I am aware that the evaporation process causes cooling in the remaining liquid, but why?
If anyone can give me the maths and or equations to prove / show this I would be most appreciative.
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This phenomenon is explained in countless sources on the web and elsewhere. It comes down to the fact that it takes a certain kinetic energy for the molecules of liquid to escape and, for any given temperature, only some percentage of the molecules have sufficient kinetic energy to do so.
Naturally, as most energetic molecules leave, the average kinetic energy of the remaining molecules and, therefore, the temperature of the remaining liquid, decreases.
For illustration purposes, the kinetic energy distribution of molecules in liquids could be roughly described using Maxwell-Boltzmann distribution developed for ideal gases. The MB distribution curves for two different temperatures are shown on the diagram below (copied form this page).
Here you can see that, at (presumably) moderate temperatures, molecules capable of escaping liquid represent only a small fraction of the total number of molecules and that fraction increases with temperature due to the shift of the distribution curve toward higher energies.
The existing answers (V.F.; fhorrobin) mention energy but energy alone is not enough to explain this. We must consider entropy as well.
First let me repeat briefly the statement about energy that you can find already in the other answers. The molecules attract one another. Therefore when a molecule in the liquid moves away from the others and enters the vapour, it is climbing out of a potential well. The slower molecules in the liquid can't do this: they can't escape the attraction to the body of the liquid. The faster molecules are the ones that preferentially escape. Hence after they depart the average energy of the molecules in the liquid is lower. This causes the fall of temperature.
The trouble with this energy argument is that it also implies that molecules entering the liquid from the vapour will speed up as they fall into the potential well caused by attraction to other molecules in the liquid, so they will warm the liquid. The energy argument is not wrong, but it is incomplete. It does not tell us why the evaporation is winning overall compared with condensation. After all, wouldn't we expect the cooling liquid to be immediately warmed up again via the reverse process? (as well as by heat conduction).
A falling temperature here is going against the usual tendency for heat to be transported from a hotter to a colder place, tending to equalized temperatures. Our explanation is not complete until we have shown why evaporative cooling does not break the Second Law of Thermodynamics. This, to me, is the main reason why the cooling associated with evaporation is counter-intuitive. I want to know why a spontaneous process which causes lowering temperature even happens in the first place!
In the case of evaporation it is highly relevant that the space available for the molecules to move in contributes to the calculation of the entropy. As the liquid cools, its entropy goes down, but the set of escaping molecules get to move about in a larger volume (in the vapour) and this corresponds to a larger set of microstates and therefore a larger entropy. This is how the entropy increase in the vapour can more than compensate for the entropy decrease in the liquid during such evaporation.
(A final comment for anyone wanting to go further. The thermodynamic details can be calculated via the properties called Gibbs function and chemical potential. These are closely related and quantify entropy change as matter moves from one place or state to another.)
No need to use equations to show this. You just need to think about what is happening thermodynamically. To evaporate water, you must add energy to it and this energy must come from somewhere. The energy is initially in the liquid but once it evaporates, the liquid carries the energy away with it (otherwise it would condense if it lost the energy).