Quantum Theory and Atomic Structure | General Chemistry 1

Heisenberg uncertainty principle:

It is impossible to measure simultaneously the position x and the momentum p = mv of a particle. The more accurately we know one of these values, the less accurately we know the other
 

The Schrödinger equation:

The 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 energies
 

Atomic orbital:

The wave function Ψ of an electron in an atom. An atomic orbital has a characteristic energy as well as a characteristic electron density distribution. The electron density is the relative probability of finding an electron at a particular point in space. For one-electron systems, the electron density of an electron in a certain orbital is described by the square of the wave function Ψ² associated with that orbital

An atomic orbital is defined by 3 quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), and the magnetic quantum number (m_l)

Principal quantum number (n):

n = 1, 2, 3, ...
It describes the size of the orbital. The larger n is, the larger the orbital is
n = 1 ⇒ first shell = lowest energy allowed = ground state

Azimuthal quantum number (l):

l = 0, 1, 2, … n-1
It describes the shape of the atomic orbital:

l = 0 ⇒ s orbital (spherical symmetry)
l = 1 ⇒ p orbital (cylindrical symmetry around its long axis)
l = 2 ⇒ d orbital
l = 3 ⇒ f orbital
 

Atomic orbitals are characterized by sets of quantum numbers:

n = 2, l = 1 ⇒ orbital 2p
n = 3, l = 0 ⇒ orbital 3s

Magnetic quantum number (m_l):

m_l = -l, -l + 1, … , -1, 0, 1, … , l - 1, l
It describes the orientation of the orbital in space

s orbital: l = 0 ⇒ m_l = 0 ⇒ 1 possible orientation
p orbital: l = 1 ⇒ m_l = -1, 0, 1 ⇒ 3 different orientations: p_x, p_y, and p_z
d orbital: l = 2 ⇒ m_l = -2, -1, 0, 1, 2 ⇒ 5 different orientations: d_xy, d_yz, d_xz, d_x²-y², and 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

Spin quantum number (m_s):

Shell vs. subshell

Electron shell: a group of atomic orbitals with the same principal quantum number n (n = 1 ⇒ first shell)
Electron subshell: a group of atomic orbitals with the same principal quantum number n and azimuthal quantum number l
 

Number of subshells in the first 2 electron shells:

First shell: n = 1 ⇒ l = 0 ⇒ 1 subshell (s subshell)
Second shell: n = 2 ⇒ l = 0, 1 ⇒ 2 subshells (s subshell and p subshell)

Energies of atomic orbitals

Electron configuration:

The arrangement of electrons in the atomic orbitals of an atom. An atomic orbital can hold a maximum of 2 electrons. The maximum number of electrons in each subshell is as follows:

• s subshell: 1 s orbital ⇒ 2 electrons
• p subshell: 3 p orbitals ⇒ 6 electrons
• d subshell: 5 d orbitals ⇒ 10 electrons
• f subshell: 7 f orbitals ⇒ 14 electrons

How to write the electron configuration:

• Electrons reside in the lowest energy orbitals available
• Each orbital can accommodate a maximum of 2 electrons
• The orbitals are filled in the order of the orbital energy

Electron configuration of oxygen and iron:

Oxygen (Z=8) ⇒ 8 protons + neutral ⇒ 8 electrons: 1s² 2s² 2p⁴
Iron (Z=26) ⇒ 26 protons + neutral ⇒ 26 electrons: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶

Pauli exclusion principle:

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

First shell: n = 1 ⇒ only 2 possible combinations: (1, 0, 0, +1/2) and (1, 0, 0, -1/2)
This explains why there is a maximum of 2 electrons in the 1s orbital

Aufbau principle:

Electrons fill subshells of the lowest available energy before filling subshells of higher energy. This principle, coming from Aufbau in Geman which means “building-up”, allows to build up the periodic table by successively adding one proton to the nucleus and one electron to the appropriate atomic orbital

Hund's rule:

Orbitals of equal energy, called degenerate orbitals, must all contain one electron with the same spin before they can contain 2 electrons
 

Electronic structure of carbon (Z=6):

And not:

Paramagnetic substance: a substance with unpaired electrons (weakly attracted by a magnetic field)
Diamagnetic substance: a substance without unpaired electrons (not attracted by a magnetic field) 

Ground state: the lowest energy state of an atom

Excited state: a state with higher energy than the ground state. Electrons are promoted from the ground state to an excited state by electromagnetic radiation of intensity hν. The first excited state corresponds to the promotion of the highest energy electron from the ground state to the next available orbital
 

First excited state of the lithium atom:

Li (1s² 2s¹) + hν → Li* (1s² 2p¹)

Core electrons: electrons of the inner energy levels. They do not participate in chemical bonding and form the atomic core with the nucleus
Valence electrons: electrons of the outermost occupied shell of an atom. They are furthest from the positive charge of the nucleus and therefore tend to react more easily than the core electrons
 

Number of valence electrons in an oxygen atom:

Electron configuration of oxygen: 1s² 2s² 2p⁴
Outermost occupied shell: n = 2. There are 2 e^- in 2s, 4 e^- in 2p ⇒ 6 valence electrons

For the electron configuration, we can use an abbreviated form by replacing the electronic configuration of the core electrons by [previous nearest noble gas]
 

Iron (Z = 26): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶ = [Ar] 4s² 3d⁶