As the approximation, the saturated vapour pressure does not depend on the total pressure, only on the substance and temperature. (*)
If there is nothing but the vapor in the gaseous phase under the piston then the amount of vapor and its pressure accomodate to external pressure. (**)
- If the external pressure is kept greater than saturated vapor pressure for given temperature, all vapour will eventually condense and released heat will dissipate, assuming constant temperature is maintained.
- If the external pressure is kept lower than saturated vapor pressure, all liquid will eventually evaporate and needed evaporation heat will be extracted from surrounding.
- If the external pressure is by chance equal to the saturated vapor pressure, liquid and vapor will maintain the equilibrium with the same vapour pressure as the external one.
The piston stays in rest if and only if the net force on it is zero, therefore if actual vapour pressure ( saturated in equilibrium ) is equal to the total external pressure caused by external, non-vapor based forces acting on the piston. Otherwise it starts moving toward the lower pressure.
There is assumed the lifting of the piston is slow enough for liquid to have time to establish saturated vapour pressure at any time. Fast piston lifting can end with higher volume and less than saturated vapor pressure, which may not be possible to be topped again due lack of remaining liquid.
(*) - Finer analysis tells us that for gas+vapor mixtures, liquid saturated vapor pressure increases little with the total pressure. This is proportional to liquid molar volume and such an increase is usually negligible. Unless temperature and pressure are near the liquid critical point where it can make the difference.
(**) - Atmospheric pressure is just one of possible sources or components of external pressure on the piston. Atmospheric pressure can be
- the only part of external pressure, like in idealized scenario with a free moving, massless, frictionless piston
- missing, e.g. if the system is placed in vacuum
- a part of total external pressure, like if the weight caused by the piston mass cannot be neglected and/or if explicit mechanical external force is applied.