 MORE PRECISELY 4-2 The Energy Levels of the Hydrogen Atom |
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| By observing the emission spectrum of hydrogen, Niels Bohr determined early in the twentieth century what the energy differences between the various energy levels must be. Using that information, he was then able to infer the actual energies of the excited states of hydrogen.
A unit of energy often used in atomic physics is the electron volt (eV). Its name derives from the amount of energy imparted to an electron by accelerating it through an electric potential of 1 volt. For our purposes, however, it is just a convenient quantity of energy, numerically equal to 1.6 Thus, the ground state has energy E1 = 0 (by our definition), the first excited state has energy E2 = 10.2 eV, the second excited state has energy E3 = 12.1 eV, and so on. Notice that there are infinitely many excited states between the ground state and the energy at which the atom is ionized, crowding closer and closer together as n becomes large and En approaches 13.6 eV. Knowing the energy of each electron orbital, we can calculate the energy associated with a transition between any two given states. For example, to boost an electron from the second state to the third, an atom must be supplied with E3 - E2 = 1.9 eV of energy, or 3.0 Similarly, the jump from level 3 to level 4 requires E4 - E3 = 0.66 eV of energy, corresponding to an infrared photon with a wavelength of 1890 nm (18,900 Å), and so on. The accompanying diagram summarizes the structure of the hydrogen atom. The various energy levels are shown as a series of circles of increasing radius, representing increasing energy (but remember that the electron orbitals are actually rather fuzzy and do not really have well-defined radii). The electronic transitions between these levels (indicated by arrows) are conventionally grouped into families, named after their discoverers, that define the terminology used to |
identify specific spectral lines. (Note that the spacings of the energy levels are not drawn to scale here, to provide room for all labels on the diagram. In reality, the circles should become more and more closely spaced as we move outward.)
Transitions down to (or up from) the ground state (level 1) from (or to) higher, excited levels form the Lyman series. The first is Lyman alpha (Ly The next series of lines, the Balmer series, involves transitions down to (or up from) level 2, the first excited state. All the Balmer series lines lie in or close to the visible portion of the electromagnetic spectrum. Because they form the most easily observable part of the hydrogen spectrum and were the first to be discovered, the Balmer series lines are often referred to simply as the "Hydrogen" series and denoted by the letter H. As with the Lyman series, the individual transitions are labeled with Greek letters. An H The classification continues with the Paschen series (transitions down to or up from the second excited state), the Brackett series (third excited state), and the Pfund series (fourth excited state). Beyond that point, infinitely many other families exist, moving farther and farther into the infrared and radio regions of the spectrum, but they are not commonly referred to by any special names. A few of the transitions making up the Lyman, Balmer, and Paschen series are marked on the figure. Astronomically, the most important series are the Lyman and Balmer (Hydrogen) sequences. |
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10-19 joule—roughly half the energy carried by a single photon of red light. The minimum amount of energy needed to ionize hydrogen from its ground state is 13.6 eV. Bohr numbered the energy levels of hydrogen, with level 1 the ground state, level 2 the first excited state, and so on. He found that by assigning zero energy to the ground state, the energy of any state (the nth, say) could be written as follows: 
), corresponding to the transition between the first excited state (level 2) and the ground state. As we have seen, the energy difference is 10.2 eV, and the Ly
(beta) transition, between level 3 (the second excited state) and the ground state, corresponds to an energy change of 12.1 eV and a photon of wavelength 102.6 nm (1026 Å). Ly
(gamma) corresponds to a jump from level 4 to level 1, and so on. You can calculate the energies, frequencies, and wavelengths of all the photons in the Lyman series using the formulas given in the text. All Lyman series energies lie in the ultraviolet region of the spectrum.