Periodic Table & Periodicity
Advanced University‑Level Study Notes (MIT‑style)
Subject: Chemistry
Program: INSPIRE Scholarship
Topic: Periodic Table & Periodicity
1. Introduction
- The Periodic Law – “Elements with the same properties recur at regular intervals when arranged by increasing atomic number.”
- Historical Milestones – Mendeleev (1869) → modern quantum‑mechanical understanding (1920s–1930s).
- Why it matters – Predicts electronic structure, reactivity, bonding, and physical properties of every known element and the ones yet to be discovered.
Goal of the notes: Derive periodicity from first‐principles quantum mechanics, explain trends, discuss edge cases, and provide a challenging problem to test deep understanding.
2. Quantum Mechanical Foundations
2.1 The Atomic Hamiltonian
For a nucleus of charge (+Ze) and (N) electrons:
[ \hat{H} = \sum_{i=1}^{N}\left(-\frac{\hbar^2}{2m_e}\nabla_i^2 - \frac{Ze^2}{4\pi\varepsilon_0 r_i}\right)+ \sum_{i<j}\frac{e^2}{4\pi\varepsilon_0 r_{ij}} ]
- First term: kinetic energy of electrons.
- Second term: Coulomb attraction to nucleus.
- Third term: electron–electron repulsion – the source of many‑body complexity.
2.2 Separation of Variables & Quantum Numbers
Solving the single‑electron Schrödinger equation for the Coulomb potential gives four quantum numbers:
| Symbol | Meaning | Allowed values |
|---|---|---|
| (n) | Principal quantum number | (1,2,3,\dots) |
| (l) | Orbital angular momentum | (0\le l \le n-1) |
| (m_l) | Magnetic quantum number | (-l,\dots,l) |
| (m_s) | Spin | (\pm \tfrac12) |
Energy depends only on (n) in the hydrogenic case:
[ E_n = -\frac{Z^2}{n^2},R_\infty ]
with (R_\infty = 13.6,\text{eV}).
2.3 Many‑Electron Atom – The Aufbau Principle
- Electron filling: Lowest energy orbitals fill first.
- Pauli Exclusion Principle: No two electrons share identical quantum numbers.
- Hund’s Rule: Maximize spin multiplicity within a subshell.
This gives the Aufbau diagram:
[ 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p ]
The filling order is not a strict rule; relativistic and exchange effects cause occasional deviations (e.g., (\mathrm{Cu}), (\mathrm{Zn})).
3. Derivation of Periodicity
3.1 Effective Nuclear Charge ((Z_{\text{eff}}))
Electrons experience an attraction to the nucleus reduced by shielding from inner electrons. Slater’s rules provide a semi‑empirical estimate:
[ Z_{\text{eff}} = Z - S ]
[ S = \sum_{\text{shielding electrons}} s_i ]
where (s_i) depends on the orbital of the shielding electron.
Key Insight: Elements in the same period have the same principal quantum number (n) and thus similar electron shells, but varying (Z_{\text{eff}}) due to increasing nuclear charge. This drives trends.
3.2 Periodic Trends as a Function of (Z_{\text{eff}})
| Property | Trend across a period | Underlying quantum‑mechanical cause | |----------|