Explain the 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 substances 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 CaCO₃ and CaO and a gas phase of CO₂. Assume the reaction CaCO₃(s) → CaO(s) + CO₂ (g) is at equilibrium. We could prepare the CaCO₃ 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 CaCO₃ 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 tem- perature 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 Σ_ix_i = 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 in- dependent. 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 1 independent relations for each component (μ^β_i = μ^α_i, μ^γ_i = μ^α_i etc.) and a total of C(P-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

F = [2 + P(C-1)] - C(P-1) = 2 + C- P

If the equilibrium 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 Σ_iv_iμ_i = 0 for the reaction; again the changes cancel.

We conclude that, regardless of the kind of system, the expression for F based on components is given by F = 2 + C - P. By comparing this expression and F = 2 + s - r- P, we see that the number of components is related to the number of species by

C = s - r