Activity coefficients, function

Studies of the reaction between various substituted benzenes and nitric acid, which can take place in a number of different acid systems, have been reported in an extensive series of papers by the Marziano group, using correlations with the Me activity coefficient function. This reaction is a two-stage process,... 

This activity coefficient function was applied to toluene on the retentate/feed side in the model. The activity coefficient of toluene on the permeate side was assumed to be unity, because the solute mole fraction is sufficiently low. For simplicity, aU the TOABr activity coefficients were assumed to be unity since the solute mole fraction on the permeate side is close to zero and so this term does not contribute significantly to the results. 

The design model was then apphed to the docosane system. The results were calculated Brsdy assuming that the activity coefficients of the solvent and solute were equal to unity, and subsequently by applying the activity-coefficient functions derived from the UNIFAC data (Eqs. (18) and (19)). The comparisons of the model results with the experimental values for the permeate flux and docosane rejection are shown in Fig. 4.5 (A, A, B, B ). 

Calculated flux with activity coefficient functions, Eqs. (18), (19)... 

Since the activity coefficient has been shown to have an important role in this system, the activity coefficient function (Eq. (20)), was included in the model. 

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. 

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. 

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). ... 

In typical situations, we do not have the necessary experimental data to find constants b... To obtain these constants, we need experimental vapor-liquid equilibria (i.e. activity coefficients) as a function of temperature. 

Derive the equation of state, that is, the relationship between t and a, of the adsorbed film for the case of a surface active electrolyte. Assume that the activity coefficient for the electrolyte is unity, that the solution is dilute enough so that surface tension is a linear function of the concentration of the electrolyte, and that the electrolyte itself (and not some hydrolyzed form) is the surface-adsorbed species. Do this for the case of a strong 1 1 electrolyte and a strong 1 3 electrolyte. 

The alternative approach is to treat the film as a nonideal two-dimensional gas. One may use an appropriate equation of state, such as Eq. Ill-104. Alternatively, the formalism has been developed for calculating film activity coefficients as a function of film pressure [192]. 

The following data (for 25°C) were obtained at the pzc for the Hg-aqueous NaF interface. Estimate and plot it as a function of the mole fraction of salt in solution. In the table,/ is mean activity coefficient such that a = f m , where m is mean molality. 

Figure A2.4.6. Mean activity coefFicient for NaCl solution at 25 °C as a function of the concentration full curve from ((A2A61 ) dashed curve from ((A2A63 ) dot-dashed curve from (A2.4.64). The crosses denote experimental data. From [2],...

The true thermodynamic equilibrium constant is a function of activity rather than concentration. The activity of a species, a, is defined as the product of its molar concentration, [A], and a solution-dependent activity coefficient, Ya. 

In most quantitative analyses we are interested in determining the concentration, not the activity, of the analyte. As noted earlier, however, the electrode s response is a function of the analyte s activity. In the absence of interferents, a calibration curve of potential versus activity is a straight line. A plot of potential versus concentration, however, may be curved at higher concentrations of analyte due to changes in the analyte s activity coefficient. A curved calibration curve may still be used to determine the analyte s concentration if the standard s matrix matches that of the sample. When the exact composition of the sample matrix is unknown, which often is the case, matrix matching becomes impossible. 

Accuracy and Interpretation of Measured pH Values. The acidity function which is the experimental basis for the assignment of pH, is reproducible within about 0.003 pH unit from 10 to 40°C. If the ionic strength is known, the assignment of numerical values to the activity coefficient of chloride ion does not add to the uncertainty. However, errors in the standard potential of the cell, in the composition of the buffer materials, and ia the preparatioa of the solutioas may raise the uacertaiaty to 0.005 pH unit. 

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

The solubihty parameter, 5, is a function of temperature, but the difference 6 — 6) is only weaMy dependent on temperature. By convention, both 5 and IV are evaluated at 25°C and are treated as constants independent of both T and P. The activity coefficients given by equation 30 are therefore functions of Hquid composition and temperature, but not of pressure. 

Fiend s Constant. Henry s law for dilute concentrations of contaminants ia water is often appropriate for modeling vapor—Hquid equiHbrium (VLE) behavior (47). At very low concentrations, a chemical s Henry s constant is equal to the product of its activity coefficient and vapor pressure (3,10,48). Activity coefficient models can provide estimated values of infinite dilution activity coefficients for calculating Henry s constants as a function of temperature (35—39,49). 

UNIFAC andASOG Development. Pertinent equations of the UNIQUAC functional-group activity coefficient (UNIFAC) model for prediction of activity coefficients including example calculations are available (162). Much of the background of UNIFAC involves another QSAR technique, the analytical solution of groups (ASOG) method (163). 

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]).

Throughout this section the hydronium ion and hydroxide ion concentrations appear in rate equations. For convenience these are written [H ] and [OH ]. Usually, of course, these quantities have been estimated from a measured pH, so they are conventional activities rather than concentrations. However, our present concern is with the formal analysis of rate equations, and we can conveniently assume that activity coefficients are unity or are at least constant. The basic experimental information is k, the pseudo-first-order rate constant, as a function of pH. Within a senes of such measurements the ionic strength should be held constant. If the pH is maintained constant with a buffer, k should be measured at more than one buffer concentration (but at constant pH) to see if the buffer affects the rate. If such a dependence is observed, the rate constant should be measured at several buffer concentrations and extrapolated to zero buffer to give the correct k for that pH. 

There is a third experimental design often used for studies in electrolyte solutions, particularly aqueous solutions. In this design the reaction rate is studied as a function of ionic strength, and a rate variation is called a salt effect. In Chapter 5 we derived this relationship between the observed rate constant k and the activity coefficients of reactants l YA, yB) and transition state (y ) ... 

Because it is impossible to vary single ion concentrations independently, the activity coefficient of an electrolyte is a function of activity coefficients of the cation and anion of the electrolyte. For example, for 1 1 electrolytes the relationship is... 

The proliferation of acidity functions is a consequence of the activity coefficient cancellation assumption. According to Eq. (8-89), a plot of log(cB/cBH+) against Hq should be linear with unit slope. Such plots are usually linear (for bases of closely related structure), but the slopes often differ from unity. - This behavior is an indication that the cancellation assumption (also called the zero-order approximation) is not valid, and several groups have devised alternatives. We will use the symbolism of Cox and Yates. ... 

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