Vapour pressure at the steady state

Question

I came across a question today about vapour pressure, it was as follows :

$2$ flasks $X$ and $Y$, each containing liquid water, joined by a tube connecting their mouths (no vapour could escape) having a valve in the middle were immersed in water baths of $300K$ and $350K$ respectively, which were used to ensure that the flasks are maintained at a constant temperature. It is given that the vapour pressure of water is $22\ \text {torr}$ at $300K$ and $40 \ \text {torr} $ at $350 K$, and that the valve was initially closed. Now, the valve is opened, comment on the final pressure in both the flasks.

I was able to deduce that some amount of water would evaporate from $Y$ and enter $X$, and this would continue until pressure in both the flasks is the same, however, I am not able to visualize the steady state of the system, and confused on whether or not the vapour pressures can ever be the same, seeing vapour pressure is only dependent on temperature, and the temperatures of the $2$ flasks will always be different.

So, what would happen after the valve is closed, and after the steady state is attained ?

Answer 1

There will not be a steady state solution for this problem. When the valve is opened, water vapor will flow from the hotter flask to the cooler flask because the hotter flask has a higher pressure. When this happens, the pressure in the 350K flask will drop, more water will vaporize, the temperature of the water in the 350K flask will drop, and heat transfer will take place from the environment into the flask that tends to raise the water temperature back to 350K. In the cooler flask, the pressure will rise above the equilibrium pressure, water vapor will condense, heat will be released as a result, the water temperature will rise a bit, and heat transfer will take place from the flask to the environment outside that flask. Assuming that the environment outside each flask continues to push one flask towards 350K and the other flask towards 300K, this process will continue until there is no more water in the 350K flask.

Answer 2

It's my guess thanthe end result will have the higher temperature flask empty, and the remaining colder one with water in equilibrium with the vapour at 22 torr. The pressure will be the same everywhere, but there will be a density gradient in the vapour as the hotter vapour will be less dense.