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, energies of atomic orbitals and electron configuration, Pauli exclusion principle, Hund’s rule, excited states, core vs. valence electrons

Quantum Theory

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

(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:

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

Quantum Numbers

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)

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 (ml):

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

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 one specific orientation (px or py or pz) different than those of the orbitals = 1, ml = 0 and 1

Electron Spin

Spin quantum number (ms):

2 electrons that occupy the same atomic orbital have different electron spin quantum numbers, either - $\frac{1}{2}$ or + $\frac{1}{2}$. We can differentiate the spin of an electron by representing an arrow pointing down (spin down = - $\frac{1}{2}$) or up (spin up = + $\frac{1}{2}$). 1s orbital with 2 electrons is represented by:

Arrangement of Electrons

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

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 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:

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

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 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)

Excited States

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 (1s2s1) + hν → Li* (1s2p1)

Core vs. Valence Electrons

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: 1s2s2p4
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): 1s2s2p3s3p6 4s3d6 = [Ar] 4s3d6

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:

• 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

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

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

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

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

• The core electrons are the electrons of the inner energy levels. They do not participate in chemical bonding and form the atomic core with the nucleus
• The valence electrons are the 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