Phase Transition Temperature Part 1

Last weekend was lazy and unproductive so I didn't write anything - I will end that ignominy with immediate effect.

Today I will try to answer a question I had no satisfactory answer for until I came across this website, and even now have a poor appreciation for. The question is: why is temperature constant during a phase change of a single-component system? Most people with a basic grounding in science are familiar with the standard answer: energy transfer is required for some process in the phase change (like bond-breaking or forming, particle rearrangement), and thus doesn't contribute to changing the temperature. If this answer satisfies your sensibilities, please skip this article and I will see you back in a few weeks. If you're like me and feel like a lot of wool is being pulled over my eyes and dust swept under my carpet, please stick around - your insights will be very helpful in a topic I am uncertain about.

The explanation from the Temperature and Phase Change website linked to in the previous paragraph relies on Gibbs' Phase Rule, which says the number of independent intensive variables is

F = c - p + 2,

where c is the number of unreacting components and p the number of phases in the system. We can get this rule by adding the number of intensive variables needed to describe a system and subtracting the number of equations relating these variables:

Intensive Variables
cp densities
cp temperatures
cp pressures

Equations
Thermal equilibrium: cp-1 temperature relations
Pressure equilibrium: cp-1 pressure relations
Chemical equilibrium: c(p-1) stoichiometric chemical potential relations
Equations of state: p

F = 3cp - 2(cp-1) - c(p-1) - p = c - p + 2.

We can now apply this rule to analyse phase change in a single-component system where c=1.

When both phases are present, p=2 and F=1 - if we specify 1 intensive, all other intensives will follow. Suppose we fix the pressure at which the phase transition occurs, then the temperature is uniquely determined and constant. On the other hand, if we don't fix the pressure, the temperature can vary, dependent on the pressure at any point during the phase transition. So the question we initially asked wasn't quite right - it should have been: why is temperature constant during a constant-pressure phase change of a single-component system?

If at this point you have the nagging feeling something's missing, you're not alone. I thought: if the pressure and temperature are fixed, and the densities of each phase are thus uniquely specified, how can the proportion of either phase change during the transition? The answer must be that the system volume is not constant as I'd implicitly assumed, so if the proportion of the less dense phase increases, the system volume must increase. For example, if one of the phases is an ideal gas, with pV = NkT, V increases with N when p and T are constant until all particles are in the ideal gas phase.

And here is the missing piece to the energy puzzle! The so-called latent heat is actually the energy needed to expand the system at constant pressure, and given by pressure times the change in volume. Since this expansion does involve some "bond-breaking" or "particle rearrangement," there is some merit in the textbook basis for latent heat.

Moreover, we can now see how 2 systems with different proportions of either phase can have the same temperature and thus be in thermal equilibrium with each other. To do so, we start with a system with some proportion of either phase. Since the system temperature is uniform, we can form a new system at the same temperature with selected portions of the former. In particular, the new system can have a different proportion of either phase, and will be in equilibrium because the temperature, pressure and densities are unchanged. Thus there will be no net heat transfer between systems with different proportions of coexisting phases.

Finally, when there is only 1 phase, p=1 and F=2. So even if the pressure is fixed, the temperature can vary. Only when both the pressure and the density are specified is the temperature uniquely determined. This is evident for an ideal gas with p=nkT.

So Gibbs does a good job of explaining why temperature is constant during a fixed-pressure single-component phase transition. To get any more mileage on this problem, we really need concrete examples, which I will go into later. Meanwhile, I am still considering an alternative explanation based on free energies. I'll let you know if that bears fruit.