Ideal systems chemical equilibrium

Some earlier thermodynamic studies on rutile reported expressions involving simple idealized quasi-chemical equilibrium constants for point defect equilibria (see, e.g., Kofstad 1972) by correlating the composition x in TiOx with a function of AGm (O2), which is the partial molar free energy of oxygen. However, the structural effects were not accounted for in these considerations. Careful measurements of AGm (O2) in the TiOjc system (Bursill and Hyde 1971) have indicated that complete equilibrium is rarely achieved in non-stoichiometric rutile. 

For high Da the column is dose to chemical equilibrium and behaves very similar to a non-RD column with n -n -l components. This is due to the fact that the chemical equilibrium conditions reduce the dynamic degrees of freedom by tip the number of reversible reactions in chemical equilibrium. In fact, a rigorous analysis [52] for a column model assuming an ideal mixture, chemical equilibrium and kinetically controlled mass transfer with a diagonal matrix of transport coefficients shows that there are n -rip- 1 constant pattern fronts connecting two pinches in the space of transformed coordinates [108]. The propagation velocity is computed as in the case of non-reactive systems if the physical concentrations are replaced by the transformed concentrations. In contrast to non-RD, the wave type will depend on the properties of the vapor-liquid and the reaction equilibrium as well as of the mass transfer law. 

Here AG = Ai/ — TA5 represents a standard change in the Gibbs free energy in the course of a realized chemical reaction. We have seen above that in the case of so-called ideal systems the equilibrium constant, coincides with the kinetically introduced equilibrium constant, and can easily be expressed as the ratio of the corresponding equilibrium concentrations of reagents to the power of their stoichiometric coefficients, i.e., as the left side of (2.4). In principle, this allows us to determine experimentally the equilibrium constant, by measuring the equilibrium concentrations of the reagents. 

The approach outlined here will describe a viewpoint which leads to the standard calculational rules used in various applications to systems in themiodynamic (themial, mechanical and chemical) equilibrium. Some applications to ideal and weakly interacting systems will be made, to illustrate how one needs to think in applying statistical considerations to physical problems. 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

Harvie, C.E., J.P. Greenberg and J. H. Weare, 1987, A chemical equilibrium algorithm for highly non-ideal multiphase systems free energy minimization. Geochimica et Cosmochimica Acta 51, 1045-1057. 

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... 

The need to abstract from the considerable complexity of real natural water systems and substitute an idealized situation is met perhaps most simply by the concept of chemical equilibrium in a closed model system. Figure 2 outlines the main features of a generalized model for the thermodynamic description of a natural water system. The model is a closed system at constant temperature and pressure, the system consisting of a gas phase, aqueous solution phase, and some specified number of solid phases of defined compositions. For a thermodynamic description, information about activities is required therefore, the model indicates, along with concentrations and pressures, activity coefficients, fiy for the various composition variables of the system. There are a number of approaches to the problem of relating activity and concentrations, but these need not be examined here (see, e.g., Ref. 11). 

Fig. 4.2. Potential singular point surfaces (dashed-dotted curve) for an ideal ternary system with single reaction A + B C. (a) Ellipse-type system (b) hyperbola-type system. RA = reactive azeotrope solid curve = chemical equilibrium surface.

If the liquid mixture is extremely non-ideal, liquid phase splitting will occur. Here, we first consider the hypothetical ternary system. The physical properties are adopted from Ung and Doherty [17] and Qi et al. [10]. The catalyst is assumed to have equal activity in the two liquid phases. The corresponding PSPS is depicted in Fig. 4.5, together with the liquid-liquid envelope and the chemical equilibrium surface. The PSPS passes through the vertices of pure A, B, C, and the stoichiometric pole Jt. The shape of the PSPS is affected significantly by the liquid phase non-idealities. As a result, there are three binary nonreactive azeotropes located on... 

However, we would like to point here not to the differences between the equilibrium tunneling mechanism and the above examples of mechanisms of the nonequilibrium type in low-temperature chemical conversions, but, on the contrary, to a simplifying assumption which relates them but which has to be rejected in a number of cases—and that is the subject matter of this chapter. In the above models the solid matrix itself was considered, in essence, from a special point of view, namely, as an ideal system, devoid of defects, which is in mechanical equilibrium. In other words, the fact that the systems in question are significantly out of equilibrium with respect to their mechanoenergetic state was ignored. This property of the experimentally studied samples was the result of both their preparation conditions and the ionizing radiation. 

Interphase — A spatial region at the interface between two bulk phases in contact, which is different chemically and physically from both phases in contact (also called interfacial region). The plane that ideally marks the boundary between two phases is called the interface. Particles of a condensed phase located near a newly created (free) surface are subject to unbalanced forces and possibly to a unique surface chemistry. Modifications occurring to bring the system to equilibrium or metastability generally extend somewhat into one of the phases, or into both. 

Equations (6.1) and (6.2) pertain to ideal systems, that is, systems where there are no interactions between the molecules. In a real system the pressure effect on p. in the vapor phase has to be modified by a fugacity coefficient < >, and the effect of mixing on the chemical potential in the liquid phase has to be modified by an activity coefficient 7,. The more general expression for equilibrium (called the 4>-y representation) then becomes... 

We now provide a preliminary discussion of chemical equilibrium in ideal heterogeneous systems. This treatment is provisional since the generalized approach of Section 7 of Chapter 3 is more systematic and does not require a distinction to be made... 

The quantity of primary interest in our thermodynamic construction is the partial molar Gihhs free energy or chemical potential of the solute in solution. This chemical potential depends on the solution conditions the temperature, pressure, and solution composition. A standard thermodynamic analysis of equilibrium concludes that the chemical potential in a local region of a system is independent of spatial position. The ideal and excess contributions to the chemical potential determine the driving forces for chemical equilibrium, solute partitioning, and conformational equilibrium. This section introduces results that will be the object of the following portions of the chapter, and gives an initial discussion of those expected results. 

Activity and Activity Coefficient. —When a pure liquid or a mixture is in equilibrium with its vapor, the chemical potential of any constituent in the liquid must be equal to that in the vapor this is a consequence of the thermodynamic requirement that for a system at equilibrium a small change at constant temperature and pressure shall not be accompanied by any change of free energy, i.e., (d( )r. p is zero. It follows, therefore, that if the vapor can be regarded as behaving ideally, the chemical potential of the constituent i of a solution can be written in the same form as equation (7), where p,- is now the partial pressure of the component in the vapor in equilibrium with the solution. If the vapor is not ideal, the partial pressure should be replaced by an ideal pressure, or fugacity, but this correction need not be considered further. According to Raoult s... 

CHEMICAL EQUILIBRIUM IN IDEAL HETEROGENEOUS SYSTEMS Lastly, we obtain 155... 

The definitions of the equilibrium parameters for nonideal systems involve the chemical potentials of the pure constituents that undergo the chemical reaction of interest. Thus, they are either exactly the same, or differ only slightly, from those adopted for ideal systems. For this reason the methodology and the results of Section 2.11 may be taken over (with appropriate minor modifications, as necessary) and need not be repeated here. 

Contrary to the concept of the random mixing of ions, Fellner (1984), Fellner and Chrenkova (1987) proposed the molecular model for molten salt mixtures in which it is assumed that in an ideal molten mixture, molecules (ionic pairs) mix randomly. The model composition of the melt, i.e. the molar fractions of ionic pairs in the molten mixture, is calculated on the basis of simultaneous chemical equilibrium among the components of the mixture. For instance, in the melt of the system M1X-M2X-M2Y one can assume random mixing of the ionic pairs -X , mJ-X , mJ-Y , -X , and 2MJ-XY . 

In the first approach, the structure (i.e. the ionic composition) is determined by the thermodynamic equilibrium composition, after all the chemical reactions taking place in the system are over. After reaching the chemical equilibrium, the ideal mixing of components is supposed. If the obtained standard deviation of the calculated property for the given chemical reactions is comparable with the experimental error of measurement, it is reasonable to assume that the structure of the electrolyte is given by the equilibrium composition determined by the calculated equilibrium constants. Besides, also information on e.g. the thermal stability and the Gibbs energy of the present compounds may be obtained. The task is solved by means of the material balance and use of the thermodynamic relations valid for ideal solutions. 

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