Derivation of the Phase Rule

Phase relations involve a small number of carefully defined terms. 

A phase is defined as a homogeneous body of matter having distinct boundaries with adjacent phases, and so is in principle mechanically separable from them. Each mineral in a rock is therefore a single phase, as is a salt solution, or a mixture of gases. 

Each phase therefore has a definite chemical composition, and the various phases in a system may have the same (polymorphs) or different compositions. The compositions are described in terms of chemical formulas, such as Si02 or CaMgSiOj. The smallest number of chemical formulas needed to describe the composition of all the phases in a system is called the number of components of the system. The choice of components is to some extent a matter of convenience. 

To understand the phase rule and how to use it, you must first understand the concept of variance or degrees of freedom. 

A single homogeneous phase such as an aqueous salt (say NaCl) solution has a large number of properties, such as temperature, density, NaCl molality, refractive index, heat capacity, absorption spectra, vapor pressure, conductivity, partial molar entropy of water, partial molar enthalpy of NaCl, ionization constant, osmotic coefficienf ionic strength, and so on. We know, however, that these properties are not all independent of one another. Most chemists know instinctively that a solution of NaCl in water will have aU its properties fixed if temperature, pressure, and salt concentration are fixed. In other words, there are apparently three independent variables for this two-component system, or three variables that must be fixed before all variables are fixed. Furthermore, there seems to be no fundamental reason for singling out temperature, pressure, and salt concentration from the dozens of properties available - it s just more convenient any three would do. The number of variables (system properties) that must be fixed in order to fix all system properties is known as the system variance or degrees of freedom. 

The concept of freedom plays an important role in order to understand the derivation of the phase rule. Therefore, we repeat a few terms concerning this aspect here. 

The phase rule is deduced from rather formal arguments of linear algebra pointed out above. For the individual phases, there are certain relations, i.e., equations among the intensive variables in equilibrium. These include equal temperatures, equal pressures, and equal chemical potentials. Further, there are the variables that describe the system. We chose the intensive variables and subdivide these into the chemical potentials and other intensive variables, like temperature, pressure. For each component there is a chemical potential. So the number of variables is K for the chemical potential and I for the other intensive variables in a single phase. 

There are several ways to derive the phase rule. Subsequently, we present some of the methods to derive the phase rule. 

Now consider two phases at equilibrium, say solid NaCl and a saturated salt solution. Again, intuition or experience tells us that we no longer have three independent 

We emphasize that the degrees of freedom include only intensive variables, and inasmuch as there is a functional relationship, known or unknown, between any intensive variable and all the others, the quantity c — p+ 2 refers to any combination of the intensive variables of a system. Naturally, in practice, these are normally T, P and concentrations. 

Degrees of freedom can also be described as the number of intensive variables that can be changed (within limits) without changing the number of phases in a system. This point of view is perhaps more useful to someone looking at a phase diagram thus divariant, univariant, and invariant systems correspond to areas, lines, and points in a P-T projection. We prefer however to emphasize the fact that coexisting phases reduee the number of independent variables, and that some systems have all their properties determined. This fact is very useful, as we will elaborate on below, and its explanation in terms of the Phase Rule is a very beautiful example of the interface between mathematics and physical reality. 

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. 

The phase rule is expressed in terms of p, the number of phases in the system C, the number of components and F, the number of degrees of freedom or the variance of the system. 

The number of phases is the number of different homogeneous regions in the system. Thus, in a system containing liquid water and several chunks of ice, only two phases exist. The number of degrees of freedom is the number of intensive variables that can be altered freely without the appearance or disappearance of a phase. First we wUl discuss a system that does not react chemically, that is, one in which the number of components is simply the number of chemical species. 

Chemical Thermodynamics Basic Concepts and Methods, Seventh Edition. By Irving M. Klotz and Robert M. Rosenberg 

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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. 

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... 

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... 

Derivation of the Phase Rule using the Gibbs-Duhem Equation... 

One important prerequisite for the derivation of the phase rule is that the phases are distinguishable. Obviously, the phases cannot be distinguished by their intensive parameters, except composition. Therefore, other properties must hold in order to characterize a phase. In order to detect a phase, at least one property, either a direct thermodynamic property such as the extensive properties or a property that is dependent on these properties, such as diffraction, must differ in each phase. Here we annotate that there is agreement that a thermodynamic potential contains the complete information in order to describe a thermodynamic system unambiguously. Among the extensive variables of a phase, the mol numbers just tell about the size of the phase. In contrast, the mole fractions give information about the constitution of the components in the phase. 

If we state that all variables are fixed by the functional form of the energies of the subsystem and the total entropy, volume, and mol number of the combined system, then this should not be confused that there would not be any freedom from the view of the phase rule. In fact, the derivation of the phase rule starts with different prerequisites. 

We show here the derivation of the phase rule as done by Gibbs in his original paper [3, pp. 55-353]. In a multicomponent system for a homogeneous phase the... 

P = the number of phases. A phase is defined as any homogeneous part of a system, bounded by surfaces, and capable of mechanical separation from the rest of the system The definition of these terms must be made most carefully for proper application of the rule. A complete discussion is beyond the scope of this book for this and a derivation of the phase rule the reader is referred to the standard works of physical chemistry and others dealing specifically with the subject (16, 21, 53). It is important to emphasize here that the rule applies only to systems at equilibrium and that additional restrictions imposed on a system have the effect of reducing the value of F by one for each restriction. 

For systems in which there is a three-phase contact line between the phases as a result of a solid phase, the concept of contact angle is introduced. For such systems, the Phase Rule remains the same (Li et al, 1989). For highly curved interfaces where the thickness of the heterogeneous region between the phases is not small compared to r, there are other considerations in the derivation of the Phase Rule (Li et al., 1989 Li, 1994). 

The derivation of the phase rule in this section uses the concept of components. The number of components, C, is the minimum number of substances or mixtures of fixed composition from which we could in principle prepare each individual phase of an equilibrium state of the system, using methods that may be hypothetical. These methods include the addition or removal of one or more of the substances or fixed-composition mixtures, and the conversion of some of the substances into others by means of a reaction that is at equilibrium in the actual system. 

Although still squabbled about in recent literature (Godek, Gaberscek and Jamnik 2009), the precise derivation of the phase rule for capillary systems was already given by Defay in his Brussels Thesis of 1932 [Defay and Priogine, 1966/1951, 76], taking into account the presence of surfaces and related equilibria. 

If we leave the foundations of statistical theory alone, we see that the applications depend on taking the spectroscopic data and plugging them into equation 25. Thus in actual practice, statistical thermodynamics accepts the classical theory without question and provides methods for the calculation of thermodynamic properties. Many classical thermodynamic relations cannot be derived from statistical theory. For instance, the phase rule simply falls out of classical theory. Derivation of the phase rule from statistical theory, without a large set of assumptions, does not seem to be possible. Popular equations such as the Clapeyron and Clausius-Clapeyron equations cannot be derived from statistical theory without the aid of classical thermodynamics. 

This requires us to rethink our derivation of the phase rule. Consider the following example for a chemical reaction ... 

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