Where does the energy come from to lift water by osmosis?
For example in a tree, the water is lifted from the high concentration in the soil to the low concentration higher up in the tree. But at the end of that process the water has been elevated, which should take energy (=mgh), but it seems like it kind of gets lifted for free without spending any energy?
Similarly, dipping a paper towel into a bowl of water, the water "climbs" the towel (by capillary action?) and absorbs upwards, meaning the water was lifted upwards (so gained potential energy) seemingly for free?
Briefly, the water molecules are attracted to the molecules of the tube by adhesive force. The liquid molecules are also attracted to each other by cohesive force. The interplay of these forces causes capillary action.
Generally, yes, but in the case of trees there is also negative pressure (vacuum) exerted from transpiration when water leaves the top of the tree and "pulls" other water up behind it.
I think it's more like sunlight energy + adhesion energy - cohesion energy, because the capillary action is also helping to lift the water but generally yes the energy is coming from a combination of the chemical forces in the water and the pressure gradient from the sunlight/heat.