Quantum Theory and Atomic Structure | General Chemistry 1

Heisenberg uncertainty principle:

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

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 m_l.
n = principal quantum number = 1, 2, 3 …
l = azimuthal quantum number = 0, 1, 2 … n-1
m_l = magnetic quantum number = -l, -l + 1, … , -1, 0, 1, … , l - 1, l

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

l = azimuthal quantum number = 0, 1, 2 … n - 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

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

p orbitals: l = 1 ⇒ m_l = -1, 0, 1 ⇒ 3 p orbitals: same shape but different orientation (p_x, p_y, p_z)
d orbitals: l = 2 ⇒ m_l = -2, -1, 0, 1, 2 ⇒ 5 d orbitals: same shape but different orientation (d_xy, d_yz, d_xz, d_x²-y², d_z²)

Caution: l = 1, m_l = -1 does not mean p_x but p orbital with one specific orientation (p_x or p_y or p_z) different than those of the orbitals l = 1, m_l = 0 and 1.

An electron has an intrinsic spin
⇒ fourth quantum number: spin quantum number m_s (= +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).

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:

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

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 = 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.

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)

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)

Properties of atoms are dependent on the valence electrons.

Atomic radius:

distance from the nucleus to the outermost electrons
- the higher the effective nuclear charge Z_eff, the more attracted the outer electrons are to the nucleus and the closer they are to the nucleus. Z increases across a period ⇒ Z_eff 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 Z_eff, 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