Coefficient activity —.

The real behavior of systems is described by the activity coefficient y,. Instead of the concentration C of a dissolved species, one uses the activity a, = c y,. In the light of the Debye-Hiickel theory, y takes care of the electrostatic interactions of the ions. This is the main interaction for charged species in comparison with the smaller dipole and Van der Waals forces, which may be important in the case of uncharged species, but which are not included in the Debye-Huckel theory. The chemical potential p depends on the concentration according to Equation 1.37. 

This charging contains two parts, the charging of an ion without the influence of the ionic cloud and the additional effect of the ion cloud. The latter involves the potential (P(-i(f = a) given by Equation 1.27. The second part of Equation 1.38 takes into account the difference between the real situation (c 0) and the ideal situation (c = 0). Only this part is used in the further discussion for the calculahon of the activity coefficient. Equation 1.37 refers to 1 mol of ions and thus requires multiplying the charging work in Equation 1.38 with Avogadro s number which yields Equation 1.39. 

Introducing Equation 1.27 for (pa(r = a) yields Equation 1.40 for the work p,e - Pid for the transfer of 1 mol of ions from the ideally diluted to the real solution. 

Neglecting the value a P for sufficiently diluted electrolytes and introducing the expression for p according to Equation 1.22b, one obtains Equation 1.41. 

This relation is valid for diluted electrolytes only due to the above approximation and all the others made for the Debye-Hiickel theory. Introducing the values for Avogadro s constant Na = 6.022 X 10 3 molthe charge of the electron e = 1.602 x 10 C and the Boltzmann constant k = R/Nj = 1.3807 x 10 3 j one obtains the factor 1.826 x 10 (l K3-mol- ) 2 given in Equation 1.41a. Introducing the dielectric constant of water e = 78.56 and the temperature T = 298.15 K yields finally the factor 0.509 also given in Equation 1.41a. 

The difference in the behavior of the component of a real solution, as compared to that determined by Raoult s law (Eq. 1.13.2), is described by its aaivity coefficient (Y/)  

The activity coefficient is of paramount importance in chemical engineering thermodynamics applications, for it represents a key quantity in separation calculations (Example 1.8). 

For i = a -1-1. c Eq. (1,139) applies, and the components are identified as solvents. The components belonging to the indexing of I = 1. a are known as solutes. When the solutes are not in the same state as the solution, the second convention becomes useful. This is often the case for liquid solutions of noncondensable gases. Activity coefficients defined by convention II are unsymmetricai. The relationship between and y will be discussed later in Chapter 5. Note that as 0 the corresponding y which is shown by has a defined value as Xi 0, the corresponding y which is shown by y - 1, as defined above. 

Relation between y, and G. Let us differentiate Eq. (1.132) with respect to Hi while holding T, P, and n, constant. The result is, 

Equation (1,145) shows that the excess Gibbs free energy and the activity coefficients are related. An alternative form of Eq. (1.145) is given by 

Pressure and temperature derivative of yi. Let us divide Eq. (1.138) by T and take the derivative of the resulting expression with respect to temperature while holding P and x constant. 

The above equation provides the effect of temperature on the activity coefficients. The effect of pressure on can be obtained by taking the derivative of Eq. (1.138) with respect to pressure at constant T and x  

In Equation (16.4), is the molality that corresponds to p2. that is, the molality of the standard state. The latter three constraints are imposed because we want Equation (16.1) to approach Equation (14.6), Equation (15.5), or Equation (15.11) in the appropriate limit. 

The deviation of a solvent from the limiting-law behavior of Raoult s law is described conveniently by a function called the activity coefficient, which is defined (on a mole fraction scale) as 

The deviation of a solute from the limiting behavior of Hemy s law, on the mole fraction scale, is also described conveniently by the activity coefficient, which in this 

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In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

Detailed and extensive information on the UNIFAC method for estimating activity coefficients with application to vapor-liquid equilibria at moderate pressures. 

The activity coefficient y relates the liquid-phase fugacity... 

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

When we speak of the normalization of activity coefficients, we mean a specification of the state wherein the activity coef-... 

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention. ... 

Henry s constant is the standard-state fugacity for any component i whose activity coefficient is normalised by Equation (14). ... 

In a binary liquid solution containing one noncondensable and one condensable component, it is customary to refer to the first as the solute and to the second as the solvent. Equation (13) is used for the normalization of the solvent s activity coefficient but Equation (14) is used for the solute. Since the normalizations for the two components are not the same, they are said to follow the unsymmetric convention. The standard-state fugacity of the solvent is the fugacity of the pure liquid. The standard-state fugacity of the solute is Henry s constant. 

The standard-state fugacity of any component must be evaluated at the same temperature as that of the solution, regardless of whether the symmetric or unsymmetric convention is used for activity-coefficient normalization. But what about the pressure At low pressures, the effect of pressure on the thermodynamic properties of condensed phases is negligible and under such con-... 

Equation (16) is a differential equation and applies equally to activity coefficients normalized by the symmetric or unsymme-tric convention. It is only in the integrated form of the Gibbs-Duhem equation that the type of normalization enters as a boundary condition. 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

When the pressure is low and mixture conditions are far from critical, activity coefficients are essentially independent of pressure. For such conditions it is common practice to set P = P in Equations (18) and (19). Coupled with the assumption that v = v, substitution gives the familiar equation... 

Chapter 3 discusses calculation of fugacity coefficient < ). Chapter 4 discusses calculation of adjusted activity coefficient Y fugacity of the pure liquid f9 [Equation (24)], and Henry s constant H. 

It is important to be consistent in the use of fugacity coefficients. When reducing experimental data to obtain activity coefficients, a particular method for calculating fugacity coefficients must be adopted. That same method must be employed when activity-coefficient correlations are used to generate vapor-liquid equilibria. 

Equations (2) and (3) are physically meaningful only in the temperature range bounded by the triple-point temperature and the critical temperature. Nevertheless, it is often useful to extrapolate these equations either to lower or, more often, to higher temperatures. In this monograph we have extrapolated the function F [Equation (3)] to a reduced temperature of nearly 2. We do not recommend further extrapolation. For highly supercritical components it is better to use the unsymmetric normalization for activity coefficients as indicated in Chapter 2 and as discussed further in a later section of this chapter. 

Activity-coefficient data at infinite dilution often provide an excellent method for obtaining binary parameters as shown, for example, by Eclcert and Schreiber (1971) and by Nicolaides and Eckert (1978). Unfortunately, such data are rare. 

When no experimental data at all are available, activity coefficients can sometimes be estimated using the UNIFAC method (Fredenslund et al., 1977a, b). However, for many real engineering problems it is often necessary to obtain new experimental data. 

Figure 4-7. Vapor-liquid equilibria and activity coefficients in a binary system showing a weak minimum in the activity coefficient of methanol.

Figure 4-9. Vapor-liquid equilibria for a binary system where one component dimerizes in the vapor phase. Activity coefficients show only small deviations from liquid-phase ideality.

Moderate errors in the total pressure calculations occur for the systems chloroform-ethanol-n-heptane and chloroform-acetone-methanol. Here strong hydrogen bonding between chloroform and alcohol creates unusual deviations from ideality for both alcohol-chloroform systems, the activity coefficients show... 

As discussed in Chapter 2, for noncondensable components, the unsymmetric convention is used to normalize activity coefficients. For a noncondensable component i in a multicomponent mixture, we write the fugacity in the liquid phase... 

Null (1970) discusses some alternate models for the excess Gibbs energy which appear to be well suited for systems whose activity coefficients show extrema. 

When a condensable solute is present, the activity coefficient of a solvent is given by Equation (15) provided that all composition variables (x, 9, and ) are taicen on an (all) solute-free basis. Composition variables 9 and 4 are automatically on a solute-free basis by setting q = q = r = 0 for every solute. 

Figure 4-11. Activity coefficients for noncondensable solutes at infinite dilution.

In some cases, the temperature of the system may be larger than the critical temperature of one (or more) of the components, i.e., system temperature T may exceed T. . In that event, component i is a supercritical component, one that cannot exist as a pure liquid at temperature T. For this component, it is still possible to use symmetric normalization of the activity coefficient (y - 1 as x - 1) provided that some method of extrapolation is used to evaluate the standard-state fugacity which, in this case, is the fugacity of pure liquid i at system temperature T. For highly supercritical components (T Tj,.), such extrapolation is extremely arbitrary as a result, we have no assurance that when experimental data are reduced, the activity coefficient tends to obey the necessary boundary condition 1... 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the... 

Since we make the simplifying assumption that the partial molar volumes are functions only of temperature, we assume that, for our purposes, pressure has no effect on liquid-liquid equilibria. Therefore, in Equation (23), pressure is not a variable. The activity coefficients depend only on temperature and composition. As for vapor-liquid equilibria, the activity coefficients used here are given by the UNIQUAC equation. Equation (15). ... 

Liquid-liquid equilibria are much more sensitive than vapor-liquid equilibria to small changes in the effect of composition on activity coefficients. Therefore, calculations for liquid-liquid equilibria should be based, whenever possible, at least in part, on experimental liquid-liquid data. 

Evaluation of the activity coefficients, (or y for noncondensable components),is implemented by the FORTRAN subroutine GAMMA, which finds simultaneously the coefficients for all components. This subroutine references subroutine TAUS to obtain the binary parameters, at system temperature. 

The molar excess enthalpy h is related to the derivatives of the activity coefficients with respect to temperature according to... 

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