Modified Raoults law

As mentioned previously, many systems show significant deviations from Raoult s law. There are even some solutions, such as azeotropic mixtures, that are not even qualitatively described by Raoult s law. By rearranging the definition of the activity coefficient (see Eq. (6.2)), Raoult s law can be generalized to apply to a wider class of mixtures  

Raoult s law was derived under the assumptions that the vapor phase behaved as an ideal gas and 

In general, an excess property is defined as the deviation of a property from its ideal solution value  

Substituting the expression for the chemical potential given in Eq. (6.4) into Eq. (5.7), we find that the Gibbs free energy of a non-ideal solution can be written as  

While the theoretical treatment we have presented for non-ideal solutions are all formally exact, they are of no practical use until we have some kind of model for the variation of the activity coefficients with the system composition, temperature, and pressure. In the next section, we present various models that can be employed to describe the activity coefficients. 

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Example 3 Detv and Bubble Point Calculations As indicated by Example 2a, a binary system in vapor/liquid equilibrium has 2 degrees of freedom. Thus of the four phase rule variables T, P, x, and t/i, two must be fixed to allow calculation of the other two, regardless of the formulation of the equilibrium equations. Modified Raoults law [Eq. (4-307)] may therefore be applied to the calculation of any pair of phase rule variables, given the other two. 

If the vapor phase is close to an ideal phase, then Eq. (1.190) yields the modified Raoults law... 

Of the assumptions inherent in this approximate relation, that of Hquid-phase ideaHty is usually least reaHstic. Therefore, the simplest generally useful special case of equation 222 is the modified Raoult s law expression ... 

For part i, use the modified Raoult s Law. Define the pressure and vapor mole fraction yj as functions of the liquid mole fraction,... 

The final fractions represent corrections to modified Raoult s law for vapor nonidealities... 

This modified Raoult s law was used for data reduction in Sec. 11.6. Bubble- and dew-point calculations made with Eq. (11.74) are, of course, somewhat simpler than those shown by Figs. 12.12 through 12.15. Indeed, the BUBL P calculation yields final results in a single step, without iteration. The additional assumption of liquid-phase ideality (yk - 1), on the other hand, is justified only infrequently. We note that yk for ethanol in Table 12.1 is greater than 8. 

Given in what follows are values of infinite-dilution activity coefficients and pure-spedes vapor pressures for binary systems at spedfied temperatures. For one of the systems, determine the Margules parameters, and then apply the Margules equation to a suffident number of VLE calculations to allow construction of a Pxy diagram for the given temperature. Base your calculations on the modified Raoult s-law expression, i.e., Eq. (11.74). 

The vapor-liquid phase equilibrium is represented by the modified Raoult s law (Eq. 1.191)... 

Tliis is a restatement of Eq. (10.5), modified Raoult s law, and is adequate for present purposes, allowing easy calculation of activity coefficients from experimental low-pressure VLE data. Values from tins equation appear in the last two columns of Table 12.1. 

The simplest models for vapor/liquid equilibrium, based on Raoult s law and Henry s law, are presented in Chap. 10, largely from an empirical point of view. The calculations by modified Raoult s law, described in Sec. 10.5, are adequate for many purposes, but are hmited to low pressures. The initial sections of this chapter therefore present two general calculational procedures for VLB, the first by an extension of modified Raoult s law and the second by equations of state. The theoreticalfoundationforbothproceduresispresentedinChap. 11. The remainder of tliis clrapter deals more generally witlr plrase equilibria, with consideration given in separate sections to liquid/liquid, vapor/liquid/liquid, solidlliquid, solid/vapor, adsorption, and osmotic equilibria. 

The nature of dewpoint and bubblepointcalculatioiis is evidentfrom the examples of Secs. 10.4 and 10.5 for Raoult s law and modified Raoult s law. All such calculations made by the gammdplii fommlation require iteration because of its complex functionality ... 

Values of parameters for the Margules, van Laar, Wilson, NRTL, and UNIQUAC equations are given for many binary pairs by Gmehling et ai in a summary collection of the world s published VLB data for low to moderate pressures. These values are based on reduction of experimental data through application of modified Raoult s law,Eq. (10.5). On the other hand, data reduction for determination of parameters in the UMFAC method (App. H) does not include tlie ideal-gas assumption, and is carried out with Eq. (14.1). 

Tliermodynamic insight into tlie phenomenon of low-pressure VLLE is provided by tire modified Raoult s-law expression, Eq. (10.5). For temperature T and tire three-phase-equilibrium pressure P, Eq. flOSjhas a double applieation ... 

Modified Raoult s Law Of the qualifications that lead to Raoult s law, the one least often reasonable is the supposition of solution ideality for the liquid phase. Real solution behavior is reflected by values of activity coefficients that differ from unity. When y, of Eq. (4-303) is retained in the equilibrium equation, the result is the modified Raoult s law. 

When Raoult s law applies, Eq. (4-326) reduces to K, = P( /P. For modified Raoult s law, K, = yP/ /P. With K) = y,/x, these are alternative expressions of Raoult s law and modified Raoult s law. Were Raoult s valid, K values could be correlated as functions of just T and P. However, Eq. (4-326) shows that they are in general functions of T, P Xi), and y,, making convenient and accurate correlation impossible. Those correlations that do exist are approximate and severely limited in application. The nomographs for K values of light hydrocarbons as functions of T and P, prepared by DePriester Eng. 

One of the shortcomings of Raoult s law is that it cannot describe the behavior of azeotropic systems. The modified Raoult s law, however, can describe azeotropic systems as we will demonstrate below. 

In order to describe vapor-liquid equilibria, we have been using Raoult s law (see Eq. (6.1)) or the modified Raoult s law (see Eq. (6.3)). One of the required inputs to the modified Raoult s law is the vapor pressure of each component a in the pure state this is the pressure below which pure... 

If we compare Henry s law and the modified Raoult s law, then we find that Henry s law can be considered as the modified Raoult s law with 7 =... 

If the vapor phase is ideal, then the fugacity coefficients are equal to one, and we see that Eq. (11.25) becomes the modified Raoult s law that we have derived previously. 

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