The atom consists of three fundamental particles:

Key Definitions

• Atomic number ($Z$): Number of protons (defines the element)
• Mass number ($A$): Total protons + neutrons
• Isotopes: Atoms of same element with different numbers of neutrons

Notation: $^A_ZX$ where $X$ is element symbol

Example: $^{12}_6C$ (Carbon-12) has 6 protons, 6 neutrons, 6 electrons

Quantum Numbers

Four quantum numbers describe each electron in an atom:

1. Principal Quantum Number ($n$)

• Values: $n = 1, 2, 3, 4, ...$
• Meaning: Energy level (shell)
• Capacity: Maximum $2n^2$ electrons per level

2. Angular Momentum Quantum Number ($\ell$)

• Values: $\ell = 0$ to $n-1$
• Meaning: Sublevel (subshell) shape
  • $\ell = 0$: s orbital (spherical)
  • $\ell = 1$: p orbital (dumbbell)
  • $\ell = 2$: d orbital (cloverleaf)
  • $\ell = 3$: f orbital (complex)

3. Magnetic Quantum Number ($m_\ell$)

• Values: $-\ell$ to $+\ell$ (including 0)
• Meaning: Orbital orientation in space
• Number of orbitals: $2\ell + 1$

4. Spin Quantum Number ($m_s$)

• Values: $+\frac{1}{2}$ or $-\frac{1}{2}$
• Meaning: Electron spin direction
• Pauli Exclusion Principle: No two electrons can have the same four quantum numbers

Orbital Capacity

Electron Configuration

Electron configuration shows how electrons are distributed among orbitals.

Order of Filling (Aufbau Principle)

Electrons fill orbitals in order of increasing energy:

$1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p$

Memory aid: Use the diagonal rule or periodic table

Notation

Full configuration: List all sublevels with number of electrons

Example: Oxygen (8 electrons) $1s^2 2s^2 2p^4$

Noble gas configuration: Use previous noble gas in brackets

Example: Calcium (20 electrons) $[Ar] 4s^2$

Orbital diagram: Show each orbital as a box with electron spins

Three Key Rules

1. Aufbau Principle

Electrons fill lowest energy orbitals first.

2. Pauli Exclusion Principle

Maximum 2 electrons per orbital, with opposite spins.

3. Hund's Rule

When filling orbitals of equal energy (like three p orbitals), place one electron in each orbital before pairing.

Example: Nitrogen (7 electrons)

2p orbitals: ↑ ↑ ↑ (one electron in each)

NOT: ↑↓ ↑ (incorrect - violates Hund's rule)

Electron Configurations of Ions

Cations (positive ions)

Remove electrons from highest $n$ value first (usually outermost s)

Example: Fe → Fe²⁺

• Fe: $[Ar] 4s^2 3d^6$
• Fe²⁺: $[Ar] 3d^6$ (remove 4s² electrons)

Anions (negative ions)

Add electrons following normal filling order

Example: O → O²⁻

• O: $1s^2 2s^2 2p^4$
• O²⁻: $1s^2 2s^2 2p^6$ (add 2 electrons to 2p)

Exceptions to Filling Order

Some elements have anomalous electron configurations for extra stability:

Chromium (Cr):

• Expected: $[Ar] 4s^2 3d^4$
• Actual: $[Ar] 4s^1 3d^5$ (half-filled d sublevel is more stable)

Copper (Cu):

• Expected: $[Ar] 4s^2 3d^9$
• Actual: $[Ar] 4s^1 3d^{10}$ (filled d sublevel is more stable)

Why: Half-filled and fully-filled sublevels have extra stability.

Valence Electrons

Valence electrons are electrons in the outermost shell (highest $n$).

They determine:

• Chemical properties
• Bonding behavior
• Reactivity

Example: Nitrogen $[He] 2s^2 2p^3$

• Valence electrons: 5 (in $n=2$ shell)

Core vs. Valence Notation

Core electrons: Inner electrons (represented by noble gas) Valence electrons: Outer electrons (written out)

Example: Phosphorus (P)

• Full: $1s^2 2s^2 2p^6 3s^2 3p^3$
• Noble gas: $[Ne] 3s^2 3p^3$
• Core: [Ne] (10 electrons)
• Valence: $3s^2 3p^3$ (5 electrons)

(d) Subshell being filled:

• Transition metals fill the d subshell (specifically the (n-1)d subshell)

Step 2: Draw orbital diagram

1s: ↑↓

2s: ↑↓

2p: ↑ ↑ _ (three orbitals: px, py, pz)

Step 3: Apply Hund's Rule

For the 2 electrons in 2p:

• Place one electron in first orbital (↑)
• Place one electron in second orbital (↑)
• Leave third orbital empty

Complete diagram:

      ↑↓        ↑↓      ↑  ↑  _
      1s        2s      2p

Answer:

• Orbital diagram shown above
• Unpaired electrons: 2 (both in 2p sublevel)

Explanation:

• Hund's rule requires maximizing unpaired electrons in degenerate orbitals
• Carbon's 2 unpaired electrons make it chemically reactive
• These unpaired electrons form bonds in compounds like CO₂, CH₄

Verification:

• Total electrons: 2 + 2 + 2 = 6 ✓
• Hund's rule followed ✓

• When forming ions, electrons are removed from the highest n value first (4s before 3d)
• Even though Cu fills 3d last, the 4s¹ electron is removed first, then one 3d electron
• This is why Cu²⁺ is [Ar] 3d⁹, not [Ar] 4s¹ 3d⁸

Expanded: $[Ar] = 1s^2 2s^2 2p^6 3s^2 3p^6$

Step 2: Form Fe²⁺ (remove 2 electrons)

Key concept: When forming cations, remove electrons from the highest $n$ value first.

For Fe: Remove from 4s before 3d

$\text{Fe}^{2+}: [Ar] 3d^6$

NOT $[Ar] 4s^2 3d^4$ (incorrect!)

Step 3: Explain the difference

Neutral Fe: $[Ar] 4s^2 3d^6$ (26 electrons)

• 4s sublevel: 2 electrons
• 3d sublevel: 6 electrons

Fe²⁺: $[Ar] 3d^6$ (24 electrons)

• 4s sublevel: 0 electrons (both removed)
• 3d sublevel: 6 electrons (unchanged)

Why remove 4s first?

Even though 4s fills before 3d, once the 3d sublevel begins filling, it becomes lower in energy than 4s. When ionizing, electrons are removed from the highest energy orbital, which is 4s.

Answer:

• Fe: $[Ar] 4s^2 3d^6$
• Fe²⁺: $[Ar] 3d^6$
• Difference: Lost both 4s electrons

Verification:

• Fe has 26 electrons ✓
• Fe²⁺ has 24 electrons (26 - 2) ✓
• 4s electrons removed before 3d ✓