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Raoult’s law states that:

The partial pressure of any volatile constituent of a solution at a constant temperature is equal to the vapour pressure of pure constituent multiplied by the mole fraction of that constituent in the solution

My doubt is: Is the mole fraction of the constituent of the solution to be considered at the time when equilibrium between vapours dissolving into the solution and rate of evaporation has been achieved or at the beginning when the solution is just formed and evaporation has just begun change the mole fraction of the constituent which we have just calculated?

Also, experimentally, is there much difference in the mole fractions at equilibrium and at the beginning when there is apparently no vapour pressure (for zero vapour pressure in the beginning, I think we can just close the lid of the container)?

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Assume the system to be in equilibrium, otherwise you can't use the standard equations of thermodynamics. As for your second question, it's impossible to answer because it depends on the solvent and details of the experimental setup (size of container relative to volume of solvent, temperature, etc)

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