# Quantum Theory and Atomic Structure | General Chemistry 1

Quantum theory and atomic structure are studied in this chapter: Heisenberg uncertainty principle, Schrödinger equation, quantum numbers, electron spin, atomic energy states and electron configuration, Pauli exclusion principle, Hund’s rule, excited states, periodicity of atomic radii, ionization energy and electron affinity.

## Quantum Theory

Heisenberg uncertainty principle:

It is not possible to measure both the position and the momentum mv of a particle simultaneously.

(Δx)(Δp) $\ge$ $\frac{\mathrm{h}}{4\mathrm{\pi }}$

h = Planck constant
Δx = uncertainty in the position
Δp = uncertainty in the momentum

Schrödinger equation:

central equation of quantum theory ⇒ consistent with both the wave nature of particles and the Heisenberg uncertainty principle.
The Schrödinger equation provides wave functions Ψ of the position of the electron associated with allowed energy. Ψ2 = probability density = probability to find an electron in this region.

Atomic orbitals = 3D quantities = 3 quantum numbers n, l and ml.
n = principal quantum number = 1, 2, 3 …
l = azimuthal quantum number = 0, 1, 2 … n-1
ml = magnetic quantum number = -l, -l + 1, … , -1, 0, 1, … , - 1, l

n = effective size of an orbital
n = 1 ⇒ first shell = lowest energy allowed = ground state
n = 2 ⇒ second shell

## Azimuthal Quantum Number

l = azimuthal quantum number = 0, 1, 2 … - 1
⇒ shape of an orbital

l = 0 ⇒ s orbital (spherically symmetric)
l = 1 ⇒ p orbital (cylindrically symmetric about its long axis)
l = 2 ⇒ d orbital
l = 3 ⇒ f orbital
then alphabetical order from f: l = 4 ⇒ g orbital …

n = 2 and l = 1 ⇒ orbital 2p

n = 3 and l = 0 ⇒ orbital 3s

## Magnetic Quantum Number

ml  = magnetic quantum number = -l, -l + 1, … , -1, 0, 1, … , - 1, l
spatial orientation of an orbital

p orbitals: l = 1 ⇒ ml = -1, 0, 1 ⇒ 3 p orbitals: same shape but different orientation (px, py, pz)
d orbitals: l = 2 ⇒ ml = -2, -1, 0, 1, 2 ⇒ 5 d orbitals: same shape but different orientation (dxy, dyz, dxz, dx²-y², d)

Caution: l = 1, ml = -1 does not mean px but p orbital with one specific orientation (px or py or pz) different than those of the orbitals l = 1, ml = 0 and 1.

## Electron Spin

An electron has an intrinsic spin
⇒ fourth quantum number: spin quantum number m(= +1/2 or -1/2)

We can differentiate the spin of an e- by representing an arrow pointing up (spin up = + 1/2) or down (spin down = - 1/2).

1s orbital with 2 electrons is representing by:

## Atomic Energy States

The energy states of atoms with 2 or more electrons depend on the values of both n and l (electrons-nucleus + electron-electron interactions).

The ordering of the orbital energies is: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s ...
You can easily remember this order with to the mnemonic device opposite:

## Pauli Exclusion Principle

Pauli Exclusion Principle:

2 electrons in the same atom cannot have the same set of 4 quantum numbers (n, l, ml, ms).

n = 1 ⇒ only 2 possible combinations: (1, 0, 0, +1/2) and (1, 0, 0, -1/2) ⇒ maximum 2 e- in 1s orbital

shell = level designated by n (n = 1 ⇒ first shell)
subshell = group of orbitals designated by different l

n = 1 ⇒ l = 0 ⇒ 1 subshell = s subshell
n = 2 ⇒ l = 0, 1 ⇒ 2 subshells: s subshell + p subshell

## Electron Configuration

Electron configuration = occupancy of electrons in atomic orbitals

s orbital = 2 electrons (2 x 1)
p orbitals = 6 electrons (2 x 3)
d orbitals = 10 electrons (2 x 5)
f orbitals = 14 electrons (2 x 7)

Electron configuration of:
- oxygen (Z=8): 1s22s22p4
- iron (Z=26): 1s22s22p63s23p64s23d6

Core electrons = electrons in inner (lower n) energy levels
Valence electrons = electrons in the outermost occupied shell (highest n shell)

Electron configuration of oxygen (Z = 8): 1s22s22p4
Outermost occupied shell: n = 2 ⇒ 2 e- in 2s, 4 e- in 2p
⇒ 6 valence electrons

For the electron configuration, we can use an abbreviated form by using [previous nearest noble gas] for the valence electrons

Iron (Z=26): 1s22s22p63s23p64s23d6 = [Ar] 4s23d6

Elements in the same period (same column of the periodic table) = similar valence-electron configurations = same behavior in a chemical reaction.

## Hund’s Rule

The ground state electron configuration is obtained by:
- placing one electron in each orbital of the subshell with the highest energy
- giving these electrons parallel spin

Orbitals of carbon (Z=6):

And not:

paramagnetic = substances with unpaired electrons (attracted to a magnetic field)
diamagnetic = substances without unpaired electrons (not attracted to a magnetic field)

## Excited States

Electromagnetic radiation: electrons are promoted to excited states (see chapter 4)
First excited state = promoting the electron of highest energy in the ground state to the next available orbital

Li (1s22s1) + hν → Li* (1s22p1)

## Atomic Radii, Ionization Energies and Periodicity

Properties of atoms are dependent on the valence electrons.

distance from the nucleus to the outermost electrons
- the higher the effective nuclear charge Zeff, the more attracted the outer electrons are to the nucleus and the closer they are to the nucleus. Z increases across a period ⇒ Zeff increases ⇒ atomic radius decreases across a period
- n ⇒ size of an orbital (as n increases, the orbital gets bigger) ⇒ atomic radius increases across a group

Ionization Energy (IE):

energy necessary to remove an electron from the electron cloud
- the closer an electron is to the nucleus, the more difficult it is to remove
- the greater the Zeff, the more tightly an electron is held to an atom ⇒ IE increases across the period, decreases across the group
- each additional electron is more difficult to remove (less e--e- repulsion)
- core electrons are very attracted by the nucleus ⇒ high IE ⇒ significant jump in ionization energies occurs after the outer most electrons are removed

Electron Affinity (EA):

amount of energy released when an electron is attached to a neutral atom or molecule in the gaseous state