Mass action law model

An alternative to the phase separation model which has nearly the same simplicity is the mass action law model. In this model it is assumed that a single micellar species, of aggregation number n, is in equilibrium with the monomers ... 

Fig. 3.1. The relative amount of amphiphile fm that is incorporated into the micellar aggregate at an infinitesimal increase in the total concentration S as a function of total amphiphile concentration, fm = 1 -, calculated from the mass action law model (Eq. 3.2) with n = 10...

The physical properties of the micelle, e.g. average aggregation number, shape and formation constant, can be deduced from the chemical-shift data from the mass action model.48 Using the mass action law model for micelle formation, the... 

The analysis of non-linear mechanisms and corresponding kinetic models are much more difficult than that of linear ones. The obvious difficulty in this case is the follows an explicit solution for steady-state reaction rate R can be obtained only for special non-linear algebraic systems of steady-state (or pseudo-steady-state) equations. In general case it is impossible to solve explicitly a system of non-linear steady-state (or pseudo-steady-state) equations. However, in the case of mass-action-law-model it is always possible to apply to this system a method of elimination of variables and reduce it to a polynomial in one variable [4], i.e., a polynomial in terms of the steady-state reaction rate. We refer a polynomial in the steady-state reaction as a kinetic polynomial. The idea of this polynomial was firstly emphasized in [5]. 

When Reuben, Shporer, and Gabbay (42) analyzed their results with a mass action law model, the resultant quadrupolar coupling constant had this absurdly low value which led these authors (42) to reject a site-binding model. 

Based on the approximate validity of the mass action law model ... 

In the presence of autocatalytic steps, the derivatives 5/i(ci, C2)/5ci and 9/2(ci, C2)/dc can be positive in some parametric domain, whereas in the absence of autocatalytic steps these derivatives are always negative. This results in more complicated behavior of autocatalytic processes compared to traditional chemical processes. Thus, the mass-action-law model with two variables and without autocatalytic steps obeys the Poincare-Bendixson criterion ... 

This type of defect equilibrium treatment has been used extensively to model the defect chemistry and non-stoichiometry of inorganic substances and has the great advantage that it easily takes several simultaneous defect equilibria into account [22], On the other hand, the way the mass action laws are normally used they are focused on partial thermodynamic properties and not on the integral Gibbs energy. The latter is often preferred in other types of thermodynamic analyses. In such cases the following solid solution approach is an alternative. 

Empirical Models vs. Mechanistic Models. Experimental data on interactions at the oxide-electrolyte interface can be represented mathematically through two different approaches (i) empirical models and (ii) mechanistic models. An empirical model is defined simply as a mathematical description of the experimental data, without any particular theoretical basis. For example, the general Freundlich isotherm is considered an empirical model by this definition. Mechanistic models refer to models based on thermodynamic concepts such as reactions described by mass action laws and material balance equations. The various surface complexation models discussed in this paper are considered mechanistic models. 

The proper treatment of ionic fluids at low T by appropriate pairing theories is a long-standing concern in standard ionic solution theory which, in the light of theories for ionic criticality, has received considerable new impetus. Pairing theories combine statistical-mechanical theory with a chemical model of ion pair association. The statistical-mechanical treatment is restricted to terms of the Mayer/-functions which are linear in / , while the higher terms are taken care by the mass action law... 

Fisher [15] has discussed various other field-theoretical scenarios that could be profitably employed for establishing the universality class of the RPM. In particular, he has advocated the use of a four-state lattice model with lattice sites occupied by cations, anions, and neutral pairs, with the number of pairs controlled by the mass action law. Again this path has not been followed up in detail, and only a highly speculative phase diagram has been given [15],... 

An empirical distribution ratio model was first elaborated to describe the extraction of Am(III) and Ln(III) in kerosene by purified HBTMPDTP, based on mass balances and mass-action laws of HBTMPDTP dimerization in kerosene, dissociation... 

The above kinetic models are based on the surface action law that is absolutely analogous to the mass action law for volume reactions in ideal systems. In this case a model of "an ideal adsorbed layer acts, which is valid under the following assumptions ... 

Contrary to the preceding treatment the so-called mass-action model develops apparently more naturally from the application of the mass action law applied to the overall aggregation process... 

It is possible, however, to modify the model in order to account for a cooperativity and a critical concentration according to the mass action law the concentration of aggregates containing n monomers is [see Eq. (3)]... 

Besides, let us note the automatic observance (certainly with correctly set initial data) and, hence, needlessness of the formalized descriptions in equilibrium modeling of such important regularities of macroscopic system behavior as the Gibbs phase rule, the Le Chatelier-Brown principle, mass action laws, the Henry law, the Raoult law, etc. 

Two models of current use in chemical kinetics are the mass action law for the reaction rates and the Arrhenius law for the rate coefficients, viz. 

Tallarida, R., On stability and control of ligand-receptor interactions according to the mass action law A theoretical model of chaos, Drug Development Research, Vol. 19, 1990, pp. 257-274. 

Note that the physicochemical mechanisms that enables us to perform the chromatographic bioseparations are not always adsorption-like but can involve ion exchange, ion exclusion, or size exclusion. Even if it is generally possible to fit experimental data with a mathematical function derived from the adsorption theory, it is strongly advisable to refer to the proper physicochemical process before modeling the separation. For instance, ion exchange can be modeled with selectivity coefficients (derived from the mass action law) that can be constant or not,18,19 ion-exclusion can be modeled thanks to theories based on the Donnan exclusion, etc. 

Chemical reactions can be described by thermodynamics (chapter 1.1.2) and kinetics (chapter 1.2). Reactions expressed by the mass-action law (chapter 1.1.2.1), are thermodynamically reversible and independent of time. In contrast, kinetic processes are time dependent reactions. Thus, models that take into account kinetics can describe irreversible reactions such as decay processes that require finite amounts of time and cannot be reversed under a given set of conditions. 

The model is built using equations that follow two laws. The mass action law is applied to acido-basic reactions of carbonate and oxalate species with water. The equilibrium between the CO2 concentrations in the gas and the solution is described by Henry s law ... 

The TLM (Davis and Leckie, 1978) is the most complex model described in Figure 4. It is an example of an SCM. These models describe sorption within a framework similar to that used to describe reactions between metals and ligands in solutions (Kentef fll., 1988 Davis and Kent, 1990 Stumm, 1992). Reactions involving surface sites and solution species are postulated based on experimental data and theoretical principles. Mass balance, charge balance, and mass action laws are used to predict sorption as a function of solution chemistry. Different SCMs incorporate different assumptions about the nature of the solid - solution interface. These include the number of distinct surface planes where cations and anions can attach (double layer versus triple layer) and the relations between surface charge, electrical capacitance, and activity coefficients of surface species. 

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