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Consider the following graph describing a phase change from a liquid state (blue area) to a gaseous state (light blue area). enter image description here

Why does the pressure remain constant in the coexistence area (grey area)?

Say we went from a gaseous state to a liquid state ($E\rightarrow A$), shouldn't the pressure increase due to the increasing amount of liquid compared to gas in the system?

Secondly, the temperature during a phase change stays constant as far as I know, so according to the ideal gas law $pV=nRT$, the pressure should increase with decreasing volume in order to keep the temperature constant.

EDIT:I realised the ideal gas law can't be applied in this case and the Van-der-Waals-equations $(p+\frac{a}{V_M^2})\cdot (V_M-b) =R\cdot T$ should be considered instead, still the general idea of increasing pressure with decreasing volume when holding temperature constant should hold, right?

So why does the pressure still stay constant?

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The system that you are considering is not an ideal or even a real gas.
It is a system which consists of a vapour phase separated by an interface (surface) with the liquid phase.

Imagine that you have some vapour in a dynamic equilibrium with liquid at a certain temperature.
This means that the rate at which liquid molecules escape into the vapour phase is equal to the rate at which vapour molecules escape into the liquid phase.

Now reduce the volume without changing the temperature.
The density of the vapour phase has increased and so the rate at which vapour molecules go into the liquid phase is greater than the rate at which liquid molecules go into the vapour phase.

This results in a net loss of vapour molecules and a gain of liquid molecules.
Eventually the rate at which liquid molecules escape into the vapour phase becomes equal to the rate at which vapour molecules escape into the liquid phase and the pressure of the vapour is the same as before the reduction in volume - it is called the saturated vapour pressure.

Above the critical temperature $T_{\rm krit}$ there is no interface (surface) between liquid and vapour - the substance only exist in the vapour phase.

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The pressure doesn't have to remain constant. For example, think of warm moist air rising. The pressure and temperature at higher altitude is lower and colder. As air rising, it expands because of the lower pressure. As it expands, it cools. After a while, the water vapor condenses, forming a cloud.

On the other hand, thing of a teakettle on a stove. The outside pressure stays constant as the kettle is heated. Eventually, the kettle boils. The pressure is still constant.

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