Application of the Phase Rule

To begin with, recall the Mineralogical Phase Rules, (14.60) and (14.61), which can also be written 

Thus the maximum number of phases to be expected in a natural system is decreased by one for each environmental component, compared to the same system having that component as a system component. In other words, systems having / 2 will lose one phase for every component that is given an arbitrary potential. 

In a ternary system, G — X diagrams become difficult to draw, but can be imagined as having planes tangent to three free energy loops. Instead of attempting this, let s 

Suppose now one found a rock having only two of these phases, say kaolinite and quartz. You could not conclude necessarily that water was a phase during metanaor- 

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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 ... 

As a simple application of the phase rule, let us consider the M-O system described in Section 1.1, in which the number of components, c, is equal to... 

Application of the phase rule to the binary (M-O) and ternary oxide system (M1-M2-O) in a closed system ... 

The application of the phase rule does not require a knowledge of the actual constituents of a phase... 

Thus, temperature and pressure can both be varied while remaining in a region of a single phase, whereas only temperature or pressure can be varied (e.g., on a vapor-pressure curve) while retaining equilibrium between two phases. The triple point is completely invariant. More interesting applications of the phase rule are obtained with multicomponent systems, as indicated in the following examples. 

Two-dimensional phase diagrams are often displayed in the form of In [p] against 1/7 (at a constant specific amount adsorbed), which provides a convenient way of indicating the conditions for the coexistence of two phases (see Figure 4.3). Indeed, the application of the Phase Rule indicates that when two adsorbed phases coexist in equilibrium, the system has one degree of freedom therefore, at constant... 

Brittain, H. G. (1999c). Application of the phase rule to the characterization of polymorphic systems. In Drugs in the pharmaceutical sciences (ed. J. Sarbrick), Vol. 95 Polymorphism in pharmaceutical solids (ed. H. G. Brittain), pp. 35-72. Marcel Dekker, New York. [30]... 

An awkward conception is degrees of freedom. It is closely related to the number of observations, in general the number of obsen ations minus the number of constraints imposed on the system. An accurate definition is difficult, however, and we will be content with giving an explicit account for each particular application. It is interesting to note that rather similar conceptions will be. familiar to physical chemists in the application of the phase rule and in the quantum mechanics of molecules. 

Other examples of application of the phase rule are given in the following discussion of some alloy systems. 

A few examples will illustrate the application of the phase rule to chemical equilibria. Systems consisting of the various phases of a single component substance have already been discussed in the beginning of the chapter. We shall now consider systems with two independent components. An example of this kind is the system calcium chloride and water. We know by Lehrbuch, 6th edition, p. 608. 

Equation (4) illustrates the application of the phase rule to equilibria between solids and solutions. Thus the number of variable concentrations in the equilibrium equation is exactly equal to the degrees of freedom / of the system, namely, the total number n of the molecular types taking part in the reaction less the number B of the substances present in the solid phase (f=n—B) n is also the number of the independent components of the system, which is equal to the total number of molecular types present (n-f sol vent), less the number of the chemical equations (1). The number of phases is P = B+2 (solution and vapour). Hence... 

We now examine the application of the phase rule and Duhem s theorem to an osmotic system. The phase rule was deduced in chap. XIII by assuming that all phases of the system were subject to the same applied pressure p. The introduction of a semi-permeable membrane increases the variance of the system by one, and renders possible the existence of two different pressures p and p". Instead of the 2 + C(/> variables T, p,. .. we have now 3-fc< variables T, p p", x. .. x. ... 

This difference is further reinforced by application of the phase mle to the equihbrium between two strictly defined polymorphs of a compound or the equihbrium between a compound and a corresponding solvate of that compound. In the former case, there is only one component (in the phase rule sense—the compound). There are two phases (the two polymorphs) and, therefore, there is only one degree of freedom for equihbrium between two polymorphs by application of the phase rule equation... 

Application of the phase rule to a system that includes a dissolving mineral, solution, and gas phase is a complex undertaking, which can be approached using a modified statement of the phase rule... 

Or, again, take the case of pure benzene on the one hand and a saturated solution of benzene in water on the other, both systems being at the same temperature A saturated solution of benzene is necessarily in equilibrium with pure liquid benzene itself because of the fact of saturation The conclusion to he drawn from the thermodynamic criterion considered is, that under these conditions, (8A)TV = o, and therefore, if we imagine one mole of benzene transferred from the pure benzene to the saturated solution, the work must be zero That is, there must be the same vapour pressure over the pure benzene as there is over its saturated solution in water, the vapour in each case being benzene vapour In the case of a hydrated salt on the one hand and the saturated solution of the salt on the other, the conditions are more complex We shall consider this point in Chap X in connection with the application of the Phase Rule to two component systems... 

As an example of the application of the phase rule with reaction, consider a mixture of four components in which only two participate in a reaction. If all components are present in the gas phase, then F = 4 + 2- l- l=4. Hence, only four intensive variables are independently variable (e.g., T, P, yi, and y2, or P, y, y, and y ) if the vapor is ideal. For example, the independent reaction yields the following form of the equilibrium constant ... 

Before proceeding to discuss the application of the Phase Rule to the study of the iron-carbon alloys, however, the main facts with which we have to deal may be stated very briefly,... 

A short description may now be given of the application of the Phase Rule to the two-component system iron—carbon and of the diagram showing how the different systems are related, and with the help of which the behaviour of the different mixtures under given conditions can be predicted. 

Having discussed qualitatively the application of the Phase Rule to three-component systems formed by water and two salts with a common ion, one may now consider how the equilibrium diagrams, constructed on the basis of experimental data, can be employed in the quantitative study of the behaviour of such systems, and can be used to guide the practical operations of the winning of salts by crystallisation from solution. ... 

Amrtionia Soda Process.—One of the most important applications of the Phase Rule to systems of four components with reciprocal... 

Applications of the phase rule to three typical operations follow. 

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