📊 Periodic Properties & Periodic Trends

The Periodic Table is the master map of electronic configurations and atomic properties. Advanced questions test anomalous ionization enthalpy inversions, lanthanide contraction radius matches, electron affinity exceptions ($Cl>F,S>O$), and Slater effective nuclear charges.

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1. 🧲 Effective Nuclear Charge ($Z_{\text{eff}}$) & Slater's Rules

The valence electrons do not feel the full nuclear charge $Z$ due to electrostatic repulsion from inner shielding electrons:

$$Z_{\text{eff}}=Z-\sigma$$

Slater's Shielding Calculation Rules:

1. For an electron in an $ns$ or $np$ orbital:
   • Electrons in the same $(ns,np)$ group: Contribute $0.35$ each ($0.30$ for $1s$).
   • Electrons in the $(n-1)$ shell: Contribute $0.85$ each.
   • Electrons in the $(n-2)$ and deeper shells: Contribute $1.00$ each.
2. For an electron in an $nd$ or $nf$ orbital:
   • Electrons in the same $(nd)$ or $(nf)$ group: Contribute $0.35$ each.
   • All electrons in all groups to the left: Contribute $1.00$ each.

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2. 📏 Atomic & Ionic Radii

• Across a Period (Left to Right): $Z_{\text{eff}}$ increases $⟹$ Atomic radius decreases.
• Down a Group (Top to Bottom): Principal quantum number $n$ increases $⟹$ Atomic radius increases.

2.1 Critical Radius Anomalies

1. Lanthanide Contraction: Poor shielding by $14$ filled $4f$ electrons ($4f<4d<4p<4s$) causes $Z_{\text{eff}}$ to rise sharply.
   • Result: Radii of $4d$ and $5d$ transition elements are almost identical:$$\mathbf{Zr}\mathbf{(}\mathbf{4}\mathbf{d}\mathbf{)}\approx \mathbf{Hf}\mathbf{(}\mathbf{5}\mathbf{d}\mathbf{)}\approx \mathbf{1.60}\text{AA},\mathbf{Nb}\mathbf{(}\mathbf{4}\mathbf{d}\mathbf{)}\approx \mathbf{Ta}\mathbf{(}\mathbf{5}\mathbf{d}\mathbf{)},\mathbf{Mo}\mathbf{(}\mathbf{4}\mathbf{d}\mathbf{)}\approx \mathbf{W}\mathbf{(}\mathbf{5}\mathbf{d}\mathbf{)}$$
2. Transition Metal Contraction in Group 13:
   • Radii order: $B<Ga<Al<In<Tl$.
   • $Ga<Al$: Due to poor shielding by $10$ filled $3d$ electrons in Gallium ($r_{Ga}\approx 1.35\text{AA}<r_{Al}\approx 1.43\text{AA}$).
3. Isoelectronic Species Radius Rule:$$\mathbf{r}\propto \frac{\mathbf{1}}{\mathbf{Z}\mathbf{/}\mathbf{e}}⟹N^{3-}>O^{2-}>F^{-}>N{a}^{+}>M{g}^{2+}>A{l}^{3+}$$

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3. ⚡ Ionization Enthalpy (IE) Anomalies

$$\text{General Trend: Increases across period, decreases down group}$$

3.1 Crucial Period 2 & 3 Anomalies (Subshell & Exchange Stability)

1. $Be>B$ ($I{E}_1$):
   • $Be$ ($1s^{2}2s^2$): Electron removed from fully-filled stable $2s$ subshell.
   • $B$ ($1s^{2}2s^{2}2p^1$): Electron removed from higher energy $2p$ subshell ($I{E}_1(Be)=899\text{ kJ/mol}>I{E}_1(B)=801\text{ kJ/mol}$).
2. $N>O$ ($I{E}_1$):
   • $N$ ($1s^{2}2s^{2}2p_x^{1}2p_y^{1}2p_z^1$): Extra stability of exactly half-filled $2p^3$ subshell with 3 parallel exchangeable electrons.
   • $O$ ($1s^{2}2s^{2}2p^4$): Inter-electronic pairing repulsion in $2p_x^2$ makes electron removal easier ($I{E}_1(N)=1402\text{ kJ/mol}>I{E}_1(O)=1314\text{ kJ/mol}$).
3. Group 13 Ionization Enthalpy Inversion:$$\mathbf{B}\mathbf{>}\mathbf{Tl}\mathbf{>}\mathbf{Ga}\mathbf{>}\mathbf{Al}\mathbf{>}\mathbf{In}$$
   • $Tl$ has high IE due to relativistic effects and poor shielding by both $4f$ and $5d$ electrons.

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4. 🧲 Electron Gain Enthalpy (${\mathrm{\Delta }}_{eg}H$) Exceptions

$${\mathrm{\Delta }}_{eg}H\text{ becomes more negative across a period, and less negative down a group}$$

The 2nd Period vs 3rd Period Anomaly

Small atomic size and high inter-electronic charge density in compact $2p$ subshells cause incoming electrons to experience strong repulsion:

$$\text{Group 17 (Halogens): }\mathbf{Cl}\mathbf{>}\mathbf{F}\mathbf{>}\mathbf{Br}\mathbf{>}\mathbf{I}$$$${\mathrm{\Delta }}_{eg}H:Cl(-349\text{ kJ/mol})>F(-328\text{ kJ/mol})>Br(-325\text{ kJ/mol})>I(-295\text{ kJ/mol})$$$$\text{Group 16 (Chalcogens): }\mathbf{S}\mathbf{>}\mathbf{Se}\mathbf{>}\mathbf{Te}\mathbf{>}\mathbf{Po}\mathbf{>}\mathbf{O}$$

(Oxygen has the least negative electron gain enthalpy in its entire group due to extreme electron-electron crowding!)