The potential of a component in any phase depends not only on the composition of the phase, but also on the temperature and the pressure (or volume). If, therefore, we have a system of C components existing in P phases, then, in order to fix the composition of unit mass of each phase, it is necessary to know the masses of (C - 1) components in each of the phases. As regards the composition, therefore, each phase possesses (C - 1) variables. Since there are P phases, it follows that, as regards composition, the whole system possesses P(C - 1) variables. Besides these there are, however, two other variables, viz. temperature and pressure, so that altogether a system of C components in P phases possesses P(C - 1) + 2 variables. In order to define the state of the system completely, it will be necessary to have as many equations as there are variables. If, therefore, there are fewer equations than there are variables, then, according to the deficiency in the number of the equations, one or more of the variables will have an undefined value; and values must be assigned to these variables before the system is entirely defined. The number of these undefined values gives us the variability or the degree of freedom of the system. The equations by which the system is to be defined are obtained from the relationship between the potential of a component and the composition of the phase, the temperature and the pressure. Further, as has already been stated, equilibrium occurs when the potential of each component is the same in the different phases in which it is present. If, therefore, we choose as standard one of the phases in which all the components occur, then in any other phase in equilibrium with {20} it, the potential of each component must be the same as in the standard phase. For each phase in equilibrium with the standard phase, therefore, there will be a definite equation of state for each component in the phase; so that, if there are P phases, we obtain for each component (P - 1) equations; and for C components, therefore, we obtain C(P - 1) equations. But we have seen above that there are P(C - 1) + 2 variables, and as we have only C(P - 1) equations, there must be P(C - 1) + 2 - C(P - 1) = C + 2 - P variables undefined. That is to say, the degree of freedom (F) of a system consisting of C components in P phases is-- F = C + 2 - P * * * * * {21} CHAPTER III TYPICAL SYSTEMS OF ONE COMPONENT A. _Water._ For the sake of rendering the Phase Rule more readily intelligible, and at the same time also for the purpose of obtaining examples by which we may illustrate the general behaviour of systems, we shall in this chapter examine in detail the behaviour of several well-known systems consisting of only one component.