Stoichiometric equilibrium model

The very general success of the phenomenological theory in quantitatively describing the composition dependence of many chemical and physical processes arises from the treatment of solvation effects by a stoichiometric equilibrium model. It is this model that pro-... 

According to the stoichiometric displacement model, the equilibrium constant for peptide adsorption with the solvated nonpolar ligands can be expressed as follows ... 

E. The equilibrium constants for these equilibria are, respectively, Kle, Km, Keh, Kleh, and Kle/Klh- These models are termed stoichiometric because they use stoichiometric equilibrium constants instead of thermodynamic constants (see below) to describe the ion association process. 

In the case of Ni, the situation is even more complex since the stoichiometric equations reported in the literature differ widely depending on the diluents used, on the aqueous phase compositions, or on the extractant concentrations employed. A discrimination procedure of the equilibrium models corresponding to the back-extraction reactions has been reported previously by taking into account the expressions given in Table 37.1 and obtaining the best results with the following equation [59] ... 

These models, although of practical and intuitive value, are not well founded in physical chemistry. The pioneeristic, even if qualitative, work of Bidlingmayer demonstrated that IIRs adsorb onto the stationary phase. It follows that stoichiometric equilibrium constants, which depend on the change in free energy of adsorption of the analyte, cannot be considered constant if the IIR concentration in the mobile phase increases, because the stationary phase surface properties (including its charge density) are modified. The multibody interactions and long-term forces involved in IIC can better be described by a thermodynamic approach. 

The state of equilibrium in ion exchange chromatography is currently described by stoichiometric models where the solute, for example a protein, displaces a stoichiometric number of salt ions bound on the ion exchanger. A basic concept is the stoichiometric displacement model developed by Kopaciewicz et al. (1983). For monovalent counterions the reaction is described as follows ... 

In tiying to model dynamic adsorption mathematically and predict the shape and size of the MTZ, assumptions must be made about the equilibrium model, molecular diffusion, axial-flow dispersion, and isothermality. The simplest model is isothermal equilibrium operation with infinite diffusion rate and negligible dispersion, that is, a stoichiometric front. 

The importance of three-dimensional structure to chromatographic behavior is reflected in the nonmechanistic model, the stoichiometric displacement model (SDM). The central hypothesis of the SDM is that the displacement of a solute from a surface is accompanied by the adsorption of a stoichiometric amount of displacing agent. The process may be described by the equilibrium expression ... 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

Here va and va are the stoichiometric coefficients for the reaction. The formulation is easily extended to treat a set of coupled chemical reactions. Reactive MPC dynamics again consists of free streaming and collisions, which take place at discrete times x. We partition the system into cells in order to carry out the reactive multiparticle collisions. The partition of the multicomponent system into collision cells is shown schematically in Fig. 7. In each cell, independently of the other cells, reactive and nonreactive collisions occur at times x. The nonreactive collisions can be carried out as described earlier for multi-component systems. The reactive collisions occur by birth-death stochastic rules. Such rules can be constructed to conserve mass, momentum, and energy. This is especially useful for coupling reactions to fluid flow. The reactive collision model can also be applied to far-from-equilibrium situations, where certain species are held fixed by constraints. In this case conservation laws... 

The next step in formulating a kinetic model is to express the stoichiometric and regulatory interactions in quantitative terms. The dynamics of metabolic networks are predominated by the activity of enzymes proteins that have evolved to catalyze specific biochemical transformations. The activity and specificity of all enzymes determine the specific paths in which metabolites are broken down and utilized within a cell or compartment. Note that enzymes do not affect the position of equilibrium between substrates and products, rather they operate by lowering the activation energy that would otherwise prevent the reaction to proceed at a reasonable rate. 

The construction of the structural kinetic model proceeds as described in Section VIII.E. Note that in contrast to previous work [84], no simplifying assumptions were used the model is a full implementation of the model described in Refs. [113, 331]. The model consists of m = 18 metabolites and r = 20 reactions. The rank of the stoichiometric matrix is rank (N) = 16, owing to the conservation of ATP and total inorganic phosphate. The steady-state flux distribution is fully characterized by four parameters, chosen to be triosephosphate export reactions and starch synthesis. Following the models of Petterson and Ryde-Petterson [113] and Poolman et al. [124, 125, 331], 11 of the 20 reactions were modeled as rapid equilibrium reactions assuming bilinear mass-action kinetics (see Table VIII) and saturation parameters O1 1. 

Table 6.1 lists the stoichiometric yields of hydrogen and percentage yields by weight from steam reforming of some representative model compounds present in biomass pyrolysis oils, and also several biomass and related materials. The table also shows the equilibrium yield of H2, as a percentage of the stoichiometric yield, predicted by thermodynamic calculations at 750 °C and vdth a steam-to-carbon (S/C) ratio of 5 [32]. 

Extensive literature has developed related to the preferential interaction of different solvents with proteins or peptides in bulk solution.156-5X1 Similar concepts can be incorporated into descriptions of the RPC behavior of peptides and employed as part of the selection criteria for optimizing the separation of a particular peptide mixture. As noted previously, the dependency of the equilibrium association constant, /CassoCji, of a peptide and the concentration of the solvent required for desorption in RPC can be empirically described1441 in terms of nonmechanistic, stoichiometric solvent displacement or preferential hydration models, whereby the mass distribution of a peptide P, with n nonpolar ligands, each of which is solvated with solvent molecules Da is given by the following ... 

Oxidation of zinc to zinc oxide is another example whose kinetics have been interpreted in terms of the Wagner model (Wagner Grunewald, 1938). At 670 K, the reaction has been found to be independent of oxygen pressure between 0.02 and 1 atm. ZnO is a n-type semiconductor, having a stoichiometric excess of zinc accommodated as interstitials the defect equilibrium could be represented as... 

The steps for constructing and interpreting an isothermal, isobaric thermodynamic model for a natural water system are quite simple in principle. The components to be incorporated are identified, and the phases to be included are specified. The components and phases selected "model the real system and must be consistent with pertinent thermodynamic restraints—e.g., the Gibbs phase rule and identification of the maximum number of unknown activities with the number of independent relationships which describe the system (equilibrium constant for each reaction, stoichiometric conditions, electroneutrality condition in the solution phase). With the phase-composition requirements identified, and with adequate thermodynamic data (free energies, equilibrium con-... 

In applying equation 33, Cpsl (the constant-pressure molar heat capacity of the stoichiometric liquid) is usually extrapolated from high-temperature measurements or assumed to be equal to Cpij of the compound, whereas the activity product, afXTjafXT), is estimated by interjection of a solution model with the parameters estimated from phase-equilibrium data involving the liquid phase (e.g., solid-liquid or vapor-liquid equilibrium systems). To relate equation 33 to an available data base, the activity product is expressed... 

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