Activity coefficient equations

Haeany Solution Model The initial model (37) considered the adsorbed phase to be a mixture of adsorbed molecules and vacancies (a vacancy solution) and assumed that nonideaUties of the solution can be described by the two-parameter Wilson activity coefficient equation. Subsequendy, it was found that the use of the three-parameter Flory-Huggins activity coefficient equation provided improved prediction of binary isotherms (38). 

TABLE 13-3 Activity-Coefficient Equations in Binary Form for Use with Parameters and Constants in Tables 13-2 and 13-4... 

Equilibrium Data Collection, DECHEMA, Frankfurt, Germany, 1979) have reported several volumes of data that have been correlated with activity-coefficient equations. 

The surface pair activity coefficient equation alternative to conventional activity coefficient models. AICHE J. 2002, 48, 2332-2349. 

The constants in any of the activity coefficient equations can be readily calculated from experimental values of the activity coefficients at infinite dilution. For the Wilson equation ... 

With respect to parameters for activity coefficient equations, the literature referenced earlier under this topic should be consulted. 

Calculate the temperatures and vapor compositions from the vapor-liquid equilibrium data, using the subroutine BUfiPT. Raoult s law is used in the example, but nonideality can be included by adding activity coefficient equations. Newton-Raphson convergence is used. 

It can be observed from the limiting behavior of activity coefficients [Equation (19.11)] that... 

It can be shown that the virial type of activity coefficient equations and the ionic pairing model are equivalent, provided that the ionic pairing is weak. In these cases, it is in general difficult to distinguish between complex formation and activity coefficient variations unless independent experimental evidence for complex formation is available, e.g., from spectroscopic data, as is the case for the weak uranium(VI) chloride complexes. It should be noted that the ion interaction coefficients evaluated and tabulated by Cia-vatta [10] were obtained from experimental mean activity coefficient data without taking into account complex formation. However, it is known that many of the metal ions listed by Ciavatta form weak complexes with chloride and nitrate ions. This fact is reflected by ion interaction coefficients that are smaller than those for the noncomplexing perchlorate ion (see Table 6.3). This review takes chloride and nitrate complex formation into account when these ions are part of the ionic medium and uses the value of the ion interaction coefficient (m +,cio4) for (M +,ci ) (m +,noj)- Io... 

Introducing the regular solution expression for the activity coefficients (Equations 14 and 15 with superscript s replaced by m) into Equation 24 Rubingh found the following useful relations ... 

The Osmotic Coefficient and Activity Coefficient Equations for calculating the osmotic coefficient have the formd... 

If we choose to combine the activity coefficients of the ions to obtain the mean activity coefficients, Equation (11.87) would become... 

The determination of the value of SB may be illustrated by the use of Equation (12.42). We choose the standard state of silver and silver chloride to be the solid state at the experimental temperature and pressure, so that the activities are unity. We also express the activities of the hydrogen and chloride ions in terms of their molalities and activity coefficients. Equation (12.41) then becomes... 

The local composition model (LCM) is an excess Gibbs energy model for electrolyte systems from which activity coefficients can be derived. Chen and co-workers (17-19) presented the original LCM activity coefficient equations for binary and multicomponent systems. The LCM equations were subsequently modified (1, 2) and used in the ASPEN process simulator (Aspen Technology Inc.) as a means of handling chemical processes with electrolytes. The LCM activity coefficient equations are explicit functions, and require computational methods. Due to length and complexity, only the salient features of the LCM equations will be reviewed in this paper. The Aspen Plus Electrolyte Manual (1) and Taylor (21) present the final form of the LCM binary and multicomponent equations used in this work. 

Similarly, the activity coefficient equations (which can be derived from the excess Gibbs energy expression) have the general form ... 

The Lee-Erbar-Edmister method is of the same type, but uses different expressions for the fugacity and activity coefficients. The vapor phase equation of state is a three-parameter expression, and binary interaction corrections are included. The liquid phase activity and fugacity coefficient expressions were derived to extend the method to lower temperatures and to improve accuracy. Binary interaction terms were included in the liquid activity coefficient equation. 

The interfacial tension at the flat interface is calculated using eq 5.2, but with the activity coefficients equated to unity. Hence... 

This equation is useful for an ionic strength up various activity coefficient equations. 

In addition, there are frequently more than two nonideal-liquid components, which requires the use of a multicomponent activity-coefficient equation see Prausnitz [8]. Often the parameters of the... 

The combinatorial part of the activity coefficient [Equation (3C-2)1 is known as the Staverman-Guggenheim form. This term is intended to account for size and shape effects. The residual contribution accounts for interactions among groups. The Staverman-Guggenheim term has been shown to predict an exaggerated degree of nonideality for systems where the residual contribution is zero (Kikic et al., 1980). Predictions for such systems are expected to be less accurate. 

Combining Equations (4C-13) and (4C-18) with the definition of the activity coefficient [Equation (4C-10)] gives... 

Table 13-1, based on the binary-system activity-coefficient-equation forms given in Table 13-3. Consistent Antoine vapor-pressure constants and liquid molar volumes are listed in Table 13-4. The Wilson equation is particularly useful for systems that are highly nonideal but do not undergo phase splitting, as exemplified by the ethanol-hexane system, whose activity coefficients are shown in Fig. 13-20. For systems such as this, in which activity coefficients in dilute regions may...

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