# 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, the difference between shells, subshells, and orbitals, the shapes and energies of atomic orbitals, Pauli exclusion principle, Aufbau principle, Hund’s rule, electron configurations, ground state vs excited states.

## Quantum Theory

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:

(Δx)(Δp)

Δx = uncertainty in measuring the position
Δp = uncertainty in measuring the momentum
h = Planck constant = 6.63 x 10-34 kg.m2.s-1

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.

## Quantum Numbers

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 (ml). While these quantum numbers are sufficient to describe an atomic orbital, an additional quantum number, the electron spin quantum number (ms), 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 (ml):

It describes the orientation of the orbital in space.

• Values: ml = -l, -l + 1, … , -1, 0, 1, … , - 1, l
• Significance:
s orbital: l = 0 ⇒ m= 0 ⇒ 1 possible orientation
p orbital: l = 1 ⇒ m= -1, 0, 1 ⇒ 3 different orientations: px, py, and pz
d orbital: l = 2 ⇒ m= -2, -1, 0, 1, 2 ⇒ 5 different orientations: dxy, dyz, dxz, dx²-y², and d

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

Spin quantum number (ms):

It describes the intrinsic spin of an electron within an orbital.

• Values: ms $\frac{1}{2}$ (spin up) or - $\frac{1}{2}$ (spin down)
• Significance: Differentiates between the spins of electrons in the same orbital by representing an arrow pointing down (spin down = - $\frac{1}{2}$) or up (spin up = + $\frac{1}{2}$).

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

## Shells, Subshells, and Orbitals

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 ⇒ = 0 ⇒ 1 subshell (1s subshell)
• Second shell: n = 2 ⇒ = 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.

## Atomic Orbitals

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 px, py, and pz. 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 dxy, dxz, dyz, dx2-y2, and dz2.

Energies of atomic orbitals:

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 order of the orbital energies is: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s ... You can easily remember this order by using the mnemonic on the right:

## Electronic Structure Principles

Pauli exclusion principle:

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

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 Configurations

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: 1s2s2p4
Iron (Z=26) ⇒ 26 protons ⇒ 26 electrons: 1s2s2p3s3p4s3d6

## Ground State and Excited States

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 (1s2s1) + hν → Li* (1s2p1)

The uncertainty principle states that 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

(Δx)(Δp)

Δx = uncertainty in measuring the position
Δp = uncertainty in measuring the momentum
h = Planck constant = 6.63 x 10-34 kg.m2.s-1

The Schrödinger equation is the central equation of quantum theory consistent with both the wave nature of particles and the Heisenberg uncertainty principle. This equation provides wave functions Ψ of the electron’s position associated with allowed energies

An atomic orbital is 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

An atomic orbital is defined by 3 quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), and the magnetic quantum number (ml). The electrons that occupy an atomic orbital are defined by their spin quantum number (ms)

• The principal quantum number (n) describes the size of the orbital. The larger n is, the larger the orbital is
• The azimuthal quantum number (l) describes the shape of the atomic orbital
• The magnetic quantum number (ml) describes the orientation of the orbital in space
• The spin quantum number (ms) describes the spin of an electron in an atomic orbital, either - $\frac{1}{2}$ or +
• Principal quantum number: n = 1, 2, 3, ...
• Azimuthal quantum number: l = 0, 1, 2, … - 1
• Magnetic quantum number: m= -l, -l + 1, … , -1, 0, 1, … , - 1, l
• Spin quantum number: ms = $\frac{1}{2}$ or +

The azimuthal quantum number (l) describes the subshells and thus the shape of the corresponding atomic orbital:

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

The electron shell is a group of atomic orbitals with the same principal quantum number n (n = 1 ⇒ first shell) while the electron subshell is a group of atomic orbitals with the same principal quantum number and the same azimuthal quantum number l

An atomic orbital can hold a maximum of 2 electrons. Thus:

• 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

The order of orbital energies is: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s ... You can easily remember this order by using the mnemonic on the right:

The Pauli exclusion principle states that 2 electrons in an atom cannot have the same set of 4 quantum numbers (n, l, ml, ms)

The Aufbau principle states that 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

The Hund’s rule states that orbitals of equal energy, called degenerate orbitals, must all contain one electron with the same spin before they can contain 2 electrons

When we assign electrons to orbitals, we must follow a set of three rules: the Pauli exclusion principle, the Aufbau principle, and the Hund's rule

• 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

The ground state is the lowest energy state of an atom, while an excited state is 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