Why does water have potential energy?

Question

I was learning about surface tension, and came across the curious concept of surface energy. My physics textbook defines it as the potential energy of molecules which helps them remain on the surface (and thus not go inside).

However I have 2 questions about the concept of surface energy. Firstly, why would the water molecules have potential energy by just staying at the surface, I can't see the intuition on why they would have any potential energy (neglecting gravitational potential energy). Secondly, how did this energy even come about in the first place, if I spit on the ground and the droplets form spheres, where did that energy come from, my mouth? It couldn't have appeared out of nowhere since that would violate the rule that energy cannot be created or destroyed.

Answer 1

Consider a big container filled with pure water. The large permanent dipole moment of the H$_2$O molecule means that each molecule feels an attractive force towards its neighbors. Transporting a molecule from the center of the container to the exterior costs energy because you have to do work to overcome those attractive forces; let's call that energy $u$.

If we want to free all of the hydrogen molecules in some small volume $V$, we need to provide a total energy equal to $$u_{tot} = Nu = nVu$$ where $N$ is the total number of molecules in that volume and $n$ is the number density of molecules in the water. We can therefore say that the total potential energy of a volume $V$ deep within the container due to the attractive forces is equal to $U = - nuV \equiv -\alpha V$, where $\alpha \equiv nu$ and the minus sign comes from the fact that we have to provide energy to the system in order to pull the molecules apart.

Rather than considering water in the middle of a container, now imagine the water in a (not necessarily spherical) droplet with total volume $V$. How much total potential energy does it have? Naively, one would simply say $U=-\alpha V$, but this would be incorrect. The energy calculated before assumed that we would need the same amount of energy to free each molecule, but the molecules near the surface of the droplet are only being pulled from one side. As a result, it requires less energy to free them, which means that the total potential energy of our droplet is slightly higher than our estimate.

How much higher? As a crude model, imagine that our previous estimate is correct except for a thin layer of thickness $t$ near the surface of the droplet, and that in that layer the energy required to remove a molecule is cut in half. The total number of molecules in this layer is equal to $nAt$ where $A$ is the surface area of the droplet, and so our corrected estimate for the droplet's potential energy is

$$U = - \alpha V + \frac{nAtu}{2} = -\alpha V +\gamma A$$

where $\gamma \equiv \frac{ntu}{2}$. The potential energy of a droplet therefore has a negative contribution proportional to its volume and a positive contribution proportional to its surface area.

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The droplet will assume the shape which minimizes this potential energy. In the absence of additional external influences, this corresponds to minimizing the second term, since water and other liquids essentially don't change their volume. The 3D shape which minimizes surface area for a given volume is a sphere, which is why liquid droplets tend to be approximately spherical.

If we attempt to increase this area, we will increase the potential energy: $$\Delta U = \gamma \Delta A$$ we call the ratio $\frac{\Delta U}{\Delta A} = \gamma$ the surface tension.

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Firstly, why would the water molecules have potential energy by just staying at the surface

A molecule near the surface has a higher (i.e. less negative) energy than a molecule in the center because it takes less work to free it from the droplet. This corresponds to an effective inward pull on the surface, so saying that surface tension keeps water molecules from going inside the droplet is incorrect.

However, surface tension does act to keep the surface from deforming (and therefore increasing its surface area). An object which applies sufficiently little pressure will not break the surface (think of the molecules on the surface holding hands in a molecular version of red rover), and in this case the surface tension acts to effectively hold the object up.

how did this energy even come about in the first place

The potential energy of a droplet is ultimately due to attractive forces between the molecules, which exists all the time. If you spit an amorphous blob into the air, its forming into a sphere will decrease its potential energy, corresponding to an increase in the kinetic energy of molecules (in the form of vibrations and heat).

Answer 2

Your question is a very good question, and I think I have an answer (a little disclaimer, I didn't learn the answer to your question - I'm explaining how I understand this phenomenon emerges).

So the first thing we need to understand is why in the first place water in room temperature is a liquid and not a gas. When you cool down a system from a very high temperature, you're actually lowering the average energy of the molecules in it. All of the chemicals in nature interact via electric and magnetic forces (even covalent bonds are just quantum mechanics with electric pull of the electrons and nuclei). Some bonds are weak, meaning you don't need a lot of energy to break them apart and let every atom or molecule part to separate ways. On the other hand, some are really strong, taking a lot of energy to break.

When you cool down the system enough, the energy of the bonds between molecules will be higher than the average energy of a molecule, meaning they will be "tied" to each other. If the molecules still have enough energy to move around and change shape, this is called a liquid, and if all of the molecules are stuck at one place (more or less..) this is called a solid.

Helium, for example, is a noble gas which interacts very very weakly, meaning that it could be a gas down to $4^\circ $ kelvin! Water molecules, on the other hand, interact more strongly via hydrogen bonds (which are still weak comparable to other materials - notice how almost all metals are solids at room temperature) and that is why they are most commonly encountered in everyday life as a liquid.

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"Ok, yeah, we learned some of that in school when we were taught about the phases of matter, but what does that have to do with surface tension?" you're probably asking yourself. And as I understand it, the answer is "everything".

Think about a water molecule at the surface - it feels a pull from every direction from one side of the surface (where there is water) and nothing from the other side, meaning it feels a pull inward. Thus, if the surface molecules were pulled inward, that would mean that work has been done. If so, you can attribute potential energy to having molecules on the surface (which needs to be linear with the amount of molecules on the surface), so the surface tension energy is linear with the amount of surface area itself.

I hope this answers your question :)

Answer 3

Your physics textbook seems to me to be giving a very poor description at this point. In fact all you need to think about is inter-molecular forces. Every water molecule pulls on every other one. That's all there is to it. No special potential energy. Just molecules pulling on each other. But potential energy can be a useful way of quantifying the result.

Think of a liquid such as water as made of millions of tiny bits of mater. In fact we may as well talk about molecules.

Suppose you have first of all some water with a flat surface. Now introduce a wiggle in the surface (and ignore gravity for this---imagine the experiment is being done in a low-gravity environment). Make this wiggle in such a way that the volume of the water has not changed. This means the water has the same density as it did, and therefore the same average distance between molecules. So all the inter-molecular forces between neighbouring molecules are still the same.

But something has changed: the surface area. This means that there are now more molecules at the surface than before.

Those extra molecules at the surface were previously in the body of the water, surrounded by water. Now they are at the surface, with water on one side, and air (or some other gas) on the other. So they have fewer bonds to the rest of the water than they used to have. As the water surface changed shape, various molecular pairs had to move apart from one another, and energy has to be supplied to make this happen. In other words, changing the shape of the surface to one with larger area requires energy to be provided! This is at the heart of the phenomenon called surface tension. It means that, in order to find the configuration of least energy, the surface pulls back so as to try to minimize its area.

Here is a 2-dimensional picture to illustrate the idea: $$ \begin{array}{cccccccccccccccc} . &. &. &. &. &. &. &. &. &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. &. &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. &. &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. &. &. &. &. &. &. &. &. \end{array} $$

There are 64 dots altogether. 36 are at the surface; 28 in the bulk. Now imagine a line drawn between each pair of neighbours, and for simplicity just treat horizontal and vertical connections. You can find $16 \times 3 = 48$ vertical lines and $15 \times 4 = 60$ horizontal lines, making $108$ in total.

Now I will draw the same number of dots, but arranged in a shape with a smaller perimeter: $$ \begin{array}{cccccccc} . &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. \end{array} $$ We have again 64 dots, but now with 28 at the surface and 36 in the bulk. The number of vertical bonds is $8 \times 7 = 56$ and the number of horizontal bonds is also 56, making $112$ in total. That's more bonds! So if the bonds represent attractive forces, as they do here, then this second configuration is the one where the dots have "pulled themselves together" more---even though the total volume has not changed. (To agree this statement about volume, don't forget that each dot represents the centre of a blob, not a mathematical point.)

You can have some fun rearranging these dots a little further to increase the total number of bonds a bit more.

What kind of shape are you heading towards?

What kind of shape do drops of water have in zero gravity?

The "potential energy" mentioned in your book is the potential energy associated with moving the molecules apart, breaking some of the bonds, when moving from a configuration of small surface area to one of larger surface area, under conditions of fixed density. It is a useful way of expressing the net result. But the underlying reason for it is the one I have explained.

In the case of any given blob of water, the energy it contains is all provided by the source of that blob of water, together with the forces, and possibly heat exchange, exerted on it by its environment. So there is no magic and energy is conserved.

Answer 4

The water molecules on the surface are just attracted by the molecules inside the water and feel a net attraction towards the inside because there is a resultant attraction by the molecules inside.

The attraction is basically electrostatic in nature. Now suppose you need to move away 2 opposite charges you need to put in energy into the system. That is because the system's potential energy before you put in extra energy is negative.

Similarly, the water molecules being attracted inside is the cause of the potential energy. Some molecules move inside and the others from below take their place but the attraction still remains.

Now when you spit, it happens that just as a two opposite charged system tries to minimize it's potential energy by coming near ( thus changing the initial configuration), similarly when you spit the droplets form a sphere because it happens that a sphere has the minimum the potential energy. The cause is again electrostatic attraction.

Answer 5

"I can't see the intuition on why they would have any potential energy"

They do have electrostatic potential energy just by existing at a distance away from each other. Keep in mind that charged particles kept separated at a distance do have potential energy (this is the energy which gets converted into kinetic energy if the particles are allowed to move freely under the influence of each others' electric fields)

So a water drop (which is an assembly of attracting water molecules) also has potential energy.

Now why does the surface have a higher potential energy than the bulk?

Well, if you consider a molecule at the surface, the net electric field at a point on the surface (which is the vector sum of the electric fields at that point due to other molecules) points towards the bulk. This is because the molecules of the surface are attracted by the molecules in the bulk, so there's a net force towards the bulk. But electric field always points in the direction of decreasing potential energy. So the surface molecules are at a higher potential energy than the bulk molecules.