Hund's rule of maximum multiplicity

2.1.9 · Chemistry › Quantum Atomic Structure

The Core Statement

KYA decide karta hai: ek subshell ke andar electrons ka arrangement, yeh nahi ki kaunsa subshell pehle bharta hai (woh Aufbau principle ka kaam hai).

KYO kaam karta hai — do physical reasons:

Kam electron–electron repulsion. Alag orbitals mein do electrons spatially zyada door hote hain same orbital ke do electrons ki tulna mein, isliye repulsion kam hoti hai.
Exchange energy (deeper reason). Same spin wale electrons alag orbitals mein indistinguishable hote hain aur "exchange" kar sakte hain. Har aisa swappable pair energy ko $K$ (exchange integral) se lower karta hai. Zyada parallel-spin electrons ⇒ zyada exchange pairs ⇒ zyada stability.

KAISE use karte hain: pehle singly bharo, spins parallel, phir pair karo.

Deriving "Multiplicity" from Scratch

Toh Hund's rule = woh electron configuration chunno jisme $n$ sabse bada ho (sabse zyada unpaired, sab parallel).

Exchange Energy — the "WHY", made quantitative

Exchange pairs count kaise karo: agar $n$ parallel-spin electrons hain, toh distinct pairs jo exchange kar sakte hain ki sankhya hai: $pairs = (2 n) = \frac{n ( n - 1 )}{2} .$

Yeh step kyun? $n$ same-spin electrons mein se koi bhi 2 ek exchange pair banate hain; $n$ mein se 2 chunna $(2 n)$ hai.

Total exchange stabilization $= - K \cdot \frac{n ( n - 1 )}{2}$ .

Figure — Hund's rule of maximum multiplicity

Common Mistakes (Steel-manned)

Active Recall

Recall Quick self-test (answers chhupao, pehle khud socho)

$2 p^{3}$ ki multiplicity? → $4$ (n=3).
$2 p^{4}$ ki multiplicity? → $3$ (n=2).
$d^{5}$ mein exchange pairs? → $(2 5) = 10$ .
Cr $3 d^{5} 4 s^{1}$ kyun hai? → Half-filled $d^{5}$ exchange energy/multiplicity maximize karta hai.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek row mein identical khali chairs hain (orbitals), sab ek jaisi comfortable. Naye bachche (electrons) andar aate hain. Bina kisi ke saath squeeze kiye ek chair par, pehle har bachcha apni alag chair leta hai, aur sab ek hi taraf muh karke baithte hain (same spin). Akele baithne wale aur ek hi direction face karne wale bachche sabse khush hote hain — squeeze karna tab shuru hota hai jab har chair par pehle se koi baith chuka ho. Yahi "sab apni seat lo, same direction" wala rule Hund's rule hai.

Flashcards

Hund's rule kehta hai degenerate orbitals mein electrons pairing se pehle kya karte hain?

Har orbital ko singly parallel spins ke saath occupy karte hain.

Unpaired electrons n ke terms mein spin multiplicity define karo.

Multiplicity

= 2 S + 1 = n + 1

, jahan

S = n /2

Unpaired parallel-spin electrons maximize karna energy kyun lower karta hai?

Kam e–e repulsion (alag orbitals) + zyada exchange energy stabilization.

n parallel-spin electrons ke liye exchange pairs ki sankhya?

(2 n) = n (n - 1) /2

Nitrogen atom (2p³) ki multiplicity?

4 (teen unpaired electrons).

Oxygen atom (2p⁴) ki multiplicity?

3 (do unpaired electrons).

Subshells ke beech order kaunsa principle set karta hai (Hund nahi)?

Aufbau principle (n+l rule).

Cr [Ar]3d⁵4s¹ kyun hai?

Half-filled 3d⁵ maximum multiplicity/exchange energy stability deta hai.

Unpaired electrons se magnetic moment formula?

μ = n (n + 2) μ_{B}

Fully filled subshell jaise 2p⁶ ki multiplicity?

1 (n=0 unpaired).

Connections

Aufbau Principle — decide karta hai kaunsa subshell pehle bharta hai.
Pauli Exclusion Principle — 2 electrons/orbital ki cap, opposite spins ke saath.
Electron Spin Quantum Number — $m_{s} = \pm \frac{1}{2}$ aur total spin $S$ ka origin.
Exchange Energy and Half-filled Stability — Cr, Cu, Mn ke peeche ka deep "kyun".
Paramagnetism and Diamagnetism — unpaired electrons ⇒ paramagnetic.
Magnetic Moment of Atoms — $μ = n (n + 2)$ .