Why does a liquid vapor pressure correlates with its evaporation speed?

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The way I understand vapor pressure is the pressure vapor (and empty space of a closed chamber) starts applying on the liquid that hasn’t evaporated yet. How would a higher vapor pressure of a particular liquid make it evaporate faster, while it’s known that water for example evaporates more rapidly at low pressure?

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I guess you misunderstand vapor pressure a little bit. Vapor pressure, also known as equilibrium vapor pressure, is the pressure exerted by a vapor of some substance in equilibrium with the liquid phase of the same substance. That is, at vapor pressure the rate of evaporation of liquid into vapor is the same as the rate of condensation from vapor into liquid. So if you start out with some liquid and little or no vapor, then higher vapor pressure means that the system is further from equilibrium, and that the balanced rate of evaporation (rate of evaporation minus rate of condensation) will be higher, compared to a situation with low vapor pressure when the system is closer to the equilibrium to start with.