# Gibbs' Phase Rule

1. Mar 4, 2013

### cjc0117

I am having some trouble fully understanding the basics and I just wanted to see if somebody would please clarify this for me.

First, say you have a one component system in two phases: vapor and liquid. Gibbs' phase rule restricts the system to one independent, intensive variable that may be specified. Am I correct in saying that the mole or mass fraction of the system in the vapor phase does not count as an intensive variable? What about an "overall" property, like the two-phase density? These wouldn't count as intensive variables, right? Then is it impossible to find them through calculation even if the system is specified?

Second, say you have a two component system in two phases: vapor and liquid. Gibbs' phase rule restricts the system to two independent, intensive variables that may be specified. But this time, variables like chemical composition of each phase (e.g. vapor mole fraction of component 1, y1, liquid mole fraction of component 2, x2) DO count as intensive variables, right? What about overall chemical compositions (e.g. total mole fraction of component 1, z1) or total vapor fraction (yT). Are these intensive or extensive variables? Can overall mole fractions or total vapor fraction be found if the chemical compositions of each phase are known?

I know this is probably a simple question and it's kind of embarrassing to ask, but I couldn't find anything that clearly explained this for me. I find it odd (if I'm right) that vapor fraction counts as an extensive variable for a one component, two-phase system, but it counts as an intensive variable when another component is introduced. I just want to make sure this is right. Thanks.

2. Mar 25, 2013

### davidyanni10

I'm not really sure what you're asking but I'll take a crack at answering you since nobody else seems to be :)

The Gibbs phase rule can't really tell you what phases will be thermodynamically favorable in what proportions for a given system. You need a phase diagram to do that, but phase diagrams have to obey the Gibbs phase rule. But you are correct, you can't find something like two-phase density from a single component system phase diagram. The "one independent, intensive variable that may be specified" refers only to either pressure, temperature, or composition.

What this really means is that you never find a two-phase region in a single component system unless that two phase region is one-dimensional. That is what the one degree of freedom thing refers to. (Look at a phase diagram, the "boundaries" between vapor and liquid for example represent a one-dimensional, two-phase region with one degree of freedom. You cannot learn anything about the amounts of each phase present in this two phase region because the two-phase region is a line!)

So in your single component system you could vary temperature and pressure together in a precise, dependent way without changing phase (If you walk along the line between liquid and vapor for example). If you have three phases coexisting then you have zero degrees of freedom and your temperature AND pressure have been set. You are now looking at a zero-dimensional POINT on the phase diagram. You can never have 4 phases in equilibrium for a one component system. In the one component system your variables can really only be temperature or pressure (unless you're working with some weird system where an EM field can cause a phase change or something) but typically you're talking about temp or pressure.

In your two component system the compositional make-up of the system is another variable. You needn't think about things like "vapor mole fraction of component 1" because if you've specified a composition, call it $$X_a$$ then the vapor mole fraction of component 1 at $$X_a$$ = liquid mole fraction of component 1 at $$X_a$$. If you specify a composition, then there's no way a phase change can alter the composition. The composition DOES count as an intensive variable (but if you've got a two component system then the composition can only be one variable because fraction of component 2 = 1-(frac component 1) )

Total vapor fraction IS something you can get from a two component system's phase diagram (as you said, if the chemical compositions of each phase are known. And it so happens that if you know the total chemical composition then you can easily determine the chem composition of each phase given you have a phase diagram). You use a tie line to find the compositions of each phase and the lever rule to find the amount of each phase. See here for a brief overview: http://www.southampton.ac.uk/~pasr1/tielines.htm#page1

I think the heart of your question involved interpreting the Gibbs phase rule, and I'm not sure I've described it very well, but hopefully that at least helped.

3. Apr 2, 2013

### cjc0117

Thanks for the reply David. You're right; my post wasn't really structured well. Just a mess of questions. Anyways, I originally asked these questions because I was trying to solve a bigger problem that required me to get two-phase properties for a single component system. I was hoping against all thermodynamics that there was some way of finding the vapor or liquid fraction, but as you said, it can't be done without already knowing an overall two-phase property. I also found it kind of non-intuitive that the total vapor fraction is possible to find (via lever rule) for a multicomponent system but not for a single component system. But that is indeed the case. The website by the way was a very quick and straightforward review. Thanks!

4. Apr 3, 2013

### davidyanni10

No worries, good luck with your problem. Just because the phase diagram can't give you the info you need doesn't mean it can't be got.

Last edited: Apr 3, 2013