Ideal solution equilibrium relations

An ideal gas obeys Dalton s law that is, the total pressure is the sum of the partial pressures of the components. An ideal solution obeys Raoult s law that is, the partial pressure of the ith component in a solution is equal to the mole fraction of that component in the solution times the vapor pressure of pure component i. Use these relationships to relate the mole fraction of component 1 in the equilibrium vapor to its mole fraction in a two-component solution and relate the result to the ideal case of the copolymer composition equation. 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

The fugaeity in Equation 2-39 is that of the eomponent in the equilibrium mixture. However, fugaeity of only the pure eomponent is usually known. It is also neeessary to know something about how the fugaeity depends on the eomposition in order to relate the two, therefore, assumptions about the behavior of the reaetion mixture must be made. The most eommon assumption is that the mixture behaves as an ideal solution. In this ease, it is possible to relate the fugaeity, f, at equilibrium to the fugaeity of the pure eomponent, f, at the same pressure and temperature by... 

We can find the reaction s equilibrium point from Equation 3.3 as soon as we know the form of the function representing chemical potential. The theory of ideal solutions (e.g., Pitzer and Brewer, 1961 Denbigh, 1971) holds that the chemical potential of a species can be calculated from the potential pg of the species in its pure form at the temperature and pressure of interest. According to this result, a species chemical potential is related to its standard potential by... 

Results of the ideal solution approach were found to be identical with those arrived at on the basis of a simple quasichemical method. Each defect and the various species occupying normal lattice positions may be considered as a separate species to which is assigned a chemical potential , p, and at equilibrium these are related through a set of stoichiometric equations corresponding to the chemical reactions which form the defects. For example, for Frenkel disorder the equation will be... 

The phenomena of surface precipitation and isomorphic substitutions described above and in Chapters 3.5, 6.5 and 6.6 are hampered because equilibrium is seldom established. The initial surface reaction, e.g., the surface complex formation on the surface of an oxide or carbonate fulfills many criteria of a reversible equilibrium. If we form on the outer layer of the solid phase a coprecipitate (isomorphic substitutions) we may still ideally have a metastable equilibrium. The extent of incipient adsorption, e.g., of HPOjj on FeOOH(s) or of Cd2+ on caicite is certainly dependent on the surface charge of the sorbing solid, and thus on pH of the solution etc. even the kinetics of the reaction will be influenced by the surface charge but the final solid solution, if it were in equilibrium, would not depend on the surface charge and the solution variables which influence the adsorption process i.e., the extent of isomorphic substitution for the ideal solid solution is given by the equilibrium that describes the formation of the solid solution (and not by the rates by which these compositions are formed). Many surface phenomena that are encountered in laboratory studies and in field observations are characterized by partial, or metastable equilibrium or by non-equilibrium relations. Reversibility of the apparent equilibrium or congruence in dissolution or precipitation can often not be assumed. 

The previous summary of activities and their relation to equilibrium constants is not intended to replace lengthier discussions [1,18,25,51], Yet it is important to emphasize some points that unfortunately are often forgotten in the chemical literature. One is that the equilibrium constants, defined by equation 2.63, are dimensionless quantities. The second is that most of the reported equilibrium constants are only approximations of the true quantities because they are calculated by assuming the ideal solution model and are defined in terms of concentrations instead of molalities or mole fractions. Consider, for example, the reaction in solution ... 

C-t, which means, of course, that the ideal solution model is adopted, no matter the nature or the concentrations of the solutes and the nature of the solvent. There is no way of assessing the validity of this assumption besides chemical intuition. Even if the activity coefficients could be determined for the reactants, we would still have to estimate the activity coefficient for the activated complex, which is impossible at present. Another, less serious problem is that the appropriate quantity to be related with the activation parameters should be the equilibrium constant defined in terms of the molalities of A, B, and C. As discussed after equation 2.67, A will be affected by this correction more than A f//" (see also the following discussion). 

Transport-related non-equilibrium behavior (e. g., physical non-equilibrium) is excluded, which plays an important role in non-ideal solute transport in the field and in some experimental column systems. Physical non-equilibrium is due to slow exchange of solute between mobile and less mobile water, such as may exist between particles or between zones of different hydraulic conductivities in the subsurface soil column, and occurs for sorbing and non-sorbing molecules alike. 

The mixture CMC is plotted as a function of monomer composition in Figure 1 for an ideal system. Equation 1 can be seen to provide an excellent description of the mixture CMC (equal to Cm for this case). Ideal solution theory as described here has been widely used for ideal surfactant systems (4.6—18). Equation 2 can be used to predict the micellar surfactant composition at any monomer surfactant composition, as illustrated in Figure 2. This relation has been experimentally confirmed (ISIS) As seen in Figure 2, for an ideal system, if the ratio XA/yA < 1 at any composition, it will be so over the entire composition range. In classical phase equilibrium thermodynamic terms, the distribution coefficient between the micellar and monomer phases is independent of composition. 

The choice of equilibrium constant for measuring the stability of a carbocation depends partly on experimental accessibility and partly on the choice of solvent. A desire to relate measurements to the majority of existing equilibrium constants implies the use of water as solvent. Water has the advantage and disadvantage that it reacts with carbocations. It follows that the most widely used equilibrium constant is that for the hydration reaction shown in Equation (1), which is denoted KR (or pAR). A simple interpretation of AR is that it measures the ratio of concentrations of unionized alcohol to carbocation in an (ideal) solution of aqueous acid of concentration 1 M. 

Martin has considered the chemical and physical factors affecting partition coefficients.20 Restricting his discussion to ideal solutions, he considered a solute, A, distributed between two phases in equilibrium with each other. The partition coefficient, a, of the solute A is related to the free energy required to transport one mole of A from one phase to... 

The properties of mixtures of ideal gases and of ideal solutions depend solely on the properties of the pure constituent species, and are calculated from them by simple equations, as illustrated in Chap. 10. Although these models approximate the behavior of certain fluid mixtures, they do not adequately represent the -behavior of most solutions of interest to chemical engineers, and Raoult s law is not in general a realistic relation for vapor/liquid equilibrium. However, these models of ideal behavior—the ideal gas, the ideal solution, and Raoult s law— provide convenient references to which the behavior of nonideal solutions may be compared. 

In Chap. 6 we treated the thermodynamic properties of constant-composition fluids. However, many applications of chemical-engineering thermodynamics are to systems wherein multicomponent mixtures of gases or liquids undergo composition changes as the result of mixing or separation processes, the transfer of species from one phase to another, or chemical reaction. The properties of such systems depend on composition as well as on temperature and pressure. Our first task in this chapter is therefore to develop a fundamental property relation for homogeneous fluid mixtures of variable composition. We then derive equations applicable to mixtures of ideal gases and ideal solutions. Finally, we treat in detail a particularly simple description of multicomponent vapor/liquid equilibrium known as Raoult s law. 

The application of Eq. (10.3) to specific phase-equilibrium problems requires use of models of solution behavior, which provide expressions for G or for the Hi as functions of temperature, pressure, and composition. The simplest of such expressions are for mixtures of ideal gases and for mixtures that form ideal solutions. These expressions, developed in this chapter, lead directly to Raoult s law, the simplest realistic relation between the compositions of phases coexisting in vapor/liquid equilibrium. Models of more general validity are treated in Chaps. 11 and 12. 

By contrast, Gibbs monolayers are continually at equilibrium with the subphase which, in this case, may be called a solution. Surface tension and concentration are related through Gibbs law, so that the surface pressure can be related to the solution concentration. By analogy with [II. 1.1.6 and 7J we have for a single adsorbed component in an ideal solution... 

Next the equations that we can write are for calculating system properties. Because equilibrium is assumed, the rate equations and, therefore, the transport and transfer properties are of no concern. In general, the thermodynamic properties of mixtures will depend on temperature, pressure, and composition, we will assume that the mixture is an ideal solution to simplify the computation of thermodynamic properties. Thus, we can write the enthalpies of the mixtures as mole fraction averages of the pure component enthalpies, without an enthalpy of mixing term. We can also write the phase equilibrium relations as functions of temperature and pressure only and not composition. The pure component enthalpies of liquids generally do not depend strongly on pressure, but there may be some effect of pressure on the vapor-phase enthalpy. We will neglect this effect for simplicity. 

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. 

All of the equilibrium relations quoted above are ideal expressions, where quotients of concentrations are constant. But in fact all the solutions will be non-ideal and the observed equilibrium constant will have to be corrected for non-ideality using activity coefficients for each species appearing in the equilibrium expression. In practice, extrapolation procedures... 

For ideal solutions in which solute interactions can be neglected, the mole fraction xs of a substance dissolved in water and its equilibrium vapor pressure p in the gas phase above the solution are related by Henry s law,... 

Ideal solution is often chosen as reference in analysis. In this case we may write for the fugacity of a component the relations f/ (HL) = XjHj and fj LR) = X/fk vhere HL stands for Henry law and L-R for Lewis-Randall assumption. The problem is now how to express the phase equilibrium A possible approach would be to use Henry law for solute, and Lewis-Randall rule for solvent. For this reason such definition of Ai-values is considered asymmetric. 

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