Evaluation of Fugacity Coefficients

Evaluation of Fugacity Coefficients Combining Eq. (4-200) with Eq. (4-45) gives... 

Eq.9.11.4 can be used along with experimental PVT data for the evaluation of fugacity coefficients through graphical integration. As with the evaluation of enthalpy and entropy departures, however, the integration is carried out by first fitting the Fl tiata to an accurate equation of state. And since such EoS express P =f(V,T), rather than V = f P,T)y direct use of Eq.9.11.4 is not possible. 

The most often used equations of state for the evaluation of fugacity coefficients are the virial and cubic ones. 

The fugacities are evaluated at tire pressures indicated iir parentheses, where P is the equilibrium nrixed-gas pressure and P° is the equilibrium pure-gas pressure that produces the same spreading pressure. If the gas-phase fugacities are elinrinated hr favor of fugacity coefficients [Eqs. (11.32)snd (11.48)], then ... 

Mansson and Andresen (1986) tried a more sophisticated thermodynamic calculation for evaluating the fugacity coefficients along the length of the catalyst bed, but the improved accuracy does not seem to justify the excessive effort associated with this technique. 

For areal gas at temperature T and pressure />, Eq. 7.8.16 or 7.8.18 allows us to evaluate the fugacity coefficient from an experimental equation of state or a second virial coefficient. We can then find the fugacity from / = (pp. 

This expression shows that an equation of state can be used to evaluate the fugacity coefficient. Similar expressions hold for a pure component i ... 

Evaluate the fugacity coefficient ij> and compare the value you get to the value of found in Example 4.14. 

Determination, thus, of the fugacity coefficients through the virial equation and of the Poynting effect through Eq.9.11.14 allows the evaluation of activity coefficients from the experimental data. An appropriate computer program (GAMMA) is presented in Appendix E. 

The fugacity coefficient departure from nonideaHty in the vapor phase can be evaluated from equations of state or, for approximate work, from fugacity/compressibiHty estimation charts. References 11, 14, and 27 provide valuable insights into this matter. 

The fugacityy) of pure compressed liqiiid i must be evaluated at the T and P of the equilibrium mixture. This is done in two steps. First, one calculates the fugacity coefficient of saturated vapor 9i = by an integrated form of Eq. (4-161), written for pure species i and evalu-atea at temperature T and the corresponding vapor pressure P = Equation (4-276) written for pure species i becomes... 

The binary interaction parameters are evaluated from liqiiid-phase correlations for binaiy systems. The most satisfactoiy procedure is to apply at infinite dilution the relation between a liquid-phase activity coefficient and its underlying fugacity coefficients, Rearrangement of the logarithmic form yields... 

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. 

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. 

For the evaluation of the solubility it is necessary to know the pure component properties and to use an equation-of-state model for the evaluation of the fugacity coefficients. In general, two problems arise ... 

Both of these integrands are infinite at their lower limit, and the integrals cannot be evaluated. To circumvent this difficulty, we define the fugacity coefficient, , so that... 

Because of the equality of fugacities of saturated liquid and vapor, calculation of fugacity for species i as a compressed liquid is done in two ste First, one calculates the fugacity coefficient of saturated vapor = by integrated form of Eq. (11.20), evaluated for P = P . Then by Eqs. (11. and (11.23),... 

The appropriate expression for the equilibrium equation is Eq. (15.23). equation requires evaluation of the fugacity coefficients of the species prese equilibrium. Although the generalized correlation of Sec. 11.4 is applicable calculations involve iteration, because the fugacity coefficients are functions of sition. For purposes of illustration, we carry out only the first iteration, based assumption that the reaction mixture is an ideal solution. In this case Eq. (1 reduces to Eq. (15.24), which requires fugacity coefficients of the pure reacting at the equilibrium T and P. Since v = = -1, this equation becomes... 

In this equation is the fugacity coefficient of pure saturated i (either H vapor) evaluated at the temperature of the system and at A, the vapor pr pure i. The assumption that the vapor phase is an ideal solution allows sub of < CiH4 for < csh,. where 4>cth s tit fugacity coefficient of pure ethylene system T and P. With this substitution and that of Eq. (F), Eq. ( ) becomes... 

Fugacity coefficients (and therefore fugadties) are evaluated by this equation from PVT data or from an equation of state. For example, when the compressibility factor is given by Eq. (3.31), we have... 

The integrals in these equations may be evaluated numerically or graphically for various values of Tr and Pr from the data for and Z given in Tables E.l through E.4 (App. E). Another method, and the one adopted by Lee and Kesler to extend their correlation to fugacity coefficients, is based on an equation of state. 

For purposes of illustration we evaluate the pure-species fugacity coefficients by Eq. (4-206), written here as... 

The data necessary for the evaluation of the activity coefficients (or fugacities) of the individual gases in a mixture ( 30e) are not usually available, and an alternative treatment for allowing for departure from ideal behavior is frequently adopted. 

Fugacity coefficient 2 is evaluated from en appropriate equation of state by procedures described In Section 1.3. The fogacity of the solid is given by an analogue of Eq- (1.2-33), namely,... 

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