In the more complex cases of concurrent or consecutive reactions the composition of the system will depend on a simultaneous knowledge of the amounts of two or more species present as a function of time. The number of composition variables to be studied will in general be equal to the number of independent chemical equations that describe the reaction. 

In more complex cases when several reactions are occurring simultaneously in the system under observation, calculations of the composition of the system as a function of time will require knowledge of a number of independent composition variables equal to the number of independent chemical equations used to characterize the reactions involved. In principle, one can use any of the many tools and methods of the analytical chemist in carrying out these determinations. In practice, analytical methods are chosen on the basis of their specificity, accuracy, ease of use, and rapidity of measurement. For purposes of discussion, these methods may be classified into two groups physical and chemical. Regardless of the method chosen, however, it must meet the following criteria ... 

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. 

In the previous examples we have found the number of independent chemical equations by intuition or knowledge of the system. This works well for simple systems, but is harder and more umeliable for complex ones. Of the various algorithms for doing this, the simplest reliable one seems to be the following ... 

In principle, Chen, given the flux relations there is no difficulty in constructing differencial equations to describe the behavior of a catalyst pellet in steady or unsteady states. In practice, however, this simple procedure is obstructed by the implicit nature of the flux relations, since an explicit solution of usefully compact form is obtainable only for binary mixtures- In steady states this impasse is avoided by using certain, relations between Che flux vectors which are associated with the stoichiometry of Che chemical reaction or reactions taking place in the pellet, and the major part of Chapter 11 is concerned with the derivation, application and limitations of these stoichiometric relations. Fortunately they permit practicable solution procedures to be constructed regardless of the number of substances in the reaction mixture, provided there are only one or two stoichiomeCrically independent chemical reactions. 

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

Equation (4-274) for each independent chemical reaction, giving r equations. 

The Bom-Oppenheimer approximation shows us the way ahead for a polyelec-tronic molecule comprising n electrons and N nuclei for most chemical applications we want to solve the electronic time-independent Schrodinger equation... 

A catalyst speeds up both the forward and the reverse reactions by the same amount. Therefore, the dynamic equilibrium is unaffected. The thermodynamic justification of this observation is based on the fact that the equilibrium constant depends only on the temperature and the value of AGr°. A standard Gibbs free energy of reaction depends only on the identities of the reactants and products and is independent of the rate of the reaction or the presence of any substances that do not appear in the overall chemical equation for the reaction. 

Example 7.17 illustrates the utility of the reaction coordinate method for solving equilibrium problems. There are no more equations than there are independent chemical reactions. However, in practical problems such as atmospheric chemistry and combustion, the number of reactions is very large. A relatively complete description of high-temperature equilibria between oxygen and... 

The style of the book is to present only a small amount of information on each page with a slide-like illustration using short descriptions and easily understood chemical equations and structures. Under each illustration is additional information or comments with room for the reader to make notes if desired. Although there is obvious continuity, an attempt has been made to make each page subject somewhat independent so that readers can study the contents of the book one page at a time at their own pace. Of necessity, because of this format, there is considerable repetition. We do not consider this bad. 

The application of the time-independent Schrodinger equation to a system of chemical interest requires the solution of a linear second-order homogeneous differential equation of the general form... 

If the process involves chemical reaction the number of independent components will not necessarily be equal to the number of chemical species, as some may be related by the chemical equation. In this situation the number of independent components can be calculated by the following relationship ... 

The first step in the analysis is to determine if the chemical equations A to C are independent by applying the test described above. When one does this one finds that only two of the reactions are independent. We will choose the first two for use in subsequent calculations. Let the variables a and B represent the equilibrium degrees of advancement of reactions A and B, respectively. A mole table indicating the mole numbers of the various species present at equilibrium may be prepared using the following form of equation 1.1.6. 

The purpose of most quantum chemical methods is to solve the time-independent Schrodinger equation. Given that the nuclei are much more heavier than the electrons, the nuclear and electronic motions can generally be treated separately (Born-Oppenheimer approximation). Within this approximation, one has to solve the electronic Schrodinger equation. Because of the presence of electron repulsion terms, this equation cannot be solved exactly for molecules with more than one electron. 

The foregoing example illustrates the importance of combining independent chemical and isotopic data for maximum resolving power and reliability. Several cautions are evident, however. As indicated in figure 4 (matrix equation) all significant carbonaceous sources must be represented in the model and uncertainties for both the sample (y.) and the source matrix (A.j) must be... 

To derive the governing equations we need to identify each independent chemical reaction that can occur in the system. It is possible to write many more reactions than are independent in a geochemical system. The remaining or dependent reactions, however, are linear combinations of the independent reactions and need not be considered. 

If we had chosen to describe composition in terms of elements, we would need to carry the elemental compositions of all species, minerals, and gases, as well as the coefficients of the independent chemical reactions. Our choice of components, however, allows us to store only one array of reaction coefficients, thereby reducing memory use on the computer and simplifying the forms of the governing equations and their solution. In fact, it is possible to build a complete chemical model (excluding isotope fractionation) without acknowledging the existence of elements in the first place ... 

The implication is that chemical phenomena are determined by the laws of quantum mechanics, as expressed in the fundamental time-independent Schrodinger equation... 

Attempts to define operationally the rate of reaction in terms of certain derivatives with respect to time (r) are generally unnecessarily restrictive, since they relate primarily to closed static systems, and some relate to reacting systems for which the stoichiometry must be explicitly known in the form of one chemical equation in each case. For example, a IUPAC Commission (Mils, 1988) recommends that a species-independent rate of reaction be defined by r = (l/v,V)(dn,/dO, where vt and nf are, respectively, the stoichiometric coefficient in the chemical equation corresponding to the reaction, and the number of moles of species i in volume V. However, for a flow system at steady-state, this definition is inappropriate, and a corresponding expression requires a particular application of the mass-balance equation (see Chapter 2). Similar points of view about rate have been expressed by Dixon (1970) and by Cassano (1980). 

A proper set of chemical equations for a system is made up of R linearly independent equations. 

I of reaction as a reaction path). The important consequence is that the maximum / number of steps in a kinetics scheme is the same as the number (R) of chemical equations (the number of steps in a kinetics mechanism is usually greater), and hence stoichiometry tells us the maximum number of independent rate laws that we must obtain experimentally (one for each step in the scheme) to describe completely the macroscopic behavior of the system. 

Equation 1.5-1 used as a mass balance is normally applied to a chemical species. For a simple system (Section 1.4.4), only one equation is required, and it is a matter of convenience which substance is chosen. For a complex system, the maximum number of independent mass balance equations is equal to R, the number of chemical equations or noncomponent species. Here also it is largely a matter of convenience which species are chosen. Whether the system is simple or complex, there is usually only one energy balance. 

The study described above for the water-gas shift reaction employed computational methods that could be used for other synthesis gas operations. The critical point calculation procedure of Heidemann and Khalil (14) proved to be adaptable to the mixtures involved. In the case of one reaction, it was possible to find conditions under which a critical mixture was at chemical reaction equilibrium by using a one dimensional Newton-Raphson procedures along the critical line defined by varying reaction extents. In the case of more than one independent chemical reaction, a Newton-Raphson procedure in the several reaction extents would be a candidate as an approach to satisfying the several equilibrium constant equations, (25). 

For a system undergoing R independent chemical reactions among N chemical species, R equilibrium expressions are to be added to the relationships among the intensive variables. From Equation (13.1), the total number of intensive variables in terms of N becomes... 

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