Fugacity and equations of state

TABLE 13.1. The Soave Equation of State and Fugacity Coefficients... 

Limiting L ws. Simple laws that tend to describe a narrow range of behavior of real fluids and substances, and which contain few, if any, adjustable parameters are called limiting laws. Models of this type include the ideal gas law equation of state and the Lewis-RandaH fugacity rule (10). 

The chemical literature is rich with empirical equations of state and every year new ones are added to the already large list. Every equation of state contains a certain number of constants which depend on the nature of the gas and which must be evaluated by reduction of experimental data. Since volumetric data for pure components are much more plentiful than for mixtures, it is necessary to estimate mixture properties by relating the constants of a mixture to those for the pure components in that mixture. In most cases, these relations, commonly known as mixing rules, are arbitrary because the empirical constants lack precise physical significance. Unfortunately, the fugacity coefficients are often very sensitive to the mixing rules used. 

In Section II, we discussed the fugacity coefficient, which relates the vapor-phase fugacity to the total pressure and to the composition. The fugacity coefficient can be calculated exactly from an equation of state and, therefore, the problem of calculating vapor-phase fugacities reduces to the problem of... 

For the monomers in the polymerization under consideration the fugacity coefficients were estimated by Redlich-Kwong equation of state and were found to be close to unity. The activity coefficients (8) for the monomers were estimated by Scatchard-Hildebrand s method (5) for the most volatile monomer there was a temperature dependence but none for the other monomer. These were later confirmed by applying the UNIFAC method (6). The saturation vapor pressures were calculated by Antoine coefficients (5). 

An initial guess for the pressure is assumed and the fugacity coefficient of each component in the liquid phase ( ) can be calculated. An initial guess is also assumed for the fugacity coefficient of each component in the vapour phase ( v), and consequently a first estimate of the vapour composition is evaluated. With this value of y, the fugacity coefficients in the vapour phase are recalculated using the equation of state and a second estimate for y,- is evaluated. This iterative procedure is continued until the difference between two successive values of the composition are below a predetermined error. At this point, the sum of y, is checked if the sum is different from unity a new value of the pressure is assumed for a new iteration. The iterative procedure ends when the y, differs from unity by less then a given value. 

Combining the Peng-Robinson equation of state and Equation 15-22 results in an equation for fugacity coefficient of each component.3... 

The vapor fugacity coefficient, It accounts for the effect of vapor nonideality on vapor fugacity, It is usually estimated from an equation of state and is based on system temperature, pressure, and vapor mole fraction. 

If it is assumed that the specific volume, u , of the phase is very nearly constant over a large pressure increase, then when the pressure difference in the exponential is small enough, the exponential will be nearly one. Thus, at moderate pressures the fugacity of a condensed phase is nearly equal to the vapor pressure. This approximation was used in Eq. (11) to write Raoult s Law. When the pressure is low enough that the fugacity coefficient is nearly 1.00, Eq. (13) reduces to Raoult s Law extended with the activity coefficient included or Eq. (11). The fugacity coefficient can be calculated from equations of state, and the activity coefficient can be found from various correlations as discussed earlier. 

The fugacity coefficient can be calculated from equations of state and the activity coefficient can be found from various correlations discussed in a later section. 

From the relation between the fugacity, the Gibbs free energy, and an equation of state, the fugacity in a vapor can be computed from... 

The /<-values are calculated by Equation 1.25, with the vapor phase fugacity coefficients calculated from the equation of state and the liquid phase fugacity coefficients for an ideal solution calculated as... 

Experimental VLE data, namely equilibrium temperature, pressure, and vapor and liquid compositions, can be used to calculate the activity coefficients from Equation 1.29. The X-values are calculated from the composition data, /f = T/X . The fugacities and fugacity coefficients, /T, are calculated from the compositions, temperature, and pressure, using, for instance, an equation of state and liquid density data as described earlier. The activity coefficients are then calculated by rearranging Equation 1.29 ... 

Other approaches to the computation of solid-liquid equilibria are shown in Table 11.2-3. The Soave-Redlich-Kwong equation of state evaluates fugacities to calculate solid-liquid equilibria,7 while Wenzel and Schmidt developed a modified van der Waals equation of state forthe representation ofphase equilibria. The Wenzel-Scbmidt approach generates fugacities, from which the authors developed a trial-and-error approach to compute solid-liquid equilibrium. Unno et a .9 recently presented a simplification of the solution of groups model (ASOG) that allows prediction of solution equilibrium from limited vapor-liquid equilibrium data. 

The use of the virial equation of state and the correlations of Pitzer and Curl15,48,49 for the prediction of fugacity coefficients for nonpolar mixtures is presented first and then the method is extended to polar gas mixtures. 

To compute the fugacity of a pure gaseous species we will always use a volumetric-equation of state and Eq. 7.4-6b or... 

It should be evident from the examples in Chapters 10, 11, and 12 that the evaluation of species fugacities or partial molar Gibbs energies (or chemical potentials) is central to any phase equilibrium calculation. Two different fugacity descriptions have been used, equations of. state and activity coefficient models. Both have adjustable parameters. If the values of these adjustable parameters are known or can be estimated, the phase equilibrium state may be predicted. Equally important, however, is the observation that measured phase equilibria can be used to obtain these parameters. For example, in Sec. 10.2 we demonstrated how activity coefficients could be computed directly from P-T-x-y data and how activity coefficient models could be fit to such data. Similarly, in Sec. 10.3 we pointed out how fitting equation-of-state predictions to experimental high-pressure phase equilibrium data could be used to obtain a best-fit value of the binary interaction parameter.. /"... 

Nonideal solution effects can be incorporated into /f-value formulations in two different ways. Chapter 4 described the use of the fugacity coefficient, in conjunction with an equation of state and adequate mixing rules. This is the method most frequently used for handling nonidealities in the vapor phase. However, tv reflects the combined effects of a nonideal gas and a nonideal gas solution. At low pressures, both effects are negligible. At moderate pressures, a vapor solution may still be ideal even though the gas mixture does not follow the ideal gas law. Nonidealities in the liquid phase, however, can be severe even at low pressures. In Section 4.5, il was used to express liquid-phase nonidealities for nonpolar species. When polar species are present, mixing rules can be modified to include binary interaction parameters as in (4-113). 

Fugacities were calculated from the Redlich-Kwong equation of state and there were differences of 30% between experiment and prediction, which reflects the uncertainties in calculating fugacities. 

In some earlier work the shift reaction was assumed always at equilibrium, Fugacities were calculated with the SRK and Peng-Robinson equations of state, and correlations were made of the equilibrium constants. 

Route A requires an equation of state and sophisticated mixing rules for calculating the fugacity coefficient for both the vapor and the liquid phase. The advantage of using equations of state is that other information (e.g. molar heat capacities, densities, enthalpies, heats of vaporization), which is necessary for designing and optimizing a sustainable distillation process, is also obtained at the same time. 

In 5.3 we showed how excess properties, which are difference measures for deviations from ideal-solution behavior, can be obtained from residual properties, which are difference measures for deviations from ideal-gas behavior. In this section we establish a similar set of equations that relate activity coefficients to fugacity coefficients. As a result, the equations given here, together with those in 5.3, establish a complete connection between the description of mixtures based on models for PvTx equations of state and the description based on models for and y. ... 

For the classes of binary-mixture stability behavior discussed in 8.4.2 make a table that tells whether the equation of state and the fugacity equation bifurcate. Your table should contain five rows, one for each class (I, II, IIIA, IIIB, IV), and it should have four columns, one for each equation (pure-1 equation of state, pure-2 equation of state, mixture equation of state, and mixture fugacity equation). 

Numerical calculations of phase equilibria require thermodynamic data or correlations of data. For pure components, the requisite data may include saturation pressures (or temperatures), heat capacities, latent heate, and volumetric properties. For mixtures, one requires a PVTx equation of state (for determination of d/), and/or an expression for the molar excess Gibbs energy (fw determination of yt). We have discussed in Sections 1.3 and 1.4 the correlating capabilities of selected equations of state and expressions for g, and the behavior of the fugacity coefficients and activity coefficients derived ftom them. 

In Approach A, the fugacity coefficients of the liquid (pf and vapor phase are needed. They describe the deviation from ideal gas behavior and can be calculated with the help of equations of state, for example, cubic equations of state and reliable mixing rules. In Approach B, besides the activity coefficients s value for the standard fugacity is required. In the case ofVLE usually the fugacity of the pure liquid at system temperature and system pressure is used as standard fugacity. For the calculation of the solubilities of supercritical compounds Henry constants are often applied as standard fugacity (see Section 5.7). 

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