Quantum Mechanics

Spectroscopy

The study of how matter absorbs, emits, or scatters light

Spectra

The range of wavelengths or frequencies of electromagnetic radiation emitted, absorbed, or scattered by a substance/substances

Light

A form of electromagnetic radiation that behaves like a wave

Wave

A vibrating disturbance that carries energy from one place to another

Electromagnetic Wave

• Have a magnetic field, which vibrates perpendicular to the electric field

  • These two fields work in harmony to create the electromagnetic waves that make up light, radio waves, X-rays, and more

Photoelectric Effect

Occurs when light, or more specifically photons, strikes a material and ejects electrons from its surface

Three Parameters of EM radiation

Wavelength, Frequency, and Amplitude

Wavelength

A distance between two consecutive peaks or troughs in a wave

Frequency

Number of waves that pass a given point in space in a second

Amplitude

The height of a wave crest or depth of a trough

• Reciprocal/inverse relationship between wavelength and frequency

Diffraction

Wave bends when encountering obstacles or passing through small openings

Refraction

Bending of light as it moves from one medium to another

Constructive Interference

Two waves overlap and combine to make a stronger wave

Destructive Interference

Two waves overlap and cancel each other

Problem with the Rutherford Model

• Rutherfords model views atoms as being centred around the nucleus

  • However, charged particles move in an electric fielder known to emit EM radiation

Bohr Model

1. The electron can only occupy certain allowed stationary states, with fixed energies

2. The electron does not lose energy (in the form of radiation) while in a stationary state

3. The electron undergoes a transition from one stationary state to another only by absorbing or emitting photons

Ground state

n=1

Excited state

n >1, starts at n =2

When is an electron free from the nucleus

n = infinity

Electronic Transitions

The energy of the photon equals the difference in the energies of the two states

Absorption

Excitation from a lower allowed energy level to a higher allowed energy level

Emission

Relaxation from a higher allowed energy level to a lower allowed energy level

Key Concepts of the Bohr Model

• Energy is quantized

  • Light can exhibit wave and particle properties

Limitations of the Bohr Model

Does not accurately predict the emission spectra of atoms or ions with more than one electron

de Broglie

Matter is wavelike

• wavelength = Plank’s constant/(massXvelocity)

Heisenberg Uncertainty Principle

It is impossible to measure the exact position and momentum of a particle at the same time

• delta x times delta p >_ h/4pi

Schrodinger Theory of Quantum/Wave Mechanics

Describes particles as waves

• Standing waves provide physical basis for quantization for energy levels

Quantization

Electrons can only exist at specific discrete energy levels

Bohr Equation for the energy levels of a Hydrogen-like Atom

E =2.718-2.178×10^-18J(Z²/n²)

c = Speed of light, what is c?

2.998×10^-9 m

Formula for delta E of a photon

(h times c) / nm

Nodes

Points with zero movement

• Form standing waves

Antinodes

Points with maximum movement

Orbital

Mathematical function that defines the probability distribution of an electron within an atom

n

Principal Quantum Number tells us how far the electron is from the nucleus and how much energy it has

• Larger n means further distance, larger orbital size, and more energy

l

Angular Momentum Quantum/Azimuthal Quantum number tells us the shape of the orbital where an electron is found

• Determines number of angular nodes in an orbital

Types of nodes

• s-orbital (l=0, angular nodes = 0)

• p-orbital (l=1, angular nodes = 1)

• d-orbital (l=2, angular nodes = 2)

• f-orbital (l=3, angular nodes = 3)

Radial/Spherical Nodes

• Occur at certain distances from the nucleus where the probability of finding an electron is zero

  • Found in s, p, d, and f orbitals

Angular Nodes

• Flat planes or cones where the electron probability is zero

  • p orbitals always have one angular node

Magnetic Quantum Number

• Specifies the orientation of an orbital in space relative to an external magnetic field

• Determines how an orbital is positioned around the nucleus

Magnetic Quantum Number relation to l

1. The possible ranges of MQN range from -l to l, including 0

2. Formula: MQN = -l,…,0,….+1

   1. This means each sub level (s, p, d, f) contains multiple orbitals

How Magnetic Quantum Number Determines the Number of Orbitals in a Sub-level

1. Each orbital within a sub level has a unique MQN value

2. The number of orbitals in a sub level is 2l + 1

MQN values for s, p, d, and f sub-levels

• s: l=0, range=0, 1 orbital, n = 1

• p: l=1, range= -1 to 1, 3 orbitals, n = 2

• d: l=2, range= -2 to 2, 5 orbitals, n = 3

• f: l=3, range= -3 to 3, 7 orbitals, n = 4

S orbital shape and rate

• Spherical

• One s-orbital for each value of n

P orbital shape and rate

• Two identical lobes

• 3 p-orbitals for each value of n (except n=1)

D orbital shape and rate

• Two angular nodes, which divide into four lobes for each n greater than or equal to 3

F orbital shape and rate

• Three angular modes and can exhibit 4, 6, or 8 lobes

Electron Spin Quantum Number (ms)

• Intrinsic characteristic

• Two possible values:

  • +1/2 (spin up) and -1/2 (spin down)

Kinetic Energy of Electrons

Electrons move around the nucleus, contributing kinetic energy

Potential Energy of Attraction

The nucleus attracts the electrons, generating electrostatic potential energy

Potential Energy of Electron-Electron Repulsion

Unlike hydrogen, helium has two electrons that repel each other, making the system more complex

• Schrodinger equation cannot be solved exactly for helium due to electron-electron repulsion

Effective Nuclear Charge Formula

Zeff = Z - S

• Z = actual nuclear charge (number of protons in nucleus)

• S = shielding constant

Shielding Effect

Inner electrons repel outer electrons, so outer electrons feel a weaker effective nuclear charge than the full charge of the nucleus and are easier to remove

Penetration Effect

Some orbitals allow electrons to get closer to the nucleus than others, experiencing less shielding and a stronger effective nuclear charge

• s orbitals penetrate more

• p, d, and f penetrate less, meaning they are more shielded

Aufbau Principle

• As protons are added to the nucleus to form new elements, electrons are also added to atomic orbitals

  • To minimize total electron energy, electrons always fill the lowest-energy orbitals first before occupying higher-energy orbitals

Pauli Exclusion Principle

Each orbital can hold a maximum of two electrons with opposite spins

Hund’s Rule

When filling degenerate orbitals, each orbital receives one electron first before any electrons paired

• Minimizes repulsion and stabilizes atom

Electron-Electron Repulsion Orbital relation

• Repulsions are stronger when two electrons occupy the same orbital even if they are degenerate

• Repulsions are weaker when they occupy different orbitals

D-block

Transition metals have full 4s orbitals, so electrons enter at 3d

(n-1)d Orbitals

Period 4-7 all have the same number of d orbitals as their principal quantum number subtracted by 1

Penetration and Shielding effect in d-block

• Low penetration

• Greater shielding

• Initial high energy that drops rapidly

Aufbau exception - Chromium

• Energy gap between 4s and 3d orbitals narrows

  • Cr promotes one electron from 4s orbital to 3d orbital

Aufbau exception - Copper

• Between nickel and copper, energy of 3d orbitals drops below that of the 4s orbital due to increasing effective nuclear charge and poor shielding by 4-electrons

  • Most stable electron configuration for Cu is the one that maximizes electron occupancy in the now lower-energy 3d orbital

Post-Transition Elements

• (n-1)d orbitals have much lower energy than the ns and np orbital so they are not considered part of the valence shell

• (n-1)d orbitals should be listed before the ns orbitals