Phase rule limitations

It is tempting to place significance on the relative magnitudes of the saturation indices calculated for various minerals and then to relate these values to the amounts of minerals likely to precipitate from solution. The data in Table 6.6, however, suggest no such relationship. Thirteen minerals are supersaturated in the initial fluid, but the phase rule limits to ten the number of minerals that can form only two (dolomite and quartz) appear in the final phase assemblage. 

The results of the fractionation model (Fig. 18.9) differ from the equilibrium model in two principal ways. First, the mineral masses can only increase in the fractionation model, since they are protected from resorption into the fluid. Therefore, the lines in Fig. 18.9 do not assume negative slopes. Second, in the equilibrium calculation the phase rule limits the number of minerals present at any point along the reaction path. In the fractionation calculation, on the other hand, no limit to the number of minerals present exists, since the minerals do not necessarily maintain equilibrium with the fluid. Therefore, the fractionation calculation ends with twelve minerals in the system, whereas the equilibrium calculation reaches an invariant point at which only six minerals are present. 

The phase rule is a mathematical expression that describes the behavior of chemical systems in equilibrium. A chemical system is any combination of chemical substances. The substances exist as gas, liquid, or solid phases. The phase rule applies only to systems, called heterogeneous systems, in which two or more distinct phases are in equilibrium. A system cannot contain more than one gas phase, but can contain any number of liquid and solid phases. An alloy of copper and nickel, for example, contains two solid phases. The rule makes possible the simple correlation of very large quantities of physical data and limited prediction of the behavior of chemical systems. It is used particularly in alloy preparation, in chemical engineering, and in geology. 

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. 

A phase is defined as a state of matter that is uniform throughout in terms of its chemical composition and physical state in other words, a phase may be considered a pure substance or a mixture of pure substances wherein intensive properties do not vary with position. Accordingly, a gaseous mixture is a single phase, and a mixture of completely miscible liquids yields a single hquid phase in contrast, a mixture of several solids remains as a system with multiple solid phases. A phase rule therefore states that, if a limited number of macroscopic properties is known, it is possible to predict additional properties. 

When applying the Gibbs phase rule, it must be remembered that the choice of components is not arbitrary the number of components is the minimum number compatible with the compositional limits of the system. 

This relationship holds for any chemical system which is subject to variations in temperature, pressure, and proportions of its basic components and describes the number of phases P present in terms of the system s degrees of freedom F and the number of component species C. Even though the phase rule is simple in form, it is not limited in its ability to describe very complex systems. Equilibrium effects arising from the presence of surface tension, stress, magnetic fields, etc. can be accounted for by the incorporation of additional degrees of freedom into the phase rule. Such effects, however, will not be considered in this discussion. 

Before moving on, it is wise here to note two important limitations of the phase rule. The criteria for components only prescribe that they be able to represent each phase in the system. The phase rule says nothing about how these components may combine to give other species and, thus, does not define the number or nature of other species in the system. That is, given the components CaO and C02, the phase rule cannot predict the existence of the intermediate compound CaCOs. 

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 basic asymmetry between intensive and extensive vectors can also be recognized in the Gibbs phase rule. This establishes the dimensionality of Ms in terms of the number of independent intensities, as expressed in (11.9b) in terms of rank(M). An alternative extensity-based (or M-based) description necessarily diverges at points where M becomes singular, i.e., at critical limits, where dimensionality changes, as shown by (11.24). 

The variables (or rather, intensive variables ), are p (pressure), T (temperature), and the concentrations (e.g., mole fractions of the components) in each separate phase. P is the number of phases present at equilibrium, and C is the minimum number of components necessary to duplicate any system that represents the equilibrium in question. (The components may sometimes be chosen in several ways). Finally, F is the number of degrees of freedom or the number of independent variables. With P phases present, one can (within limits) assign values independently of F variables, but then all other variables are fixed by the conditions for equilibrium. [For example, one may apply the phase rule to the system CaO—CCL—H20, with one, two, three, four, or five phases, determine how many independent variables result, and decide what will be the most practical choice of variables. (Five phases might be CaC03(s), Ca(OH)2(s), ice, an aqueous solution, and a gas phase.)]... 

This is the usual relation given for the phase rule. The difference V is called the number of variances or the number of degrees of freedom. (Note that some texts use F = C — P + 2, where F is the same as V.) These are the number of intensive variables to which values may be assigned arbitrarily but within the limits of the condition that the original number of P phases exist. It is evident from Equations (5.63) and (5.56) that the intensive variables are the temperature, pressure, and mole fractions or other composition variables in one or more phases. Note that the Gibbs phase rule applies to intensive variables and is not concerned with the amount of each component in each phase. 

This relation is called the Gibbs phase rule. However, as indicated next, c should be limited to the number of independent components. 

The system of equations (2.27) is seen to be rather complicated. Its solution, if obtainable at all in quadratures, must probably be even more complicated. However, in experiments certain conditions which enable the initial equations to be simplified are usually fulfilled. Consider limiting cases of particular interest from both theoretical and practical viewpoints.134,136,139,140 The process of growth of the ApBq and ArBs layers will be analysed in its development with time from the start of the interaction of initial substances A and B up to the establishment of equilibrium at which, according to the Gibbs phase rule (see Refs 126-128), no more than two phases should remain in any two-component system at constant temperature and pressure. 

The phase rule(s) can be used to distinguish different types of equilibria based on the number of degrees of freedom. For example, in a unary system, an invariant equilibrium (/ = 0) exists between the liquid, solid, and vapor phases at the triple point, where there can be no changes to temperature or pressure without reducing the number of phases in equilibrium. Because / must equal zero or a positive integer, the condensed phase rule (/ = c — p + 1) limits the possible number of phases that can coexist in equilibrium within one-component condensed systems to one or two, which means that other than melting, only allotropic phase transformations are possible. Similarly, in two-component condensed systems, the condensed phase rule restricts the maximum number of phases that can coexist to three, which also corresponds to an invariant equilibrium. However, several invariant reactions are possible, each of which maintains the number of equilibrium phases at three and keeps / equal to zero (L represents a liquid and S, a solid) ... 

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