[Thermodynamics] How much mass is extracted to decrease pressure

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Homework Statement:
I have a pressurized cold vessel with a saturated liquid and saturated vapor. Due slow heat transfer, the vessel increases in pressure and needs to be relieved. I want to estimate the mass of vapor that needs to be extracted in order to depressurize the vessel to a desired value.

I know the initial pressure (P1) and temperature (T1), I know the desired pressure (P2), the vessel geometry and liquid volume. I'm assuming no heat transfer as the mass extraction process will be quick.

Any idea how to approach this problem?
Relevant Equations:
First Law of Thermodynamics
I have a feeling some integration will be required, I just want to see if anyone know of a quick way.
 

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