QUANTUM MECHANICS

Erwin Schrödinger developed equations to describe the behavior of electrons in complicated atoms.

Werner Heisenberg proposed the uncertainty principle (Heisenberg’s Uncertainty Principle), which stated that it was not possible to accurately know both the position and momentum (p = m x v) of a small particle like an electron because in measuring it with light you change the position and momentum of the particle.

This leaves us with the probability of finding an electron.

Schrödinger’s wave equation is used to determine the probability of finding an electron in any given place. Four principal quantum numbers are used in describing electron behavior.

Each electron within an atom can be described using the four quantum numbers: n, l, m, and s.

n

  • the “principal quantum number”
  • corresponds to the energy level or shell
  • n = 1, 2, 3, 4, … (any positive integer)
  • Electrons may be found in each energy level of an atom. The greatest number of electrons in any one energy level is 2n2. For example, energy level #2 can hold 8 electrons.

The energy of the energy level increases as the number of the energy level increases.

l

  • l is the number for a sublevel of energy level n. Not all electrons in the same energy level have the same energy. An energy level is actually made up of many energy states called sublevels.
  • Defines the shape
  • “angular quantum number”
  • l = 0, 1, 2, … (n-1)
  • l = 0 sublevel = s (holds 2e-)

= 1 = p (holds 6e-)

= 2 = d (holds 10e-)

= 3 = f (holds 14e-)

As the value for l increases, the energy of the sublevel increases.

m

  • “magnetic quantum number”
  • m = 0, +/-1, +/-2, +/-3, … +/-l
  • represents the number of an orbital within a sublevel

The Shapes of the Orbitals

s orbital – is spherical with the greatest electron density near the nucleus.

p orbital – is dumbbell shaped with a nodal plane through the center.

d orbital – complicated shape, 2 nodal planes

f orbital – complicated shapes, 3 nodal planes

s

  • the electron spin quantum number
  • s = + ½ (clockwise spin) or – ½ (counterclockwise spin)

In 1925, Wolfgang Pauli fit quantum theory with experiment. The result was the Pauli Exclusion Principle.

The Pauli Exclusion Principle states that no two electrons have the same set of 4 quantum numbers. This means that no orbital can have more than 2 electrons, and in an orbital with 2 electrons the spins must be paired (one cw and one ccw).

Example #1

What values of l are permitted for an electron with n =3?

Example #2

What are the permitted values for m in an electron which has l = 4?