Does Water Evaporate Differently in a Beaker vs. a Pan Under a Bell Jar?

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SUMMARY

In the discussion regarding water evaporation in a beaker versus a pan under a bell jar, it is established that the potential energy difference between the water in the beaker and the pan does not influence evaporation rates. The thermal energy at room temperature (4 * 10^-21 J) significantly outweighs the gravitational potential energy (3 * 10^-25 J) of water molecules, making the latter negligible. Consequently, both the beaker and the pan will experience similar evaporation rates, as the energy of vaporization (7 * 10^-20 J) dictates the process rather than potential energy differences.

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Mike_In_Plano
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The following is something that puzzles me a bit, and I'd appreciate some insight in the matter.
Thank You :)

Given the following:
A shallow pan with a smaller diameter beaker sitting in it's center
Both the pan and the beaker contain distilled water
The level of the water in the beaker is substantially high than that in the pan
The pan and beaker are additionally enclosed by a bell jar (with an id = to the od of the pan)
The bell jar is evacuated until only water vapor remains
The system starts at a uniform temperature, say room temperature
The system ends at the same temperature via thermal conductivity of a plate that the pan sets on.

Here's my question:
Does the beaker loose water to evaporation while the pan gains the water? After all, the water at the top of the beaker is at a higher potential energy than that at the top of the pan.

Thanks for your considerations,

Mike
 
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Maybe it would help to consider the following:

Thermal energy (kT) at room temperature is 4 * 10^-21 J.
Gravitational potential energy (mgh) for water atoms at the top of a 1-meter high column is 3*10^-25 J.
Energy of vaporization for a water molecule: 7 *10^-20 J.

The amount of potential energy is negligible in comparison to thermal energy, so there is no tendency for water to preferentially evaporate from the column.