The Gibbs Phase Rule and Duhems Theorem

Thus far we have considered the equilibrium between two phases of a single compound. When many compounds or components and more than two phases are in equilibrium, the chemical potential of each component should be the same in every phase in which it exists. When we have a single phase, such as a gas, its intensive variables, pressure and temperature, can be varied independently. However, when we consider equilibrium between two phases, such as a gas and a liquid, p and T are no longer independent. Since the chemical potentials of the two phases must be equal (p (p, T) = T)), only one of the two intensive 

The independent intensive variables that specify a state are called its degrees of freedom. Gibbs observed that there is a general relationship between the number of degrees of freedom, /, the number of phases, P, and the number of components C  

This can be seen as follows. At a given T, specifying p is equivalent to specifying the density in terms of mole number per unit volume (through the equation of state). For a given density, the mole fractions specify the composition of the system. Thus, for each phase, p, T and the C mole fractions jc[ (in which the superscript indicates the phase and the subscript the component) are the intensive variables that specify the state. Of the C mole fractions in each phase i, there are (C— 1) independent mole fractions x] because = 1- In a 

If a component a does not exist in one of the phases b , then the corresponding mole fraction x = 0, reducing the number of independent variables by one. However, this also decreases the number of constraining equations (7.2.2) by one. Hence there is no overall change in the number of degrees of freedom. 

As an illustration of the Gibbs phase rule, let us consider the equilibrium between solid, liquid and gas phases of a pure substances, i.e. one component. In this case we have C = 1 and P = 3, which gives / = 0. Hence for this equilibrium, there are no free intensive variables there is only one pressure and temperature at which they can coexist. This point is called the triple point (Fig. 7.1). At the triple point of H2O, T — 213A6K = 0.01 °C and p = 611 pa = 6.11 X 10 bar. (This unique coexistence between the three phases of water may be used in defining the Kelvin scale.) 

---