Bohr’s Theory of the Hydrogen Atom

Atomic and molecular emission and absorption spectra have been known for over a century to be discrete (or quantized). Well before they were understood from first principles, chemists have been using the emission and absorption spectra for identification of elements. Figure 12.7.2 shows iron emission spectrum, for example. No other elements emit the exactly the same set of frequencies of light. With the discovery of substructure of the atom and the discovery of photon (or more precisely, refined understanding of the particle nature of electromagnetic waves where the particle energy is proportional to the frequency of electromagnetic waves), these resonant frequencies of light emitted by atoms could be used to infer an atomic model.

For the hydrogen atom, the lightest element with the simplest atom, a pattern for its line spectrum was noticed by experimentalists (see Figure 12.7.3). All wavelengths of the line spectrum could be described by a following formula, for the suitable choice of two integers $n_{\mathrm{i}}$ and $n_{\mathrm{f}}$:

where $\lambda$ is the wavelength of the emitted EM radiation and $R$ is the Rydberg constant, determined by the experiment to be

The $n_{\text{f}}$ is a positive integer associated with a specific series, which are named after their discoverers. For the Lyman series, $n_{\text{f}}=1$; for the Balmer series, $n_{\text{f}}=2$; for the Paschen series, $n_{\text{f}}=3$; and so on. The Lyman series is entirely in the UV, while part of the Balmer series is visible with the remainder UV. The Paschen series and all the rest are entirely IR. There are apparently an unlimited number of series, although they lie progressively farther into the infrared and become difficult to observe as $n_{\text{f}}$ increases. The $n_{\text{i}}$ is a positive integer greater than $n_{\text{f}}$. So for example, for the Balmer series, $n_{\text{f}}=2$ and $n_{\text{i}}=\mathrm{3,\; 4,\; 5,\; 6,\; ...}$.

So, before Bohr’s model of the hydrogen atom, such was the picture of atomic theory—full of suggestive (and even well-organized) data and no unifying explanation. Ernest Rutherford is quoted as saying, “All science is either physics or stamp-collecting.” What he meant is, there are branches of science whose practitioners would be satisfied with a collection of interesting facts (i.e. “stamp-collecting”). But what makes physics physics is the search for the theoretical framework providing explanations based on fundamental principles, not idiosyncratic descriptions. Bohr’s model brought the science of spectroscopy into physics.

The planetary model of the atom suggested by Rutherford was in trouble. While the model provided a possible picture of how the very small atomic nucleus might be arranged with the electrons in a stable arrangement, it did not provide for the size of electron orbits (which would be related to the size of the atom), and the arrangement was not actually stable—an orbiting electron is an oscillating charge; an oscillating charge emits electromagnetic waves; electromagnetic waves carry away energy; so as the electron loses energy, it would fall into the proton. By some estimates, this would occur in as short a time as ${10}^{\mathrm{-7}}s$!

Bohr’s starting point for his successful model was this: he proposed that the orbits of electrons in atoms are quantized. To fully understand this statement, we can compare the orbits of electrons in atoms to the orbits of planets in the solar system. The orbits of planets are not quantized. While laws of physics govern how planets move in the solar system (see for example, Kepler’s laws, or their derivation by Newton starting with the inverse-square law of gravitation), there is no law of physics dictating how far each body in the solar system must be from the Sun. So the orbits of planets are not quantized.

So what Bohr was proposing was an entirely new law of physics no one had known before. In one sense, it was not completely new (Planck and Einstein already enjoyed some successes from suggesting quantization of energy in thermal oscillators and EM radiation); in another sense, it was a big break from centuries of classical mechanics. This was Bohr’s quantization rule: angular momentum of an electron in its orbit is quantized. In mathematical form,

where $n$ could take on any positive integer value ($n=\text{1, 2, 3, ...}$), and $\hslash$ is known as the reduced Planck constant ($\hslash =h/\mathrm{2\pi }$). And angular momentum, $L$, as you might remember from earlier chapter, is given by the following for a particle in a uniform circular orbit: $L=mvr$, where $m$ is the mass of the particle, $v$ is the speed of the particle in orbit, and $r$ is the radius of circular orbit. Using this as the starting point, semiclassical analysis of orbital motion yields a whole array of quantized (i.e. allowed) values of orbital distance ($r_n$), orbital speed ($v_n$), and orbital energy ($E_n$), among others (see: Table 12.7.1 for a summary).

With the quantized orbital energies for the electron, we have a ready explanation for the features of atomic spectra. EM radiation is emitted when an electron transitions from a higher energy level ($E_{\mathrm{i}}$) to a lower energy level ($E_{\mathrm{f}}$), with the photon carrying away the energy difference,

where $f$ is the frequency of the photon. Figure 12.7.4 shows a schematic representation of this relationship. With only discrete values of energy $E_n$ allowed, there are only discrete values of frequency ($f$) and wavelength ($\lambda$) allowed also, as shown in the line spectra.

Energy-level diagram, shown in Figure 12.7.5, is another convenient way to illustrate these relationships. Allowed energy levels for the atom are plotted vertically with the lowest state (or ground state) at the bottom and with excited states above that. The energies of the lines in an atomic spectrum correspond to the differences in energy levels in the level diagram (figure illustrates a transition from $E_4$ to $E_2$, which would show up in the atomic spectrum as one line).

Two key results are worth highlighting. The first is the Bohr radius, the radius of the lowest-energy electron orbit in the Bohr model of hydrogen. Its value is $a$, given for $n=1$,

This is the Bohr model’s prediction for the size of the atom, made with nothing more than electric constants, mass of the electron, and the Planck’s constant, and this theoretical prediction matches experimentally measured sizes of atoms fairly well.

The second is the derivation of the Rydberg formula, first given in Equation 12.7.1. To derive this, we start out with Equation 12.7.4 and substitute in expressions for hydrogen energies from Table 12.7.1:

Frequency $f$ is equal to $c/\lambda$. Plugging this in and solving for $1/\lambda$ while also replacing all instances of $\hslash$ with $h/2\pi$, we get,

which yields an analytical expression for the Rydberg constant,

Figure 12.7.6 shows an energy-level diagram for hydrogen that also illustrates how the various spectral series for hydrogen are related to transitions between energy levels.

We see that Bohr’s theory of the hydrogen atom answers the question as to why this previously known formula describes the hydrogen spectrum. It is because the energy levels are proportional to $1/n^2$, where $n$ is a positive integer. A downward transition releases energy, and so $n_{\mathrm{i}}$ must be greater than $n_{\text{f}}$. The various series are those where the transitions end on a certain level. For the Lyman series, $n_{\text{f}}=1$ — that is, all the transitions end in the ground state (see also Figure 12.7.6). For the Balmer series, $n_{\text{f}}=2$, or all the transitions end in the first excited state; and so on. What was once a recipe is now based in physics, and something new is emerging—angular momentum is quantized.

Bohr did what no one had been able to do before. Not only did he explain the spectrum of hydrogen, he correctly calculated the size of the atom from basic physics. Some of his ideas are broadly applicable. Electron orbital energies are quantized in all atoms and molecules. Angular momentum is quantized. The electrons do not spiral into the nucleus, as expected classically. These are major triumphs.

Bohr’s model also applies to hydrogen-like ions: atoms with all but one electron removed, such as He⁺ and Li²⁺. But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is a semiclassical model. The orbits are quantized (quantum mechanical) but are assumed to be simple circular paths (classical). As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are “clouds” of probability. Bohr’s theory also did not explain that some spectral lines are doublets (split into two) when examined closely. These deficiencies are addressed in later, fully-quantum-mechanical atomic models, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation.