The Origin of Elements
From the Big Bang to the formation of atoms - the cosmic journey of matter
This chapter lays the foundations for explaining trends in physical and chemical properties of all inorganic compounds. To understand molecules and solids, we must first understand atoms.
About 14 billion years ago, the visible universe was concentrated into a point-like region that exploded. With initial temperatures of ~10⁹ K, fundamental particles had too much kinetic energy to bind together.
As the universe cooled and expanded, particles began adhering under various forces, eventually forming nuclei and atoms.
About two hours after the Big Bang, most matter existed as:
A short-range but powerful attractive force between nucleons (protons and neutrons) that binds particles together into nuclei.
A relatively weak but long-range force between electric charges that binds electrons to nuclei to form atoms.
| Particle | Symbol | Mass/mᵤ | Mass Number | Charge/e | Spin |
|---|---|---|---|---|---|
| Electron | e⁻ | 5.486×10⁻⁴ | 0 | −1 | ½ |
| Proton | p | 1.0073 | 1 | +1 | ½ |
| Neutron | n | 1.0087 | 1 | 0 | ½ |
| Photon | γ | 0 | 0 | 0 | 1 |
| Positron | e⁺ | 5.486×10⁻⁴ | 0 | +1 | ½ |
* mᵤ = 1.6605 × 10⁻²⁷ kg (atomic mass constant), e = 1.602 × 10⁻¹⁹ C (elementary charge)
Atomic Number (Z)
The number of protons in the nucleus of an atom.
Mass Number (A)
Total number of protons and neutrons (nucleon number).
Isotopes
Atoms with same Z but different mass numbers.
Spectroscopic Information
How light reveals the secrets of atomic structure
Spectroscopic observations on hydrogen atoms suggest that an electron can occupy only certain energy levels and that the emission of discrete frequencies of electromagnetic radiation occurs when an electron makes a transition between these levels.
When electromagnetic radiation from hydrogen is passed through a prism, it separates into distinct series:
Johann Rydberg found that all wavelengths in hydrogen's spectrum can be described by:
Variables:
- R = Rydberg constant = 1.097 × 10⁷ m⁻¹
- n₁ = lower energy level (1, 2, 3, ...)
- n₂ = higher energy level (n₁+1, n₁+2, ...)
- λ = wavelength of emitted light
Spectral Series:
- • Lyman (n₁=1): Ultraviolet
- • Balmer (n₁=2): Visible
- • Paschen (n₁=3): Infrared
- • Brackett (n₁=4): Infrared
The allowed energies of a hydrogenic atom are given by:
Principles of Quantum Mechanics
The revolutionary theory that describes electron behavior in atoms
Electrons can behave as particles or as waves. Solution of the Schrödinger equation gives wavefunctions (ψ), which describe the location and properties of electrons. The probability of finding an electron is proportional to ψ².
In 1924, Louis de Broglie suggested that electrons, like photons, exhibit both wave and particle properties.
This dual nature means we cannot know both position and momentum simultaneously.
The product of uncertainty in momentum and position cannot be less than a fundamental limit:
This is a fundamental property of nature, not a limitation of measurement.
Erwin Schrödinger formulated an equation that accounts for wave-particle duality:
The wavefunction ψ contains all information about an electron. The probability of finding an electron at a location is proportional to ψ².
- High ψ² → high probability
- ψ² = 0 → node (electron not found)
- ψ² is the probability density
Physically acceptable solutions to the Schrödinger equation exist only for certain values of E.
This naturally explains why electrons can only possess discrete (quantized) energies in atoms.
✓ Constructive Interference
When wavefunctions have the same sign, they add to give enhanced amplitude.
✗ Destructive Interference
When wavefunctions have opposite signs, they cancel to give reduced amplitude.
Atomic Orbitals
The three-dimensional regions where electrons are most likely to be found
Each orbital is uniquely labeled by three quantum numbers: n (principal), l (orbital angular momentum), and mₗ (magnetic). The wavefunction of an electron in an atom is called an atomic orbital.
Specifies energy & size
Specifies shape
Specifies orientation
Spin up ↑ or down ↓
| Value of l | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Subshell | s | p | d | f | g |
| Number of orbitals (2l+1) | 1 | 3 | 5 | 7 | 9 |
| Max electrons | 2 | 6 | 10 | 14 | 18 |
The boundary surface defines the region with ~90% probability of finding the electron.
Nodes are regions where the wavefunction passes through zero (ψ = 0). An electron will not be found at a node.
Radial Nodes
Spherical surfaces where R(r) = 0
Angular Nodes (Nodal Planes)
Planes through nucleus where Y(θ,φ) = 0
| Orbital | n | l | Radial Nodes | Angular Nodes | Total Nodes |
|---|---|---|---|---|---|
| 1s | 1 | 0 | 0 | 0 | 0 |
| 2s | 2 | 0 | 1 | 0 | 1 |
| 2p | 2 | 1 | 0 | 1 | 1 |
| 3s | 3 | 0 | 2 | 0 | 2 |
| 3p | 3 | 1 | 1 | 1 | 2 |
| 3d | 3 | 2 | 0 | 2 | 2 |
Many-Electron Atoms
Penetration, shielding, and effective nuclear charge
For atoms with more than one electron, we use the orbital approximation: each electron occupies an atomic orbital resembling those in hydrogenic atoms. Due to penetration and shielding, the order of energy levels becomes s < p < d < f.
The reduction of nuclear charge experienced by an electron due to repulsion from other electrons.
σ is the shielding constant, determined empirically (Slater's rules).
The potential for an electron to be found inside shells of other electrons, experiencing more nuclear charge.
s orbitals penetrate most effectively (nonzero at nucleus). p, d, f orbitals have nodes at the nucleus.
Due to penetration and shielding effects:
s orbitals are most penetrating, f orbitals are least penetrating
The Building-Up Principle
Aufbau, Pauli Exclusion, and Hund's Rule
No more than two electrons may occupy a single orbital and, if two do occupy a single orbital, then their spins must be paired (↑↓).
Equivalently: No two electrons can have the same four quantum numbers (n, l, mₗ, mₛ).
When more than one orbital has the same energy (degenerate), electrons occupy separate orbitals with parallel spins (↑↑).
✓ Correct (Carbon 2p²)
✗ Incorrect
Orbitals are filled in order of increasing energy:
⚠️ Exceptions occur to maximize exchange energy with half-filled (d⁵) or filled (d¹⁰) subshells.
Classification of Elements
The periodic table reflects electronic structure
The layout of the periodic table reflects the electronic structure of atoms. A block indicates the type of subshell being filled. The period number equals the principal quantum number n of the valence shell.
s-Block (Groups 1-2)
Last electron enters an s orbital. Includes alkali metals and alkaline earth metals.
Valence electrons = Group number
p-Block (Groups 13-18)
Last electron enters a p orbital. Includes metalloids, nonmetals, halogens, noble gases.
Valence electrons = Group − 10
d-Block (Groups 3-12)
Transition metals where (n−1)d orbitals are being filled.
Valence shell: ns + (n−1)d
f-Block
Lanthanoids and actinoids where (n−2)f orbitals are being filled.
Shown separately below the main table.
Atomic Properties
Periodic trends in radii, ionization energy, electron affinity, and electronegativity
Metallic Radius
Half the distance between centers of neighboring atoms in a metal.
Covalent Radius
Half the internuclear distance in a molecule.
Ionic Radius
Related to cation-anion distance in ionic compounds.
Higher n, more electron shells
Increasing Zeff
Lanthanoid Contraction
Elements after the f-block have smaller radii than expected because 4f orbitals have poor shielding. This makes 3rd row d-block elements similar in size to 2nd row (e.g., Mo ≈ W).
The minimum energy needed to remove an electron from a gas-phase atom:
Higher Zeff, electrons more tightly bound
Valence electrons farther from nucleus
Notable Exceptions:
- B < Be: 2p electron is easier to remove than 2s
- O < N: Electron pairing in O causes extra repulsion
The energy change when a gaseous atom gains an electron:
Positive Ea means A⁻ is more stable than A. Elements near F (especially halogens) have highest electron affinities.
| Element | Ea (kJ/mol) | Notes |
|---|---|---|
| F | 328 | High due to small size |
| Cl | 349 | Highest! |
| O | 141 | |
| N | −8 | Half-filled p subshell stable |
| Be | ≤0 | Filled 2s subshell |
The power of an atom in a molecule to attract electrons to itself.
Pauling Scale (χP)
Based on bond energies
Mulliken Scale (χM)
χ = ½(I + Ea)
Allred-Rochow
Based on Zeff/r²
Electronegativity increases left → right and decreases top → bottom
The ability of an atom to be distorted by an electric field. Related to size and electron configuration.
Fajan's Rules
- Small, highly charged cations have polarizing ability
- Large, highly charged anions are easily polarized
- Cations without noble-gas configuration are easily polarized