Quantum Theory and Atomic Structure | General Chemistry 1
Quantum Theory
Heisenberg uncertainty principle:
It is not possible to measure both the position and the momentum mv of a particle simultaneously.
(Δx)(Δp)
h = Planck constant
Δx = uncertainty in the position
Δp = uncertainty in the momentum
Schrödinger equation:
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. Ψ2 = probability density = probability to find an electron in this region.
Atomic orbitals = 3D quantities = 3 quantum numbers n, l and ml.
n = principal quantum number = 1, 2, 3 …
l = azimuthal quantum number = 0, 1, 2 … n-1
ml = magnetic quantum number = -l, -l + 1, … , -1, 0, 1, … , l - 1, l
n = effective size of an orbital
n = 1 ⇒ first shell = lowest energy allowed = ground state
n = 2 ⇒ second shell
Azimuthal Quantum Number
l = azimuthal quantum number = 0, 1, 2 … n - 1
⇒ shape of an 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
then alphabetical order from f: l = 4 ⇒ g orbital …
n = 2 and l = 1 ⇒ orbital 2p
n = 3 and l = 0 ⇒ orbital 3s
Magnetic Quantum Number
ml = magnetic quantum number = -l, -l + 1, … , -1, 0, 1, … , l - 1, l
⇒ spatial orientation of an orbital
p orbitals: l = 1 ⇒ ml = -1, 0, 1 ⇒ 3 p orbitals: same shape but different orientation (px, py, pz)
d orbitals: l = 2 ⇒ ml = -2, -1, 0, 1, 2 ⇒ 5 d orbitals: same shape but different orientation (dxy, dyz, dxz, dx²-y², dz²)
Caution: l = 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 l = 1, ml = 0 and 1.
Electron Spin
An electron has an intrinsic spin
⇒ fourth quantum number: spin quantum number ms (= +1/2 or -1/2)
We can differentiate the spin of an e- by representing an arrow pointing up (spin up = + 1/2) or down (spin down = - 1/2).
1s orbital with 2 electrons is representing by:
Atomic Energy States
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 with to the mnemonic device opposite:
Pauli Exclusion Principle
Pauli Exclusion Principle:
2 electrons in the same atom cannot have the same set of 4 quantum numbers (n, l, ml, ms).
n = 1 ⇒ only 2 possible combinations: (1, 0, 0, +1/2) and (1, 0, 0, -1/2) ⇒ maximum 2 e- in 1s orbital
shell = level designated by n (n = 1 ⇒ first shell)
subshell = group of orbitals designated by different l
n = 1 ⇒ l = 0 ⇒ 1 subshell = s subshell
n = 2 ⇒ l = 0, 1 ⇒ 2 subshells: s subshell + p subshell
Electron Configuration
Electron configuration = occupancy of electrons in atomic orbitals
s orbital = 2 electrons (2 x 1)
p orbitals = 6 electrons (2 x 3)
d orbitals = 10 electrons (2 x 5)
f orbitals = 14 electrons (2 x 7)
Electron configuration of:
- oxygen (Z=8): 1s22s22p4
- iron (Z=26): 1s22s22p63s23p64s23d6
Core electrons = electrons in inner (lower n) energy levels
Valence electrons = electrons in the outermost occupied shell (highest n shell)
Electron configuration of oxygen (Z = 8): 1s22s22p4
Outermost occupied shell: n = 2 ⇒ 2 e- in 2s, 4 e- in 2p
⇒ 6 valence electrons
For the electron configuration, we can use an abbreviated form by using [previous nearest noble gas] for the valence electrons
Iron (Z=26): 1s22s22p63s23p64s23d6 = [Ar] 4s23d6
Elements in the same period (same column of the periodic table) = similar valence-electron configurations = same behavior in a chemical reaction.
Hund’s Rule
The ground state electron configuration is obtained by:
- placing one electron in each orbital of the subshell with the highest energy
- giving these electrons parallel spin
Orbitals of carbon (Z=6):
And not:
paramagnetic = substances with unpaired electrons (attracted to a magnetic field)
diamagnetic = substances without unpaired electrons (not attracted to a magnetic field)
Excited States
Electromagnetic radiation: electrons are promoted to excited states (see chapter 4)
First excited state = promoting the electron of highest energy in the ground state to the next available orbital
Li (1s22s1) + hν → Li* (1s22p1)
Atomic Radii, Ionization Energies and Periodicity
Properties of atoms are dependent on the valence electrons.
Atomic radius:
distance from the nucleus to the outermost electrons
- the higher the effective nuclear charge Zeff, the more attracted the outer electrons are to the nucleus and the closer they are to the nucleus. Z increases across a period ⇒ Zeff increases ⇒ atomic radius decreases across a period
- n ⇒ size of an orbital (as n increases, the orbital gets bigger) ⇒ atomic radius increases across a group
Ionization Energy (IE):
energy necessary to remove an electron from the electron cloud
- the closer an electron is to the nucleus, the more difficult it is to remove
- the greater the Zeff, the more tightly an electron is held to an atom ⇒ IE increases across the period, decreases across the group
- each additional electron is more difficult to remove (less e--e- repulsion)
- core electrons are very attracted by the nucleus ⇒ high IE ⇒ significant jump in ionization energies occurs after the outer most electrons are removed
Electron Affinity (EA):
amount of energy released when an electron is attached to a neutral atom or molecule in the gaseous state