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Is it possible for a cup of water to completely evaporate in a room with a constant temperature of 21ºC???

Yes, of course. Why would you think it would not? A wet towel will dry, right?

but why is that?

The Kinetic Molecular Theory, distributed velocity /energy in molecules as a result of collisions. Some have sufficient energy after a collision, and if they are at the surface, and the velocity is upwards, they can escape the attractive forces of cohesion (ie, the water pulling them backwards).

No, because if the room temperature is kept constant than the energy (heat) from the water is absorbed into the air than the air temperature would have to increase, but if the temperature of the room is kept constant than the water will not be able to evaporate because there could not be enough energy to make the phase change happen.

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