Fugacity equilibrium

For most of the situations encountered in solvent extraction the gas phase above the two liquid phases is mainly air and the partial (vapor) pressures of the liquids present are low, so that the system is at atmospheric pressure. Under such conditions, the gas phase is practically ideal, and the vapor pressures represent the activities of the corresponding substances in the gas phase (also called their fugacities). Equilibrium between two or more phases means that there is no net transfer of material between them, although there still is a dynamic exchange (cf. Chapter 3). This state is achieved when the chemical potential x as... 

Generally speaking, there are two main problems in learning thermodynamics. One, of course, is to learn the details of the specialized procedures in one s disciphne, which in our case involves chemical reactions, activities, fugacities, equilibrium constants, and so on. The other, either more or less important depending on your point of view, is to understand thermodynamics as a whole - what is it, what do the variables mean, and how does it relate to other branches of science In this book, the first four chapters deal mostly with this latter problem, and the rest of the book with the details. 

According to the deviations obtained, it can be concluded that a satisfactory prediction of the solubility of solid phenolic acids in ethyl lactate can be achieved using the equi-fugacity equilibrium condition and UNIFAC group contribution approach to calculate the activity coefficient of the phenolic acid in the liquid ethyl lactate-rich phase. 

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. 

Two additional illustrations are given in Figures 6 and 7 which show fugacity coefficients for two binary systems along the vapor-liquid saturation curve at a total pressure of 1 atm. These results are based on the chemical theory of vapor-phase imperfection and on experimental vapor-liquid equilibrium data for the binary systems. In the system formic acid (1) - acetic acid (2), <() (for y = 1) is lower than formic acid at 100.5°C has a stronger tendency to dimerize than does acetic acid at 118.2°C. Since strong dimerization occurs between all three possible pairs, (fij and not... 

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

However, when carboxylic acids are present in a mixture, fugacity coefficients must be calculated using the chemical theory. Chemical theory leads to a fugacity coefficient dependent on true equilibrium concentrations, as shown by Equation (3-13). ... 

Subroutine MULLER. MULLER iteratively solves the equilibrium relations and computes the equilibrium vapor composition when organic acids are present. These compositions are used by subroutine PHIS2 to calculate fugacity coefficients by the chemical theory. 

IF BINARY SYSTEM CONTAINS NO ORGANIC ACIDS. THE SECOND VIRTAL coefficients ARE USED IN A VOLUME EXPLICIT EQUATION OF STATE TO CALCULATE THE FUGACITY COEFFICIENTS. FOR ORGANIC ACIDS FUGACITY COEFFICIENTS ARE PREDICTED FROM THE CHEMICAL THEORY FOR NQN-IOEALITY WITH EQUILIBRIUM CONSTANTS OBTAINED from METASTABLE. BOUND. ANO CHEMICAL CONTRIBUTIONS TO THE SECOND VIRIAL COEFFICIENTS. 

The computer subroutines for calculation of vapor-phase and liquid-phase fugacity (activity) coefficients, reference fugac-ities, and molar enthalpies, as well as vapor-liquid and liquid-liquid equilibrium ratios, are described and listed in this Appendix. These are source routines written in American National Standard FORTRAN (FORTRAN IV), ANSI X3.9-1978, and, as such, should be compatible with most computer systems with FORTRAN IV compilers. Approximate storage requirements and CDC 6400 execution times for these subroutines are given in Appendix J. 

The amounts of each phase and their compositions are calculated by resolving the equations of phase equilibrium and material balance for each component. For this, the partial fugacities of each constituent are determined ... 

For mixtures, the calculation is more complex because it is necessary to determine the bubble point pressure by calculating the partial fugacities of the components in the two phases at equilibrium. 

At equilibrium, a component of a gas in contact with a liquid has identical fugacities in both the gas and liquid phase. For ideal solutions Raoult s law applies ... 

One of the simplest cases of phase behavior modeling is that of soHd—fluid equilibria for crystalline soHds, in which the solubility of the fluid in the sohd phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component are equal for all phases at equilibrium under constant temperature and pressure (51). The soHd-phase fugacity,, can be represented by the following expression at temperature T ... 

Thermodynamics of Liquid—Liquid Equilibrium. Phase splitting of a Hquid mixture into two Hquid phases (I and II) occurs when a single hquid phase is thermodynamically unstable. The equiUbrium condition of equal fugacities (and chemical potentials) for each component in the two phases allows the fugacitiesy andy in phases I and II to be equated and expressed as ... 

The chemical potential pi plays a vital role in both phase and chemical-reaction equilibria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence pi, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, pi approaches negative infinity when either P or Xi approaches zero. While these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a quantity that takes the place of p. but which does not exhibit its less desirable characteristics. 

In apphcatious to equilibrium calculations, the fugacity coefficients of species iu a mixture are required. Given au expression for G /RT as aetermiued from Eq. (4-158) for a coustaut-compositiou mixture, the corresponding recipe for In is found through the partial-property relation... 

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient at given T and P reqmres prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. 

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... 

Equilibrium Constants For practical application, Eq. (4-336) must be reformulated. The initial step is elimination of the in favor of fugacities. Equation (4-74) for species i in its standard state is subtracted from Eq. (4-77) for species i in the equilibrium mixture, giving... 

The phase-distribution restrictions reflect the requirement that ff =ff at equilibrium where/is the fugacity. This may be expressed by Eq. (13-1). In vapor-hquid systems, it should always be recognized that all components appear in both phases to some extent and there will be such a restriction for each component in the system. In vapor-liquid-hquid systems, each component will have three such restrictions, but only two are independent. In general, when all components exist in all phases, the uumDer of restricting relationships due to the distribution phenomenon will be C(Np — 1), where Np is the number of phases present. 

Hydrogen Electrode an electrode at which the equilibrium (aq.) + jHj, is established. By definition, at unit activity of hydrogen ions and unit fugacity of hydrogen gas the potential of the standard hydrogen electrode h+/y//2 =... 

Vapor-phase fugacity coefficients are needed not only in high-pressure phase equilibria, but are also of interest in high-pressure chemical equilibria (D6, K7, S4). The equilibrium yield of a chemical reaction can sometimes be strongly influenced by vapor-phase nonideality, especially if reactants and products have small concentrations due to the presence in excess of a suitably chosen nonreactive gaseous solvent (S4). 

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