# 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 both the position x and the momentum p = mv of a particle

(Δx)(Δp) $\ge$ $\frac{\mathrm{h}}{4\mathrm{\pi }}$

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 energy

The electron density gives the probability of finding an electron in a particular region of an atom. An atomic orbital is the region of 3D space, defined by Ψ2, where the probability of finding an electron is high. An atomic orbital can accommodate a maximum of 2 electrons

## 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, the larger the orbital
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 (spherically symmetric)
l = 1 ⇒ p orbital (cylindrically symmetric about 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 value of the 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 ordering of the orbital energies is: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s ... You can easily remember this order using the mnemonic device opposite:

Electron configuration:

The arrangement of electrons in the atomic orbitals of an atom. An atomic orbital can accommodate a maximum of 2 electrons:

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 electron configuration:

• Electrons will reside in the available orbitals of the lowest possible energy
• Each orbital can accommodate a maximum of 2 electrons
• Orbitals will fill 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:

The process by which the periodic table can be built up 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 any can contain 2 electrons

Electronic structure of carbon (Z=6):

And not:

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

## Excited States

Ground state: the lowest energy state of an atom

Excited state: a state that is higher in energy than the ground state. The 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 in 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 in inner energy levels
Valence electrons: electrons in 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