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Here is the situation. Let's say I have have a mass of a given liquid and I heat it to create a gas. The gas has a lower density than air, so it will move higher and higer in the atmosphere. Then, I recondensate this gas into liquid.

Suppose both process are ideal, so the energy I put in the system to cause evaporation is the same energy released by the system from the condensation. We now have the same liquid as before, but since it moved up in the atmosphere, it now has a muche bigger gravitationnal potential energy than it previously had.

Of course, the energy is conserved, so this additionnal potential energy must come from somewhere, but where?

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    $\begingroup$ Suppose you burned fuel to heat the liquid? You don't have the same fuel. $\endgroup$
    – mmesser314
    Commented Oct 10, 2023 at 15:20
  • $\begingroup$ Rising is accompanied by cooling; see lapse rate. Is this what you’re asking about? $\endgroup$
    – Chemomechanics
    Commented Oct 10, 2023 at 15:56
  • $\begingroup$ The main source of the change in potential energy is the work done by the buoyant force, buoyant force times vertical displacement. This is work done by the surroundings. $\endgroup$
    – Chet Miller
    Commented Oct 10, 2023 at 16:30
  • $\begingroup$ I think the gravitational potential energy is going down when the gas falls up. You could assign it a negative mass in $m_{eff}gh = V(\rho-\rho_{air})gh $ with $m_{eff} < 0$, though there may be pitfalls to that approach. $\endgroup$
    – JEB
    Commented Oct 10, 2023 at 17:26

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When we heat up a liquid to turn it into a gas, there are three things we need to keep track of concerning the energy. First, we need heat to raise the temperature of the liquid (this depends on its specific heat capacity). Second, there is the latent heat of evaporation, which we need to supply to turn the liquid into a gas. Third, we need to consider that once we start turning the liquid into a gas, it expands and pushes away the air around it. This expansion against atmospheric pressure takes work, which corresponds to extra heat that we need to supply.

This last contribution for the total heat needed to vaporize the liquid is the key to your question, as the amount of work that the gas does when it expands depends on the surrounding pressure. Let's also assume that we add even more heat to raise the temperature of the gas well above its boiling point because the temperature of the gas starts to drop as soon as it starts rising (and we don't want the gas to start condensing right away).

(If you wonder why the gas cools down as it rises up, that's because with increasing altitude, the surrounding air pressure decreases and therefore the gas expands, which means that it is doing work. Therefore, its energy decreases.)

So after the gas has risen in the atmosphere, it has a lower temperature and a lower pressure. The lower temperature isn't really of our concern—we can just count the energy that the gas gave off when cooling down towards the energy that we get back now as we start to cool down the gas. The latent heat and the heat from cooling down the liquid (after the gas condenses) are also the same as when we first heated the liquid. But the last contribution to the energy has changed. When the gas starts to condense, it contracts so that now the surrounding air does work on our gas. However, as the pressure is smaller than at ground level, this work is smaller. Therefore, we get less energy back from the atmosphere compared to what we supplied when we evaporated the gas. This cancels the change in potential energy of the gas.

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  • $\begingroup$ Thank you for your clear answer! So if I understand correctly, the work the gas does on the environment by going up is equal to the energy it loses by cooling down. One thing I would like to clarify tho: Why is the gas going up in the first place? I guess the pressure at the bottom has to be higher than the one on top, all along the trajectory of the gas. Therefore, could we say there is a pressure gradient, associated to a potential, acting on the gas, such that, the gravitational potential it is gaining by going up is compensated by the "pressure potential" it is losing? $\endgroup$
    – benjamichon
    Commented Oct 18, 2023 at 20:31
  • $\begingroup$ @benjamichon I would phrase it a little differently: As the gas goes up, it expands and therefore does work on the environment. This work is equal to the energy it loses by cooling down on its way up. $\endgroup$
    – WillHallas
    Commented Mar 22 at 10:46
  • $\begingroup$ The gas is going up in the first place, because it is hotter and therefore less dense than the environment. Therefore there is a buoyancy force. This has nothing directly to do with pressure! Notice that our gas even at the bottom has the same pressure as the environment (if it didn't, it would quickly expand/contract until it had the same pressure as the environment). But because it has a higher temperature, it has a lower density than the environment, which leads to the buoancy force. With rising altitude, the ambient pressure decreases, which is why the gas expands in the first place. $\endgroup$
    – WillHallas
    Commented Mar 22 at 10:54
  • $\begingroup$ In total, the liquid, does gain net energy! It is in the same state in the beginning and in the end with the only difference being that it has a higher gravitational potential in the end. The missing energy (because energy in total is conserved) comes from the atmosphere, which becomes ever so slightly hotter and also has a slightly smaller gravitational potential afterwards, as some air goes down to take the place at the bottom where the liquid once had been. $\endgroup$
    – WillHallas
    Commented Mar 22 at 11:01

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