Atomic Structure
Bohr model, dual nature, quantum numbers, orbitals, Aufbau, Hund, Pauli.
Bohr model
Quantized orbits, energy levels, hydrogen spectrum.
HISTORY OF ATOMIC MODELS:
- Dalton (1808): atoms are indivisible spheres.
- Thomson (1898): plum pudding model — electrons embedded in positive sphere. Discovered electron via cathode ray (e/m = 1.76 × 10¹¹ C/kg).
- Rutherford (1911): gold-foil α-scattering experiment → nucleus discovered. Most of atom is empty space; nucleus is small, dense, positive. Problem: electrons should spiral in (classical EM).
- Bohr (1913): electrons in fixed orbits with quantized energy. Worked for H atom; failed for multi-electron atoms.
- Quantum mechanical model (Schrödinger 1926): electrons described by wavefunctions; we get probability (orbitals), not exact paths. Heisenberg uncertainty: cannot know both position and momentum exactly.
SUBATOMIC PARTICLES:
| Particle | Mass | Charge | Discoverer |
|---|---|---|---|
| Electron | 9.11 × 10⁻³¹ kg | −1.6 × 10⁻¹⁹ C | Thomson |
| Proton | 1.673 × 10⁻²⁷ kg | +1.6 × 10⁻¹⁹ C | Goldstein |
| Neutron | 1.675 × 10⁻²⁷ kg | 0 | Chadwick |
Atomic number (Z) = number of protons.
Mass number (A) = protons + neutrons.
Isotopes: same Z, different A (¹H, ²H, ³H).
Isobars: same A, different Z (⁴⁰Ar, ⁴⁰K, ⁴⁰Ca).
Isotones: same number of neutrons.
BOHR MODEL (for H):
- Energy: Eₙ = −13.6 / n² eV (n=1: −13.6 eV, n=2: −3.4 eV).
- Radius: rₙ = 0.529 × n² Å.
- Velocity: vₙ ∝ 1/n.
- Hydrogen spectrum series:
- Lyman (n→1) — UV.
- Balmer (n→2) — visible.
- Paschen (n→3), Brackett (n→4), Pfund (n→5) — IR.
- Rydberg formula: 1/λ = R(1/n₁² − 1/n₂²), R = 1.097 × 10⁷ m⁻¹.
QUANTUM NUMBERS:
| QN | Symbol | Values | Significance |
|---|---|---|---|
| Principal | n | 1, 2, 3, ... | shell (energy level) |
| Azimuthal | l | 0 to n−1 | subshell (s=0, p=1, d=2, f=3); shape |
| Magnetic | m | −l to +l | orientation |
| Spin | s | +½, −½ | spin direction |
Max electrons in shell = 2n². Max electrons in subshell = 2(2l+1).
Orbital shapes:
- s — spherical (1 orbital).
- p — dumbbell (3 orbitals: px, py, pz).
- d — double dumbbell + ring (5 orbitals).
- f — complex (7 orbitals).
RULES FOR FILLING ORBITALS:
- Aufbau principle: fill lowest energy first. Order (n+l rule): 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p.
- Pauli exclusion: no two electrons in an atom have all 4 quantum numbers identical. So each orbital holds max 2 electrons (with opposite spins).
- Hund's rule: electrons occupy degenerate orbitals singly first (with parallel spins) before pairing.
EXAMPLES:
- C (Z=6): 1s² 2s² 2p² → 2px¹ 2py¹ (Hund).
- N (Z=7): 1s² 2s² 2p³ → half-filled p (extra stability).
- O (Z=8): 1s² 2s² 2p⁴.
- Cr (Z=24): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d⁵ (half-filled d gives extra stability — anomaly).
- Cu (Z=29): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d¹⁰ (fully-filled d — anomaly).
dual nature of electron (de Broglie 1924): λ = h/(mv). Confirmed by Davisson-Germer experiment.
Heisenberg uncertainty: Δx · Δp ≥ h/(4π). Cannot simultaneously know exact position and momentum.
EXAM HOOKS:
- Bohr orbit radii ratio: 1 : 4 : 9 : 16 (∝ n²).
- 3d orbital fills AFTER 4s (Aufbau).
- Cr and Cu are classic exceptions (half-filled / fully-filled stability).
- Number of electrons with l=2 in Z=24 (Cr): five (in 3d⁵).
Quantum numbers and orbitals
n, l, m_l, m_s; shapes of s, p, d orbitals.
Every electron in an atom is uniquely identified by four quantum numbers:
1. Principal (n) — the energy level / shell. n = 1, 2, 3, 4, ... corresponds to K, L, M, N shells.
2. Azimuthal / orbital angular momentum (l) — the subshell shape. l = 0, 1, 2, ..., (n−1).
- l = 0 → s orbital (spherical)
- l = 1 → p orbital (dumbbell)
- l = 2 → d orbital (cloverleaf)
- l = 3 → f orbital (complex)
3. Magnetic (m_l) — orientation of the orbital in space. m_l ranges from −l to +l, including 0. So an l=1 (p) subshell has 3 orbitals (m_l = −1, 0, +1). An l=2 (d) subshell has 5 orbitals.
4. Spin (m_s) — intrinsic angular momentum. m_s = +½ or −½.
Pauli exclusion principle: no two electrons in an atom share all four quantum numbers. Combined with the rules above, this means each orbital holds at most 2 electrons (one spin up, one spin down).
Capacity of subshells: s = 2, p = 6, d = 10, f = 14.
Total electrons in shell n = 2n². Shell K (n=1): 2. L (n=2): 8. M (n=3): 18. N (n=4): 32.
Worked example. For an electron in 3d_xy with spin up:
- n = 3, l = 2 (d), m_l ∈ {−2,−1,0,1,2} (one specific value per orbital), m_s = +½.
HISTORY OF ATOMIC MODELS:
- Dalton (1808): atoms are indivisible spheres.
- Thomson (1898): plum pudding model — electrons embedded in positive sphere. Discovered electron via cathode ray (e/m = 1.76 × 10¹¹ C/kg).
- Rutherford (1911): gold-foil α-scattering experiment → nucleus discovered. Most of atom is empty space; nucleus is small, dense, positive. Problem: electrons should spiral in (classical EM).
- Bohr (1913): electrons in fixed orbits with quantized energy. Worked for H atom; failed for multi-electron atoms.
- Quantum mechanical model (Schrödinger 1926): electrons described by wavefunctions; we get probability (orbitals), not exact paths. Heisenberg uncertainty: cannot know both position and momentum exactly.
SUBATOMIC PARTICLES:
| Particle | Mass | Charge | Discoverer |
|---|---|---|---|
| Electron | 9.11 × 10⁻³¹ kg | −1.6 × 10⁻¹⁹ C | Thomson |
| Proton | 1.673 × 10⁻²⁷ kg | +1.6 × 10⁻¹⁹ C | Goldstein |
| Neutron | 1.675 × 10⁻²⁷ kg | 0 | Chadwick |
Atomic number (Z) = number of protons.
Mass number (A) = protons + neutrons.
Isotopes: same Z, different A (¹H, ²H, ³H).
Isobars: same A, different Z (⁴⁰Ar, ⁴⁰K, ⁴⁰Ca).
Isotones: same number of neutrons.
BOHR MODEL (for H):
- Energy: Eₙ = −13.6 / n² eV (n=1: −13.6 eV, n=2: −3.4 eV).
- Radius: rₙ = 0.529 × n² Å.
- Velocity: vₙ ∝ 1/n.
- Hydrogen spectrum series:
- Lyman (n→1) — UV.
- Balmer (n→2) — visible.
- Paschen (n→3), Brackett (n→4), Pfund (n→5) — IR.
- Rydberg formula: 1/λ = R(1/n₁² − 1/n₂²), R = 1.097 × 10⁷ m⁻¹.
QUANTUM NUMBERS:
| QN | Symbol | Values | Significance |
|---|---|---|---|
| Principal | n | 1, 2, 3, ... | shell (energy level) |
| Azimuthal | l | 0 to n−1 | subshell (s=0, p=1, d=2, f=3); shape |
| Magnetic | m | −l to +l | orientation |
| Spin | s | +½, −½ | spin direction |
Max electrons in shell = 2n². Max electrons in subshell = 2(2l+1).
Orbital shapes:
- s — spherical (1 orbital).
- p — dumbbell (3 orbitals: px, py, pz).
- d — double dumbbell + ring (5 orbitals).
- f — complex (7 orbitals).
RULES FOR FILLING ORBITALS:
- Aufbau principle: fill lowest energy first. Order (n+l rule): 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p.
- Pauli exclusion: no two electrons in an atom have all 4 quantum numbers identical. So each orbital holds max 2 electrons (with opposite spins).
- Hund's rule: electrons occupy degenerate orbitals singly first (with parallel spins) before pairing.
EXAMPLES:
- C (Z=6): 1s² 2s² 2p² → 2px¹ 2py¹ (Hund).
- N (Z=7): 1s² 2s² 2p³ → half-filled p (extra stability).
- O (Z=8): 1s² 2s² 2p⁴.
- Cr (Z=24): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d⁵ (half-filled d gives extra stability — anomaly).
- Cu (Z=29): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d¹⁰ (fully-filled d — anomaly).
dual nature of electron (de Broglie 1924): λ = h/(mv). Confirmed by Davisson-Germer experiment.
Heisenberg uncertainty: Δx · Δp ≥ h/(4π). Cannot simultaneously know exact position and momentum.
EXAM HOOKS:
- Bohr orbit radii ratio: 1 : 4 : 9 : 16 (∝ n²).
- 3d orbital fills AFTER 4s (Aufbau).
- Cr and Cu are classic exceptions (half-filled / fully-filled stability).
- Number of electrons with l=2 in Z=24 (Cr): five (in 3d⁵).
Aufbau, Hund and Pauli
Electronic configuration rules; exceptions (Cr, Cu).
Three rules govern how electrons fill orbitals in a neutral atom:
1. Aufbau principle: lower-energy orbitals fill first. Order:
1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s → 5f → 6d → 7p
The (n+l) rule predicts this: lower (n+l) fills first; if equal, lower n fills first.
2. Pauli exclusion: an orbital holds at most 2 electrons, and they must have opposite spins.
3. Hund's rule: electrons fill degenerate orbitals (same energy) one at a time with parallel spins before pairing up. Why: maximum multiplicity gives lower energy via exchange interaction.
Famous exceptions — Cr and Cu (and similar in Mo, Ag, Au, Pd):
| Element | Expected | Actual |
|---|---|---|
| Cr (24) | [Ar] 3d⁴ 4s² | [Ar] 3d⁵ 4s¹ |
| Cu (29) | [Ar] 3d⁹ 4s² | [Ar] 3d¹⁰ 4s¹ |
Why? Half-filled (d⁵) and fully-filled (d¹⁰) subshells are particularly stable due to exchange energy and symmetric distribution. The atom borrows one electron from 4s to achieve this.
The Pd (46) case is even weirder: [Kr] 4d¹⁰ — no 5s electron. This is a real exam favourite.
HISTORY OF ATOMIC MODELS:
- Dalton (1808): atoms are indivisible spheres.
- Thomson (1898): plum pudding model — electrons embedded in positive sphere. Discovered electron via cathode ray (e/m = 1.76 × 10¹¹ C/kg).
- Rutherford (1911): gold-foil α-scattering experiment → nucleus discovered. Most of atom is empty space; nucleus is small, dense, positive. Problem: electrons should spiral in (classical EM).
- Bohr (1913): electrons in fixed orbits with quantized energy. Worked for H atom; failed for multi-electron atoms.
- Quantum mechanical model (Schrödinger 1926): electrons described by wavefunctions; we get probability (orbitals), not exact paths. Heisenberg uncertainty: cannot know both position and momentum exactly.
SUBATOMIC PARTICLES:
| Particle | Mass | Charge | Discoverer |
|---|---|---|---|
| Electron | 9.11 × 10⁻³¹ kg | −1.6 × 10⁻¹⁹ C | Thomson |
| Proton | 1.673 × 10⁻²⁷ kg | +1.6 × 10⁻¹⁹ C | Goldstein |
| Neutron | 1.675 × 10⁻²⁷ kg | 0 | Chadwick |
Atomic number (Z) = number of protons.
Mass number (A) = protons + neutrons.
Isotopes: same Z, different A (¹H, ²H, ³H).
Isobars: same A, different Z (⁴⁰Ar, ⁴⁰K, ⁴⁰Ca).
Isotones: same number of neutrons.
BOHR MODEL (for H):
- Energy: Eₙ = −13.6 / n² eV (n=1: −13.6 eV, n=2: −3.4 eV).
- Radius: rₙ = 0.529 × n² Å.
- Velocity: vₙ ∝ 1/n.
- Hydrogen spectrum series:
- Lyman (n→1) — UV.
- Balmer (n→2) — visible.
- Paschen (n→3), Brackett (n→4), Pfund (n→5) — IR.
- Rydberg formula: 1/λ = R(1/n₁² − 1/n₂²), R = 1.097 × 10⁷ m⁻¹.
QUANTUM NUMBERS:
| QN | Symbol | Values | Significance |
|---|---|---|---|
| Principal | n | 1, 2, 3, ... | shell (energy level) |
| Azimuthal | l | 0 to n−1 | subshell (s=0, p=1, d=2, f=3); shape |
| Magnetic | m | −l to +l | orientation |
| Spin | s | +½, −½ | spin direction |
Max electrons in shell = 2n². Max electrons in subshell = 2(2l+1).
Orbital shapes:
- s — spherical (1 orbital).
- p — dumbbell (3 orbitals: px, py, pz).
- d — double dumbbell + ring (5 orbitals).
- f — complex (7 orbitals).
RULES FOR FILLING ORBITALS:
- Aufbau principle: fill lowest energy first. Order (n+l rule): 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p.
- Pauli exclusion: no two electrons in an atom have all 4 quantum numbers identical. So each orbital holds max 2 electrons (with opposite spins).
- Hund's rule: electrons occupy degenerate orbitals singly first (with parallel spins) before pairing.
EXAMPLES:
- C (Z=6): 1s² 2s² 2p² → 2px¹ 2py¹ (Hund).
- N (Z=7): 1s² 2s² 2p³ → half-filled p (extra stability).
- O (Z=8): 1s² 2s² 2p⁴.
- Cr (Z=24): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d⁵ (half-filled d gives extra stability — anomaly).
- Cu (Z=29): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d¹⁰ (fully-filled d — anomaly).
dual nature of electron (de Broglie 1924): λ = h/(mv). Confirmed by Davisson-Germer experiment.
Heisenberg uncertainty: Δx · Δp ≥ h/(4π). Cannot simultaneously know exact position and momentum.
EXAM HOOKS:
- Bohr orbit radii ratio: 1 : 4 : 9 : 16 (∝ n²).
- 3d orbital fills AFTER 4s (Aufbau).
- Cr and Cu are classic exceptions (half-filled / fully-filled stability).
- Number of electrons with l=2 in Z=24 (Cr): five (in 3d⁵).