The Quantum Atom

Niels Bohr, On the Constitution of Atoms and Molecules, 1913

Rutherford's discovery of atomic nuclei raised some difficult questions. If all the positive charge is concentrated in the nucleus, why don't the negatively charged electrons get sucked into the center of the atom? The existing physics implied that even orbiting electrons would slowly radiate energy and proceed to spiral into the nucleus. Obviously, this isn't what actually happens, because the radius of an atom is much larger than the radius of its nucleus. So, what is it that keeps electrons in a stable orbit?

In 1911-12, Rutherford mentored Niels Bohr, a 26-year-old Danish physicist. Bohr soon began applying the earlier work of Planck and Einstein on the quantization of light to the new model of the atom. Recall that their great discovery was that atoms can only absorb or radiate light in tiny fixed amounts, called quanta. Bohr guessed that it was, in fact, electrons that could only absorb or radiate light in quanta. In working out the implications to the atomic model, he managed to explain a surprising number of things.

For simplicity's sake, Bohr assumes an electron makes a circular orbit around the nucleus. Given the charge of an electron and the charge on the nucleus, the radius of the orbit completely determines the number of orbits per second the electron will make. So, what determines the orbital radius?

The answer is intimately related to energy. With a little work you can figure out the amount of energy needed to remove an electron to an arbitrarily large (infinite) distance from the nucleus. This value is effectively a measure of how much energy an electron loses as it gets closer to a nucleus. If you are given the escape energy of an electron, you can work out exactly what its orbital distance is (again, assuming a circular orbit).

The only way an electron can lose energy is to emit light, one quantum at a time.

The energy of a quantum of light isn't fixed; it's a function of the frequency of the light. If an electron could emit any frequency of light, it could emit a quantum of exactly the right energy to allow it to reach the nucleus. However, Bohr figured out that electrons emit light with a frequency equal to one half the frequency of the electron's orbit after the emission. (an intriguing fact that was not to be fully explained for some time).

So, imagine an electron at an infinite distance from a nucleus, but nonetheless in a circular orbit about it. At this point the escape energy of the electron is zero, and the orbital frequency is also zero (infinity is, once again, making my head hurt). The escape energy of the electron can only increase one quantum at a time, which, in turn, implies that the orbital frequencies the electron can reach are also quantized. In turn, this implies that an electron in a circular orbit can only reach certain distances from the nucleus. An electron could, in theory, jump from any orbit to any other orbit, if the light emitted as a result were in some multiple of a quantum. In practice, since only some quanta are allowed, only some orbits are allowed.

Now, at some point (easily calculated), the orbital frequency is so high, and the electron is so close to the nucleus, that the electron can no longer emit a complete quantum of light: it would hit the nucleus before emitting a full quantum. So it doesn't. This is the smallest possible orbit.

By plugging in the mass of an electron, the charge of an electron, the charge of a proton, and Planck's constant, you can work out the closest distance an electron in a circular orbit can approach a proton. This number is the same as the observed radius of a hydrogen atom.

Given that distance, you can also work out the orbital frequency, and therefore the frequency (colour) of light most commonly emitted by hydrogen.

Given that distance, you can also work out how much energy you need to rip an electron away from a hydrogen atom, which scientists call the ionization-potential. Again the value is confirmed by observation.

You can also work out the colour of light emitted as electrons jump closer to a hydrogen atom. These colours create the characteristic atomic spectrum of hydrogen. Indeed, Bohr not only showed how to compute the hydrogen spectrum from first principles, he also predicted some bits of it that hadn't been observed yet.

By plugging in the charge of two protons instead of one, you can work out all the same information for singly ionized helium (that is, a helium atom that is missing an electron). Beyond this, the presence of multiple electrons requires a more complicated model, one that is, however, based on Bohr's basic model of the atom.