10.08 - Concepts for Generalized Phase Equilibria

The concept behind multicomponent equilibrium is practically the same as that for pure component equilibrium: minimize the total Gibbs energy by setting the derivative equal to zero. The notation involved in taking that derivative is more complicated than in Chapter 6 because we have a new partial derivative in our chain rule for every component that is added to the mixture. This live video (10min, uakron.edu) portrays students wrestling with why the derivatives must be expressed with respect to constant mole number instead of mole fraction, leading to a better appreciation of each term in the expansion. 

When expressing the derivative of the total Gibbs energy by chain rule, there is one particular partial derivative that relates to each component in the mixture: the "chemical potential." By adapting the derivation from Chapter 9 of the equilibrium constraint for pure fluids, we can show that the equilibrium constraint for mixtures is that the chemical potential of each component in each phase must be equal. That is fine mathematically but it is not very intuitive. By translating the chemical potential into a rigorous definition of fugacity of a component in a mixture, we recognize that an equivalent equilibrium constraint is that the fugacity of each component in each phase must be equal. (8min, Live, uakron.edu) This offers the intuitive perspective of, say, molecules from the liquid escaping to the vapor and molecules from the vapor escaping to the liquid; when the "escaping tendencies" are equal, the phases experience no net change and we call that equilibrium. 

What is the entropy of a "mixture" that is unmixed? (ie. What is the entropy of the overall system when two separate beakers are considered as one total system?) At first glance it may seem like an impossible riddle, but the simple answer to this puzzle holds the key to all of phase equilibria. This video breaks down the entropy calculation for an ideal mixture (20min, uakron.edu) into a series of simple questions (below). By answering these questions, we build an approach for formulating mixture properties in general, not just ideal solutions, and not just for entropy, but for any property (FYI, G is of particular interest.)

Quickly estimate the entropy (J/molK) of a stream that is equimolar in ethane and propane at 25°C and 26 bars relative the ideal gas elements at 25°C and 1 bar.(a) Quickly estimate ΔS for a component going from ideal gas to liquid.(b) Develop a formula for the S of a "mixture" that is NOT mixed (ie. total S of separate beakers). (c) Develop a general formula for the S of a "mixture" that IS mixed.(d) BTW, develop a general formula for the H of a "mixture" that IS mixed.(e) Perform the numerical calculation.

Comprehension Questions:
1. Prepare a graph of G vs. x_E  for ethane+propane at 298K and sufficient pressure to remain liquid at all compositions. 
2. Suppose you had two beakers, one that was 75mol% ethane and another that was 75%propane. Develop a formula to describe the G of this overall system as the size of the ethane-rich beaker goes from overwhelmingly dominant to negligible relative to the propane-rich beaker. Plot this result on the graph from part 1.