Nonideal solutions, calculating

For nonideal solutions, the thermodynamic equilibrium constant, as given by Equation (7.29), is fundamental and Ei mettc should be reconciled to it even though the exponents in Equation (7.28) may be different than the stoichiometric coefficients. As a practical matter, the equilibrium composition of nonideal solutions is usually found by running reactions to completion rather than by thermodynamic calculations, but they can also be predicted using generalized correlations. 

Here BLS is the second virial coefficient of the polymeric solute in the original solution before ultracentrifugation. BLs is a quantity which can be obtained in light-scattering experiments (17, 25, 30) or in Archibald experiments (31), provided it is calculated from a plot of l/MWapp° vs. c. Here 1/Mw pp° is obtained from values of Mw pp (at rm or rb) that have been extrapolated to zero time. The reason for using Equation 75 is that it leads to a simple method of estimating the MWD in nonideal solutions. 

In order to use Donnelly s (11,12) method, we must convert the observed values of cr and d In c/d(r2) in nonideal solutions to ideal values. Then we can use the ideal value of d In c/d(r2) to calculate MWD s as has been described previously here and elsewhere (11, 12). One must also remember to use Equation 49 for non-sector-shaped centerpieces. 

The calculations shown in Fig. 4 represent an upper bound, but not an unreasonable one. Firstly, although the void was assumed to remain at constant volume, compaction of the laminate is certainly occurring, and unless resin flow into the bleeder is significant, not much resin volume decrease will occur. Furthermore, if volume were not kept constant, then the voids would grow and likely coalesce, a condition of stabilization just as bad, if not worse, than that considered in the calculation. Secondly, it is assumed that Raoult s Law provides the relationship between the water partial pressure in the void and the water dissolved in the resin. It is more likely that Henry s Law, or some other nonideal solution rule, would hold, since the water-resin solution is certainly not an ideal solution unfortunately there is no way to estimate a priori what nonideal rule should apply. Thus, using Raoult s Law provides an upper bound on the partial pressure, and the calculation represents a worst-case estimate. 

Jefferson Tester I would like to shift gears and direct a question to John Prausnitz regarding his comments. You didn t talk very much about some of your own contributions and those of your students, for instance, the NRTL equation and UNIQUAC-UNIFAC models for nonideal solutions now in widespread use throughout the chemical industry and certainly employed by many people making practical calculations. There have been extensions of that local composition approach, in particular to electrolyte systems, by C. C. Chen and others. I wonder how you personally feel about that and... 

The properties of mixtures of ideal gases and of ideal solutions depend solely on the properties of the pure constituent species, and are calculated from them by simple equations, as illustrated in Chap. 10. Although these models approximate the behavior of certain fluid mixtures, they do not adequately represent the -behavior of most solutions of interest to chemical engineers, and Raoult s law is not in general a realistic relation for vapor/liquid equilibrium. However, these models of ideal behavior—the ideal gas, the ideal solution, and Raoult s law— provide convenient references to which the behavior of nonideal solutions may be compared. 

Compute the Gfj parameters for the Wilson equation. General engineering practice is to establish liquid-phase nonideality through experimental measurement of vapor-liquid equilibrium. Models with adjustable parameters exist for adequately representing most nonideal-solution behavior. Because of these models, the amount of experimental information needed is not excessive (see Example 3.9, which shows procedures for calculating such parameters from experimental data). 

Related Calculations. If the gas is not ideal, the fugacity coefficients , will not be unity, so the activities cannot be represented by the mole fractions. If the pressure is sufficient for a nonideal solution to exist in the gas phase, , will be a function of y, the solution to the problem. In this case, the y, value obtained for the solution with Lewis-Randall rule for... 

Figure 21. Computer-calculated distribution of alkali-containing vapor species as a junction of temperature for the nonideal solution glass-combustion gas system...

We conclude this discussion with one final reminder. The vapor-liquid equilibrium calculations we have shown in Section 6.4c are based on the ideal-solution assumption and the corresponding use of Raoult s law. Many commercially important systems involve nonideal solutions, or systems of immiscible or partially miscible liquids, for which Raoult s law is inapplicable and the Txy diagram looks nothing like the one shown for benzene and toluene. 

To perform energy balance calculations on processes involving nonideal solutions, take the... 

In order to perform quantitative thermodynamic calculations using the Gibbs free energy for a nonideal solution (see Eqs. (6.9)—(6.11)), we need explicit expressions for the activity coefficients. A few empirical expressions that are typically employed are ... 

The two liquid phases are necessarily nonideal solutions and their component equilibrium coefficients are best calculated from the activity coefficients in each phase (Section 1.3.5) ... 

However, it should be understood that, because of the assumptions and approximations used in the nonideal solution theory upon which these relations are based, the calculated values for conditions at the point of maximum synergism may only approximate the values found under experimental conditions and should be used mainly for estimation purposes. This is especially true when commercial surfactants are used that may contain surface-active materials (impurities) of a type different from that of the nominal surfactant. These may cause the molecular interaction parameters to have values somewhat different from those listed in Table 11-1 for the nominal surfactant. When such impurities are suspected, it is advisable to determine experimentally the values of the interaction parameters. 

Most foods show marked nonideality, and calculation of aw from the composition is generally not feasible. For a mixed solution, calculation may be done according to the so-called Ross equation,... 

The difference between the calculated results for nonideal solutions attains a maximum value at a 0.3-0.4 for all isotherms within the temperature range 373 < T < 623 K. The maximum difference in the total pressure predicted by the approaches of pages 344-346 is 13% at T = 373 K,... 

Most problems involving the separation of nonideal solutions may be solved by use of the 6 method of convergence. When used to solve such problems, the 6 method does become slower and it may be necessary to place certain restraints on the calculational procedure. [The Almost Band Algorithm which is presented in Chap. 5 may be used to solve any problem for which the 9 method fails.]... 

Few liquid mixtures are ideal, so vapor-liquid equilibrium calculations can be more complicated than is the case for the hexane-triethylamine system, and the system phase diagrams can be more structured than Fig. 10.1-6. These complications arise from the (nonlinear) composition dependence of the species activity coefficients. For example, as a result of the composition dependence of y, the equilibrium pressure in a fixed-temperature experiment will no longer be a linear function of mole fraction. Thus nonideal solutions exhibit deviations from Raoult s law. We will discuss this in detail in the following sections of this chapter. However, first, to illustrate the concepts and some of the types of calculations that arise in vapor-liquid equilibrium in the simplest way, we will assume ideal vapor and liquid solutions (Raoult s law) here, and then in Sec. 10.2 consider the calculations for the more difficult case of nonideal solutions.. ... 

The equilibrium solubility refers to the solubility that would be obtained with very large particles (therefore, small surface areas). As in all thermodynamic calculations, activity coefficients should be used in nonideal solutions. 

In 10.1 we present the basic thermodynamic relations that are used to start phase-equilibrium calculations we discuss vapor-liquid, liquid-liquid, and liquid-solid calculations. We have seen that the most interesting phase behavior occurs in nonideal solutions, but when we describe nonidealities using an ideal solution as a basis, we must select an appropriate standard state. Common options for standard states are discussed in 10.2 they include pure-component standard states and dilute-solution standard states. 

Calculate the phase diagram of nonideal solutions using Raoult s law. 

The ideal solution assumes equal strength of self- and cross-interactions between components. When this is not the case, the solution deviates from ideal behavior. Deviations are simple to detect upon mixing, nonideal solutions exhibit volume changes (expansion or contraction) and exhibit heat effects that can be measured. Such deviations are quantified via the excess properties. An important new property that we encounter in this chapter is the activity coefficient. It is related to the excess Gibbs free energy and is central to the calculation of the phase diagram. 

Equation 11.3 assumes that the solution is sufficiently dilute that it can be considaed to be ideal. For more concentrated solutions (> 0.05 M for singly charged species), ion-pair formation and other types of intermolecular interactions can lead to nonideal solution behavior, and the ion activities ( effective concentrations ) that go into Equation 11.2 will differ somewhat from the molarities. In that case, concentrations calculated from Equation 11.3 will deviate from the actual concentrations in the solution. 

Often semiempirical models for determination of the activity coefficients that account for nonideal solution behavior are used. An application-oriented introduction is given in Ref [12]. Here, several easy-to-use approximate calculations for solubility estimation are discussed. In principle, such methods are recommended to be applied as complementary tools to experimental solubility determination that can help to reduce the experimental efforts required. 

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