Why does liquid evaporation occur at any temperature?

There is always a number of rapidly moving molecules in a liquid at any temperature that can leave the surface of the liquid, but the higher the temperature, the more and faster their evaporation.
Evaporation is explained by the fact that there is always some fraction of “fast” molecules in the liquid, the kinetic energy of which is sufficient for them to escape from the liquid, overcoming the attraction of neighboring molecules. Such “fast” molecules exist at any temperature of the liquid, so evaporation occurs at any temperature.

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