Equilibrium between phases

In distillation work for binary systems with constant relative volatility, the equilibrium between phases for a given component can be expressed by the following equation ... 

A useful relationship between the temperature and pressure of phases in equilibrium can be derived from the condition for equilibrium. We start with equilibrium between phases A and B written as... 

In this section we limit our discussion to the phase equilibria involved with pure substances. In this case, the condition for equilibrium between phases A, B, C,..., becomes... 

The concept of mole fraction of a component used in Equation (4.1) is a convenient measure of concentration when dealing with trace quantities and dilute solutions, often experienced in environmental systems. This is especially the case with transport phenomena and equilibrium between phases, where it results in simple quantitative expressions. The phenomena of interest when dealing with the exchange of odorous compounds and oxygen between wastewater and a sewer atmosphere are, in this respect, relevant examples. 

Consider a sensor in dynamic equilibrium between phases or states A and B as illustrated in Figure 9.5. This situation may occur in pH sensors in which the lifetime of the ionized molecule is different from the lifetime of the neutral molecule. In this case a distinction needs to be made between the lifetimes of the individual states and the lifetimes measured. Let xA and xb be the lifetimes of states A and B, respectively. The measured lifetime values, t and n, are then given by the following equation 12 ... 

The condition of Equation (13.7) can be met only if p,j = p,n, which is the condition of transfer equilibrium between phases. Or, to put the argument differently, if the chemical potentials (escaping tendencies) of a substance in two phases differ, spontaneous transfer will occur from the phase of higher chemical potential to the phase of lower chemical potential, with a decrease in the Gibbs function of the system, until the chemical potentials are equal (see Section 10.5). For each component present in aU p phases, (p 1) equations of the form of Equation (13.7) provide constraints at transfer equilibrium. Furthermore, an equation of the form of Equation (13.7) can be written for each one of the C components in the system in transfer equUibrium between any two phases. Thus, C(p — 1) independent relationships among the chemical potentials can be written. As chemical potentials are functions of the mole fractions at constant temperamre and pressure, C(p — 1) relationships exist among the mole fractions. If we sum the independent relationships for temperature. 

Three-Phase Transformations in Binary Systems. Although this chapter focuses on the equilibrium between phases in binary component systems, we have already seen that in the case of a entectic point, phase transformations that occur over minute temperature fluctuations can be represented on phase diagrams as well. These transformations are known as three-phase transformations, becanse they involve three distinct phases that coexist at the transformation temperature. Then-characteristic shapes as they occnr in binary component phase diagrams are summarized in Table 2.3. Here, the Greek letters a, f), y, and so on, designate solid phases, and L designates the liquid phase. Subscripts differentiate between immiscible phases of different compositions. For example, Lj and Ljj are immiscible liquids, and a and a are allotropic solid phases (different crystal structures). 

Nucleation and Growth (Round 1). Phase transformations, such as the solidification of a solid from a liquid phase, or the transformation of one solid crystal form to another (remember allotropy ), are important for many industrial processes. We have investigated the thermodynamics that lead to phase stability and the establishment of equilibrium between phases in Chapter 2, but we now turn our attention toward determining what factors influence the rate at which transformations occur. In this section, we will simply look at the phase transformation kinetics from an overall rate standpoint. In Section 3.2.1, we will look at the fundamental principles involved in creating ordered, solid particles from a disordered, solid phase, termed crystallization or devitrification. 

A phase is a region of space in which the intensive properties vary continuously as a function of position. The intensive properties change abruptly across the boundary between phases. For equilibrium between phases, the chemical potential of any species is the same in all phases in which it exists. 

The advantages of the fluctuation theory are that it does not require that clusters be spheres, they need not have sharply defined bounding surfaces, nor is an equilibrium between phases assumed. The disadvantage is a practical one how can the work term (defined later) be evaluated ... 

Equations describing equilibriums between phases a and p, P and y, y and solution obtain as previously, and by the same reasoning one arrives at an equation analogous to (A6.8) ... 

The condition for thermodynamic equilibrium between phases is that the species chemical potentials are equal in each of the phases. Thus, at equilibrium,... 

The defining features of phase diagrams are the phase boundaries that delineate phase domains and mark the conditions of coexistence with adjacent phases. Theoretical description of a phase diagram is therefore tantamount to finding the equations of coexistence that describe these phase boundaries. For a simple phase equilibrium between phases a and /3, as shown below, the a + /3 coexistence curve is described by an equation of the form P = P(T), whose form we now wish to determine ... 

In its strictest sense the phase rule assumes that the equilibrium between phases is not influenced by gravity, electrical or magnetic forces, or by surface action. Thus, the only variables are temperature, pressure, and concentration if two are fixed, then the third is easily determined (another reason for the constant 2 in Equation 2.3). 

If the rate constants for the sorption-desorption processes are small equilibrium between phases need not be achieved instantaneously. This effect is often called resistance-to-mass transfer, and thus transport of solute from one phase to another can be assumed diffusional in nature. As the solute migrates through the column it is sorbed from the mobile phase into the stationary phase. Flow is through the void volume of the solid particles with the result that the solute molecules diffuse through the interstices to reach surface of stationary phase. Likewise, the solute has to diffuse from the interior of the stationary phase to get back into the mobile phase. 

FIGURE 11.18 A phase diagram of temperature versus composition (mole fraction) for a mixture of benzene and toluene. Liquid composition is given by the lower curve, and vapor composition is given by the top curve. The thin region between curves represents an equilibrium between phases. Liquid and vapor compositions at a given temperature are connected by a horizontal tie line, as explained in the text. 

The fundamental condition for equilibrium between phases, the equality of the chemical potential of a component in every phase in which the component is present still applies. We then substitute for the chemical potential of a component the equivalent chemical potential of the same substance in terms of species or appropriate sums of chemical potentials of the species, as determined by the methods of Section 8.15 and used in the preceding sections. Several examples are discussed in the following paragraphs. 

Most speciation modelling is based on the assumption of thermodynamic equilibrium between phases, so it is necessary to describe the various equations that are used to quantify these chemical reactions. 

The chemical potential of a homogeneous material (a phase) is a function of two intensive variables, usually chosen as temperature and pressure. We say that such a material has two degrees of freedom (i.e., we are free to set two intensive variables). (Note that only intensive variables count as degrees of freedom.) In addition to being able to specify a number of intensive variables equal to the number of degrees of freedom of a system, we are also at liberty to specify the size of the phase with one extensive variable. The chemical potential can be represented as a surface on a plot of p versus P and T. The condition for equilibrium between phase a and phase p is, according to Eq. (24),... 

Assuming thermodynamic equilibrium between phase ( ) and ("), the two-phase model in Eq. (1) can be further simplified. The resulting pseudohomogeneous model is obtained from the sum of the balances of both phases according to... 

Statistical mechanics, the science that should yield parameters like A/x , is hampered by the multibody complexity of molecular interactions in condensed phases and by the failure of quantum mechanics to provide accurate interaction potentials between molecules. Because pure theory is impractical, progress in understanding and describing molecular equilibrium between phases requires a combination of careful experimental measurements and correlations by means of empirical equations and approximate theories. The most comprehensive approximate theory available for describing the distribution of solute between phases—including liquids, gases, supercritical fluids, surfaces, and bonded surface phases—is based on a lattice model developed by Martire and co-workers [12, 13]. 

We examine a solute zone migrating under linear conditions. For the moment we confine our attention to the solute in the mobile phase. If complete equilibrium between phases existed at all points, the mobile-phase solute would form a Gaussian-like concentration profile as indicated by the shaded profile in Figure 10.7. However the actual profile for the mobile phase, shown by the dashed line, is shifted ahead of the shaded (equilibrium) profile due to solute migration. The basis of the profile shift is explained as follows. 

The phase rule of Josiah Willard Gibbs (1839-1903) gives the general conditions for chemical equilibrium between phases in a system. At equilibrium, AG = 0, there is no further change with time in any of the system s macroscopic properties. It is assumed that surface, magnetic, and electrical forces may be neglected. In this case, the phase rule can be written as... 

The Latent Heats and Clapeyron s Equation.—There is a very important thermodynamic relation concerning the equilibrium between phases, called Clapeyron s equation, or sometimes the Clapeyron-Clausius equation. By way of illustration, let us consider the vaporization of water at constant temperature and pressure. On our P-V-T surface, the process we consider is that in which the system is carried along an isothermal on the ruled part of the surface, from the state whore it is all liquid, with volume Fz, to the state where it is all gas, with volume F . As we go along this path, we wash to find ihe amount of heat absorbed. We can find this from one of Maxwell s relations, Eq. (4.12), Chap. II ... 

Clapeyron s equation holds, as we can see from its method of derivation, for any equilibrium between phases. In the general ease, the difference of volumes on the right side of the equation is the volume after absorbing the latent heat L, minus the volume before absorbing it. 

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