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Question: Consider two glass bottles filled with humid air, connected with a tube. One of the glasses is hot (heated), the other is cold (cooled) and the situation is steady state. How will humidity be distributed among the two bottles? Will the relative humidity be the same between them? Or will it be the absolute humidity? Or something inbetween?

EDIT: Suppose that these is no condensed water anywhere in the system, because the temperatures are high and the total amount of water is low.

Commentary 1: The question is inspired by humidity in cellars. If the cellar (cold bottle) and the outside air (hot bottle) are in steady state, what will be the relation between their relative humidities? (The steady state assumption reduces pretty much to the air circulation being faster than weather changes.) If the relative humidity is the same, then the cellar will be dry if there's sunshine outside and wet if it rains. If the absolute humidity is the same, then the cellar will be dry if it's cold outside and wet if it's hot outside.

Commentary 2: Ultimately, I'd like to expand this question to: I know the temperatures and relative humidities both in the cellar and outside, will opening the window result in a drop of relative humidity in the cellar? (By opening the window I cause a transition from two isolated bottles to two connected bottles in steady state. To make the calculations, however, I need to know whether it's the relative or absolute humidities which move toward equilibrium.)

Commentary 3: Physics explanation based on molecule movement will be greatly appreciated. While you are at it - I have a hypothesis why humidity of the atmosphere is seldom 100 %. Water molecules are lighter than nitrogen (or average air) molecules and thus water vapour rises upward due to buoyancy. Depending on the temperature and vicinity of open water surfaces, the rate at which water molecules float upward may be faster than the rate of evaporation, resulting in a temporary steady state where relative humidity is below 100 %. The water vapour in the atmosphere is not lost to the space, however. At some point it cools down so much that it condenses. Thus clouds are made, which float until they can't support their own weight (by whatever means, up-drafts, buoyancy...) at which point the liquid water obeys gravity and falls down again. This greatly enhances evaporation rates while cooling the atmosphere down, and thus relative humidity rises to near 100 % while it's raining. Once the rain is done and the excess water is either absorbed or evaporates again, the cycle starts up again. Water molecules are lost to the upper layers of the atmosphere faster than evaporation can supply them, and thus relative humidity falls below 100 % again. How far below depends on the temperature, major air currents, open bodies of water, and so on.

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  • $\begingroup$ There is too much left unspecified to allow a universal answer. The result will depend on whether there is or is not convective motion between the volume and whether there is or is not condensation (or evaporation in the warm volume) in the cool volume and whether or not the condensed volume is left in contact with the cooled air or is removed from the system. $\endgroup$ – dmckee --- ex-moderator kitten Jul 16 '17 at 21:26
  • $\begingroup$ @dmckee Thank you, I have added specification that there is no condensed water in the system. As for convective motion, why should it make a difference? $\endgroup$ – Ross Jul 17 '17 at 7:12
  • $\begingroup$ I'm not sure this qualifies as "steady state" in the thermodynamic sense since heat is being absorbed on one side and added on the other. So it will depend on the heat conductivity. $\endgroup$ – Señor O Jul 17 '17 at 7:18
  • $\begingroup$ @SeñorO I know that there is a flow of heat involved, that's why I say the situation is steady state and not in (thermodynamic) equilibrium. Eventually if you set the system up this way, it will reach a stable gradient of temperature between the two bottles. The gradient will drive heat exchange by conduction, and if it is too great, also by convection. The latter will create some turbulence, I suppose, but let's assume that in average this simply boosts the heat transfer rate. $\endgroup$ – Ross Jul 17 '17 at 9:22
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With neither evaporation nor condensation is the system then you can regard the water vapor and the dry air as distinct systems each subject to the same boundary conditions and conclude that they will have the same behavior.

Thus the absolute fraction of water will be the same at both ends and the relative humidity will vary.

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  • $\begingroup$ Thank you for your answer - thanks to you I think I finally figured it out. As you say, water vapour and air can be treated as separate systems. In steady state then, the pressures of water vapour in both bottles must be the same. For ideal gas (which works well enough here), $p=nk_BT$ (where $n$ is the amount of molecules in a cubic metre). This means that, since mass density $\rho=mn$, different temperatures mean different mass densities as well. If this approach isn't wrong, then both absolute and relative humidities in the two bottles will be different. $\endgroup$ – Ross Jul 20 '17 at 8:10

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