Water vapor / energy question

[Mike_In_Plano]

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

 

[Andy Resnick]

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.