The Phase Rule and Its Applications. Alexander Findlay

cannot be varied independently. A variation in the amount of hydrogen necessitates a definite variation in the amount of oxygen.

      In the case, already referred to, in which hydrogen and oxygen are present along with water at the ordinary temperature, we are not dealing with a condition of true equilibrium. If, however, the temperature is raised to a certain point, a state of true equilibrium between hydrogen, oxygen, and water-vapour will be possible. In this case hydrogen and oxygen will be components, because now they do take part in the equilibrium; also, they need no longer be present in definite proportions, but excess of one or the other may be added. Of course, if the restriction be arbitrarily made that the free hydrogen and oxygen shall be present always and only in the proportions in which they are combined to form water, there will be, as before, only one component, water. From this, then, we see that a change in the conditions of the experiment (in the present case a rise of temperature) may necessitate a change in the number of the components.

      It is, however, only in the case of systems of more than one component that any difficulty will be found; for only in this case will a choice of components be possible. Take, for instance, the dissociation of calcium carbonate into calcium oxide and carbon dioxide. At each temperature, as we have seen, there is a definite state of equilibrium. When equilibrium has been established, there are three different substances present—calcium carbonate, calcium oxide, and carbon dioxide; and these are the constituents of the system between which equilibrium exists. Now, although these constituents take part in the equilibrium, they are not all to be regarded as components, for they are not mutually independent. On the contrary, the different phases are related to one another, and if two of these are taken, the composition of the third is defined by the equation

      CaCO₃ = CaO + CO₂

      Now, in deciding the number of components in any given system, not only must the constituents chosen be capable of independent variation, but a further restriction is imposed, and we obtain the following rule: As the components of a system there are to be chosen the smallest number of independently variable constituents by means of which the composition of each phase participating in the state of equilibrium can be expressed in the form of a chemical equation.

      Applying this rule to the case under consideration, we see that of the three constituents present when the system is in a state of equilibrium, only two, as already stated, are independently variable. It will further be seen that in order to express the composition of each phase present, two of these constituents are necessary. The system is, therefore, one of two components, or a system of the second order.

      When, now, we proceed to the actual choice of components, it is evident that any two of the constituents can be selected. Thus, if we choose as components CaCO₃ and CaO, the composition of each phase can be expressed by the following equations:—

      CaCO₃ = CaCO₃ + 0CaO CaO = CaO + 0CaCO₃ CO₂ = CaCO₃ - CaO

      As we see, then, both zero and negative quantities of the components have been introduced; and similar expressions would be obtained if CaCO₃ and CO₂ were chosen as components. The matter can, however, be simplified and the use of negative quantities avoided if CaO and CO₂ are chosen; and it is, therefore, customary to select these as the components.

      While it is possible in the case of systems of the second order to choose the two components in such a way that the composition of each phase can be expressed by positive quantities of these, such a choice is not always possible when dealing with systems of a higher order (containing three or four components).

From the example which has just been discussed, it might appear as if the choice of the components was rather arbitrary. On examining the point, however, it will be seen that the arbitrariness affects only the nature, not the number, of the components; a choice could be made with respect to which, not to how many, constituents were to be regarded as components. As we shall see presently, however, it is only the number, not the nature of the components that is of importance.

      After the discussion of the conditions which the substances chosen as components must satisfy, another method may be given by which the number of components present in a system can be determined. Suppose a system consisting of several phases in equilibrium, and the composition of each phase determined by analysis. If each phase present, regarded as a whole, has the same composition, the system contains only one component, or is of the first order. If two phases must be mixed in suitable quantities in order that the composition of a third phase may be obtained, the system is one of two components or of the second order; and if three phases are necessary to give the composition of a fourth coexisting phase, the system is one of three components, or of the third order.^[16]

      Although the examples to be considered in the sequel will afford sufficient illustration of the application of the rules given above, one case may perhaps be discussed to show the application of the method just given for determining the number of components.

      Consider the system consisting of Glauber's salt in equilibrium with solution and vapour. If these three phases are analyzed, the composition of the solid will be expressed by Na₂SO₄, 10H₂O; that of the solution by Na₂SO₄ + xH₂O, while the vapour phase will be H₂O. The system evidently cannot be a one-component system, for the phases have not all the same composition. By varying the amounts of two phases, however (e.g. Na₂SO₄, 10H₂O and H₂O), the composition of the third phase—the solution—can be obtained. The system is, therefore, one of two components.

      But sodium sulphate can also exist in the anhydrous form and as the hydrate Na₂SO₄, 7H₂O. In these cases there may be chosen as components Na₂SO₄ and H₂O, and Na₂SO₄, 7H₂O and H₂O respectively. In both cases, therefore, there are two components. But the two systems (Na₂SO₄, 10H₂O—H₂O, and Na₂SO₄, 7H₂O—H₂O) can be regarded as special cases of the system Na₂SO₄—H₂O, and these two components will apply to all systems made up of sodium sulphate and water, no matter whether the solid phase is anhydrous salt or one of the hydrates. In all three cases, of course, the number of components is the same; but by choosing Na₂SO₄ and H₂O as components, the possible occurrence of negative quantities of components in expressing the composition of the phases is avoided; and, further, these components apply over a much larger range of experimental conditions. Again, therefore, we see that, although the number of the components of a system is definite, a certain amount of liberty is allowed in the choice of the substances; and we also see that the choice will be influenced by the conditions of experiment.

      Summing up, now, we may say—

      (1) The components are to be chosen from among the constituents which are present when the system is in a state of true equilibrium, and which take part in that equilibrium.

      (2) As components are to be chosen the smallest number of such constituents necessary to express the composition of each phase participating in the equilibrium, zero and negative quantities of the components being permissible.

      (3) In any given system the number of the components is definite, but may alter with alteration of the conditions of experiment. A certain freedom of choice, however, is allowed in the (qualitative, not quantitative) selection of the components, the choice being influenced by considerations of simplicity, suitability, or generality of application.^[17]

      Degree of Freedom. Variability of a System.—It is well known that in dealing with a certain mass of gas or vapour, e.g. water vapour, if only one of the independently variable factors—temperature, pressure, and concentration (or volume)—is fixed, the state of the gas or vapour is undefined; while occupying the same volume (the concentration, therefore, remaining unchanged), the temperature and the pressure may be altered; at a given temperature, a gas can exist under different pressures and