Thomson’s electron and Rutherford’s nucleus made it clear that the atom has a structure of some sort. The challenge at the beginning of the 20th century was to deduce, from experimental evidence, the correct structure. The difficulty of this task cannot be exaggerated. The evidence about atoms, such as observations of atomic spectra, was very indirect, and experiments were carried out with only the simplest measuring devices.

Rutherford’s nuclear model was the most successful of various proposals, but Rutherford’s model failed to explain why atoms are stable or why their spectra are discrete. A missing piece of the puzzle, although not recognized as such for a few years, was Einstein’s 1905 introduction of light quanta. If light comes in discrete packets of energy, which we now call photons, and if atoms emit and absorb light, what does that imply about the structure of the atoms?

This was the question posed by the Danish physicist Niels Bohr. In 1913, Bohr proposed a new model of the atom in which he added quantization to Rutherford’s nuclear atom. The basic assumptions of the Bohr model of the atom are as follows:

MODEL 38.2 The Bohr model of the atom

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1. The electrons in an atom can exist in only certain allowed orbits. A particular arrangement of electrons in these orbits is called a stationary state.

2. Each stationary state has a discrete, well-defined energy $E_n$. That is, atomic energies are quantized. The stationary states are labeled by the quantum number n in order of increasing energy: $E_1<E_2<E_3<\cdots$.

3. An atom can undergo a transition or quantum jump from one stationary state to another by emitting or absorbing a photon whose energy is exactly equal to the energy difference between the two stationary states.

   Atoms can also move from a lower energy state to a higher energy state by absorbing energy in a collision with an electron or other atom in a process called collisional excitation.

   The excited atoms soon transition to lower states, eventually ending in the stable ground state.

Bohr’s model builds upon Rutherford’s model, but it adds two new ideas that are derived from Einstein’s ideas of quanta. The first, expressed in assumption 1, is that only certain electron orbits are “allowed” or can exist. The second, expressed in assumption 3, is that the atom can undergo a transition from one state to another—a quantum jump—by emitting or absorbing a photon of just the right frequency to conserve energy.

According to Einstein, a photon of frequency $f$ has energy $E_{\text{photon}}=hf$. If an atom jumps from an initial state with energy $E_{\text{i}}$ to a final state with energy $E_{\text{f}}$, energy will be conserved if the atom emits or absorbs a photon with $E_{\text{photon}}=\text{Δ}{E}_{\text{atom}}=|E_{\text{f}}-E_{\text{i}}|$. This photon must have frequency

if it is to add or carry away exactly the right amount of energy. The total energy of the atom-plus-light system is conserved.

The implications of Bohr’s model are profound. In particular:

1. Matter is stable. An atom in its ground state has no states of any lower energy to which it can jump. It can remain in the ground state forever.

2. Atoms emit and absorb a discrete spectrum. Only those photons whose frequencies match the energy intervals between the stationary states can be emitted or absorbed. Photons of other frequencies cannot be emitted or absorbed without violating energy conservation.

3. Emission spectra can be produced by collisions. In a gas discharge tube, the current-carrying electrons moving through the tube occasionally collide with the atoms. A collision transfers energy to an atom and can kick the atom to an excited state. Once the atom is in an excited state, it can emit photons of light—a discrete emission spectrum—as it jumps back down to lower-energy states.

4. Absorption wavelengths are a subset of the wavelengths in the emission spectrum. Recall that all the lines seen in an absorption spectrum are also seen in emission, but many emission lines are not seen in absorption. According to Bohr’s model, most atoms, most of the time, are in their lowest energy state, the $n=1$ ground state. Thus the absorption spectrum consists of only those transitions such as $1\to 2\text{,}1\to 3\text{,}\dots$ in which the atom jumps from $n=1$ to a higher value of $n$ by absorbing a photon. Transitions such as $2\to 3$ are not observed because there are essentially no atoms in $n=2$ at any instant of time. On the other hand, atoms that have been excited to the $n=3$ state by collisions can emit photons corresponding to transitions $3\to 1$ and $3\to 2$. Thus the wavelength corresponding to $\text{Δ}{E}_{\text{atom}}=E_3-E_1$ is seen in both emission and absorption, but transitions with $\text{Δ}{E}_{\text{atom}}=E_3-E_2$ occur in emission only.

5. Each element in the periodic table has a unique spectrum. The energies of the stationary states are the energies of the orbiting electrons. Different elements, with different numbers of electrons, have different stable orbits and thus different stationary states. States with different energies emit and absorb photons of different wavelengths.