Raising water using thermal energy (bottle)

In the pictured container, points A and B are at the same altitude and we assume a slow process so that the velocities are insignificant.  Thus `d(rho g h) and `d(1/2 rho v^2) are insignificant, and the pressure at A must equal the pressure at B.

If we increase the pressure at A, the only way pressure at B can equalize is for water to rise in the tube.

Suppose that we wish to lift water from one large reservoir to another which lies at altitude yf relative to the first.   If the tube ends at altitude yf relative to the lower water level the the following observations hold:

• If the tube is very thin compared to the bottle then there will be very little expansion of the gas at the liquid rises in the tube as the result of added thermal energy.
• Once the liquid has reached the top of the tube the gas will be free to expand as additional thermal energy is added.  The volume change during expansion will be equal to the volume of water moved from the low reservoir to the high reservoir.
• Once the gas in the bottle is as hot possible, given the temperature of the thermal energy source, no more expansion is possible until the system is somehow cooled. 

The process of lifting the water increases the PE of the system.  This required the addition of thermal energy to the system.

• We have therefore done work against gravity using thermal energy.

As the principle of energy conservation leads us to believe, we expect that the work done to lift the water will be less than the thermal energy put into the system.