Basic Phase-Equilibrium Relations

Consider any two homogeneous phases a and P in equilibrium with one another at temperature T and pressure P. Each phase contains any number of nonreacting components C that can freely cross the phase boundary. No other constraints apply, so the intensive state can be identified by specifying values for T properties, with the number tF given by the phase rule (9.1.13), 

For example, if in addition to T (or P), we know values for the (C -1) mole fractions in one phase, then the state is specified, and we should be able to compute values for P (or T) plus the (C -1) mole fractions in the other phase. The computation requires us to solve the C phase-equilibrium conditions 

If we choose to use FFF 1 for all components in both phases, then we have selected the phi-phi method, and the equilibrium conditions (7.3.12) become 

Since the pressures are the same in the two phases, this reduces to 

The fugacity coeffidents are to be obtained from a model for the PvTx equation of state. The volume explicit form, v P, T, x ), should not be used for multiphase systems instead, a pressure explicit model, P(T, v, x ), should be chosen. Then the fugacity coefficients would be computed from the integral in (4.4.23). 

---

BASIC PHASE-EQUILIBRIUM RELATIONS 421 10.1 BASIC PHASE-EQUILIBRIUM RELATIONS... 

Basic equations for mass and enthalpy balances, phase equilibrium relations, and the formulation of the concentration differ-... 

In discussing chemical systems, one must be aware of the rules which determine the chemical species that are permitted to occur for a given set of conditions. The basic rule governing systems which are considered to be in thermodynamic equilibrium was first stated by J. Willard Gibbs as early as 1876. The Gibbs phase rule relates the physical state of a mixture with the chemical species of which it is composed and is given in its simplest form as... 

As discussed in Section 3.10.3, in the gas phase the basicity of simple amines follows the order NMe3 > NHMe2 > NH2Me > NH3 because of the electron donating effect of the methyl (Me) groups. In solution, however, we can define a basicity constant as the equilibrium constant for the reaction shown in Equation 3.4. Note it is important to specify temperature, solvent (usually water) and solution ionic strength, 1 Basicity constants are related to the acid dissociation constants (/Q of the base s conjugate acid via the dissociation constant of water, K = 10 14 at 25 °C. Thus Kbx K = Kw. 

This chapter describes basic physico-chemical relations between the gas phase transport of atoms and molecules and their thermochemical properties, which are related to the adsorption-desorption equilibrium. These methods can either be used to predict the behavior of the adsorbates in the chromatographic processes, in order to design experiments, or to characterize the absorbate from its experimentally observed behavior in a process. While Part I of this chapter is devoted to basic principles of the process, the derivation of thermochemical data is discussed in Part n. Symbols used in the following sections of Part I are described in Section 5. For results, which were obtained applying the described evaluation methods in gas-adsorption chromatography, see Chapters 4 and 7 of this book. 

If an equation of state is used to describe both phases, the basic equilibrium relation becomes... 

In 10.1 we present the basic thermodynamic relations that are used to start phase-equilibrium calculations we discuss vapor-liquid, liquid-liquid, and liquid-solid calculations. We have seen that the most interesting phase behavior occurs in nonideal solutions, but when we describe nonidealities using an ideal solution as a basis, we must select an appropriate standard state. Common options for standard states are discussed in 10.2 they include pure-component standard states and dilute-solution standard states. 

The phase relations for quasi-binary solutions outlined in Section 1 are general and exact under the basic assumptions made. However, the computational work with them becomes exponentially difficult as the number of components increases. In fact, it is virtually impossible to solve the phase equilibrium equations for solutions of actual synthetic polymers, which contain an almost infinite number of components. We thus need a novel approach to analyze phase equilibrium data on such systems. The discipline called continuous thermodynamics has emerged to meet this requirement. It deals with mixtures of molecules whose physical properties such boiling point, molecular weight, and so forth vary continuously, and is the correct method for treating solutions of a truly polydisperse polymer (see Section 1.1 of this chapter for its definition). 

This example also illustrates the use of the three basic concepts on which the analysis of more complex mass transfer problems is based namely, conservation laws, rate expressions, and equilibrium thermodynamics. The conservation of mass principle was implicitly employed to relate a measured rate of accumulation of sugar in the solution or decrease in undissolved sugar to the mass transfer rate ftom the crystals. The dependence of the rate expression for mass transfer on various variables (area, stirring, concentration, etc.) was explored experimentally. Phase-equilibrium thermodynamics was involved in setting limits to the final sugar concentration in solution as well as providing the value of the sugar concentration in solution at the solution-crystal interface. 

Methods I and 2 are commonly used for regeneration of adsorbents used for gaseous phase adsorption. Naturally, method 2 can be applied for liquid phase adsorption if the equilibrium relation allows in specific cases. Fig. 9.1 shows these schemes of desorption. Desorption using an inert stream free of adsorbent is essentially the same operation as adsorption, which can be analyzed by the same basic equation with different initial, and boundary conditions. The same is true of desorption at high temperature (thermal desorption) except that the equilibrium relation is very different. Also, in the actual operation of thermal desorption, nonisothermal treatment becomes important in most cases. The combination of desorption at low pressure and adsorption at high pressure is the principle of pressure swing operation (PSA), which is discussed in Chapter II. 

The basic model equations for coimtercurrent continuous mass transfer are the differential mass balances over each phase, which were derived in Illustration 2.3, and the companion equilibrium relation (Equation 2.12f). They represent a complete model for tiie system and are reproduced below ... 

The first three chapters of the book are an introduction and review of basic thermodynamics and of very simple equilibrium. Chapters 4—7 set out the basic thermodynamics of equilibrium. Chapters 8-10 deal with the most common type of problem, vapor-liquid equilibrium. Chapter 11 deals with other kinds of phase equilibrium. Chapters 12-13 deal with chemical equilibrium, and Chapters 14,15 and 16 deal with a variety of related topics. Appendix A contains the data tables that are used for examples and homework problems. Appendixes B-G contain derivations and other material that supports the material in main text. It is placed there to keep the treatment in the texi as simple as possible. Appendix H contains answers to some of the problems. 

The Volta potential is defined as the difference between the electrostatic outer potentials of two condensed phases in equilibrium. The measurement of this and related quantities is performed using a system of voltaic cells. This technique, which in some applications is called the surface potential method, is one of the oldest but still frequently used experimental methods for studying phenomena at electrified solid and hquid surfaces and interfaces. The difficulty with the method, which in fact is common to most electrochemical methods, is lack of molecular specificity. However, combined with modem surface-sensitive methods such as spectroscopy, it can provide important physicochemical information. Even without such complementary molecular information, the voltaic cell method is still the source of much basic electrochemical data. 

Kinetic Acidities in the Condensed Phase. For very weak acids, it is not always possible to establish proton-transfer equilibria in solution because the carbanions are too basic to be stable in the solvent system or the rate of establishing the equilibrium is too slow. In these cases, workers have turned to kinetic methods that rely on the assumption of a Brpnsted correlation between the rate of proton transfer and the acidity of the hydrocarbon. In other words, log k for isotope exchange is linearly related to the pK of the hydrocarbon (Eq. 13). The a value takes into account the fact that factors that stabilize a carbanion generally are only partially realized at the transition state for proton transfer (there is only partial charge development at that point) so the rate is less sensitive to structural effects than the pAT. As a result, a values are expected to be between zero and one. Once the correlation in Eq. 13 is established for species of known pK, the relationship can be used with kinetic data to extrapolate to values for species of unknown pAT. 

How is an equation of state related to an adsorption isotherm What is the basic thermodynamic principle that governs the equilibrium between the surface phase and bulk phase ... 

---