a. In vapor-liquid equilibrium, mixtures sometimes occur in which the compositions of the coexisting vapor and liquid phases are the same. Such mixtures are called azeotropes. Show that a binary azeotropic mixture has only one degree of freedom. 

b. In osmotic equilibrium, two mixtures at different pressures and separated by a rigid membrane permeable to only one of the species present attain a state of equilibrium in which the two phases have different compositions. How many degrees of freedom are there for osmotic equilibrium in a binary mixture? 

c. The phase equilibrium behavior of furfural (C₅H₄O₂)–water mixtures is complicated because furfural and water are only partially soluble in the liquid phase. 

(i) How many degrees of freedom are there for the vapor-liquid mixture if only a single liquid phase is present? 

(ii) How many degrees of freedom are there for the vapor-liquid mixture if two liquid phases are present?