Chemical equilibrium in a mixture

The volume of mixing is obtained by differentiating the Gibbs energy of mixing with respect to pressure, the temperature and composition being constant. 

However, inspection of Eq. (11.16) shows that the Gibbs energy of mixing is independent of pressure, so the derivative is zero hence, 

Ideal mixtures are formed without any volume change. 

Consider a closed system at a constant temperature and under a constant total pressure. The system consists of a mixture of several chemical species that can react according to the equation 

Since T and p are constant, as the reaction advances the change in Gibbs energy of the system is given by Eq. (11-7), which becomes 

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White, W.B., Johnson, S.M., and Dantzig, G.B. Chemical equilibrium in complex mixtures. J. Chem. Phys. 28, 751-755 (1958), Warga, J. A convergent procedure for solving the thermo-classical equilibrium problem. J,. Soc. Ind. Appl. Math. 11, 594-606 (1963). 

Chemical equilibrium in a closed system at constant temperature and pressure is achieved at the minimum of the total Gibbs energy, min(G) constrained by material-balance and electro-neutrality conditions. For aqueous electrolyte solutions, we require activity coefficients for all species in the mixture. Well-established models, e.g. Debye-Htickel, extended Debye-Hiickel, Pitzer, and the Harvie-Weare modification of Pitzer s activity coefficient model, are used to take into account ionic interactions in natural systems [15-20]. 

The relations given above are valid for any -compon-ent system. As pointed out in Section 2.4, the mixture may be a continuous chemical mixture, where the composition is described by a molar density functionyfr whose independent variable r is some characterizing property, e.g. molecular weight, carbon number, etc. The criterion for chemical equilibrium in a two-phase (or two-region) system of 7 = 1, 2 is, for all values of r, (Cotterman etal, 1985)... 

However, when carboxylic acids are present in a mixture, fugacity coefficients must be calculated using the chemical theory. Chemical theory leads to a fugacity coefficient dependent on true equilibrium concentrations, as shown by Equation (3-13). ... 

Some chemical reactions are reversible and, no matter how fast a reaction takes place, it cannot proceed beyond the point of chemical equilibrium in the reaction mixture at the specified temperature and pressure. Thus, for any given conditions, the principle of chemical equilibrium expressed as the equilibrium constant, K, determines how far the reaction can proceed if adequate time is allowed for equilibrium to be attained. Alternatively, the principle of chemical kinetics determines at what rate the reaction will proceed towards attaining the maximum. If the equilibrium constant K is very large, for all practical purposes the reaction is irreversible. In the case where a reaction is irreversible, it is unnecessary to calculate the equilibrium constant and check the position of equilibrium when high conversions are needed. 

The use of kinetic inhibitors and/or anti-agglomcrators in actual fieid operations is a new and evolving technology. These are various formulations of chemicals that can be used in a mixture of one or more kinetic inhibitors and/or anti-agglomerators. At the current time, to get an optimum mixture for a specific application it is necessary to set up a controlled bench test using the actual fluids to be inhibited and determine the resulting equilibrium phase line. As the mixture of chemicals is changed, a family of equilibrium phase lines will develop. This will result m an initial determination of a near optimum mixture of chemicals. 

It is found that after the elapse of a sufficient time interval, all reversible reactions reach a state of chemical equilibrium. In this state the composition of the equilibrium mixture remains constant, provided that the temperature (and for some gaseous reactions, the pressure also) remains constant. Furthermore, provided that the conditions (temperature and pressure) are maintained constant, the same state of equilibrium may be obtained from either direction of a given reversible reaction. In the equilibrium state, the two opposing reactions are taking place at the same rate so that the system is in a state of dynamic equilibrium. 

Intelligent engineering can drastically improve process selectivity (see Sharma, 1988, 1990) as illustrated in Chapter 4 of this book. A combination of reaction with an appropriate separation operation is the first option if the reaction is limited by chemical equilibrium. In such combinations one product is removed from the reaction zone continuously, allowing for a higher conversion of raw materials. Extractive reactions involve the addition of a second liquid phase, in which the product is better soluble than the reactants, to the reaction zone. Thus, the product is withdrawn from the reactive phase shifting the reaction mixture to product(s). The same principle can be realized if an additive is introduced into the reaction zone that causes precipitation of the desired product. A combination of reaction with distillation in a single column allows the removal of volatile products from the reaction zone that is then realized in the (fractional) distillation zone. Finally, reaction can be combined with filtration. A typical example of the latter system is the application of catalytic membranes. In all these cases, withdrawal of the product shifts the equilibrium mixture to the product. 

Equilibrium in multiphase and/or multireaction systems. If more than one phase is present in the system, a criterion of phase equilibria has to be satisfied together with the chemical equilibrium criterion. For instance, in a gas-liquid system components are in chemical equilibrium in the phase where the reaction occurs, but vapour-liquid equilibria between the gas and the liquid phases must also be taken into account. To determine the equilibrium composition of a reacting mixture in both phases, chemical equilibrium constants as well as data concerning vapour-liquid equilibria for all components of the reaction mixture should be known. In the equilibrium state ... 

One of the earliest examples of Gibbs energy minimisation applied to a multi-component system was by White et al. (1958) who considered the chemical equilibrium in an ideal gas mixture of O, H and N with the species H, H2, HjO, N, N2, NH, NO, O, O2 and OH being present. The problem here is to find the most stable mixture of species. The Gibbs energy of the mixture was defined using Eq. (9.1) and defining the chemical potential of species i as... 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modem ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic stmcture picture of molecular and supramolecular interactions. 

Remark. The idea of detailed balance first arose in chemical reaction kinetics. Suppose that in a mixture of chemical compounds a cycle of three possible reactions exists, as in fig. 9. Equation (4.2) asserts that, in equilibrium, each reaction takes place... 

This paper shows that the conditions of thermodynamic equilibrium in a mix-tine of chemically reacting ideal gases always have a solution for the concentrations of the mixture components and that this solution is unique. The paper has acquired special significance in the last few years in connection with the intensive study of systems in which this uniqueness does not occur. Such anomalies may be related either to nonideal components, or to treatment of stationary states, rather than truly equilibrium ones, in which the system exchanges matter or energy with the surrounding medium. 

Chemical equilibrium is a dynamic state in which the concentrations of reactants and products remain constant because the rates of the forward and reverse reactions are equal. For the general reaction a A + iiB cC + d D, concentrations in the equilibrium mixture are related by the equilibrium equation ... 

In a mixture, equilibrium is established when the chemical potential of each component is the same in each phase. That is,... 

In studies of competitive adsorption, the usually measured quantity is the overall composition of the adsorbed phase for a given composition of the bulk phase in equilibrium with it. It has been found that chemical shifts can provide a more detailed description. In a mixture of Xe and Kr in N 4 zeolite it was possible to observe the individual signals from XenKr mixed clusters as well as the Xen clusters under magic angle spinning (28). The absolute 129Xe chemical shifts of the XenKr mixed clusters and the increments between XenKr and the Xen+1 in various Xe-Kr mixtures in Na4 zeolite, are shown in Table I. 

The number of isomers in an isomer group is represented by Njso. At chemical equilibrium, all of the isomers have the same chemical potential, and this chemical potential is represented by iso. The amount of an isomer group is represented by niso = En . For a group of gaseous isomers at equilibrium, the chemical potential of the isomer group in a mixture of ideal gases is given by... 

The high efficiency may be better understood if one recognizes that in a mixture of silicon tetrachloride and oxygen the chemical equilibrium is totally shifted to silica at the temperatures we apply. The activation energy for the reaction prevents the thermal reaction from proceeding rapidly under the experimental conditions chosen. Thus the plasma merely acts as a catalyst to overcome the activation energy of the reaction. 

For example, the AB mixture expressed in Figure A.1 by XA and XB mole fractions on the AB edge leads at equilibrium to a mixture (xA, xB, xc) obtained by intersecting the equilibrium curve with the stoichiometric line passing through the initial mixture. Conversely, a ternary mixture where a chemical reaction at equilibrium takes place may be described only by two transformed composition variables. 

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