Components approach to the phase rule

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. 

It is not always easy to decide on the number of components of an equilibrium system. The number of components may be less than the number of substanees present, on account of the existence of reaction equilibria that produce some substances from others. When we use a reaction to prepare a phase, nothing must remain unused. For instance, consider a system consisting of solid phases of CaCOs and CaO and a gas phase of CO2. Assume the reaction CaCOsCs) CaO(s) + C02(g) is at equilibrium. We could prepare the CaC03 phase from CaO and CO2 by the reverse of this reaction, but we can only prepare the CaO and CO2 phases from the individual substances. We could not use CaCOs to prepare either the CaO phase or the CO2 phase, because CO2 or CaO would be left over. Thus this system has three substances but only two components, namely CaO and CO2. 

In deriving the phase rule by the components approach, it is convenient to consider only intensive variables. Suppose we have a system of P phases in which each substance present is a component (i.e., there are no reactions) and each of the C components is present in each phase. If we make changes to the system while it remains in thermal and mechanical equilibrium, but not necessarily in transfer equilibrium, we can independently vary the temperature and pressure of the whole system, and for each phase we can independently vary the mole fraction of all but one of the substances (the value of the omitted mole fraction comes from the relation x/ = 1). This is a total of 2 - - P(C — 1) independent intensive variables. 

When there also exist transfer and reaction equilibria, not all of these variables are independent. Each substance in the system is either a component, or else can be formed from components by a reaction that is in reaction equilibrium in the system. Transfer equilibria establish P — independent relations for each component (/jL = /xf, /xJ = etc.) and a total of C(E — 1) relations for all components. Since these are relations among chemical potentials, which are intensive properties, each relation reduces the number of independent intensive variables by one. The resulting number of independent intensive variables is 

If the equilihrium system lacks a particular component in one phase, there is one fewer mole fraction variable and one fewer relation for transfer equilibrium. These changes cancel in the calculation of F, which is still equal to 2 - - C — P. If a phase contains a substance that is formed from components by a reaction, there is an additional mole fraction variable and also the additional relation Y iViPi = 0 for the reaction again the changes cancel. 

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