Atomic Structure – KVPY (Historic)
Physics – Chapter 2 (for the 2025–26 exam)
Goal: Understand how the modern picture of the atom is built, from early models to the quantum‑mechanical description that explains its chemistry and spectroscopy.
1. Simple Explanation
An atom consists of a tiny, positively charged nucleus (protons + neutrons) surrounded by a cloud of negatively charged electrons.
The electrons occupy quantised energy levels (or shells) that can be described mathematically by wavefunctions.
The structure of these levels, and how electrons are distributed among them, determines all chemical and physical properties of an element.
2. Key Points
| Topic | What to remember |
|---|---|
| Historical Evolution | 1. Dalton’s solid sphere (1803) 2. Thomson’s “plum‑pudding” (1904) 3. Rutherford’s nuclear model (1911) 4. Bohr’s quantised orbits (1913) 5. Quantum mechanics & Schrödinger equation (1920s) |
| Quantum Numbers | Four numbers specify an electron’s state: • (n) – principal (radial) number, (n=1,2,...) • (\ell) – azimuthal (orbital angular momentum), (0\le \ell \le n-1) • (m_\ell) – magnetic quantum number, (-\ell \le m_\ell \le +\ell) • (m_s) – spin, (m_s=\pm \tfrac12) |
| Pauli Exclusion Principle | No two electrons in an atom can have the same set of four quantum numbers. |
| Hund’s Rule | For a given subshell, electrons fill separate orbitals singly (with parallel spins) before pairing. |
| Electron Configuration | Write electrons in order of increasing energy: 1s² 2s² 2p⁶ … (Aufbau principle). |
| Orbital Shapes & Angular Momentum | s (ℓ=0) – spherical; p (ℓ=1) – dumbbell; d (ℓ=2) – cloverleaf; f (ℓ=3) – complex. |
| Periodic Trends | 1. Atomic radius decreases across a period, increases down a group. 2. Ionisation energy increases across a period, decreases down a group. 3. Electron affinity generally increases across a period. |
| Spectroscopy | Transitions between energy levels produce spectral lines (Balmer series, Lyman series, etc.). |
| Spin–Orbit Coupling & Fine Structure | Interaction between spin and orbital angular momentum gives small energy splittings. |
| Relativistic Effects | For heavy elements (high Z), electrons near the nucleus move fast; corrections (Dirac equation) become important. |
3. Worked Example
Problem:
Write the electron configuration of the element with atomic number 30 (Z = 30), and determine the valence electrons and the ground‑state term symbol for the outermost electrons.
Step 1 – Fill Orbitals (Aufbau)
| n | ℓ | Orbital | Capacity | Electrons |
|---|---|---|---|---|
| 1 | 0 | 1s | 2 | 2 |
| 2 | 0 | 2s | 2 | 2 |
| 2 | 1 | 2p | 6 | 6 |
| 3 | 0 | 3s | 2 | 2 |
| 3 | 1 | 3p | 6 | 6 |
| 4 | 0 | 4s | 2 | 2 |
| 3 | 2 | 3d | 10 | 2 |
Adding up: (2+2+6+2+6+2+2 = 22).
So far we have 22 electrons. Next 8 electrons go into the 3d subshell.
| Orbital | Electrons |
|---|---|
| 3d | 8 |
Resulting configuration
[ \mathbf{1s^2,2s^2,2p^6,3s^2,3p^6,4s^2,3d^8} ]
Valence Electrons
The valence shell is the outermost filled shell: 4s.
So, 2 valence electrons (the two in the 4s² orbital).
(3d electrons are “semi‑valence” – they can participate in bonding for transition metals.)
Ground‑state Term Symbol (for 3d⁸)
For a transition‑metal ion with d⁸ configuration, the ground state term is (^3F).
Derivation sketch (for students):
- Total (L) (orbital angular momentum) for d‑electrons: choose the maximum possible (L) consistent with Pauli.
- Total (S) (spin): with 8 electrons, you can have 4 unpaired spins → (S = 2) → multiplicity (2S+1 = 5).
- But due to electron–electron repulsion, the lowest energy term is usually the one with highest multiplicity but lowest (L) among those.
- For d⁸, the ground term is (^3F) (multiplicity 3, (L=3)).
Note: Detailed derivation uses Hund’s rules and the spectroscopic notation (^{2S+1}L).
4. Quick Revision Summary
- Atomic nucleus: protons + neutrons.
- Electron shells: (n=1,2,3…) (energy increases).
- Quantum numbers: (n, \ell, m_\ell, m_s).
- Pauli Exclusion Principle: 4 numbers must differ.
- Hund’s Rule: maximize unpaired electrons in a subshell.
- Aufbau principle: fill orbitals in order of increasing energy.
- Periodic trends: radius ↓ across period,