In the determining molar volume experiment, we will use water. how will vapor pressure play a role in this experiment?

Molar volume is a property of a component in a solution. It is defined as the volume occupied by one mole of the component in the closed system. You would not expect all solutions to execute volume additivity because intermolecular forces between the components come into play. There is no such thing as conservation of volume.

Vapor pressure affects molar volume because gases are very sensitive by these process conditions. Vapor pressure is very temperature-dependent. Consequently, at a different temperature, your component could expand or compress, thus, affecting the molar volume. Moreover, the pressure affects the molecular collisions in the system.