Quantum Theory and Atomic Structure | General Chemistry 1

Heisenberg uncertainty principle:

According to this principle, it is impossible to simultaneously measure the position (x) and the momentum (p = mv) of a particle with absolute precision. The principle states that the product of the uncertainties in measuring the position (Δx) and momentum (Δp) of a particle is always greater than or equal to Planck's constant divided by 4π:
 

The Schrödinger equation:

This is the fundamental equation of quantum mechanics, describing how the quantum state of a physical system changes over time. It provides wave functions (Ψ) that represent the position of an electron associated with allowed energies.
 

Atomic orbital:

An atomic orbital refers to the wave function (Ψ) of an electron within an atom. Each atomic orbital corresponds to a specific energy level and electron density distribution. The electron density, representing the relative probability of finding an electron at a given point in space, is described by the square of the wave function (Ψ²) associated with the orbital.

An atomic orbital is defined by 3 quantum numbers: the principal quantum number (n), the angular momentum quantum number (l), and the magnetic quantum number (m_l). While these quantum numbers are sufficient to describe an atomic orbital, an additional quantum number, the electron spin quantum number (m_s), becomes necessary to describe an electron occupying the orbital.

Principal quantum number (n):

It represents the size and energy of an atomic orbital.

• Values: n = 1, 2, 3, ...
• Significance: Larger n values correspond to larger orbitals. n = 1 represents the first shell, also known as the ground state.

Angular momentum quantum number (l):

It describes the shape of the atomic orbital.

• Values: l = 0, 1, 2, … n-1
• Significance:
  l = 0 ⇒ s orbitals (spherical symmetry)
  l = 1 ⇒ p orbitals (cylindrical symmetry around their long axis)
  l = 2 ⇒ d orbitals
  l = 3 ⇒ f orbitals

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):

It describes the orientation of the orbital in space.

• Values: m_l = -l, -l + 1, … , -1, 0, 1, … , l - 1, l
• Significance:
  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 a specific orientation (p_x or p_y or p_z) different from those of orbitals l = 1, m_l = 0 and 1.

Spin quantum number (m_s):

2 electrons that occupy the same atomic orbital have different electron spin quantum numbers 

Shells:

Shells refer to the energy levels in which the electrons are located around the nucleus of an atom. They are characterized by the principal quantum number (n), which determines the energy of the shell and the distance of the electrons from the nucleus.

• Shells are organized concentrically around the nucleus and have a specific energy associated with them.
• Electrons can occupy different shells, with higher energy shells farther from the nucleus.

Subshells:

Subshells refer to a set of orbitals with the same n and l values. They are characterized by their angular momentum quantum number (l), which determines their shape and orientation in space.

• Each subshell corresponds to a specific letter designation: 's', 'p', 'd', 'f', and so on, representing different shapes of electron clouds.
• The number of subshells within a shell is equal to the principal quantum number.
• The maximum number of electrons that can occupy a subshell is 2 x (2l + 1), following the Pauli exclusion principle.

Number of subshells in the first 2 electron shells:

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

Orbitals:

Orbitals are regions within a subshell where electrons are likely to be found. Orbitals are characterized by their shape, size, and orientation.

• Each orbital can hold a maximum of 2 electrons with opposite spins.
• The number of orbitals in a subshell is 2l + 1.

Number of orbitals in a subshell:

For 's' subshells there is 1 orbital, for 'p' subshells there are 3 orbitals, for 'd' subshells there are 5 orbitals, and so on.

Types of atomic orbitals:

• s orbital:

Spherical in shape, centered around the nucleus. There is an s subshell in each shell, and each s subshell contains only one s orbital, designated as 1s, 2s, 3s, and so on. The size of the s orbitals increases as the principal quantum number increases.
 

• p Orbital:

Dumbbell shaped, consisting of two lobes with a node at the center. There are 3 p orbitals per energy level, oriented along the x, y, and z axes, designated as p_x, p_y, and p_z. These 3 p orbitals are identical in size, shape, and energy. Like s orbitals, p orbitals increase in size from 2p to 3p to 4p orbital and so on.
 

• d Orbital:

Complex in shape, with four lobes and additional angular nodes. There are 5 d orbitals per energy level, designated as d_xy, d_xz, d_yz, d_x²-y², and d_z².
 

Energies of atomic orbitals:

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 first occupy the lowest energy orbitals before filling higher energy orbitals. This principle, which comes from the German word "Aufbau," allows for the systematic filling of atomic orbitals and contributes to the construction of the periodic table.

Hund's rule:

Degenerate orbitals (orbitals with the same energy) must be singly occupied with parallel spins before they can contain 2 electrons.
 

Electronic structure of carbon (Z=6):

And not:

A paramagnetic substance is a substance with unpaired electrons (weakly attracted by a magnetic field), while a diamagnetic substance is a substance without unpaired electrons (not attracted by a magnetic field).

Electron configuration:

Electron configuration refers to the arrangement of electrons in the atomic orbitals of an atom. Each atomic orbital can hold a maximum of 2 electrons, following the Pauli exclusion principle. 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 fill the lowest energy orbitals available before filling higher energy ones (Aufbau principle).
• Each orbital can accommodate a maximum of 2 electrons (Pauli exclusion principle).
• Electrons will not pair in degenerate orbitals if an empty orbital is available (Hund's rule).
• The orbitals are filled in the order of the orbital energy.

Electron configuration of oxygen and iron:

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

Ground state:

The ground state is the lowest energy state of an atom in which the electrons occupy the lowest energy orbitals available to them.
 

Excited states: 

An excited state is any state of an atom in which electrons have absorbed energy and are therefore at higher energy levels than in the ground state. This absorption of energy typically occurs through the absorption of electromagnetic radiation of a specific frequency (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)