Lecture 18: Polyelectronic Atoms

• Reading: Zumdahl 12.10-12.13

• Outline:– Spin– The Aufbau Principle– Filling up orbitals and the Periodic Table

H-atom wavefunctions• The Coulombic potential can be generalized:

e-

P+r

V (r) Ze2

r Z

• Z = atomic number (= 1 for hydrogen)

H-atom wavefunctions• If we solve the Schrodinger equation using this potential, we find that the energy levels are quantized:

En Z 2

n2

me4

802h2

2.178x10 18J

Z 2

n2

• n is the principle quantum number, and ranges from 1 to infinity.

Orbitals

Quantum Numbers and Orbitals

n l Orbital ml # of Orb.

1 0 1s 0 12 0 2s 0 1

1 2p -1, 0, 1 33 0 3s 0 1 1 3p -1, 0, 1 3

2 3d -2, -1, 0, 1, 2 5

Spin

• Further experiments demonstrated the need for one more quantum number.

• Specifically, some particles (electrons in particular) demonstrated inherent angular momentum.

Spin (cont.)

• The new quantum number is ms (analagous to ml).

• For the electron, ms has two values:

+1/2 and -1/2

ms = 1/2

ms = -1/2

The Aufbau Principal• For polyelectronic atoms, a direct solution of the

Schrodinger Eq. is not possible.

• When we construct polyelectronic atoms, we use the hydrogen-atom orbital nomenclature to discuss in which orbitals the electrons reside.

• This is an approximation (and it is surprising how well it actually works).

The Aufbau Principal (cont.)• When placing electrons into orbitals in the

construction of polyelectronic atoms, we use the Aufbau Principle.

• This principle states that in addition to adding protons and neutrons to the nucleus, one simply adds electrons to the hydrogen-like atomic orbitals

• Pauli exclusion principle: No two electrons may have the same quantum numbers. Therefore, only two electrons can reside in an orbital (differentiated by ms).

The Aufbau Principal (cont.)• Finally, orbitals are filled starting from the lowest

energy.

• Example: Hydrogen

1s 2s 2p

• Example: Helium (Z = 2)

1s 2s 2p

1s1

1s2

The Aufbau Principal (cont.)• Orbital configurations …

The Aufbau Principal (cont.)• Lithium (Z = 3)

1s 2s 2p

1s 2s 2p

• Berillium (Z = 4)

• Boron (Z = 5)

1s 2s 2p

1s22s1

1s22s2

1s22s22p1

The Aufbau Principal (cont.)• Carbon (Z = 6)

1s 2s 2p

1s 2s 2p

• Nitrogen (Z = 7)

Hund’s Rule: Lowest energy configuration is the one in which the maximum number of unpaired electronsare distributed amongst a set of degenerate orbitals.

1s22s22p2

1s22s22p3

The Aufbau Principal (cont.)• Oxygen (Z = 8)

1s 2s 2p

1s 2s 2p

• Fluorine (Z = 9)

1s22s22p4

1s22s22p5

1s 2s 2p

• Neon (Z = 10)

1s22s22p6

full

The Aufbau Principal (cont.)• Sodium (Z = 11)

3s

1s22s22p63s1

3s 3p

• Argon (Z = 18)

[Ne] 3s23p6

Ne [Ne]3s1

Ne

The Aufbau Principal (cont.)• We now have the orbital configurations for the first 18 elements.

• Elements in same column have the same # of valence electrons!

The Aufbau Principal (cont.)

The Aufbau Principal (cont.)• What is the radial distribution for different orbitals?

The Aufbau Principal (cont.)• Similar to Sodium, we begin the next row of the periodic table by adding electrons to the 4s orbital.

• Why not 3d before 4s?

• 3d is closer to the nucleus

• 4s allows for closer approach; therefore, is energetically preferred.

The Aufbau Principal (cont.)• Elements Z=19 and Z= 20:

Z= 19, Potassium: 1s22s22p63s23p64s1 = [Ar]4s1

Z= 20, Calcuim: 1s22s22p63s23p64s2 = [Ar]4s2

• Elements Z=21to Z=30 have occupied d orbitals:

Z= 21, Scandium: 1s22s22p63s23p64s23d1 = [Ar] 4s23d1

Z= 30, Zinc: 1s22s22p63s23p64s23d10 = [Ar] 4s23d10

Z = 24, Chromium: [Ar] 4s13d5 exception

The Aufbau Principal (cont.)

• This orbital filling scheme gives rise to the modern periodic table.

The Aufbau Principal (cont.)

• After Lanthanum ([Xe]6s25d1), we start filling 4f.

The Aufbau Principal (cont.)

• After Actinium ([Rn]7s26d1), we start filling 5f.

The Aufbau Principal (cont.)

• Heading on column given total number of valence electrons.

The Aufbau Principal (cont.)

Summary

• Electrons go into hydrogen-like orbitals to construct polyelectronic atoms.