Stability of half-filled and fully-filled subshells

2.1.11 · Chemistry › Quantum Atomic Structure

KYA ho raha hai?

Is stability ke peechhe do physical reasons hain:

Electron density ka symmetrical distribution — ek set of orbitals jisme har ek ki occupation equal ho (sab singly ya sab doubly filled) ek spherically symmetric, evenly-spread charge cloud deta hai jo nucleus ko uniformly shield karta hai aur repulsion kam karta hai.
Exchange energy — ek purely quantum-mechanical stabilization jo parallel-spin electron pairs ki ginti ke saath badhti hai. Yahi dominant reason hai.

KYUN: Exchange energy pehle principles se

Stabilization count kaise karein (approximate model). Distinct pairs of parallel-spin electrons ki ginti karo. Agar ek subshell mein $n$ electrons sab parallel spin ke saath hain, to exchange pairs ki sankhya hai:

N_{ex} = (2 n) = \frac{n ( n - 1 )}{2}

$d^{5}$ vs $d^{4}$ argument (kyun $3 d^{5}$ jeetata hai)

$d^{5}$ ke liye paanchon electrons parallel hain (Hund's rule), to $3 d$ electrons ke beech: $N_{ex} (3 d - 3 d) = (2 5) = 10 exchange pairs$

Chromium ke do candidate configurations compare karo. Neeche hum within- $3 d$ pairs count karte hain (sabse bada, sabse comparable contribution):

Config	$3 d$ spins	$4 s$ spins	$3 d$ – $3 d$ parallel pairs
$[A r] 3 d^{4} 4 s^{2}$	↑↑↑↑ (4 up)	↑↓ (paired)	$(2 4) = 6$
$[A r] 3 d^{5} 4 s^{1}$	↑↑↑↑↑ (5 up)	↑ (1 up)	$(2 5) = 10$

$3 d^{4} 4 s^{2} \to 3 d^{5} 4 s^{1}$ jaane se 4 extra $3 d$ – $3 d$ exchange pairs milte hain ( $10 - 6 = 4$ ), yaani $\approx - 4 K_{3 d, 3 d}$ .

Similarly Cu ke liye: $3 d^{9} 4 s^{2} \to 3 d^{10} 4 s^{1}$ $d$ subshell complete karta hai (fully-filled $d^{10}$ ), symmetry aur pairing stabilization dono maximize karta hai.

Figure — Stability of half-filled and fully-filled subshells

Worked examples

Common mistake Galat ideas ko steel-man karna

Galat idea 1: "Half/full shells stable hote hain kyunki unke energy orbitals 'complete' hain." Yeh sahi kyun lagta hai: symmetry aur neatness closure suggest karti hai. Fix: real driver exchange energy hai (parallel-spin swaps) aur symmetric charge distribution — sirf "completeness" nahi.

Galat idea 2: " $4 s$ hamesha $3 d$ se lower hota hai, isliye pehle fill hota hai aur kabhi empty nahi hota." Yeh sahi kyun lagta hai: Aufbau order $4 s$ ko $3 d$ se pehle list karta hai. Fix: $3 d$ aur $4 s$ energy mein bahut close hain; ek baar occupied hone par $3 d$ , $4 s$ se neeche ja sakta hai. Isliye ek atom $3 d$ mein electron rakh sakta hai (Cr, Cu) ya ionization ke dauran pehle $4 s$ remove ho sakta hai.

Galat idea 3: "Har same-spin pair energy ko SAME amount $K$ se lower karta hai." Yeh sahi kyun lagta hai: neat $(2 n) K$ formula sab pairs ko identically treat karta hai. Fix: exchange integral $K_{ij}$ orbitals $i$ aur $j$ ke spatial overlap par depend karta hai, isliye ek $3 d$ – $3 d$ pair, ek $3 d$ – $4 s$ pair, aur ek $4 s$ – $4 s$ pair ka alag-alag $K$ hota hai. Single- $K$ rule ek approximation hai; exact energy $- \sum_{i < j} K_{ij}$ hai.

Galat idea 4: "Cr $3 d^{5} 4 s^{1}$ hai isliye iske paas $3 d^{4} 4 s^{2}$ se KAM repulsion hogi." Yeh sahi kyun lagta hai: kam paired electrons = kam repulsion. Fix: $3 d$ mein ek electron move karne se actually kuch repulsion badhti hai; phir bhi yeh jeetata hai kyunki exchange stabilization ( $\approx - 4 K_{3 d, 3 d}$ ) use compensate kar deti hai.

Active recall

Recall Khud test karo (hover/expand)

Kaunse teen $p / d / f$ occupations "half-filled" count hote hain? → $p^{3}, d^{5}, f^{7}$ .
$n$ parallel electrons ke beech exchange pairs ki formula? → $(2 n) = \frac{n ( n - 1 )}{2}$ .
Kya exchange integral $K$ har pair ke liye same hota hai? → Nahi — $K_{ij}$ orbital overlap par depend karta hai ( $3 d$ – $3 d \neq = 3 d$ – $4 s$ ); single- $K$ ek approximation hai.
Cr $3 d^{5} 4 s^{1}$ kyun adopt karta hai? → 4 extra $3 d$ – $3 d$ exchange pairs milte hain + symmetry, aur $3 d \approx 4 s$ energy.
Extra stability ke do physical reasons? → Symmetric charge distribution + exchange energy.

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho electrons swings par baithe bacche hain, aur identical bacche (same spin) swings swap karna pasand karte hain. Har possible swap se poora playground thoda zyada khush ho jaata hai (energy kam hoti hai). Lekin nearby swings wale bacchon ke beech swaps zyada help karte hain — isliye har swap ki value equal nahi hoti. Jab har swing par exactly ek bachcha same direction mein face kar raha ho, to BAHUT SAARE possible swaps hote hain, playground super khush hoti hai — yahi "half-filled" shell hai. Kabhi kabhi ek atom ek bachche ko ek nearby playground se move bhi kar leta hai taaki zyada swaps possible ho sakein. Isliye Chromium aur Copper normal seating order "cheat" karte hain!

Connections

Hund's Rule of Maximum Multiplicity — parallel spins → exchange energy.
Aufbau Principle — woh order jise anomalies todte hain.
(n+l) Rule and Orbital Energies — kyun $3 d$ aur $4 s$ close hain.
Ionization Energy Trends — half-filled $2 p^{3}$ se N vs O anomaly.
Electronic Configuration of Transition Elements — jahaan Cr, Cu, Mo, Ag, Au aate hain.
Pauli Exclusion Principle — exchange ke neeche indistinguishability.

Kaunse subshell occupations "half-filled" aur extra stable hain?

p^{3}

d^{5}

f^{7}

(har orbital mein ek electron, sab parallel spins).

Kaunse subshell occupations "fully-filled" hain?

p^{6}

d^{10}

f^{14}

(har orbital mein do electrons).

Half/full-filled stability ke DO physical reasons kya hain?

Symmetrical charge distribution + exchange energy (dominant wala).

n

parallel-spin electrons ke beech exchange pairs ki formula?

(2 n) = \frac{n ( n - 1 )}{2}

Kya exchange integral har same-spin pair ke liye same hota hai?

Nahi —

K_{ij}

orbitals

i, j

ke spatial overlap par depend karta hai; single-

K

(2 n) K

form ek approximation hai. Exact:

E = - \sum_{i < j} K_{ij}

Cr (

Z = 24

) ka electronic configuration?

[A r] 3 d^{5} 4 s^{1}

(

3 d^{4} 4 s^{2}

nahi).

Cu (

Z = 29

) ka electronic configuration?

[A r] 3 d^{10} 4 s^{1}

(

3 d^{9} 4 s^{2}

nahi).

3 d^{4} 4 s^{2} \to 3 d^{5} 4 s^{1}

jaane par kitne extra 3d–3d exchange pairs milte hain?

10 - 6 = 4

pairs, yaani

\approx - 4 K_{3 d, 3 d}

(chhote cross

3 d

–

4 s

terms ignore karte hue).

Exchange energy negative (stabilizing) kyun hoti hai?

Indistinguishable same-spin electrons positions swap kar sakte hain; har swap energy ko apne

K_{ij}

se kam karta hai.

Ek atom

4 s

3 d

mein electron promote kyun kar sakta hai?

3 d

aur

4 s

energies bahut close hain, isliye exchange + symmetry gains chhoti cost se zyada ho sakti hai.

Nitrogen ki first ionization energy oxygen se zyada kyun hoti hai?

N ke paas stable half-filled

2 p^{3}

hai; electron remove karne se yeh disrupt hoti hai, extra energy cost hoti hai.

Kya half/full-filled anomaly SARE elements ke liye hoti hai?

Nahi — sirf tabhi jab

n s

aur

(n - 1) d

energies itni close hon ki exchange gain jeet sake.