Quantum Numbers

• Quantum numbers describe the properties of electrons in atoms.

Principal Quantum Number

• The principal quantum number (n) relates to the main energy level of an electron.
• n is a positive integer (n = 1, 2, 3, 4, ...).
• It is often referred to as shells or orbits.

Secondary Quantum Number

• The second quantum number (l) relates primarily to the shape of the electron's orbit.
• Arnold Sommerfeld (1915) proposed elliptical orbits to explain line-splitting in atomic spectra.
• l describes additional electron energy sublevels or sub-shells within a main energy level.
• The number of sublevels is equal to the value of the principal quantum number (n); for n=3, there are 3 sublevels (l=0, 1, 2).

Magnetic Quantum Number

• The magnetic quantum number (ml) relates primarily to the direction of the electron's orbit in space.
• The number of values for ml is the number of independent orientations of orbits that are possible.
• In a gas discharge tube placed near a strong magnet, some single lines split due to the various orientations of the orbits.
• Orbits can exist at various angles; if orbits are oriented in space in different planes, the energies of the orbits are different when an atom is near a strong orbit.
• For each value of l, ml varies from -l to +l (including 0).
  • Each non-zero value corresponds to an orientation in space.
• If l=1, ml can be -1, 0, or +1, resulting in 3 orbits with the same energy and shape but differing in orientation.
• If l=2, ml can be -2, -1, 0, +1, +2.

Spin Quantum Number

• The spin quantum number (ms) relates to a property of an atom that can be described as its spin.
• ms can only be +1/2 or -1/2 for any electron.
• It accounts for some additional spectra line splitting in a magnetic field and different kinds of magnetism.
• Ferromagnetism is a common type of magnetism, observed in substances containing Fe or Co, and involves the magnetism of a collection of atoms.
• Paramagnetism is another kind of magnetism, involving a relatively weak attraction to a strong magnet and is the magnetism of individual atoms.
• Wolfgang Pauli (1925) explained that electrons spin on their axis and can only have 2 spin states (either clockwise or counter-clockwise, or up and down).
• Two spins are equal in magnitude but opposite in direction (spin is quantized to only 2 values), with values of either +1/2 or -1/2.

Secondary Quantum Number Values

• The secondary quantum number (l) determines the possible values of the magnetic quantum number (ml) and the shape of the orbital.

• For 1 = 0, 1, 2, 3:

• l = 0, ml = 0 (1 value) - s orbital (sharp)

• l = 1, ml = -1, 0, 1 (3 values) - p orbital (principle)

• l = 2, ml = -2, -1, 0, 1, 2 (5 values) - d orbital (diffuse)

• l = 3, ml = -3, -2, -1, 0, 1, 2, 3 (7 values) - f orbital (fundamental)

• ml numbers range from -l to +l for each (2nd quantum number).

• As the l value increases, there are more ml values possible.

• Odd number values: l = 0 => 1, 3, 5, 7...

Spin Quantum Numbers

• An opposite pair of electron spins creates a stable arrangement.
• It produces no magnetism of the substance.
• For substances to produce no magnetic field, electrons must be paired (a counter-clockwise spin with a clockwise one).
• A single unpaired electron has magnetism.
• Because if two electrons are in the same space, they would repel; however, spin takes this away.
• Each electron is like a magnet and spins the opposite way.
• Four quantum numbers are needed to fully describe an electron in an atom.