The Gibbs Phase Rule

 The Gibbs Phase Rule

That there is a definite relationship in a system between the number of degrees of freedom, the number of components and the number of phases present was first established by J. Willard Gibbs in 1876. This relation, known as the phase rule, is a principle of the widest generality, and its validity is in no way dependent on any concepts of the atomic or molecular constitution. Credit is due to Ostwald, Roozeboom, Van't Hoff, and others for showing how this generalization can be 'utilized in the study of problems in heterogeneous equilibrium. 

To arrive at a formulation of the phase rule, consider in general a system of C components in which P phases are present. The problem now is to determine the total number of variables upon which such a system depends. First, the state of the system will depend upon the pressure and the temperature. Again, in order to define the composition of each phase, it is necessary to specify the concentration of (C-1) constituents of the phase, the concentration of the remaining component being determined by difference. Since there are P phases, the total number of concentration variables will be P(C - 1). and these along with the temperature and pressure constitute a total of [P(C-1) + 2] variables.

 The student will recall from his study of algebra that when an equation in n independent variables occurs, independent equations are necessary in order to solve for the value of each of these. Similarly, in order to define the P(C-1) + 2 variables of a system, this number of equations relating to these variables would have to be available. The next question is then: How many equations involving these variables can possibly be written from the conditions obtained in the system? To answer this query recourse must be had to thermodynamics. Thermodynamics tells us that equilibrium between the various phases in a system is possible only provided the partial molal free energy of each constituent of a phase is equal to the partial molal free energy of the same constituent in every other p base. Since the partial molal free energy of the constituent of a phase is a function of the pressure, temperature, and (C-1) concentration variables, it readily follows that the thermodynamic condition for equilibrium makes it possible. . When P phases are present, (P-1) equations are available for each constituent, and for C constituents a total of C(P-1) equations

 If this number of equations is equal to the number of variables, the system is completely defined, however, generally, this will not be the case, and the number of variables will cross the number of equations by F, where 

 F = number of variables - number of equations 

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

Equation (1) is the celebrated phase rule of Gibbs. F is the number of degrees of freedom of a system and gives the number of variables whose values ​​must be arbitrarily specified before the state of the system can be fully and unambiguously characterized. According to the phase rule, the number of degrees of freedom of a system is determined by the difference between the number of components and the number of phases present, i.e., by (C-P). 

 In this derivation, it was assumed that each component is present in every phase. If a component lacks a particular phase, however, the number of concentration variables has decreased by one. But at the same time, the number of possible equations also decreased by one. Hence the value of C P, and therefore F remains the same whether each constituent is present in every phase or not. This means that the phase rule is not restricted by the assumption made, and is generally valid under all conditions of distribution provided that equilibrium exists in the system.

 The principal value of Eq. (1) is in its use as a check in the construction of various types of plots for the representation of the equilibrium conditions existing in heterogeneous systems. Before proceeding to a discussion of some specific systems and the application of the phase rule to these, it is convenient to classify all systems according to the number of components present. Thus we may have one, two, three, etc, component systems. The advisability of this approach will become apparent from what follows