Stoichiometric constraints

In unsteady states the situation is less satisfactory, since stoichiometric constraints need no longer be satisfied by the flux vectors. Consequently differential equations representing material balances can be constructed only for binary mixtures, where the flux relations can be solved explicitly for the flux vectors. This severely limits the scope of work on the dynamical equations and their principal field of applicacion--Che theory of stability of steady states. The formulation of unsteady material and enthalpy balances is discussed in Chapter 12, which also includes a brief digression on stability problems. 

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. 

Equation (2.39) is a generalization to M reactions of the stoichiometric constraints of Equation (2.35). If the vector e is known, the amounts of all N components that are consumed or formed by the reaction can be calculated. 

These stoichiometric constraints are derived from the data given in the SSN. 

Alternatively, the conservation of atomic species is commonly expressed in the form of chemical equations, corresponding to chemical reactions. We refer to the stoichiometric constraints expressed this way as chemical reaction stoichiometry. A simple system is represented by one chemical equation, and a complex system by a set of chemical equations. Determining the number and a proper set of chemical equations for a specified list of species (reactants and products) is the role of chemical reaction stoichiometry. 

A complex reacting system is defined as one that requires more than one chemical equation to express the stoichiometric constraints contained in element balances. In such a case, the number of species usually exceeds the number of elements by more than 1. Although in some cases a proper set of chemical equations can be written by inspection, it is useful to have a universal, systematic method of generating a set for a system of any complexity, including a simple system. Such a method also ensures the correct number of equations (R), determines the number (C) and a permissible set of components, and, for convenience for a very large number of species (to avoid the tedium of hand manipulation), can be programmed for use by a computer. 

Some factors do make the process easier. Several studies suggest that metabolic networks are scale-free. There are also important stoichiometric constraints and elementary nodes... 

There is of course an equation for each species, subject to conservation of total mass given by the continuily equation, and we implicitly used conservation of total mass as stoichiometric constraints, Njo — Nj)lvj are equal for all 7 in a batch (closed) system or (Fjo — Fj)/vj are equal for aU j in a steady-state continuous reactor. 

We have found recently the topological interpretation of property (34). The stoichiometric constraints (24) can be interpreted in terms of the topological object, the circuit. Existence of the circuit "explains" the appearance of the cyclic characteristic in the constant term of kinetic polynomial. Thus, we can say that in some sense the correspondence between the detailed mechanism and thermodynamics is governed by pure topology. 

Problem Formulation. The conditions of equilibrium require the equivalence in each phase of temperature, pressure, and chemical potential for each component that is transferable between the phases and are subject to constraints of stoichiometry. A statement of the equivalence of chemical potential is identical to equations 8 and 9. An example is the AJB C D quaternary system. This system contains four binary compounds, AC, BC, AD, and BD, and the conditions of equilibrium allow three equations (of the type given by equation 8) to be written. The fourth possible equation is redundant as a result of the stoichiometric constraint (i.e., equal number of atoms on each sublattice). 

Fell, D. A. Small,). R. Fat synthesis in adipose tissue. An examination of stoichiometric constraints. Biochem J1986, 238 781-786. 

The flow rates of fresh reactants can be set at arbitrary values, but within stoichiometric constraints. Then, the internal flow rates and concentrations adjust themselves in such a way that, for each reactant, the net consumption rate equals the... 

Figure 9.3 Illustration of the kind of question that can be addressed with stoichiometric constraint-based network analysis.

In this example, the unreduced network (without the side reaction to the tetracarbonyl-acyl) has twelve rate equations, one for each participant, and fourteen coefficients. Four rate equations can be replaced by stoichiometric constraints, namely, the carbon skeleton, cobalt, hydrogen, and CO balances. This still leaves eight rate equations with ten coefficients. Pathway reduction has reduced this to a single rate equation, eqn 6.12, with two coefficients. 

If a majority of the steps of a reaction are non-simple, there is at this time no substitute to traditional "brute force" modeling of the rate equations of all participants except those that can be replaced by stoichiometric constraints. This is so, for example, in hydrocarbon pyrolysis and combustion, where, fortunately, an extensive data base on rate coefficients and activation energies has been assembled [25-29], However, in a large number of non-simple reactions of practical interest, only one or a few steps out of many are non-simple. In such cases, the complexity of mathematics can be significantly reduced. In a few other instances with only one or two offending steps, additional approximations may make it possible to arrive at explicit rate equations. 

As a rule, a reduction to a single, explicit rate equation (plus algebraic equations for stoichiometric constraints and yield ratios) is not achieved. Rather, the equations for the end members of the piecewise simple network portions must be solved simultaneously. Nevertheless, The concentrations of all trace-level intermediates that do not react with one another have been eliminated by this procedure and, in many cases of practical interest, the reduction in the number of simultaneous rate equations and their coefficients is substantial. 

Stoichiometric constraints and yield ratio equation. The stoichiometric constraints are... 

Rate equations in terms of A coefficients. With the rate equations for olefin, paraffin, H2, and CO replaced by the stoichiometric constraints and the yield ratio of paraffin to hydroformylation products, the rate equations for aldehyde and alcohol remain to be established. In terms of A coefficients these are ... 

This example has shown how the procedures developed in earlier chapters can be used effectively for modeling. The reaction system has seventeen participants olefin, paraffin, aldehyde, alcohol, H2, CO, HCo(CO)3Ph, HCo(CO)2Ph, and nine intermediates. "Brute force" modeling would require one rate equation for each, four of which could be replaced by stoichiometric constraints (in addition to the constraints 11.2 to 11.4, the brute-force model can use that of conservation of cobalt). Such a model would have 22 rate coefficients (arrowheads in network 11.1, not counting those to and from co-reactants and co-products), whose values and activation energies would have to be determined. This has been reduced to two rate equations and nine simple algebraic relationships (stoichiometric constraints, yield ration equations, and equations for the A coefficients) with eight coefficients. Most impressive here is the reduction from thirteen to two rate equations because these may be differential equations. 

For this example and seven others from this book, Table 11.1 illustrates the reduction of complexity achieved, showing a comparison of the numbers of rate and other equations and their coefficients of reduced and "brute force" models. The latter are understood to consist of the rate equations for all participants except those that can be replaced by stoichiometric constraints, and the constraints used in this fashion. The greatest reductions are where it counts most in the possibly differential rate equations. Also important is the reduction in the number of coefficients. This is because the problem with brute-force modeling today is not so much the demands of the actual calculations, but the experimental work required to obtain values for all the coefficients and their activation energies. 

Does not include equations that can be replaced by stoichiometric constraints or yield ratios. If quasi-equilibrium ole + cat — X, can be assumed as in network 6.9, kh is negligible and only seven coefficients are needed. 

Plath, K., and Boersma, M. (2001). Mineral hmitarion of zooplankton Stoichiometric constraints and optimal foraging. Ecology 82(5), 1260-1269. 

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. 

One important issue that still needs attention is the objective function. It is intuitively obvious that if a separation cost is not associated with it, we will usually end up getting near-complete separations of products, and hence complete conversions to the extent possible within stoichiometric constraints. Thus the AR in concentration space can easily be the entire stoichiometric space. Unfortunately, it is difficult to get an accurate representation for the separation cost, e.specially when sharp splits are not enforced. Here, we present a simple cost model by assuming that the variable cost of separation is determined by two factors, namely, the difficulty of separation and the mass flow rate through the separator. 

The second application is the construction of catalytic reaction mechanisms out of elementary steps, involving only on type of catalytic site. A more useful way to formulate the stoichiometric constraints for these systems is to classify every chemical species as either a terminal species or an intermediate. Terminal species represent stable compounds that can be produced or consumed in significant quantities, while intermediates are short-lived unstable species that participate in the mechanism but are neither raw materials nor final products of the process. 

There are various ways in which we can interpret the idea of meeting stoichiometric constraints by combining transformations or assembling steps. We alluded to these interpretations in earlier sections, in our informal statement of the problem. As with the formulation of the constraints, each application domain accepts more naturally a different interpretation. 

Thus, in the case of biochemical pathways we have allowed a richer vocabulary for stoichiometric constraints but we can still reproduce the earlier specifications. 

Given a set of stoichiometric constraints and a database of biochemical reactions, the algorithm carries out iterative satisfaction of constraints, just like the algorithm in Section II. The algorithm proceeds as follows. 

We do not have the space here to explain completely how h is calculated, but an example will make the idea clear. Suppose that a portion of lOOg of NH4Cl(s) decomposes in an evacuated tank to NH3(g), HCl(g), N2(g), HiCg), and CUCg). Thus n — 5, but the rank of the atomic matrix is c = 3. Because all the N, H, and Cl in the gas phase came from the NH4CICS), in the gas phase you know the following additional ratios must exist (called stoichiometric constraints) among all the compounds... 

H = enthalpy per unit mass or mole h = distance above reference plane h = number of stoichiometric constraints... 

Chemical reactions may involve large numbers of steps and participants and thus many simultaneous rate equations, all with their temperature-dependent coefficients. The full set of rate equations is easily compiled as shown in Section 2.4, and to obtain solutions by numerical computation poses no serious problems. With a large number of equations, however, it may become too much of a task to verify the proposed network and obtain values for all its coefficients. Therefore, every available tool must be brought to bear to reduce the bulk of mathematics, and that without unacceptable sacrifice in accuracy. The present chapter critically reviews the principal tools for such a purpose stoichiometric constraints and the concepts of a rate-controlling step, quasi-equilibrium steps, and quasi-stationary states. Other tools useful in catalysis, chain reactions, and polymerization will be discussed in the context of those reactions (see Sections 8.5.1, 9.3, 10.3, and 11.4.1). 

The use of stoichiometric constraints is so elementary that standard texts don t even mention it. Yet, the application to large networks with many intermediates warrants a brief review. 

If all reactants, intermediates, and products are accounted for, the stoichiometric constraints are exact. They become approximations only if any participants are disregarded. This is done routinely with trace-level species, where it entails no significant loss of accuracy. 

The principal tools for reduction of mathematical bulk by elimination of rate equations are the use of stoichiometric constraints and the concepts of a rate-controlling step, quasiequilibrium steps, and quasi-stationary states of trace-level intermediates. 

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