Electron Shells

Outline

• Quantum Mechanics and Electron Orbitals
  • Spin
  • Uniqueness: The Pauli Exclusion Principle
  • Shells and Configurations
• Practice with Concepts
• Discussion Questions
• Optional Website Reading

Quantum Mechanics and Electron Orbitals

During the early decades of quantum mechanics and the development of the Bohr model of the atom, it became clear that individual electrons within an atom each had a unique quantum state, driven by energy level, angular momentum, magnetic angular momentum, and electron spin. Because the magnetic fields arising from moving electrons affect each other, at most two electrons with opposite spins could exist in the same orbital. This fundamental principle places a final limitation of electron locations around the nucleus.

Spin

Any moving charged particle generates a magnetic field. This is a fundamental property of charged matter, which means there is no other known cause for charge or the connection between electricity and magnetism: it just is. Just as matter has mass and all matter has charge (which sometimes cancels out), all moving charges result in magnetic fields and all changing magnetic fields result in changing electrical fields. We've already seen how these observations, taken together and summarized in Maxwell's equations, explain the wave behavior of light. The principle applies to any motion of a charged particle, including spin in place.

Ampere's law tells us that a series of moving electrons creates a current i. In the Bohr atom, this "current" is the flow of the electron around the nucleus in revolutions per second. If we consider rev/sec = cycles/sec, the electron "frequency" ν (the Greek letter nu) * the charge e on the electron (1.60 * 10^-19 Coulomb) gives us the current or number of electrons passing a given point per second...even if it is the same electron coming around again. In the hydrogen atom, the ground state electron has a frequency of 6.63 * 10¹⁵ Hz (cycles per second).

i = e*ν

i = 1.6 * 10^-19 Coulomb * 6.63 * 10¹⁵ Hz = 1.06 * 10^-3 Amps

The magnetic field for a circling charged particle is

B = μ_o * i/ 2 * R

where μ_o is a constant called the permitivity constant, that depends on the ability of the magnetic field to permeate through a medium (in this case, empty space), and equal to 4*pi * 10^-7 Tesla*meters/Amps. The current i is given by the previous equation, and R is the radius at which the electron orbits the nucleus. For a hydrogen atom, R is about 5.29* 10^-11m, so we can calculate the field:

B = 4 * pi * 10^-7 Tesla*meters/Amps* 1.06 * 10^-3 Amps / 2 * 5.29 * 10^-11 meters = 12.6 Tesla

Besides the wave motion of the electron's velocity around the nucleus, it also spins. This spin creates a magnetic field that can be up out of the plane of its motion or down. For any set of n, l, and m quantum numbers, there can be two spin values, up or down (represented by -1/2 and +1/2 for traditional reasons). So each orbital of the atom can hold two electrons. A single atom in an orbital is unpaired, and exhibits a local, uncancelled magnetic field. Two atoms in a single orbital will have opposite spins, and their magnetic field will cancel. As a result, an elemental substance can exhibit one of three types of magnetic response.

• Diagmagnetism, being repelled by a magnetic field. Such substances have no unpaired electrons.
• Paramagnetism, being attracted by a magnetic field. Such substances have one or more unpaired electrons.
• Ferromagnetism, having a strong magnetism of its own. Such substances have all unpaired electrons and a strong net field.

Uniqueness: The Pauli Exclusion Principle

We've already run across this: the idea that each electron in an atom has a unique set of quantum numbers, and as a result, each orbital has at most two electrons, with the same n, l, and m quantum numbers, but different spin numbers. This principle, along with the rules that l is always less than n, and m is equal to the range of -l to +l, give us enough information to predict the allowed orbitals for any atom in a ground state. By ground state, we mean the orbitals fill up from the lowest to highest energy level.

Practice with the Concepts

Determining magnetic characteristics

Which of the following is paramagentic in its ground state: He, Mg, Li, or Ar?

Magetism

Is copper magnetic?

Discussion Questions

• How does electron shell analysis help us understand energy levels for electrons?
• Are electrons always in the ground state?

Optional Readings

• The Hyperphysics Textbiij page on the Pauli Exclusion principle explains the principle in terms of the Schrödinger wave equation.
• Physics of the Universe has a less mathematical approach that discusses the ramifications of the principle for quantum mechanics and the electron configurations that govern chemical bonding.

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