# When water climbs up a piece of paper, where is the energy coming from?

Take a glass of water and piece of toilet paper. If you keep the paper vertical, and touch the surface of the water with the tip of the paper, you can see the water being absorbed and climbing up the paper. This climbing up is intriguing me. Where is the energy coming from?

Supposedly, a lot of water molecules are moving up and gaining potential energy. This has to be balanced by something. I can think of a few possibilities, but I can't tell which one it is.

• When the water molecules dilute into the paper, they are at a state of lower potential binding energy. Some molecular interaction (van der Waals?) is at lower energy when you have a water+paper solution, compared to a water-only solution. This compensates the gain in gravitational energy.
• The surface between the paper and the water is at lower pressure than the atmosphere. This causes the water to be pushed into the paper by the atmospheric pressure, up until the point that the column of water above the surface is heavy enough to counterbalance. The potential energy would be a result of work done by the atmosphere.
• Some water molecules climb up randomly and loose kinetic energy going up, and somehow get "stuck" up there.
• Something else?
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I wasn't sure what to tag this one, suggestions welcome. –  Malabarba Dec 25 '10 at 4:30
Hmm... well, I think [capillary-action] would be a good tag, but I'm not really sure which area of physics that fits into. My thought would be [mechanics] [fluid-dynamics] [capillary-action] but consider it just a tentative suggestion. –  David Z Dec 25 '10 at 5:08
–  anna v Apr 4 '12 at 3:18

## 3 Answers

The surface of any fluid has an associated energy-per-unit-area, known as the surface energy, a.k.a. surface tension. This energy is not a property of the fluid alone, but of the fluid and the medium it is in contact with.

In your case you would have associated surface energies for the water-air interface, $e_{wa}$, as well as for the water-paper interface, $e_{wp}$. The total energy of the fluid in a configuration is the sum of the potential energy, plus the product of the corresponding surface energies by their respective surface areas, $S_{wa}e_{wa} +S_{wp}e_{wp}$.

So if you want to look at it in a purely energy balance point of view, the increase in potential energy of the water climbing up the paper is compensated by a reduction of the total surface energy. When capillary action makes things rise, it is because the liquid-solid energy is lower than the liquid-air energy. By wicking into the porous material, the solid-liquid contact area is increased at the expense of the liquid-air one, resulting in an overall reduction of contact energy, which is what drives the rise in potential energy.

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+1, neat! Could you provide some reference on surface tension (and by reference I mean book or course notes, not wikipedia article)? I guess it must be quite well-known stuff but I've unfortunately never had opportunity to learn it. –  Marek Dec 28 '10 at 12:58
I have a (very very) little experience with surface tension, but I didn't quite buy this. If I take a cylindrical glass of water and touch the surface of the water with some toilet paper. As the water climbs up the paper, the area of contact with the air is not decreasing. The only thing that happens is that the water-glass area of contact decreases as the water-paper area of contact increases, but the water-air area of contact does not change. –  Malabarba Jan 3 '11 at 0:18
+1 This the correct answer, very compact formulated! –  Georg Jan 19 '11 at 17:37

The physics behind this is the same capillary action that causes water to move up narrow cylindrical channels.

Tissue paper is extremely porous, but the pores are sufficiently narrow that the cohesion between water molecules (actually driven by the Coulomb interaction, since water molecules are polar) and adhesion between the water and the surfaces of the pores combine to "lift" the water into the pores. Adhesion is, I think, most likely driven by hydrogen bonding. In the case of a SiO2 glass tube, there will be surface oxygen atoms to bind to, and though I don't know anything about the chemistry of paper, it is organic, so I'm sure there's hanging O-H chains to bond with.

The mass of water moving through each pore is actually pretty small, so if you imagine the paper instead as a set of "ideal" cylindrical capillaries, the gravitational potential for each individual column is very weak, and so the water can rise quite a bit before the gravitational potential beats out the combination of cohesion and adhesion.

(In the interests of giving a broad answer, I remember in high school doing a neat chemistry experiment with a technique based on this effect, called paper chromatography)

Edit: I've added a bit on the possible microscopic origin of adhesion.

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So you're saying the energy comes from the electric attraction between paper molecules and water molecules? –  Malabarba Dec 25 '10 at 13:52
That's roughly what I'm saying, but in addition to their being an attractive potential between the water and paper molecules, there is an attractive potential between water molecules as well, which is why the water collectively moves up a tube instead of just wetting the walls. –  wsc Dec 26 '10 at 2:15

The previous answer tells you why the water moves up but doesn't explain where the energy comes from. In order for water to move up and thereby gain gravitational potential energy, you need to have some energy loss somewhere else to compensate.

Some of the energy comes from the random molecular movement of heat which extends the edge of the water itself up the capillary tubes/surfaces. So the water cools as it climbs because of the reduction in energy required to compensate for the reduced gravitational energy.

Some of it comes from loss of gravitational energy in the glass.

This part of the problem can be seen as the same as what happens when you take a test tube and place it in a tub filled with water, fill the tube with water and then invert it and pull the closed end out of the water so that some of the water in the closed end of the tube rises above the water in the rest of the tub. Where does the energy come from for this?

Two sources, not one:

1) From your arm moving the tube upwards and pulling on the end of the tube.

2) From the water in the tub dropping down in level and thereby leaving less gravitational potential energy for the remaining water in the tub.

If you consider a long tube placed into the tub. That water in the tube will rise to the level of the water outside the tube. If you attach the tube end above the water to a closed valve that leads to a vacuum chamber, the water level will stay the same. If you open the valve, the water level will rise. The vacuum itself supplies no energy and can do no work since there are no molecules or forces involved. What does the work is exchange of potential energy from the gravity of the water in the tub. It pushes on the water in the tube causing it to rise.

So for capillary action you describe, some of the energy comes from the loss of the potential energy of the water or fluid outside the capillary tube, and some of it comes from the loss of heat in the water and toilet paper.

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I am having a very hard time imagining your tubes and tubs. If you could provide a picture (any doodle that shows what's going on), that would be very helpful. Also, as I understand the OP's question, it asks for a microscopic description of the capillary action. I can't say I see it in this answer. Still, I won't give -1 because it seems like you know what your are talking about, so I'd like you to elaborate on the issues I mentioned and you'll get my +1 :-) –  Marek Dec 25 '10 at 23:54
chaotic heat energy can't be turned into ordered work, though, can it? –  endolith Dec 26 '10 at 17:05
@endolith: there is no chaotic heat energy. There is only heat, denoting the process of transfer of energy by the means of chaotic movement of molecules. And you can definitely extract useful work from heat under certain conditions. That's how heat engines work, for example. –  Marek Dec 26 '10 at 19:14
I do get the analogy to capillary action, but I find the answer a little hard to buy. When I place a capillary tube vertically (partially submersed) on a bowl of water, the water level inside the tube rises. Are you saying that happens because the water gets colder near the top? Why doesn't that happens to wider tubes as well? –  Malabarba Dec 26 '10 at 20:05
While the described effects certainly will happen, I'm afraid they have a negligible impact on this particular situation. It is specially obvious to rule out the importance of thermal effects: the capillary configuration remains basically the same even if we wait for thermal equlibrium to be reached again, once the water at the top and bottom of the capillaries is at the same temperature. –  Jaime Dec 27 '10 at 22:59

## protected by Qmechanic♦Jul 28 '14 at 22:44

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