Q:

On inserting a thin capillary tube into a large tray of water, does the total surface energy and the total potential energy of the system increase?

In both cases, the system is the liquid in the (tray+tube).
I have made reasonably accurate calculations for the gravitational potential energy part. It follows from the calculations that the gravitational P.E. increases for both water as well as for mercury.
But I haven't been able to decipher the surface energy part of the question. At first glance, it appears that it must increase in both the cases (as surface area of contact between water and air in the capillary tube increases). However, I haven't been able to convince myself regarding the same.
Can someone please help with the surface energy as well as the total energy part? Also, what if the liquid is mercury instead of water?
Thanks in advance :)

A:

I'll try to point you in the right direction, as follows.
The atoms which compose the free surface on the exterior of a solid body are in a different state than their brethren on the interior of the solid. They have half as many nearest neighbors with which to share their interatomic forces, and this affects how they behave. 
This is also true for liquids, and these "surface effects" give rise to things like capillarity. 
For the purposes of getting a handle on your specific problem, then, the key aspect is that for a solid that is readily "wetted" by a given liquid, the solid can minimize its energy by covering itself with a film of that liquid. This then allows you to set up an energy balance between the increase in gravitational potential energy caused by that liquid climbing up a vertical pore and the decrease in energy caused by the wetout of that solid surface. 
Give this energy balance concept a shot and come back afterwards if you are still confused!

