Activity coefficients rational

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

Other references in Table in discuss applications in precipitation of metal.compounds, gaseous reduction of metals from solution, equilibrium of copper in solvent extraction, electrolyte purification and solid-liquid equilibria in concentrated salt solutions. The papers by Cognet and Renon (25) and Vega and Funk (59) stand out as recent studies in which rational approaches have been used for estimating ionic activity coefficients. In general, however, few of the studies are based on the more recent developments in ionic activity coefficients. 

We will see in chapter 8 that, in the case of aqueous solutions, it is convenient to adopt the condition of hypothetical 1-molal solution at P = 1 bar and T = 298.15 K as the standard state. Most experimental data on aqueous solutions conform to this reference condition. In this case, the resulting activity coefficient is defined as the practical activity coefficient and must not be confused with the rational activity coefficient of general relation 2.80. 

Because activity is related to molar concentration through the rational activity coefficient, from equation 3.157 we may derive... 

This is a key feature of the system for anyone who wants to understand and rationalize the effects of the microenvironment of a biocatalyst on its activity, its stability, or its specificity. Since for many years the use of thermodynamic activity was recommended for quantifying substrate availability in non-conventional media [17, 18], the replacement of concentrations of species by their thermodynamic activities in liquid non-conventional media requires a knowledge of their activity coefficients (y values). And this point is still far from being straightforward, as (a) values depend on molar ratios of other species present in the medium, and (b) methods used to estimate these values, such as UNI FAC group contribution method [19], are often called into question, and claimed to be sources of inaccuracy [20, 21]. 

Table II presents some consequences of these analogies for a vapor phase and a condensed phase together with the thermal analogs. Here rti is the concentration of the vapor in molecules per unit volume, Nt is the mole fraction in the condensed phase, Vc is the molar volume of the substrate, p< is the partial pressure, and y< is the rational activity coefficient.

In equation 3 the terms of fNa+ and 7H + are the rational activity coefficients of exchanging cations in the zeolite phase and the terms yNa+ and XM + are the molal single ion activity coefficients in the solution phase. Equation 4 can be rewritten as equation 5 when the two salts, NaX and MX2 have a common anion. The mean molal activity coefficients usually can be estimated from literature data. The corrected selectivity coefficient includes a term that corrects for the non-ideality of the solution phase. Thus any variation in the corrected selectivity coefficient is due to non-ideality in the zeolite phase (see equation 3). 

The relationship between mole fraction and activity, and, therefore, between fQds and -fQdsc, is made through the rational activity coefficients ... 

A deeper perception of the mechanistic implications of equation (9.2) can be had if the rational activity coefficients are described on the molecular level using the methods of statistical mechanics. This approach is the analogue of the statistical mechanical theory of activity coefficients for species in aqueous solution (Sposito, 1983). Fundamental to it is the prescription of surface speciation and the dependence of the rational activity coefficient on surface characteristics. Three representative molecular models of adsorption following this paradigm are summarised in Table 9.8. Each has been applied with success to describe the surface reactions of soil colloids (Goldberg, 1992). 

Table 9.8 Three molecular models of the rational activity coefficients of adsorbed species...

Molecular model Surface species assumed Rational activity coefficient model... 

The expressions for single-species activity coefficients in Eqs. 1.21-1.24 suffice to calculate activities of dissolved solutes like H+ or C02 in Eq. 1.11. For the solvent, H20, it is still necessary to define a Reference State, which is that of the pure liquid at 298.15 K under 1 atm pressure.12 The activity of the solvent is conventionally set equal to the product of a rational activity coefficient f and the mole fraction of the solvent x 12... 

Equation 4.7 invites comparison with Eq. 1.12, as does Eq. 4.8 with Eq. 1.26. (Note, however, that the rational activity coefficient f is dimensionless.)... 

Equation 4.11 shows mathematically how measurements of In K12c as a function of x2 at fixed T° and P° are used to calculate the rational activity coefficients. These measurements also can be used to compute the equilibrium constant ... 

See, for example, Chap. 9 in K. Denbigh, The Principles of Chemical Equilibrium, Cambridge University Press, Cambridge, 1981. ThelUPAC recommendation for the symbol to represent rational activity coefficients is yx, which is not used in this book in order to make the distinction between solid solutions and aqueous solutions more evident. In strict chemical thermodynamics, however, all activity coefficients are based on the mole fraction scale, with the definition for aqueous species (Eq. 1.12) actually being a variant that reflects better the ionic nature of electrolyte solutions and the dominant contribution of liquid water to these mixtures. (See, for example, Chap. 2 inR. A. RobinsonandR. H. Stokes,Electrolyte Solutions, Butterworths, London, 1970.)... 

In a mixture formed from two liquids, components 1 and 2, the chemical potentials jUi and p2 can be expressed in terms of the chemical potentials of the pure components at the same temperature and pressure, Mi and p2, the mole fractions Xj and x2 and the rational activity coefficients, fx and f2 [eqns (24) and (25)]. As Xj... 

Thus, the thermodynamic solubility product is defined in terms of the activity of the lattice ions in solution and in the solid phase. It is usual to express the activity of the solid phase in terms of the mole fraction of the component in the phase, Xit and the rational activity coefficient, e.g. 

The incorporation of the inhibitor into the crystal affects the solid phase activity, aAB [eqn.(19)]. The activities in the crystalline phase may be expressed in terms of rational activity coefficients, i.e. for solid solution formation... 

Calculate j and j/m for each of the solutions. Assuming j/m to be constant and equal to the average of the six calculated values, determine the molal and the rational activity coefficients of formic acid in the solutions for which m - 3, 5, 10. 

S(f) = limiting theoretical slope of rational activity coefficient in interionic attraction theory a function of y, D, and T as expressed by Equation 5 T = temperature in Kelvin X, Y = function of DH extended theory equation z = valence of an ion... 

The term log (1 + 0.002 m M ) arises from the conversion of the rational activity coefficient, measured on the mole-fraction scale, to the molal activity coefficient y according to the relationship... 

The rational activity coefficients cannot be evaluated in any simple manner. Following the model of Truesdell and Christ (16), a regular solution approach to the problem can lead to expressions for the rational activity coefficients. If the exchange sites have the same charge and approximately the same size, then a symmetrical solid solution will be formed where the rational activity coefficients for the two components are given by ... 

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