Real phase equilibrium

The objective here is to construct the equilibrium surface in the T-P-x space from a set of available experimental VLE data. In general, this can be accomplished by using a suitable three-dimensional interpolation method. However, if a sufficient number of well distributed data is not available, this interpolation should be avoided as it may misrepresent the real phase behavior of the system. 

Hence, depending on the rates of the various exchange and transformation processes, phase equilibrium for the chemical i may not be achieved under real-world conditions. 

Existing correlations of phase equilibrium data contain many regressed parameters, they are often semi-empirical, and they may be successful in fitting the data in parts of the phase diagram -even with high accuracy. As far as prediction is concerned, models developed for that purpose attempt to justify theoretically a link between the model parameters and real physical phenomena. However, the distinction between these two methods is often lost, since theoretically based models are forced to fit the data better by the introduction of additional adjustable parameters. 

With regard to real electrolytes, mixtures of charged hard spheres with dipolar hard spheres may be more appropriate. Again, the MSA provides an established formalism for treating such a system. The MSA has been solved analytically for mixtures of charged and dipolar hard spheres of equal [174, 175] and of different size [233,234]. Analytical means here that the system of integral equations is transformed to a system of nonlinear equations, which makes applications in phase equilibrium calculations fairly complex [235]. 

Equations of State. Equations of state having adjustable parameters are often used to model the pressure—volume—temperature (PVT) behavior of pure fluids and mixtures (1,2). Equations that are cubic in specific volume, such as a van der Waals equation having two adjustable parameters, are the mathematically simplest forms capable of representing the two real volume roots associated with phase equilibrium, or the three roots (vapor, liquid, solid) characteristic of the triple point. 

Chapters 17 and 18 use thermodynamics to describe solutions, with nonelectrolyte solutions described in Chapter 17 and electrolyte solutions described in Chapter 18. Chapter 17 focuses on the excess thermodynamic properties, with the properties of the ideal and regular solution compared with the real solution. Deviations from ideal solution behavior are correlated with the type of interactions in the liquid mixture, and extensions are made to systems with (liquid + liquid) phase equilibrium, and (fluid -I- fluid) phase equilibrium when the mixture involves supercritical fluids. 

Under real circumstances equilibrium conditions are difficult to attain, and kinetic criteria (which are often hard to predict) play a key role. To complicate matters, the stoichiometries of some of these reactions do not obey their theoretical chemical equations. In addition, the formation of a solid phase can be the result of successive reactions with varied rates. See, for example, the simplified precipitation pathway of ferric ions with hydroxide ions, depicted in Figure 5.7. 

When analyzing the deviations of phase equilibrium for real polymeric systems forming a mesophase from the theoretically calculated phase diagram, we must pay attention to one more circumstance, namely, polydisperse nature of real polymers. In all the cases the transition from an isotropic solution to anisotropic one is a result of superposition of equilibria typical of individual fractions of a polymer, which differ in molecular mass. As was shown by Flory this must result in a broadening... 

Several formalisms have been developed leading to what may be called practical thermodynamics. These treatments include the analog of solution thermodynamics, where the adsorbent and the adsorbate are considered as components in a two-phase equilibrium [6]. Another way to study the system is to use the surface excess approach, whereby the properties of the adsorbed phase are determined in terms of the properties of the real two-phase multicomponent... 

Numerous attempts have been made to develop fluid models on the basis of molecular thermodynamics, taking into account the intermolecular forces. It is beyond the scope of this book to review these theories, and, in any case, the theoretical models are not necessarily the ones that are most widely used. The success of a model rests on its ability to represent real fluids. The principle of corresponding states is another approach that provides the foundation for some of these models. A number of models, or equations of state, that have proven their practical usefulness for phase equilibrium and enthalpy departure calculations are presented in this section. 

Consequently, the continuous variation of specific volume of the vapor-liquid mixture at fixed temperature and pressure is a result of the continuous change in the fraction of the mixture that is vapor. The conclusion, then, is that an isotherm such as that shown in Fig. 7.3-2 is an approximate representation of the real phase behavior (shown in Fig. 7.3-3) by a relatively simple analytic equation of state. In fact, it is impossible to represent the discontinuities in the derivative dP/dV)T that occur at and v with any analytic equation of state. By its sigmoidal behavior in the two-phase region, the van der Waals equation of state is somewhat qualitatively and crudely exhibiting the essential features of vapor-liquid phase equilibrium historically, it was the first equation of state to do so. 

Unfortunately, very few mixtures are ideal gas mixtures, so general methods must be developed for estimating the thermodynamic properties of real mixtures. In the dis-, cussion of phase equilibrium in a. pure fluid of Sec. 7.4, the fugacity function was especially useful the same is true for mixtures. Therefore, in an analogous fashion to the derivation in Sec. 7.4. we start from... 

This example shows the real power of the equation-of-state description in that starting with relatively little information (Tc, Pc, and co of the pure components), we can obtain the phase equilibrium, phase densities, and other thermodynamic properties. 

However, unlike the case for the pure fluid, this inflection point is not the real mi.xture critical point. The mixture critical point is the point of intersection of the dew point and bubble point curves, and this must be determined from phase equilibrium calculations, more complicated mixture stability conditions, or experiment, not simply from the criterion for mechanical stability as for a pure fluid. 

This dependence on the measurement kinetics supports the view that no real phase transition occurs, but rather that the system freezes in a non-equilibrium state. 

The fugacity concept was introduced initially to account for the non-ideal behaviour of real gases. Later the concept was generalised to phase equilibrium calculation. Let us go back to the equation describing the variation of Gibbs energy with the pressure at... 

In addition to the excess properties, which are difference measures for deviations from ideal-solution behavior, we also find it convenient to have ratio measures. In particular, for phase equilibrium calculations, it proves useful to have ratios that measure how the fugacity of a real mixture deviates from that of an ideal solution. Such ratios are called activity coefficients. The activity coefficients can be viewed as special kinds of a more general quantity, called the activity so we first introduce the activity ( 5.4.1) and then discuss the activity coefficient ( 5.4.2). 

The measurement of polymer solutions with lower polymer concentrations requires very precise pressure instruments, beeause the difference in the pure solvent vapor pressure becomes very small with deereasing amount of polymer. At least, no one can really answer the question if real thermodynamie equilibrium is obtained or only a frozen non-equilibrium state. Non-equilibrium data ean be deteeted from unusual shifts of the %-function with some experience. Also, some kind of hysteresis in experimental data seems to point to non-equilibrium results. A eommon eonsisteney test on the basis of the integrated Gibbs-Duhem equation does not work for vapor pressure data of binary polymer solutions because the vapor phase is pure solvent vapor. Thus, absolute vapor pressure measurements need very careful handling, plenty of time, and an experienced experimentator. They are not the method of choiee for high-viseous polymer solutions. 

In order to describe the three-phase (Hydrate - Liquid Water - Vapor) equihbria (H-Lw-V) the theory developed by van der Waals-Platteeuw [9, 10] is traditionally used. The theory is based on Statistical Thermodynamics and according to Sloan and Koh [1] it is probably one of the best examples of using Statistical Thermodynamics to solve successfully a real engineering problem. An excellent description of the theory for the three-phase equilibrium calculations is provided in a number of publications [1,9, 10] and will not be repeated here. In addition, extensive details on the methodology for the calculation of two-phase equilibrium (H-L ) conditions can be foimd in the review papers by Holder et al. [12], and Tsimpanogiannis et al., [11]. 

The next step towards the description of phase diagrams that include macromolecules is to change from the just discussed ideal solution to the real solutions, mentioned in Sect. 2.2.5. For this purpose one can look at the phase equilibrium between a solution and the corresponding vapor. The simplest case has a negligible vapor pressure for the second component, 2. The chemical potentials for the first component, 1, in solution and in the pure gas phase are written in Fig. 7.3 as Eqs. (1-3), following the discussion given in Fig. 2.26. For component 1, the chemical potential of the solvent, Pi°, is defined in Eq. (1) for its pure state, Xj = 1.0, but at its vapor pressure, Pi°, not at atmospheric pressure. It must then be written as... 

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