Let's return to atomic spectra for a moment. As we noted earlier the simplest line spectra is owned by hydrogen. It turns out that hydrogen's spectra consists of a repeating pattern of lines. By the late nineteenth century folks had found that the line spectra of hydrogen could be described by a simple, empirical, mathematical relationship, known as the Rydberg equation:
The first set, the Lyman series, occurs in the ultraviolet (UV) and is described by the Rydberg equation with n_{1} = 1, the second, the Balmer series, occurs in the visible and is described by the Rydberg equation with n_{1} = 2,
The third set, the Paschen series, occurs in the infrared (IR) and is described by the Rydberg equation with n_{1} = 3.
In 1913 Bohr presented a very successful model for the structure of the hydrogen atom, which neatly explained the experimental observations noted before, including the line spectra and Rydberg equation. In order to account for the stability of the atom and the existence of line spectra he postulated that electrons in hydrogen have a minimum possible energy, the ground state, and that higher, excited energy states are quantized. The essence of Bohr's theory can be summarized in the equation for the energy of the electron:
E = -2.178 x ^{-18}J (Z^{2}/n^{2})
Bohr's equation can be used to derive the Rydberg equation by recalling that the energy of a photon emitted by an atom will be the difference between two electron energy levels:
Substituting Z = 1 and plugging in values gives the Rydberg equation exactly! (Of course one reason it works so well is that Bohr used the data from spectroscopy to find the value of his constant - he "fudged it.")
This is a wonderful result, and it rocked the scientific community. It said that at the microscopic level the world is discontinuous. The only trouble is, Bohr's model only works precisely for hydrogen! For most other atoms it is a dismal failure, though with a bit of tune-up it works reasonably well for the low energy spectra of the Alkali metals. (Note that like hydrogen these metals have only one electron that participates in chemistry. The Bohr model's partial success implies that this active electron "sees" the rest of the atom as a single charge at the nucleus - the active electron is somehow "outside" of the remaining electrons and thus behaving much like hydrogen's single electron.)
For some time the scientific community struggled with the failure of the Bohr model and the strangeness of the microscopic world. In the mid 1920's Werner Heisenberg and Erwin Schrödinger independently came up with models of the atom which accurately predicted the behavior of all atoms in principle.
A basic assumption of these treatments is that electrons behave like waves in some sense, and their locations in an atom are described by equations for waves. A second assumption is that of Heisenberg's Uncertainty Principle. This states that we cannot simultaneously know the position and momentum of a particle. Mathematically:
x = (mv) ≥ h/4π
Schrödinger assumed that the electron's behavior could be described by a three dimensional standing wave. He derived an equation which described the amplitude of this wave. The simplest solution for the Schrödinger Equation for the ground state (1s) energy of a hydrogen atom is:
where A & B are constants, e is the base of the natural logs, and r is the radial distance from the nucleus.
This equation has little real meaning. However, the square of the value of psi () tells the probability of finding an electron at any given location.
Finally, we need to note one additional observation.
The Pauli exclusion principle and electron spin mean that a maximum of two electrons may occupy a single orbital.
This is an alternate way of designating the electrons in an atom. Each electron will have a unique set of quantum numbers.
Quantum Number | Symbol | Characteristic specified | Information provided | Possible values |
Principle quantum number | n | Shell | Average distance from nucleus (r) | 1, 2, 3, 4, ... |
Angular momentum (Azimuthal) quantum number | l | Subshell | Shape of orbital | 0 (s), 1 (p), 2 (d), 3 (f), ...n - 1 |
Magnetic quantum number | m_{l} | Orbital | Orientation of orbital | - l ... 0 ... +l |
Spin quantum number | m_{s} | Electron spin | Spin direction | ± 1/2 |
Syllabus / Schedule |
© R A Paselk
Last modified 23 March 2015