Phase rule

The phase rule is a general principle governing systems in thermodynamic equilibrium. If F is the number of degrees of freedom, C is the number of components and P is the number of phases, then The phase rule is a general principle governing systems in thermodynamic equilibrium. If F is the number of degrees of freedom, C is the number of components and P is the number of phases, then It was derived by Josiah Willard Gibbs in his landmark paper titled On the Equilibrium of Heterogeneous Substances, published in parts between 1875 and 1878.The rule assumes the components do not react with each other. The number of degrees of freedom is the number of independent intensive variables, i.e. the largest number of thermodynamic parameters such as temperature or pressure that can be varied simultaneously and arbitrarily without determining one another. An example of one-component system is a system involving one pure chemical, while two-component systems, such as mixtures of water and ethanol, have two chemically independent components, and so on. Typical phases are solids, liquids and gases. The basis for the rule (Atkins and de Paula, justification 6.1) is that equilibrium between phases places a constraint on the intensive variables. More rigorously, since the phases are in thermodynamic equilibrium with each other, the chemical potentials of the phases must be equal. The number of equality relationships determines the number of degrees of freedom. For example, if the chemical potentials of a liquid and of its vapour depend on temperature (T) and pressure (p), the equality of chemical potentials will mean that each of those variables will be dependent on the other. Mathematically, the equation μliq(T, p) = μvap(T, p), where μ = chemical potential, defines temperature as a function of pressure or vice versa. (Caution: do not confuse p = pressure with P = number of phases.) To be more specific, the composition of each phase is determined by C − 1 intensive variables (such as mole fractions) in each phase. The total number of variables is (C − 1)P + 2, where the extra two are temperature T and pressure p. The number of constraints is C(P − 1), since the chemical potential of each component must be equal in all phases. Subtract the number of constraints from the number of variables to obtain the number of degrees of freedom as F = (C − 1)P + 2 − C(P − 1) = C − P + 2.