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Probably the most famous equation in all of physics comes, as should be no surprise, from the crazy-hair-covered head of Albert Einstein. He theorized, in 1905, that energy was really just the same thing as mass. In Albert's words, "the mass of a body is a measure of its energy-content."1 You've heard the formula before:
E = mc2
Einstein was saying that the amount of energy in a small amount of mass is massive (pun most definitely intended). The speed of light is 3 × 108 m/s, a huge number—and squaring it means 1 kg of mass has 9 × 1016 J of energy.2
To put this in perspective, on December 26, 2004 there was a gargantuan earthquake in the Indian Ocean, with a magnitude of over 9.0 on the Richter scale. It dropped 1,000 miles of Earth's crust down about 50 feet and caused catastrophic destruction in Indonesia.
The earthquake produced, according to the US Geological Survey, right around 11 × 1016 J of energy, or right around what you'd get from all the energy stored in the mass of a decently sized meatloaf.
Einstein's discovery was a tremendous success for the early pioneers of conservation of energy (like our friend Joule), showing that these conservation laws extend beyond one quantity and could connect multiple aspects of our universe. Physicists are always trying to unite what seem like disparate phenomena in an overarching idea—just ask any particle physicist about the Theory of Everything, the coveted Holy Grail of modern physics. Einstein firmly took science one step in that direction with his paper, titled "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?'' ("Does the Inertia of a Body Depend Upon Its Energy Content?'' for those of us who aren't Swiss patent clerks).
For better or for worse, one of the major innovations that Einstein's derivation of E = mc2 paved the way for was the atomic bomb. While Einstein had only a very minor role in the Manhattan Project (the US effort to develop the bomb), it was his idea that proved the physics behind the bomb were sound.
Einstein's discovery of mass-energy equivalence led to the law of conservation of mass and energy, a grander version of the law of conservation of energy. It says that, since mass and energy are really two forms of the same quantity, a system really conserves the total amount of mass plus energy. In all the examples we've seen in this chapter, the masses of our objects—blocks, balls, cars, and so on—aren't changing, so we only need apply the law of conservation of energy. But nuclear fission works very differently than blocks and balls.
In a uranium bomb (or a uranium power plant, if that makes you feel better), uranium-235 is bombarded with free neutrons to form uranium-236, which splits into smaller elements, such as barium and krypton. But the mass of these smaller elements is less than the starting mass of 236U—the missing mass is converted into energy via the law of conservation of mass and energy and Einstein's formula, E = mc2. One fission of a 236U atom creates 3.2 × 10-11 J of energy.3
Admittedly, that isn't very much energy. But a bomb with just 1 kg of 236U can create 83 × 1012 J of energy—the energy of almost 20,000 tons of TNT, twenty-five Airbus A330s, or every single explosion ever depicted in a Michael Bay film—excluding Armageddon, which has a nuclear bomb in it (spoiler: it explodes) and so doesn't count.
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...