Problem 1: Phase Equilibria and Degrees of Freedom
(a) In class, we argued that to describe a binary mixture in VLE, we need four thermodynamic variables: T, P, $x_1$ and $y_1$. Then using the VLE criteria, we were able to show that only two out of these four variables can be specified independently.
In this part, we will perform this exercise for a ternary mixture in VLE. First, list all the pertinent thermodynamic variables. How many variables are in your list?
Now list all the VLE criteria, i.e., equations that must be true at VLE, explicitly representing the pertinent thermodynamic variables in these equations. How many equations are in your list?
For a ternary mixture in VLE, how many variables can we independently specify?
(b) Now repeat the exercise in part a for a mixture with $n_c$ components in VLE, and show that the number of independent variables is $n_c$.
(c) In this part, we will generalize the result from part b to more than 2 phases to obtain the Gibbs' phase rule, which states that for a mixture with $n_c$ components, which are distributed across $\pi$ phases in equilibrium with one another, $n_c - \pi + 2$ variables can be independently specified.
(d) Consider a mixture containing water (W), an oil (O), and two compounds of interest (A and B). Due to the nature of the components, this mixture will form two liquid phases (one water-rich and one oil-rich) in equilibrium with its vapor. How many independent variables (often referred to as degrees of freedom) can be specified?
