Phase rule derivation

Gibb s Phase Rule. The phase rule derived by W. J. Gibbs applies to multiphase equilibria in multicomponent systems, in the absence of chemical reactions. It is written as... 

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. 

The most broadly recognized theorem of chemical thermodynamics is probably the phase rule derived by Gibbs in 1875 (see Guggenheim, 1967 Denbigh, 1971). Gibbs phase rule defines the number of pieces of information needed to determine the state, but not the extent, of a chemical system at equilibrium. The result is the number of degrees of freedom Np possessed by the system. 

The phase rule, derived from thermodynamics by Gibbs, is an important ordering principle, which establishes in equilibrium systems (or models) the relationship between the number of components, the number of phases, and the degrees of freedom ... 

The vapour-liquid critical point is non-variant because for this unique state there is an extra condition to be added to the phase rule derivation, namely the complete identity of the two phases. 

The two degrees of freedom for this system may be satisfied by setting T and P, or T and t/j, or P and a-j, or Xi and i/i, and so on, at fixed values. Thus, for equilibrium at a particular T and P, this state (if possible at all) exists only at one liquid and one vapor composition. Once the two degrees of freedom are used up, no further specification is possible that would restrict the phase-rule variables. For example, one cannot m addition require that the system form an azeotrope (assuming this possible), for this requires Xi = i/i, an equation not taken into account in the derivation of the phase rule. Thus, the requirement that the system form an azeotrope imposes a special constraint and reduces the number of degrees of freedom to one. 

While the Gibbs phase rule provides for a qualitative explanation, we can apply the Clapeyron equation, derived earlier [equation (5.71)], in conjunction with studying the temperature and pressure dependences of the chemical potential, to explain quantitatively some of the features of the one-component phase diagram. 

A part of the chemical consequences of the cyclic orbital interactions in the cyclic conjngation is well known as the Hueckel rule for aromaticity and the Woodward-Hoffmann rule for the stereoselection of organic reactions [14]. In this section, we describe the basis for the rnles very briefly and other rules derived from or related to the orbital phase theory. The rules include kinetic stability (electron-donating and accepting abilities) of cyclic conjugate molecules (Sect. 2.2.2) and discontinnity of cyclic conjngation or inapplicability of the Hueckel rule to a certain class of conjngate molecnles (Sect. 2.2.3). Further applications are described in Sect. 4. 

The phase rule has been applied more conveniently to ESP measurements taken as a function of temperature. Again, Gershfeld (1982) has shown that a plateau or discontinuity in the ESP versus temperature plot may be indicative of a three-way equilibrium between the floating crystal and the separate monolayer phases that have spread from this crystal. This treatment has been used to argue for the existence of surface bilayers of phosphatidylcholine derivatives (Gershfeld, 1986, 1988). 

The proof of the phase rule is actually implicit in the derivation of the governing equations (Eqns. 3.32-3.35), and is not repeated here. It is interesting, nonetheless, to compare this well-known result with the governing equations, if only to demonstrate that we have reduced the problem to the minimum number of independent variables. 

The phase rule is often used in the form t = c - p + 2 to ascertain the number of degrees of freedom of a system even when the concentration units in the aqueous phase are molal (m) or molar. This is not correct because the phase rule is derived 1n terms of mole fractions (X). Thus, an additional quantity, the total number of moles, is required to convert X into m. This is illustrated by equations below which we shall find useful later on. 

The mathematical basis of classic thermodynamics was developed by J. Willard Gibbs in his essay [1], On the Equilibrium of Heterogeneous Substances, which builds on the earlier work of Kelvin, Clausius, and Helmholtz, among others. In particular, he derived the phase mle, which describes the conditions of equilibrium for multiphase, multicomponent systems, which are so important to the geologist and to the materials scientist. In this chapter, we will present a derivation of the phase rule and apply the result to several examples. 

When we derived the phase rule, we assumed that all phases are at the same pressure. In mineral systems, fluid phases can be at a pressure different from the solid phases if the rock column above them is permeable to the fluid. Under these circumstances, the system has an additional degree of freedom and the equilibrium at any depth depends on both the fluid pressure Pp and the pressure on the solid Ps at that level. Each pressure is determined by p, the density of the phase, and h, the height of the column between the surface and the level being studied. 

In this chapter, we discuss classical non-stoichiometry derived from various kinds of point defects. To derive the phase rule, which is indispensable for the understanding of non-stoichiometry, the key points of thermodynamics are reviewed, and then the relationship between the phase rule, Gibbs free energy, and non-stoichiometry is discussed. The concentrations of point defects in thermal equilibrium for many types of defect structure are calculated by simple statistical thermodynamics. In Section 1.4 examples of non-stoichiometric compounds are shown referred to published papers. 

Non-stoichiometry, which originates from various kinds of lattice defect, can be derived from the phase rule. As an introduction, let us consider a trial experiment to understand non-stoichiometry (this experiment is, in principle, analogous to the one described in Section 1.4.8). Figure 1.1 shows a reaction vessel equipped with a vacuum pump, pressure gauge for oxygen gas, pressure controller for oxygen gas, thermometer, and chemical balance. The temperature of the vessel is controlled by an outer-furnace and the vessel has a special window for in-situ X-ray diffraction. A quantity of metal powder... 

The relation between the non-stoichiometry and the equilibrium oxygen pressure mentioned in Section 1.1 can be deduced from the phase rule. For the purpose of the derivation of the phase rule, we shall review fundamental thermodynamics. Gibbs free energy G is defined by the relation... 

After the conditions of equilibrium have been determined, we can derive the phase rule and determine the number and type of variables that are necessary to define completely the state of a system. The concepts developed in this chapter are illustrated by means of graphical representation of the thermodynamic functions. 

The derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

The Gibbs-Duhem equation is applicable to each phase in any heterogenous system. Thus, if the system has P phases, the P equations of Gibbs-Duhem form a set of simultaneous, independent equations in terms of the temperature, the pressure, and the chemical potentials. The number of degrees of freedom available for the particular systems, no matter how complicated, can be determined by the same methods used to derive the phase rule. However, in addition, a large amount of information can be obtained by the solution of the set of simultaneous equations. 

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