Given a system at pressure p_sat, is the vapor mass fraction unique?

[AndersFK]

Assume you have a (closed) system with only one component (for instance water) which is in a pressure-temperature-point (p,T) *on the saturation curve* (i.e. liquid and vapor/gas can coexist). Are the mass fractions in the vapor and liquid phases unique? Or is it possible to change the mass fractions without moving around in the (p,T)-plane?

First I thought the answer was that they're unique (Gibb's phase rule, Duhem's theorem) but on the other hand I have never seen a plot of for instance the vapor mass fraction vs. temperature, which should be possible to produce if they're unique. Or is the reason just that the saturation curve is infinitesimal, so in reality vapor and liquid will never coexist if the system only consist of one component?

 

[TobyC]

I don't think the mass fractions are unique. If you have a container containing water vapor and liquid water in equilibrium, and you isothermally change the volume, I'm pretty sure that both the temperature and the pressure stay constant and it is the ratios of the substances which changes instead. For example, if you reduce the volume, some vapor turns to liquid.

I'm not sure how this reconciles with Gibbs' phase rule.

 

[AndersFK]

Thanks for your reply TobyC! I think I agree with your conclusion. It seems odd if you can't change the mass fractions...

However, I'm not sure either how this is in agreement with Gibb's phase rule. Something has to be wrong about my first reasoning:

The system consists of C=1 component and P=2 phases. Hence, the system has only F=2+C-P=1 degree of freedom. I.e., if one intensive variable (for instance temperature or pressure) is given, all other intensive variables are determined. Since the vapor mass fraction is an intensive variable, the mass fraction is unique.

Maybe the vapor mass fraction isn't an intensive variable? I want to believe it is scale invariant...

• Source, Gibb's phase rule: http://en.wikipedia.org/wiki/Gibbs%27_phase_rule"
• Source, intensive/extensive variables: http://en.wikipedia.org/wiki/Intensive_and_extensive_properties"

 

[TobyC]

I've been thinking about this some more.

Where I have seen Gibbs' phase rule introduced, the degrees of freedom which you can have are the pressure, the temperature, and then the ratios of the different components in your system.

The relative mass fractions in each phase are not degrees of freedom, since you don't have the ability to set them directly like you do with the ones mentioned above, they depend on the variables above. It is still puzzling that they are not unique for a system with its degrees of freedom specified though, because this suggests another degree of freedom (which in the case of the example I posted above would be the volume).

In a single component system, your only possible degrees of freedom are the pressure and the temperature, but in a single component with two phases coexisting you only get 1 degree of freedom (from Gibbs' phase rule), and this is either the pressure or the temperature (if you set one the other is determined).

This doesn't appear to agree with the fact that you can also change the volume of the system, and hence change the mass fractions in each phase, but this might be because the volume isn't considered as a degree of freedom in the derivation of Gibbs' phase rule.

 

[AndersFK]

I asked one of the professors today (by mail) and he confirmed that you need the volume in addition to the point (p,T) to determine the mass fractions. He also said that "Gibb's phase rule still holds because it does not concern itself about the relative mass fractions in each phase".

In my textbook, the derivation for Gibb's phase rule is just a simple counting "proof" that starts with the number of variables (as you say; they usually are assumed to be pressure, temperature and the mass component ratios in each phase) and subtracts the number of equations. So I agree with you that it looks like volume is not regarded as a degree of freedom. I find that a little bit strange, since the volume (with (p,T) specified) determines the density. Maybe there are historical reasons behind...

Another way of viewing the problem is that the equation of state f(p,T,rho)=0 exactly on the saturation curve has two solutions for the density, i.e. with just (p,T) specified you are free to choose one or a weighted average of the two densities...