Ionic crystals — NaCl, CsCl, ZnS, fluorite, antifluorite structures

2.4.15 · Chemistry › States of Matter (Quantitative)

YE STRUCTURES EXIST HI KYUN KARTE HAIN

Radius Ratio Rule — scratch se derive kiya hua

HUM limiting ratio kaise derive karte hain (octahedral ke liye kaam kiya, CN = 6):

Ek cation ko 4 anions ke square ke centre mein socho (ek octahedral hole ka slice). Limiting (minimum) case mein, anions bas ek dusre ko touch karte hain AUR cation bas har anion ko touch karta hai.

Yeh step kyun? Geometric limit exactly tab hoti hai jab dono contacts simultaneously hote hain — iske neeche cation "rattle" karta hai, iske upar anions alag ho jaate hain.

Square ki edge ke along: do anions touch karte hain → edge length $= 2 r_{-}$ .
Diagonal ke along: anion–cation–anion → diagonal $= 2 r_{-} + 2 r_{+}$ .

Ek square ke liye, diagonal $= 2 \times$ edge: $2 r_{-} + 2 r_{+} = 2 (2 r_{-})$

Yeh step kyun? Square face par Pythagoras geometry ko exactly fix karta hai.

$2 r_{-}$ se divide karo: $1 + \frac{r _{+}}{r _{-}} = 2 \Rightarrow \frac{r _{+}}{r _{-}} = 2 - 1 = 0.414$

Toh octahedral (CN 6) ke liye $ρ \geq 0.414$ chahiye.

Formula Radius ratio → coordination number

$ρ = r_{+} / r_{-}$	CN	Geometry	Example
$0.155 - 0.225$	3	trigonal planar	—
$0.225 - 0.414$	4	tetrahedral	ZnS
$0.414 - 0.732$	6	octahedral	NaCl
$0.732 - 1.000$	8	cubic	CsCl

Cutoffs usi trick se aate hain: tetrahedral deta hai $3/2 - 1 = 0.225$ , cubic deta hai $3 - 1 = 0.732$ .

Cubic (CN 8) limit derive karo — "Yeh step kyun?" version: Cation 8 anions ke cube mein. Anions cube edge ke along touch karte hain ( $= 2 r_{-}$ ); cation body diagonal par baith ta hai ( $= 2 r_{+} + 2 r_{-}$ ). Body diagonal $= 3 \times$ edge: $2 r_{-} + 2 r_{+} = 3 (2 r_{-}) \Rightarrow ρ = 3 - 1 = 0.732.$

Paanch structures

Figure — Ionic crystals — NaCl, CsCl, ZnS, fluorite, antifluorite structures

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho oranges (bade negative ions) ek box mein pack hain. Oranges ke beech ke gaps mein tum chote marbles (positive ions) daal sakte ho. Agar marble bahut chota hai toh woh sirf wahan fit hota hai jahan 4 oranges milte hain (ZnS). Ek bada marble ek bade gap mein fit hota hai jahan 6 oranges usse gherte hain (NaCl). Ek bahut bada marble almost orange jaisa hota hai, toh woh 8 oranges ke ek pure cube ke beech mein baithta hai (CsCl). Yeh koi har gap fill karta hai ya aadhe gaps, yeh decide karta hai recipe: 1 marble per orange (NaCl), ya 2 marbles per orange (Na₂O). Fluorite bas yahi ulta hai.

Flashcards

Ek ionic crystal mein kaun se do sublattices hote hain?

Ek cations ka aur ek anions ka, interpenetrating taaki opposite ions touch karein.

CN 6 (octahedral) ke liye radius ratio range?

0.414 \leq r_{+} / r_{-} < 0.732

Octahedral limiting ratio derive karo.

Diagonal

= 2 \times

edge:

2 (r_{+} + r_{-}) = 2 (2 r_{-}) \Rightarrow r_{+} / r_{-} = 2 - 1 = 0.414

Cubic (CN 8) limiting radius ratio?

3 - 1 = 0.732

(body diagonal =

3 \times

edge).

NaCl structure description + Z?

Cl⁻ FCC, Na⁺ saare octahedral holes mein, CN 6:6, Z = 4.

CsCl BCC kyun NAHI hai?

Corner aur centre ions alag hain (Cl⁻ vs Cs⁺); yeh simple cubic hai with a 2-ion basis, Z = 1.

ZnS (zinc blende) structure?

S²⁻ FCC, Zn²⁺ alternate (half) tetrahedral holes mein, CN 4:4, Z = 4.

Fluorite CaF₂ structure + CN?

Ca²⁺ FCC, F⁻ saare tetrahedral holes mein, CN 8:4, stoichiometry MX₂.

Antifluorite structure + example?

Anion FCC, cations saare tetrahedral holes mein, CN 4:8, M₂X jaise Na₂O, Li₂O.

NaCl ke liye edge–radius relation?

a = 2 (r_{+} + r_{-})

(cube edge ke along contact).

CsCl mein contact direction?

Body diagonal:

3 a = 2 (r_{+} + r_{-})

Cell data se density formula?

ρ = Z M / (N_{A} a^{3})

Radius ratio badhne par, CN…?

Badhta hai (bada cation apne around zyada anions fit kar sakta hai).

Connections

Close packing FCC HCP and voids — octahedral & tetrahedral holes wahi hain jahan cations baithte hain.
Unit cell and Z calculation — cell mein ions count karna.
Density of crystals formula — $ρ = Z M / N_{A} a^{3}$ use karta hai.
Lattice energy and Born-Haber cycle — energetics kyun ye structures bante hain.
Defects in ionic solids — Schottky/Frenkel inhii lattices par build karte hain.
Coordination number — 4/6/8 ka geometric meaning.