Why have phase transitions to occure at fixed T and P?

Question

It is clear that the temperatures and pressures of the different phases are equal, due to the maximum entropy requirement. Though I can't proof why both are known to remain constant during transition.

That apparently holds, if the P(T) curve intersects the curve of phase coexistence at one point only. Then the transition happens at this very point of fixed T and P.

But couldn't one imagine the P(T) curve moving towards the coexistence curve, then moving ON it for a while and then leaving it. The part where both curves show congruence would represent the transition, during which temperature and pressure would change.

Answer 1

It has been a while since I took thermo, but I'll give this a try.
T and P remain constant at state transition because it takes energy to change the T and P. During the transition, the energy is going into (or being released) to cause the state transition. Water, in a liquid, has more energy then water in a solid. So the energy that was increasing the temperature is now being used to melt the ice.
I suppose that you could alter the T and P along the phase curve and they could coexist. The matter would then remain in the state that it was in.

Answer 2

You can think of heat as the average kinetic energy of the particles in a material. Kinetic energy is 1/2*m*v^2.

When you heat up liquid water, its molecules vibrate back and forth, faster and faster, but they're still essentially stuck together (they're a liquid). As you get closer to 100 Celsius, the molecules vibrate at higher speeds.

Let's pretend the average molecular velocity is 800 miles per hour at 100 Celsius. When you add a little bit more heat energy to the system, some of the molecules just barely break free of the forces that hold them in liquid state. It takes a substantial amount of energy to break these bonds. The energy used to break the bonds cannot, by definition, remain as kinetic energy, so the molecule escapes into the gas phase, but it's still going 800 miles per hour, on average.

Therefore, the gas has the same temperature as the liquid, despite requiring a substantial addition of heat to induce a phase transition. It's a similar situation when going from solid to liquid.