Phase rule, generalized

Often we ignore the relative sizes of the phases when describing the intensive state doing so removes T - 1) variables from the number in (9.1.12), leaving the generalized phase rule,... 

We may also ask for an T -specification of the final equilibrium state. In this case we ignore the relative amounts in the phases and we ignore the extents of reaction. However, the (R reaction-equilibrium constraints (7.6.3) still apply, so the generalized phase rule (9.1.13) becomes... 

Is the problem well-posed This issue concerns whether we have enough information to compute the required unknowns. In phase and reaction-equilibrium computations, this issue is resolved by a proper application of the generalized phase rule it might not be properly resolved by a routine application of the Gibbs phase rule. In particular, we have discussed two kinds of subtleties that are often overlooked. 

But, first, we must mention a slight modification of the regular phase rule. Equation (11.1). As shown in Figure 17.9, the experiments we are discussing at a fixed pressure of 1 bar can be represented on a plane or section through P-T-X space. The general phase rule (11.1) applies to this P-T-X space. The fact that we confine ourselves to a fixed P plane within this space means that we have used one of our degrees of freedom - we have chosen P = 1 bar, and the same would be true for any other constant P section (or constant T section, for that matter). Therefore on our T-X plane the phase rule is... 

The general XT E problem involves a multicomponent system of N constituent species for which the independent variables are T, P, N — 1 liquid-phase mole fractions, and N — 1 vapor-phase mole fractions. (Note that Xi = 1 and y = 1, where x, and y, represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to estabhsh the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by siiTUiltaneous solution of the N equihbrium relations ... 

Generally, the rate of reaction depends on three principal functions temperature, pressure, and composition. However, as a result of phase rule and thermodynamics, there is a relationship between temperature, pressure, and composition. This relationship can be expressed as ... 

The material in this section is divided into three parts. The first subsection deals with the general characteristics of chemical substances. The second subsection is concerned with the chemistry of petroleum it contains a brief review of the nature, composition, and chemical constituents of crude oil and natural gases. The final subsection touches upon selected topics in physical chemistry, including ideal gas behavior, the phase rule and its applications, physical properties of pure substances, ideal solution behavior in binary and multicomponent systems, standard heats of reaction, and combustion of fuels. Examples are provided to illustrate fundamental ideas and principles. Nevertheless, the reader is urged to refer to the recommended bibliography [47-52] or other standard textbooks to obtain a clearer understanding of the subject material. Topics not covered here owing to limitations of space may be readily found in appropriate technical literature. 

The general thermodynamic treatment of binary systems which involve the incorporation of an electroactive species into a solid alloy electrode under the assumption of complete equilibrium was presented by Weppner and Huggins [19-21], Under these conditions the Gibbs Phase Rule specifies that the electrochemical potential varies with composition in the single-phase regions of a binary phase diagram, and is composition-independent in two-phase regions if the temperature and total pressure are kept constant. 

For the purpose of classifying heterogeneous equilibria we shall make use of a very general law, called the Phase Rule of Willard Gibbs (1876), the proof of which is deferred to a later chapter. 

An equation that usually fits experimental data belter than equations (38) or 39) is the general mixture rule for two-component mixtures.- m which there is a single phase that is. the components are miscible (97)... 

Many derivatives of quinones, cinnamic acids, and mucconic acids photodimerize in solid phases to give results 16> that in many cases are not in agreement with the general PMO rule of head-to-head reaction. However, it is clear that those reactions are controlled by topochemical effects, i.e. the geometry and proximity of the reactants in the solid phase. 135> Consequently, PMO theory will not be useful for calculating reactions of that type. 

The phase rule says that for each phase beyond the first that occurs at equilibrium in a system, N. decreases by one. Expressed in general form, the phase rule is,... 

A substance that can be added to a system independently (or removed from it, say by precipitation or vaporization) is called a component of such a system. The phase rule, summarizing a general behavior of nature, says ... 

Equation (2.23) is a very important result. It is known as the Gibbs Phase Rule, or simply the phase rule, and relates the number of components and phases to the number of degrees of freedom in a system. It is a more specific case of the general case for N independent, noncompositional variables... 

This observation is the simplest case of the Gibbs phase rule (to be discussed in Section 7.1). It implies, for example, that pressure P = P(V, T) is uniquely specified when V and T are chosen, and similarly, that V = V(P, T) or T = T(P, V) are uniquely determined when the remaining two independent variables are specified. Such functional relationships between PVT properties are called equations of state. We can also include the quantity of gas (as measured, for example, in moles n) to express the equation of state more generally as... 

Some phenomenological features of a representative phase diagram (for C02) were previously described in Section 2.5. In the present section, we shall first review key topological features of the phase diagram for H20 from the perspective of the phase rule (Section 7.2.1). The general theory of phase boundaries will then be developed (Section 7.2.2) and illustrated (Section 7.2.3) for some simple elemental and molecular substances. These representative examples will serve to illustrate the bewildering multiplicity of phase forms and properties that are possible even in the simple c = 1 limit. 

The dimensionality/of Ms was introduced in (10.7) to agree with the Gibbs phase rule. In general, this feature of a space is uniquely determined by the rank (number of... 

The Gibbs phase rule is the basis for organizing the models. In general, the number of independent variables (degrees of freedom) is equal to the number of variables minus the number of independent relationships. For each unique phase equilibria, we may write one independent relationship. In addition to this (with no other special stipulations), we may write one additional independent relationship to maintain electroneutrality. Table I summarizes the chemical constituents considered as variables in this study and by means of chemical reactions depicts independent relationships. (Throughout the paper, activity coefficients are calculated by the Debye-Hiickel relationship). Since there are no data available on pressure dependence, pressure is considered a constant at 1 atm. Sulfate and chloride are not considered variables because little specific data concerning their equilibria are available. Sulfate may be involved in a redox reaction with iron sulfides (e.g., hydrotroilite), and/or it may be in equilibrium with barite (BaS04) or some solid solution combinations. Chloride may reach no simple chemical equilibrium with respect to a phase. Therefore, these two ions are considered only to the... 

Since we have briefly introduced the consideration of mixed systems, we will mention, in passing, Gibbs1 famed Phase Rule. Its basic equation, which is generally applicable, is ... 

Many choices of independent variables such as the energy, volume, temperature, or pressure (and others still to be defined) may be used. However, only a certain number may be independent. For example, the pressure, volume, temperature, and amount of substance are all variables of a single-phase system. However, there is one equation expressing the value of one of these variables in terms of the other three, and consequently only three of the four variables are independent. Such an equation is called a condition equation. The general case involves the Gibbs phase rule, which is discussed in Chapter 5. 

General Considerations Involving Multicomponent and Multiphase Equilibrium The Gibbs Phase Rule... 

The Gibbs phase rule allows /, the number of degrees of freedom of a system, to be determined. / is the number of intensive variables that can and must be specified to define the intensive state of a system at equilibrium. By intensive state is meant the properties of all phases in the system, but not the amounts of these phases. Phase equilibria are determined by chemical potentials, and chemical potentials are intensive properties, which are independent of the amount of the phase that is present. The overall concentration of a system consisting of several phases, however, is not a degree of freedom, because it depends on the amounts of the phases, as well as their concentration. In addition to the intensive variables, we are, in general, allowed to specify one extensive variable for each phase in the system, corresponding to the amount of that phase present. 

Equation (15) is called the general Gibbs—Duhem relation. Because it tells us that there is a relation between the partial molar quantities of a solution, we will learn how to use it to determine a Xt when all other X/ il have been determined. (In a two-component system, knowing Asolvent determines Asolute.) This type of relationship is required by the phase rule because, at constant T, P, and c components, a single-phase system has only c — 1 degrees of freedom. 

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