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took shape in 1925. The various series of spectroscopic lines that have energies related to integers through the Rydberg formula can be understood as optical transitions between discreet energy levels that are associated with the hydrogen atom. A few of the energy levels that give rise to the Lyman series and the Balmer series are shown in Figure 9.3. In the figure, the downward arrows indicate the emission of light that would come from a hydrogen arc lamp. The hydrogen atom starts in a higher energy level and ends up in a lower energy level. Energy is conserved by the emission of a photon. To conserve energy, the photon must have the energy difference between the initial higher energy level and the final lower energy level. In the Rydberg formula, the smallest value n1 can have is 1, and n2 must be bigger than n1 . The arrow labeled 2-1 represents emission from the n = 2 level to the n = 1 level. The next higher energy emission in the Lyman series is for emission from the n = 3 level to the n = 1 level. In the Rydberg formula, the next possible value for n1 is 2, and n2 must be bigger than n1. Therefore, the lowest energy emission line in the Balmer series is labeled 3-2. The hydrogen atom begins in the n = 3 level and ends in the n = 2 level, and energy is conserved by emission of a photon with wavelength 656 nm. When light shines on hydrogen atoms, absorption occurs, which would be indicated in the diagram by up arrows. The Balmer series absorptions are shown in Figure 9.2.

FIGURE 9.3. Schematic of some of the energy levels that give rise to the Lyman and Balmer series of hydrogen atom emission lines. The down arrows indicate that light is being emitted from a hydrogen arc lamp, for example. For absorption, shown by the black lines in Figure 9.2, the arrows would point up. The level spacings are indicative but not to the true scale.

BOHR’S HYDROGEN ATOM THEORY—NOT QUITE THERE

The first detailed description of the energy levels of the hydrogen atom was developed by Niels Bohr (1885-1962) in 1913. Bohr won the Nobel Prize in Physics in 1922 “for his services in the investigation of the structure of atoms and of the radiation emanating from them.” Bohr’s theory of the hydrogen atom is referred to as the old quantum theory. Bohr made many advances and in fact was able to calculate precisely the energy levels of the hydrogen atom, and therefore obtain the Rydberg relation and predict all of the hydrogen atom spectral lines. Bohr was also the first to propose two ideas we have already been using. He said an atomic system can only exist in certain states, which he called “stationary” states. Now, we usually refer to these as energy eigenstates. Each of these states has a well-defined energy, E. Transitions from one stationary state to another can occur by absorption or emission of light or other means that can give or take energy from the system, and the amount of energy must be equal to the difference in energy of the two states. This idea is the basis for Figures 9.3 and 8.7, where the arrows represent transitions between states that occur by absorption or emission of light.

Bohr also put forward what came to be known as the Bohn Frequency Rule. The frequency of light emitted or absorbed in making a transition from an initial state with energy E1 to a final state with energy E2 is the difference in the energies divided by Planck’s constant, that is,

ν is the frequency and h is Planck’s constant (h = 6.6 × 10-34 J-s). The vertical lines are the absolute value. For absorption E1 is less than E2, so E1 - E2 would be negative. The absolute value means that you make the number positive even if the difference is negative. The frequency, ν, has to be a positive number. Multiplying both sides by h gives E, the energy difference between the energy levels (stationary states) as E = hν, which is the Planck relationship that Einstein used to explain the photoelectric effect discussed in Chapter 4.

What is a hydrogen atom and what are the failures of the Bohr method? A hydrogen atom is composed of two charged particles,