Activity coefficient derivatives

Figure 4-2 displays plots of AH, AS, and AG as functions of composition for 6 binary solutions at 50°C. The corresponding excess properties are shown in Fig. 4-3 the activity coefficients, derived from Eq. (4-119), appear in Fig. 4-4. The properties shown here are insensitive to pressnre, and for practical pnrposes represent sohition properties at 50°C (122°F) and low pressnre (P 1 bar [14.5 psi]).

The working equations for osmotic and activity coefficients, derived from equation (3) are given as equations (4) and (5), respectively. The various secondary relationships are defined in several additional equations stated and briefly described thereafter. Additional details and derivations of equations for the entropy, the heat capacity, and other related functions can be found in various published papers (11, 20, 23-29, 32-34). 

Utilize the results obtained from the data of Saxton and Waters, given in Problem 7 of Chap. Ill, together with the activity coefficients derived from the Debye-Htickel limiting equation, to evaluate the dissociation constant of a-crotonic acid. 

This expression is seen to resemble the first two terms of equation (39.24) with hf defined by (39.51), replacing j, defined by (39.20). Although h and j become identical at infinite dilution, as will be shown below, there is an important difference between these two functions and hence between the activity coefficients derived from them. Whereas j applies to the freezing point of the solution, h refers to the particular temperature, e.g., 25 C, at which the osmotic coefficient is determined, e.g., from vapor pressure measurements. 

Show that the expression for the mean ionic activity coefficient, derived from equation (40.11), can be written in the form... 

Numerical calculations of phase equilibria require thermodynamic data or cotestations of data. For pure componeats, the requisite data may include saturation pressures (or temperatures), hem capacities, latent hems, and volumetric properties. For mixtures, one requires a PVTx equation of state (for determitiation of 4>j), and/or en expression for the molar excess Gibbs eenrgy g (for deiermiention of y,). We have disoussed in Sections 1.3 and 1.4 the correlating capabilities of selected equations or mite and expressions for gR, and the behavior of the fogacily cnefficients and activity coefficients derived from them. 

All activity coefficients derived from an excess Gibbs energy expression that satisfies the boundary conditions of being zero at. V = 0 and 1 will satisfy the Gibbs-Duhem equation. Can you prove this 0... 

Figure 9.6-2 UNIFAC predictions for the activity coefficients of the benzene-2,2,4-trimethyl pentane system. The points are activity coefficients derived from experimental data.

Note that in the activity coefficient derivatives, all variables from the set 7, P, x, other than the one... 

In order to use solubility data for salts of moderate solubility in the calculation of thermodynamic values, one must also have the corresponding activity coefficients. Such data, particularly at the higher concentrations, are exceedingly scarce. Most activity coefficient data which now exist are primarily for dilute solutions and have been derived from electrochemical measurements. This subject is covered elsewhere in this chapter, and some of the activity coefficients derived from this source are listed in the appendices. Some interesting data obtained from other sources, particularly from freezing point measurements, are now beginning to appear. ... 

Mean activity coefficients derived from cryoscopic measurements have been tabulated by C. M. Criss in Appendices 2.4.15-19. These values should be used with caution as considerable uncertainties can arise from (a) the experimental method, fb) the value used for the cryoscopic constant, (c) integration using the Gibbs-Duhem equation (eqn. 2.8.13). 

Finally, composition derivatives of chemical potentials are related to the activity coefficient derivative by... 

Figure 6.10 shows activity coefficient derivatives over the whole composition range for experiment from three correlations and the Verlet method. A procedure for experimental data analysis was described by Wooley and O Connell (1991), in which one extracts the isothermal compressibility, partial molar volumes, and activity coefficient derivatives from experimental data. The activity coefficient derivatives are obtained by fitting mixture vapor-liquid equilibrium data to obtain parameters for at least two different models. Wooley and O Connell employed the Wilson, non-random, two liquid (NRTL) and modified Margules (mM) models. Partial molar volumes are obtained from correlations of mixture densities (Handa and Benson 1979). Isothermal compressibilities are either taken from measurements or estimated with... 

FIGURE 6.11 TCFIs for (a) water/water, (b) water/t-butanol, and (c) t-butanol/t-butanol obtained from simulation of water/t-butanol mixtures using the Verlet method (crosses) versus the water mole fraction Xj, compared with TCFIs obtained from experimental data nsing the Wooley/O Connell procedure, where either the Wilson (black line), NRTL (red Une), or mM (green line) models were employed for obtaining the activity coefficient derivatives. Note that the NRTL and mM model approaches infinity since they predict a phase split. (Calculated values from R. J. Wooley and J. P. O Connell, 1991, A Database of Flnctuation Thermodynamic Properties and Molecular Correlation-Function Integrals for a Variety of Binary Liquids, Fluid Phase Equilibria, 66, 233.) (See color insert.)... 

The g -values can directly be calculated from the activity coefficients derived from the experimental VLE-data. The values for g /RT were already listed in Table 5.3. At a concentration of = 0.252 the following value for the excess Gibbs energy is obtained ... 

Pitzer and co-workers have developed an ion interaction model and published a series of papers (Pitzer, 1973a-b, 1974a-b, 1975, 1977, 1995, 2000 Pabalan Pitzer, 1987) which gave a set of expressions for osmotic coefficients of the solution and mean activity coefficient of electrolytes in the solution. Expressions of the chemical equilibrium model for conventional single ion activity coefficients derived are more convenient to use in solubility calculations (Harvie Weare, 1980 Harvie et al.l984 Felmy Weare, 1986 Donad Kean, 1985). 

Equations (66)-(68) express the activity coefficient effects for the proton adsorption processes represented by mass action Eqs (20)-(22), respectively. Since general explicit equations for activity coefficients cannot be derived solely from thermodynamics [13, 14], Eqs (66)-(68) have been derived by combining electrostatic theory with Gibbs-Lewis thermodynamics. For solute ions, the procedure is commonly referred to as the Debye-Hiickel model (Chapter 4) [8, 9, 15], while for surface site ions the activity coefficient derivations are based on the primitive interfacial model (chapter 4) [16]. 

Activity coefficients have been derived and discussed for ions and eleetrolytes and for activity coefficient quotients in mass action equations. The following is a summary reference for all the activity coefficient derivations divided into three tables. 

Since the activity coefficient derives from the relative amount of unlike vs. like interactions, its value changes with changing composition. The value also depends on the specific choice of reference state. Similarly the activity of species i, Oi compares the fugacity of species i in the liquid to the fugacity of the pure species at its reference state. The Gibbs—Duhem equation allows us to relate the activity coefficients of different species in a mixture and, thereby, to test for thermodynamic consistency. 

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