Question

Do atoms lose energy during evaporation

Answer 1

Answer: yes because the gain energy when turning into a liquid.

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Answer 2

Atoms don't undergo a change in state during phase change. In fact phases or states can only be defined for groups of atoms. It is the energy state of these groups of atoms that decides the state and defines temperature.
In the process of evaporation molecules at the surface of the liquid gain higher energy when they are 'hit' with molecules of the gas surrounding it during their constant Brownian motion which gives it the energy to break away from the surface. This happening over a large scale of the liquid surface is what is evaporation.