Physics

# The Physics Philes, lesson 125: Timmy Fell Down the Potential Well

Everything we come into contact with on a regular basis – computers, door knobs, blankets, cats – are all made of molecules. Some molecules consist of only one atom, others are made up of thousands and thousands of atoms. It’s the interactions between molecules of a substance that give matter its various properties. In liquids and solids, these molecules are held together by electrical forces. Gravity, by comparison, has very little effect on individual molecules.

Obviously, stuff we can see with the naked eye are generally made up of a lot of molecules. But that’s too complicated to think about right now. Let’s think first about the interaction of two point charges. We can describe that interaction as a force. The magnitude of this force is proportional to the inverse square of the distance between the two points. This is called the inverse square law.

However, molecules are not point charges. They have positive and negative charges. When molecules are far apart – like a gas – the intermolecular forces are small and the force is usually attractive. However, as you compress the gas, the molecules get closer together and the attractive force gets bigger. The forces between molecules become zero when the molecules are about as far apart as molecules in liquids or solids, known as the equilibrium spacing. In liquids or solids we need to relatively large pressures to compress the substance noticeably. What that means is that, when the distance between molecules is even a little less than the equilibrium spacing, the forces between the molecules become repulsive.

The potential energy of the molecules can be expressed as a function of the distance between them, denoted with r. The function is at its minimum where the force is zero. These can be related by taking the derivative of the potential energy U with respect to the distance r. A potential energy function such as this is called a potential well. Say we have a particle at rest in a potential well at a distance from another particle where the force is zero. In that case, we need to add some additional energy equal to the “depth” of the potential well to get very, very far away.

If you’re like me, this is kind of hard to wrap your head around. Maybe think about it like this: Say we have an electron bound to an atom. The electron has a certain amount of energy, but not enough escape the atom. That’s like the potential well. Now say we add enough energy for the electron to be stripped away from the atom. We’ve added enough energy for the electron to escape the well. It’s not a perfect analogy, but it helps me to visualize something concrete when I’m thinking about things that I can’t see, like electrical forces. Feel free to ignore this if you find it more confusing. (And, as always, please feel free to tell me if my analogy is completely off base.)

OK, so molecules are always in motion. Always, and their kinetic energies are linked tightly with temperature. At very low temperatures, the kinetic energy of a molecule might be much less than the depth of the potential well. But at high temperatures, there might be enough kinetic energy for the molecule to escape the potential well. That is, the kinetic energy might be more than the amount of energy needed to escape the intermolecular forces that bind it and become a gas.

In solids, molecules vibrate about fixed points. More or less. This vibration might be simple harmonic if potential well is parabolic close to the equilibrium spacing. But if the potential well rises more gradually, the average position shifts to a larger distance r with increasing amplitude. This is the basis for thermal expansion.

This distances between molecules in liquids are really only slightly bigger than those distances in solids of the same substance. However, molecules in a liquid have a lot more freedom to move around. Liquids only have a regular structure in the immediate vicinity of a few molecules. Liquids, we say, have short-range order. Solids have long-range order.

Unlike solids and liquids, gases are so far apart that they only have very small attractive forces. A molecule of gas moves in a straight line until it hits another molecule or some other barrier. An ideal gas is a gas whose molecules exert no attractive forces on each other, therefore have no potential energy.

So, at low temperatures, most things we know from our day to day lives are solid. At high temperatures, they melt or vaporize. If we look at it from the point of view of a molecule, these phase transitions have to do with the amount of kinetic energy the molecule has.

Understanding how kinetic energy effects individual molecules isn’t just an intellectual exercise. It’s important in understanding the kinetic-molecular model of gases, which we’ll start to tackle next week.

Featured image credit: Kashif Mardani via Flickr