Chemical potential phase equilibria

The thermodynamic treatment of equilibrium is in terms of chemical potentials. Phase equilibria and phase changes are dictated by the leverage between enthalpic and entropic terms implicit in equations such as 7.49 and 7.55. Such equations, however, hold exactly for infinite and homogenous systems, but in real systems the infiuence of size, termination, and defects cannot be neglected. The microscopic texture of the system may then become of paramount importance, and it must be said at once that this is... 

The liquid-liquid extraction process is based on the specific distribution of dissolved components between two immiscible fluids, for instance, between aqueous and organic liquids. The process refers to a mass exchange processes in which the mass transport of component (j) from phase (1) to phase (2) by means of convection or molecular diffusion acts to achieve the chemical potential (p) equilibrium (134) ... 

This does not imply that all components will have the same chemical potential at equilibrium, only that the chemical potential for each component is the same in all phases. xThe independent components are those that are not related through chemical equilibrium reactions. See Section 1.2c for a discussion of components in a system. 

The concept of chemical potentials, the equilibrium criterion involving chemical potentials, and the various relationships derived from it (including the Gibbs phase rule derived in Chapter 5) can be used to explain the effect of pressure and temperature on phase equilibria in both a qualitative and quantitive way. 

It can readily be shown (see Box 3.5) that the chemical potential of a component in a two-phase system (for example, oil and water), at equilibrium at a fixed temperature and pressure, is identical in both phases. Because of the need for equality of chemical potential at equilibrium, a substance in a system which is not at equilibrium will have a tendency to diffuse spontaneously from a phase in which it has a high chemical potential to another in which it has a low chemical potential. In this respect the chemical potential resembles electrical potential hence its name is an apt description of its nature. 

It must be stated clearly that such results have been derived in a completely general manner and are thus independent from the particular EoS model used to describe the Helmholtz free energy or the penetrant chemical potential under equilibrium conditions. Non-equilibrium free energy functions can thus be obtained starting from different EoS such as LF, SAFT, PHSC, just to mention the relevant models considered in this chapter. The non-equilibrium information entering equations 2.11 and 2.12 is represented by the actual value of polymer density in the glassy phase, which must be known from a separate source of information, experimental data, or correlation, and cannot be calculated from the equilibrium EoS. 

Note that in (4.194), we use the nonconventional standard chemical potential. At equilibrium between the two phases, we have... 

If the higher-concentration phase lies in the postgel regime, the postgel form of v must be employed in the chemical potentials. These equilibrium conditions determine the total volume fractions and in each phase as well as the molecular distributions pj and in them. 

In the first two chapters, we learned about thermodynamics (free energy, osmotic pressure, chemical potential, phase diagram) of polymer solutions at equilibrium and static properties (radius of gyration, static structure factor, density correlation function) of dissolved polymer chains. This chapter is about dynamics of polymer solutions. Polymer solutions are not a dead world. Solvent molecules and polymer chains are constantly and vigorously moving to change their positions and shapes. Thermal energy canses these motions in a microscopic world. 

When the system is quenched into the metastable region between the coexistence curve and the spinodal, the resulting metastable phase may not spontaneously decay into two-phase equilibrium. The transformation must be activated by some perturbation, such as thermal fluctuations. Consider a spherical liquid droplet, of radius R, immersed in the uniform metastable vapour. The diflerence of the chemical potential in the metastable state and the chemical potential in equilibrium is A/x = p-pcxc- The total Gibbs energy of the droplet is given by the sum of the bulk and surface energy contributions. 

It is possible to calculate the free energy of both isotropic (high y) and anisotropic phases (low y) and to plot them as a function of v. From this plot it is possible to obtain the biphasic region by drawing a common tangent (the two phases have the same chemical potential at equilibrium see construction of the spinodal in Chapter 4). 

The chemical potential is the key property in the most important application of chemical engineering thermodynamics, chemical and phase equilibrium. The ease of separation, for example, of methanol from its mixture with water by distillation is determined by the relationship between the concentration of methanol in the liquid and vapor phases, which - as Gibbs showed (Section 4.12) - is dictated by the chemical potential of methanol in the two phases. 

It follows that, because phase equilibrium requires that the chemical potential p. be the same in the solution as in the gas phase, one may write for the chemical potential in the solution ... 

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

The problem has been discussed in terms of chemical potential by Everett and Haynes, who emphasize that the condition of diffusional equilibrium throughout the adsorbed phase requires that the chemical potential shall be the same at all points within the phase and since, as already noted, the interaction energy varies wtih distance from the wall, the internal pressure must vary in sympathy, so as to enable the chemical potential to remain constant. 

The criterion for phase equilibrium is given by Eq. (8.14) to be the equality of chemical potential in the phases in question for each of the components in the mixture. In Sec. 8.8 we shall use this idea to discuss the osmotic pressure of a... 

Experimental results describing limited mutual solubility are usually presented as phase diagrams in which the compositions of the phases in equilibrium with each other at a given temperature are mapped for various temperatures. As noted above, the chemical potentials are the same in the equilibrium phases, so Eqs. (8.53) and (8.54) offer a method for calculating such... 

This is an expression of Raoult s law which we have used previously. Freezing point depression. A solute which does not form solid solutions with the solvent and is therefore excluded from the solid phase lowers the freezing point of the solvent. It is the chemical potential of the solvent which is lowered by the solute, so the pure solvent reaches the same (lower) value at a lower temperature. At equilibrium... 

Chemical Potential. Equilibrium calculations are based on the equaHty of individual chemical potentials (and fiigacities) between phases in contact (10). In gas—soHd adsorption, the equiHbrium state can be defined in terms of an adsorption potential, which is an extension of the chemical potential concept to pore-filling (physisorption) onto microporous soHds (11—16). 

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. 

---