Coordination number, voids (tetrahedral, octahedral)

2.4.14 · Chemistry › States of Matter (Quantitative)

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1. Coordination Number (CN)

WHAT decide karta hai: packing arrangement, element nahi.

Structure	CN	Kyun
Simple cubic	6	6 face-direction ke neighbours
Body-centred cubic (bcc)	8	body-centre 8 corners ko touch karta hai
Hexagonal / cubic close packed (hcp / ccp/fcc)	12	equal spheres ke liye sabse dense possible

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2. Voids — baaki bache holes

Jab aap ek close-packed layer (A) rakhte hain aur agli layer (B) ko uske dimples mein daalte hain, do tarah ke holes aate hain.

HOW MANY voids per atom (golden numbers)

Derivation-from-scratch (fcc unit cell), WHY ye numbers:

Ek fcc (ccp) unit cell lo.

Atoms ki sankhya $N$ fcc mein:

• 8 corners × $\frac{1}{8}$ = 1
• 6 faces × $\frac{1}{2}$ = 3 $N=1+3=4$

Octahedral voids:

• 1 body centre par (poora andar) = 1
• 12 edge centres par, har ek 4 cells share karta hai → $12\times \frac{1}{4}=3$ $\text{Oct}=1+3=4=N✓$ Yeh step kyun? Sirf yahi positions 6 atoms se equidistant hain.

Tetrahedral voids:

• Cube ko 8 chhote cubes mein todо; har chhote cube ka centre ek tetrahedral void hai, sab cell ke andar. $\text{Tet}=8=2N✓$ Yeh step kyun? Har small-cube centre ke aas-paas 4 nearest fcc atoms hote hain jo ek tetrahedron banate hain.

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3. Void SIZE — radius ratio (derived, memorised nahi)

Hum chahte hain radius $r$ ka sabse bada sphere jo bade spheres of radius $R$ ke banaye hole mein snugly fit ho, unhe touch karte hue.

3a. Octahedral void

Us square ke face diagonal ke saath, spheres touch karte hain:

• Bade spheres square ke edge ke saath touch karte hain: edge $=2R$.
• Chhota sphere centre mein; diagonal ke saath hame $R+2r+R$ milta hai.

$2R$ side ke square ka diagonal $2R\sqrt{2}$ hai. Toh: $2R+2r=2R\sqrt{2}$ $r+R=R\sqrt{2}⟹r=R(\sqrt{2}-1)$ $\boxed{\frac{r}{R}=\sqrt{2}-1\approx 0.414}$

3b. Tetrahedral void

Maano cube edge $=a$.

• Bade spheres face diagonal ke paas touch karte hain: $a\sqrt{2}=2R\Rightarrow R=\frac{a\sqrt{2}}{2}$.
• Chhota + bada body diagonal ke saath: $a\sqrt{3}=2R+2r$.

Doosre ko $2R$ se divide karo: $\frac{a\sqrt{3}}{2R}=1+\frac{r}{R}$ $2R=a\sqrt{2}$ substitute karo: $\frac{a\sqrt{3}}{a\sqrt{2}}=1+\frac{r}{R}⟹\sqrt{\frac{3}{2}}=1+\frac{r}{R}$ $\boxed{\frac{r}{R}=\sqrt{\frac{3}{2}}-1\approx 0.225}$

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4. Worked examples

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5. Common mistakes (Steel-man + fix)

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Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek box mein oranges tightly packed hain. Har orange snugly 12 doosron ko touch karta hai — yeh uska "coordination number" hai. Oranges ke beech chhote khaali pockets hote hain. Chhote pockets 4 oranges se ghire hote hain (tetrahedral) aur bade pockets 6 oranges se (octahedral). Agar aap in pockets mein tiny marbles daalte ho, toh bilkul aise hi tiny ions bade ions ke crystal ke andar chhupta hain! Har orange ke liye 1 bada pocket aur 2 chhote pocket hote hain.

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Flashcards

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Connections

• Close packing in solids (hcp, ccp, bcc)
• Packing efficiency and unit cell dimensions
• Radius ratio rule and ionic structures
• NaCl, ZnS, CaF2 crystal structures
• Density of a unit cell
• States of Matter (Quantitative)

Concept Map