Equilibrium mass-action

Let us start with the nucleated assembly that is not self-catalyzed. It turns out useful to distinguish between the mean aggregation number of all the material in the solution, N, from that in which only the activated species is considered and that we denote by Na. If we define the equilibrium constant K = exp [—.%] with g as the binding free energy, and introduce the nucleation constant Ka = exp[—/3ya], then under conditions of thermodynamic equilibrium, mass action gives (Aggeli, 2001 Ciferri, 2005 Nyrkova et al., 2000 Tobolsky and Eisenberg, 1960)... 

Aggregation of surfactants in apolar solvents, e.g., aliphatic or aromatic hydrocarbons, occurs provided that small amounts of water are present [1,126,127], These aggregates are often called reverse micelles, although the solutions do not always appear to have a critical micelle concentration, and surfactant association is often governed by a multiple equilibrium, mass action, model vith a large spread of aggregate sizes [130,131], It has recently been suggested that the existence of a monomer f -mer equilibrium should be used as a criterion of micellization, and that this term should not be applied to self-associated systems which involve multiple equilibria [132],... 

The Equilibrium (Mass Action) Expression Gas Phase Equilibria Kp vs. Kp Homogeneous and Heterogeneous Equilibria Numerical Importance of the Equilibrium Expression Mathematical Manipulation of Equilibrium Constants Reversing the Chemical Equation Adjusting the Stoichiometry of the Chemical Reaction Equilibrium Constants for a Series of Reactions Units and the Equilibrium Constant... 

Write equilibrium (mass action) expressions for each of the following reactions ... 

The first theoretical model of surfactant adsorption from micellar solutions, proposed by Lucassen [142], uses the simplifying assumptions that the micelles are monodisperse and that the micellization happens as a single step, which is described as a reversible reaction of order n (the micelle aggregation number). Later, more realistic models, which account for the multi-step character of the micellar process, were developed [143-145]. The assumption for a complete local dynamic equilibrium between monomers and micelles makes possible to use the equilibrium mass action law for the micellization reaction [142,146,147]. In such a case, the surfactant transfer corresponds to a conventional diffusion-limited adsorption characterized by an effective diffusion coefficient, Deff, which depends on the micelle diffusivity, concentration, and aggregation number. Dgff is independent of the rate constants of the fast and slow demicellization processes and k. Joos et al. [146,147] confirmed experimentally that in some cases the adsorption from micellar solutions could be actually described as a diffusion-limited process characterized by an apparent diffusivity,... 

Subsequently, the model has been extended [37, 38] to the case of associated electrolytes by using a recent model for associating electrolytes[39]. Unlike the classic chemical model of the ion pair the effect of the pairing association is included in the computation of the MSA screening parameter F. Simple formulas for the thermodynamic excess properties have been obtained in terms of this parameter when a new EXP approximation is used. The new formalism based on closures of the Wertheim-Ornstein-Zernike equation (WOZ)[40, 41 does accommodate all association mechanisms (coulombic, covalent and solvation) in one single association parameter, the association constant. The treatment now includes the fraction of particles that are bonded, which is obtained by imposing the chemical equilibrium mass action law. This formalism was shown to be very successful for ionic systems, both in the HNC approximation and MSA [42, 43, 44, 45, 46, 47]. 

Chemical kinetics and chemical equilibrium Mass action law, Arrhenius law, stoichiometric relations, yield, conversion, and equilibrium constant. 

Micelle formation can be treated as a mass action equilibrium, for example. 

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

For a closed chemical system witli a mass action rate law satisfying detailed balance tliese kinetic equations have a unique stable (tliennodynamic) equilibrium, In general, however, we shall be concerned witli... 

Ion Exchange A useful tool is provided by the mass action law for describing the general exchange equilibrium in fully ionized exchanger systems as... 

Mass Action Here the equilibrium reladons, consistent with Eq. (16-25), are... 

FK . 16-8 Ideal mass-action equilibrium for three-component ion exchange with unequal valences. K a,c — 3.06 K b,c = 3.87. Diiolite C-20 polystyrenesiil-fonate resin, with Ca as A, Mg as B, and Na as C. [Klein et al, Ind. Eng. Chem. Fund., 6, 339 (1967) repiinted with permission.)... 

Hundreds of metabohc reac tions take place simultaneously in cells. There are branched and parallel pathways, and a single biochemical may participate in sever distinct reactions. Through mass action, concentration changes caused by one reac tion may effect the kinetics and equilibrium concentrations of another. In order to prevent accumulation of too much of a biochemical, the product or an intermediate in the pathway may slow the production of an enzyme or may inhibit the ac tivation of enzymes regulating the pathway. This is termed feedback control and is shown in Fig. 24-1. More complicated examples are known where two biochemicals ac t in concert to inhibit an enzyme. As accumulation of excessive amounts of a certain biochemical may be the key to economic success, creating mutant cultures with defective metabolic controls has great value to the produc tion of a given produc t. 

If n is the concentration of defects (cation vacancies or positive holes) at equilibrium, then, applying the law of mass action to equation 1.157... 

Guldberg and Waage (1867) clearly stated the Law of Mass Action (sometimes termed the Law of Chemical Equilibrium) in the form The velocity of a chemical reaction is proportional to the product of the active masses of the reacting substances . Active mass was interpreted as concentration and expressed in moles per litre. By applying the law to homogeneous systems, that is to systems in which all the reactants are present in one phase, for example in solution, we can arrive at a mathematical expression for the condition of equilibrium in a reversible reaction. 

In the deduction of the Law of Mass Action it was assumed that the effective concentrations or active masses of the components could be expressed by the stoichiometric concentrations. According to thermodynamics, this is not strictly true. The rigorous equilibrium equation for, say, a binary electrolyte ... 

The equilibrium concentration of defects is obtained by applying the law of mass action to Eq. (7) or (8). This leads in the case of Frenkel disorder to... 

Effects of Cold Gas Recycle and Approach to Equilibrium. Product gases resulting from various CGR ratios were analyzed (Table XI). For the experiments tabulated, a decrease in the cold recycle ratio resulted consistently in increases in the product gas concentrations of water vapor, hydrogen, and carbon dioxide and a decrease in methane concentration. These trends may be noted in experiment HGR-12 as the CGR ratio decreased from 8.7 1 to 1.2 1, in experiment HGR-13 as it increased from 1.0 1 to 9.1 1, and in experiment HGR-14 as it decreased from 3.0 1 to 1.0 1. These trends indicate that the water-gas shift reaction (CO + H20 —> C02 + H2) was sustained to some degree. Except for the 462-hr period in experiment HGR-14, the apparent mass action constants for the water-gas shift reaction (based on the product gas compositions in Table XI) remained fairly constant at 0.57-1.6. These values are much lower than the value of 11.7 for equilibrium conversion at 400°C. In... 

The deduction adopted is due to M. Planck (Thermodynamik, 3 Aufl., Kap. 5), and depends fundamentally on the separation of the gas mixture, resulting from continuous evaporation of the solution, into its constituents by means of semipermeable membranes. Another method, depending on such a separation applied directly to the solution, i.e., an osmotic process, is due to van t Hoff, who arrived at the laws of equilibrium in dilute solution from the standpoint of osmotic pressure. The applications of the law of mass-action belong to treatises on chemical statics (cf. Mel lor, Chemical Statics and Dynamics) we shall here consider only one or two cases which serve to illustrate some fundamental aspects of the theory. 

To each equilibrium there corresponds an equation of mass-action ... 

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