Chemistry and Physics
So far, we've looked at how mostly big things relate to energy. It's hard not to when we talk about car crashes and explosions and anything Bruce Willis touches (but we repeat ourselves). But the same principles we've learned here apply at much smaller scales, too. Potential energy, in fact, is a key component of the description of atomic bonds in molecules.
We say two atoms are "bound'' together if the potential energy between them is negative—essentially, if the potential for them to repulse each other and fly away (energy represented in the positive direction of our axis) is outweighed by the potential for them to be drawn together (energy in the negative direction). A physicist named John Lennard-Jones came up with a mathematical description for this that other people started calling the Lennard-Jones potential.4
The repulsive half of the formula (the positive fraction raised to the twelfth power) represents the effect two atoms feel when their electron orbitals overlap. Atoms like to keep their orbitals to themselves, so any time there's overlap, they'll tend to move further apart from each other.
The attractive half of the formula (the negative fraction raised to the sixth power that's wearing a stunning dress and has done something wonderful with her hair) represents the van der Waals force, which is the force between atoms or molecules that draws them close together without making a distinct ionic or covalent bond.
When you draw the potential, it looks like a big valley.
This is, believe it or not, a molecule. (Chemists aren't very artistic.) When atoms are far away from each other, they're attracted via the van der Waals force and the distance between them, r, gets smaller—they roll down the hill on the right, into the valley. When atoms get too close, they're pushed away from each other, increasing r—and they roll down the hill on the left, into the valley again. Eventually, the two atoms will settle into an equilbrium at r0. If they're pulled beyond some minimum separation distance rm, the system's energy becomes positive and the atoms will fly apart. Yikes.
When the two atoms are bobbing happily around r0, the graph of energy versus distance looks almost exactly like something we've seen before—the spring's parabola.
In fact, scientists will often model atom molecules as being held together by springs—like H2O here, for example.
But if we have two atoms flying around with positive kinetic energy, how do we make a molecule in the first place? In order to bind atoms, the system must end with a negative potential energy, so we need something to take energy away from our system. Oftentimes, this something is what we call a catalyst. Catalysts speed up chemical reactions in many ways, but one key way is by providing a sink for excess energy.
An example is the reaction of hydrogen and oxygen atoms in the presence of platinum. Two H atoms that collide with oxygen won't form an H2O molecule alone—they have too much kinetic energy. But if you add platinum, the excess energy in the collision is transferred to the platinum surface, which glows bright orange as water is formed, and binds the water molecule together.
To recap: science can make molten metal using water instead of fire. And yet there are still no flying cars...