Magnetic moments of complexes — Hinglish notes

3.4.12 · Chemistry › Coordination Chemistry

HUM KYA measure kar rahe hain?

YEH KYUN MATTER KARTA HAI? Ek transition-metal complex ke liye, $μ$ mainly unpaired $d$ -electrons ki sankhya $n$ par depend karta hai. $μ$ measure karo → $n$ deduce karo → geometry, spin state, aur oxidation state deduce karo.

FORMULA KAISE AATA HAI (derivation scratch se)

Ek electron mein magnetism ke do sources hote hain:

Spin angular momentum (yeh ek spinning charge ki tarah act karta hai).
Orbital angular momentum (yeh nucleus ke around circulate karta hai).

Spin contribution. Quantum mechanics total spin quantum number $S$ wale particle ka magnetic moment deta hai:

$μ_{S} = g S (S + 1) μ_{B}$

jahan $g \approx 2$ electron ke liye hai.

$S (S + 1)$ kyun aur sirf $S$ kyun nahi? Kyunki quantum mechanics mein quantum number $S$ ke angular momentum vector ki magnitude $S (S + 1) ℏ$ hoti hai, $S ℏ$ nahi. Magnetic moment is magnitude ke proportional hota hai.

Ab $S$ ko unpaired electrons ki sankhya $n$ se relate karo. Har unpaired electron spin $\frac{1}{2}$ contribute karta hai, sab aligned hain, toh

$S = \frac{n}{2} .$

Substitute karo ( $g = 2$ ke saath):

$μ_{S} = 2 \frac{n}{2} (\frac{n}{2} + 1) = 2 \frac{n}{2} \cdot \frac{n + 2}{2} = 2 \cdot \frac{1}{2} n (n + 2)$

$μ_{S} = n (n + 2) μ_{B}$

Quick reference table

$n$	$μ_{so} = n (n + 2)$	$μ_{B}$
0	$0$	0 (diamagnetic)
1	$3$	1.73
2	$8$	2.83
3	$15$	3.87
4	$24$	4.90
5	$35$	5.92

HIGH-SPIN vs LOW-SPIN KYUN SAB KUCH BADAL DETA HAI

Wahi metal ion alag $n$ de sakta hai ligand field ke hisaab se:

Strong field ligand (jaise $CN^{-}$ , CO) → bada $Δ_{o}$ → electrons neeche wale $t_{2 g}$ mein pair ho jaate hain → low spin, kam unpaired electrons → chhota $μ$ .
Weak field ligand (jaise $H_{2} O$ , $F^{-}$ ) → chhota $Δ_{o}$ → electrons spread out ho jaate hain (Hund) → high spin, zyada unpaired electrons → bada $μ$ .

Toh $μ$ measure karne se directly spin state pata chalta hai.

Worked examples

Common mistakes (steel-manned)

Spin-only magnetic moment formula

μ = n (n + 2) μ_{B}

, jahan

n

= unpaired electrons ki sankhya

n (n + 2)

kyun aur

n

kyun nahi

Yeh

g S (S + 1)

se aata hai jahan

S = n /2

aur

g = 2

hain; angular-momentum magnitude

S (S + 1) ℏ

hoti hai,

S ℏ

nahi

n = 4

unpaired electrons ke liye

μ

24 = 4.90 μ_{B}

Magnetic moment ki unit

Bohr magneton,

μ_{B}

High-spin vs low-spin: kis ka

μ

bada hota hai

High-spin ka (zyada unpaired electrons, weak field ligand)

[Fe(CN)_{6}]^{3 -}

[Fe(H_{2} O)_{6}]^{3 +}

d^{5}

low-spin (

n = 1

, 1.73) vs high-spin (

n = 5

, 5.92)

3d metals ke liye spin-only formula kyun achha hai

Orbital contribution largely ligand field ke through quenched ho jaata hai

Spin-only formula kab fail karta hai

Lanthanides (4f) aur heavy metals ke liye jahan orbital angular momentum quenched nahi hota

[Ni(CN)_{4}]^{2 -}

magnetism

d^{8}

square planar,

n = 0

, diamagnetic

Ek complex

μ = 2.83 μ_{B}

dikhata hai.

n

find karo

n (n + 2) = 8 \Rightarrow n = 2

Recall Feynman: ek 12-saal ke bacche ko samjhao

Ek spinning top jo electric charge ke saath hai woh ek tiny magnet ki tarah act karta hai. Ek metal atom ke andar, electrons tops ki tarah spin karte hain. Agar do electrons saath baithte hain toh woh opposite directions mein spin karte hain aur unki magnetism cancel ho jaati hai — boring. Lekin ek akela electron ke paas koi partner nahi hota cancel karne ke liye, toh woh ek chhota magnet bana rehta hai. Jitne zyada akele electrons ek atom mein hote hain, utna stronger magnet woh banta hai. Hum substance ko ek bade magnet ki taraf kheenchte hain aur dekhte hain ki woh kitna zyada khicha jaata hai: ek strong tug matlab bahut saare akele electrons hain. Formula $n (n + 2)$ bas "akele electrons ki sankhya" ko "magnet ki strength" mein convert karta hai.

Connections

Crystal Field Theory — $Δ_{o}$ set karta hai, jo high-spin vs low-spin decide karta hai.
Spectrochemical Series — ligand field strength ko rank karta hai.
Electronic Configuration of d-block ions — $d$ -count deta hai.
Square Planar vs Tetrahedral Geometry — structural proof ke roop mein diamagnetism.
Hund's Rule and Pairing Energy — spin state ke peeche competition.
Color of Coordination Compounds — wahi $Δ_{o}$ color aur magnetism dono control karta hai.