Mass-Action representation

There are a number of representations that are considered fundamental descriptions of the basic entities in various fields. The Mass-Action representation and the Michaelis-Menten representation provide two common examples. It has been demonstrated (Savageau, 1995) that these are, in fact, restricted special cases of the Generalized-Mass-Action (GMA) representation. 

The Mass-Action representation is clearly a special case of the GMA representation in which all exponents are positive integers. The Michaelis-Menten representation is, in turn, a special case of the traditional Mass-Action representation in which two important restrictions have been imposed (Savageau, 1992). First, it is assumed that the mechanism is in quasi-steady state. The derivatives of the dependent state variables in the Mass-Action Formalism can then be set to zero, thereby reducing the description from differential equations to algebraic equations. Second, it is assumed that complexes do not occur between different forms of an enzyme or between different enzymes. The algebraic equations will then be linear in the concentrations of the various enzyme forms, and one can derive the rational function that is the representation of the rate law within the Michaelis-Menten Formalism. 

Equation (1), of course, is simply a representation of the law of mass action, stating that the reaction rate is proportional to a concentration driving force. Such an equation has been applied to many reacting systems, particularly those of considerable complexity. 

The primary objective of this chapter is to develop low-dimensional representations of chemically reacting flow situations. Specifically these include batch reactors (corresponding to homogeneous mass-action kinetics), plug-flow reactors (PFR), perfectly stirred reactors (PSR), and one-dimensional flames. The derivations also serve to illustrate the approach that is taken to derive appropriate systems of equations for other low-dimensional circumstances or flow situations. 

Effect of H2S on Reaction Rate. As the desulfurization reaction proceeds, H2S is produced. This material, although mainly in the vapor phase, is in equilibrium with a concentration of dissolved H2S in the liquid. Under certain conditions the mass action effect of this material can strongly influence the overall rate of the desulfurization reaction. Figure 3 shows the effect for one set of circumstances of H2S partial pressure on the pseudo second-order reaction rate constant. Again the constant shown is not a true reaction rate constant— which would be independent of such parameters— but is an overall representation of several simultaneously occurring forward and reverse desulfurization reac-... 

These simplifying assumptions allow elimination of some reaction steps, and representation of free radical and short-lived intermediates concentrations in terms of the concentration of the stable measurable components, resulting in complex non-mass action rate expressions. 

As a local representation, the S-system variant within the Power-Law Formalism is more accurate than the GMA variant, which is generally more accurate than the conventional Linear Formalism. As a fundamental representation, the Power-Law Formalism includes as special cases the Mass-Action, Michaelis-Menten, and Linear Formalisms that are considered to accurately represent natural phenomena... 

Determination of general kinetic features of each accounted elementary reaction (or groups of analogous reactions, e.g., simple homogeneous, pressure-dependent, heterogeneous, reactions in adsorbed layers, etc.). This includes, first of all, types of applicable kinetic description (e.g., mass action law, topochemical equations, probability of interaction on phase boundaries, etc.) and adequate form of representation of kinetic parameters. 

In this representation, all curves end up on the left side in parallel straight fines with the slope RT. In Sect. 6.5 we set up the following equation to describe mass action [Eq. (6.28)] ... 

Write complex reactions using the General Mass Action (GMA) representation Calculate equilibrium constants from kinetic data Understand the concept of steady state ... 

The General Mass Action (GMA) representation for complex reactions... 

Willamowski, K.-D. (1978a). Contributions to the theory of mass action kinetics, II. Representation of closed and open kinetics. Z. Natuforsch., 33a, 983-8. 

Note that the stoichiometric coefficients in a balanced chemical equation like eq. (2.5) bear no necessary relationship to the orders that appear in the empirical (i.e., experimentally derived) rate law. This statement becomes obvious if one considers that the chemical equation can be multiplied on both sides by any arbitrary number and remain an accurate representation of the stoichiometry even though all the coefficients will change. However, the orders for the new reaction will remain the same as they were for the old one. There are cases in which the rate law depends only on the reactant concentrations and in which the orders of the reactants equal their molecularity. A reaction in which the order of each reactant is equal to its molecularity is said to obey the Law of Mass Action or to behave according to mass action kinetics. 

Figure 5. Mass action law representation of the ion exchange for Ludox HS at low pH in the presence of l.OTO mol dm La(N03)3 (o), l.OTO mol dm Ca(N03)2 (Cl), 0.56 mol dm KCl ( O ), and 0.40 mol dm LiCl (A). The maximum possible error is represented by the vertical bars. The solid lines drawn through the data for the salts with + 1. +2. and +3 charged cations are the best linear fits with slopes +1, +2 and +3, respectively [10].

Chapter 3 discusses the different aspects of the law of mass action and the equilibrium constants associated therewith. A number of graphical representations used in the study of chemical equilibria are presented. In turn, we examine pole diagrams and equilibrium representations with temperature, with the generalization of Ellingham diagrams. The ehapter ends with a presentation of binary, ternary and quaternary diagrams of chemical equilibria. 

Sire, E.-O., 1986, On topological-dynamical equivalent representation of reaction networks The omega-equation and a canonical class of mass action kinetics, Ber. Bunsen-ges. Phys. Chem.. 90 1087. 

Both the mass-action and phase-separation models, despite their limitations, are useful representations of the micellar process and may be used to derive equations relating the CMC to the various factors that determine it. Some insight into the role of the hydrocarbon chain in the micellization process may be gained from determinations of the free energy of micellization, AGS. A convenient method of determining AGS of ionic surfactants is from measurements of the effect of electrolyte on the CMC [172]. 

Historically, the use of the law of mass action was first attempted to give a representation of the rate of a global reaction, that is, when the primary reactants are assumed to immediately form the final products. However, this was followed by the subsequent reahsation that the behaviour of a reactive system was controlled by a number of reaction steps with reaction intermediates playing a key role as discussed in Sect. 2.1. Experimental and theoretical studies were then performed to determine the rate coefficients for individual reaction steps motivated by a number of different application fields. 

The fiigacity has the dimension of pressure. Often we want a nondimensional representation of the fugacity, for example, in mass-action (chemical equilibrium) calculations. We will see in Chapter 12 that this requirement leads naturally to the definition of the activity. Furthermore, when we apply the ideal solution idea to nonideal solutions, we will need a measure of departure from ideality, just as the compressibility factor z is a measure of departure from ideal gas behavior. The logical choice for that measure is the activity coefficient, defined below. We will see that the activity and activity coefficient are dimensionless, and that for ideal solutions and many practical solutions the activity is equal to the mol fraction. 

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