In 1913, Niels **Bohr** attempted to resolve the atomic paradox by ignoring classical electromagnetism’s prediction that the orbiting electron in hydrogen would continuously emit light. Instead, he incorporated into the classical mechanics description of the atom Planck’s ideas of quantization and Einstein’s finding that light consists of photons whose energy is proportional to their frequency.

Bohr assumed that the electron orbiting the nucleus would not normally emit any radiation (the stationary state hypothesis), but it would emit or absorb a photon if it moved to a different orbit. The energy absorbed or emitted would reflect differences in the orbital energies according to this equation:

\(\mid \text{Δ}E\mid =\mid {E}_{\text{f}}-{E}_{\text{i}}\mid =h\nu =\phantom{\rule{0.2em}{0ex}}\frac{hc}{\lambda }\)

In this equation, *h* is Planck’s constant and *E _{i}* and

*E*are the initial and final orbital energies, respectively. The absolute value of the energy difference is used, since frequencies and wavelengths are always positive. Instead of allowing for continuous values for the angular momentum, energy, and orbit radius, Bohr assumed that only discrete values for these could occur (actually, quantizing any one of these would imply that the other two are also quantized). Bohr’s expression for the quantized energies is:

_{f}\({E}_{n}=\text{−}\frac{k}{{n}^{2}}\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.2em}{0ex}}n=1,\phantom{\rule{0.2em}{0ex}}2,\phantom{\rule{0.2em}{0ex}}3,\phantom{\rule{0.2em}{0ex}}\dots \)

In this expression, *k* is a constant comprising fundamental constants such as the electron mass and charge and Planck’s constant. Inserting the expression for the orbit energies into the equation for Δ*E* gives

\(\text{Δ}E=k\left(\phantom{\rule{0.2em}{0ex}}\frac{1}{{n}_{1}^{2}}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{1}{{n}_{2}^{2}}\phantom{\rule{0.2em}{0ex}}\right)=\phantom{\rule{0.2em}{0ex}}\frac{hc}{\lambda }\)

or

\(\phantom{\rule{0.2em}{0ex}}\frac{1}{\lambda }\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{k}{hc}\phantom{\rule{0.2em}{0ex}}\left(\phantom{\rule{0.2em}{0ex}}\frac{1}{{n}_{1}^{2}}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{1}{{n}_{2}^{2}}\phantom{\rule{0.2em}{0ex}}\right)\)

which is identical to the Rydberg equation for \({R}_{\infty }=\phantom{\rule{0.2em}{0ex}}\frac{k}{hc}.\) When Bohr calculated his theoretical value for the Rydberg constant, \({R}_{\infty },\) and compared it with the experimentally accepted value, he got excellent agreement. Since the Rydberg constant was one of the most precisely measured constants at that time, this level of agreement was astonishing and meant that Bohr’s model was taken seriously, despite the many assumptions that Bohr needed to derive it.

The lowest few energy levels are shown in the figure below.

One of the fundamental laws of physics is that matter is most stable with the lowest possible energy. Thus, the electron in a hydrogen atom usually moves in the *n* = 1 orbit, the orbit in which it has the lowest energy. When the electron is in this lowest energy orbit, the atom is said to be in its ground electronic state (or simply ground state).

If the atom receives energy from an outside source, it is possible for the electron to move to an orbit with a higher *n* value and the atom is now in an excited electronic state (or simply an excited state) with a higher energy. When an electron transitions from an excited state (higher energy orbit) to a less excited state, or ground state, the difference in energy is emitted as a photon.

Similarly, if a photon is absorbed by an atom, the energy of the photon moves an electron from a lower energy orbit up to a more excited one. We can relate the energy of electrons in atoms to what we learned previously about energy. The law of conservation of energy says that we can neither create nor destroy energy.

Thus, if a certain amount of external energy is required to excite an electron from one energy level to another, that same amount of energy will be liberated when the electron returns to its initial state (see figure below).

In effect, an atom can “store” energy by using it to promote an electron to a state with a higher energy and release it when the electron returns to a lower state. The energy can be released as one quantum of energy, as the electron returns to its ground state (say, from *n* = 5 to *n* = 1), or it can be released as two or more smaller quanta as the electron falls to an intermediate state, then to the ground state (say, from *n* = 5 to *n* = 4, emitting one quantum, then to *n* = 1, emitting a second quantum).

Since Bohr’s model involved only a single electron, it could also be applied to the single electron ions He^{+}, Li^{2+}, Be^{3+}, and so forth, which differ from hydrogen only in their nuclear charges, and so one-electron atoms and ions are collectively referred to as hydrogen-like atoms. The energy expression for hydrogen-like atoms is a generalization of the hydrogen atom energy, in which *Z* is the nuclear charge (+1 for hydrogen, +2 for He, +3 for Li, and so on) and *k* has a value of 2.179 \(\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\) 10^{–18} J.

\({E}_{n}=\text{−}\frac{k{Z}^{2}}{{n}^{2}}\)

The sizes of the circular orbits for hydrogen-like atoms are given in terms of their radii by the following expression, in which \({\text{α}}_{0}\) is a constant called the Bohr radius, with a value of 5.292 \(×\) 10^{−11} m:

\(r=\phantom{\rule{0.2em}{0ex}}\frac{{n}^{2}}{Z}\phantom{\rule{0.2em}{0ex}}{a}_{0}\)

The equation also shows us that as the electron’s energy increases (as *n* increases), the electron is found at greater distances from the nucleus. This is implied by the inverse dependence on *r* in the Coulomb potential, since, as the electron moves away from the nucleus, the electrostatic attraction between it and the nucleus decreases, and it is held less tightly in the atom.

Note that as *n* gets larger and the orbits get larger, their energies get closer to zero, and so the limits \(n\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\infty \) \(n\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\infty ,\) and \(r\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\infty \) \(r\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\infty \) imply that *E* = 0 corresponds to the ionization limit where the electron is completely removed from the nucleus. Thus, for hydrogen in the ground state *n* = 1, the ionization energy would be:

\(\text{Δ}E={E}_{n\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\infty }-{E}_{1}=0\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}k=k\)

With three extremely puzzling paradoxes now solved (blackbody radiation, the photoelectric effect, and the hydrogen atom), and all involving Planck’s constant in a fundamental manner, it became clear to most physicists at that time that the classical theories that worked so well in the macroscopic world were fundamentally flawed and could not be extended down into the microscopic domain of atoms and molecules.

Unfortunately, despite Bohr’s remarkable achievement in deriving a theoretical expression for the Rydberg constant, he was unable to extend his theory to the next simplest atom, He, which only has two electrons. Bohr’s model was severely flawed, since it was still based on the classical mechanics notion of precise orbits, a concept that was later found to be untenable in the microscopic domain, when a proper model of quantum mechanics was developed to supersede classical mechanics.