Ionic bonding — Born-Haber cycle, lattice energy (Kapustinskii equation)

2.3.3 · Chemistry › Chemical Bonding

1. Lattice Energy Kya Hai?

YEH ITNA BADA KYUN HOTA HAI: har ion kai neighbours ki taraf attract hota hai har direction mein, sirf ek partner ki taraf nahi. Poore infinite lattice pe sum karne se pairwise attraction multiply ho jaata hai.

2. First Principles Se Lattice Energy Derive Karna

HUMEIN KYA CHAHIYE: pure crystal ki total electrostatic potential energy per mole.

Step 1 — ek pair. Do ions ke beech Coulomb potential energy jinka charge $z_{+} e$ aur $z_{-} e$ hai aur separation $r_{0}$ hai: $E_{p ai r} = \frac{1}{4 π ε _{0}} \frac{( z _{+} e ) ( z _{-} e )}{r _{0}}$ Yeh step kyun? Yeh fundamental law hai; sab kuch isi se scale hota hai.

Step 2 — poora lattice sum karo. Ek given ion nearest neighbours (attraction), next-nearest (repulsion), etc. feel karta hai. Yeh infinite alternating sum ek geometric constant pe converge karta hai jise ==Madelung constant $A$ == kehte hain (e.g. NaCl structure ke liye $A = 1.748$ ). $E_{a tt r} = - \frac{N _{A} A z _{+} z _{-} e ^{2}}{4 π ε _{0} r _{0}}$ Yeh step kyun? $A$ "poore crystal ki geometry" ko ek number mein pack kar deta hai, taaki hume baar baar sum na karna pade.

Step 3 — short-range repulsion add karo. Electron clouds squeeze hone par repel karte hain (Pauli). Born ne isse $\propto r^{- n}$ model kiya, jisse Born exponent $n$ (7–12) milta hai. Equilibrium $r_{0}$ pe total energy minimize karne par ek correction factor $(1 - \frac{1}{n})$ bachta hai:

ISSE KAISE PADHEIN: $U$ charge product $z_{+} z_{-}$ ke saath badhta hai (sabse bada lever!) aur ionic separation $r_{0}$ ke saath ghatata hai. Chhote, highly-charged ions ⇒ enormous lattice energy (MgO ≈ –3800 kJ/mol vs NaCl ≈ –788).

3. Kapustinskii Equation — ek shortcut jab $A$ unknown ho

YEH KYUN CHAHIYE: Kisi random compound ke liye aap crystal structure nahi jaante, toh $A$ available nahi hota. Kapustinskii ne notice kiya ki $A / ν$ (Madelung per ion) roughly constant hai, aur usne saare constants ko ek number mein fold kar diya.

KAISE: aapko sirf charges, ion count, aur ionic radii chahiye — koi Madelung constant nahi, koi crystal structure nahi. $(1 - 34.5/ (r_{+} + r_{-}))$ term Born repulsion correction ko mimic karta hai.

4. The Born–Haber Cycle

KYUN: Lattice energy directly measure nahi ki ja sakti. Lekin kisi bhi closed loop ke around total enthalpy $= 0$ hoti hai (Hess's Law). Toh hum elements → compound ka ek loop do routes se banaate hain aur unknown $U$ solve karte hain.

Figure — Ionic bonding — Born-Haber cycle, lattice energy (Kapustinskii equation)

NaCl ke steps (sab per mole):

Step	Process	Symbol
Atomisation Na	$Na (s) \to Na (g)$	$Δ H_{a t o m} (Na)$ (+)
Ionisation	$Na (g) \to Na^{+} (g) + e^{-}$	$I E_{1}$ (+)
Atomisation Cl	$\frac{1}{2} Cl_{2} (g) \to Cl (g)$	$\frac{1}{2} D_{C l - C l}$ (+)
Electron affinity	$Cl (g) + e^{-} \to Cl^{-} (g)$	$E A$ (–)
Lattice formation	$Na^{+} + Cl^{-} \to NaCl (s)$	$U$ (–)
Overall	$Na (s) + \frac{1}{2} Cl_{2} \to NaCl (s)$	$Δ H_{f}$ (–)

5. Common Mistakes (Steel-manned)

6. Born–Haber vs Kapustinskii: kaunsa aur kyun

Born–Haber: experimental, exact, thermodynamic data chahiye. Yeh covalent character bhi batata hai: agar Born-Haber $U$ (experimental) Born-Landé/Kapustinskii prediction se zyada negative hai, toh extra covalent bonding present hai (Fajans' rules).
Kapustinskii: theoretical estimate, sirf radii & charges chahiye. Tab great hai jab aapke paas data nahi ho.

Flashcards

Lattice energy define karo (formation convention).

Woh energy jo release hoti hai jab 1 mol solid ionic compound apne gaseous ions se banta hai; exothermic (negative).

Madelung constant kya hai?

Ek geometric constant jo crystal mein saare attractive/repulsive Coulomb interactions ko sum karta hai (NaCl ≈ 1.748).

Born–Landé equation likho.

U = - \frac{N _{A} A z _{+} z _{-} e ^{2}}{4 π ε _{0} r _{0}} (1 - \frac{1}{n})

n

= Born exponent.

(1 - 1/ n)

factor kyun introduce kiya?

Electron clouds ki short-range Pauli/Born repulsion account karne ke liye, equilibrium

r_{0}

pe evaluate ki gayi.

Kapustinskii equation batao.

U \approx - \frac{1.202 \times 1 0 ^{5} ν z _{+} z _{-}}{r _{+} + r _{-}} (1 - \frac{34.5}{r _{+} + r _{-}})

kJ/mol, radii pm mein.

Kapustinskii Born–Landé se kab prefer ki jaati hai?

Jab crystal structure (isliye Madelung constant) unknown ho; sirf charges, ion count, radii chahiye.

Born–Haber ko U ke liye rearrange karo.

U = Δ H_{f} - Δ H_{a t o m} - I E_{1} - \frac{1}{2} D - E A

MgO ki lattice energy NaCl se itni badi kyun hai?

Charge product 4 vs 1 aur chhote ions (chhota

r_{0}

);

U \propto z_{+} z_{-} / r_{0}

Born–Haber covalent character kaise reveal karta hai?

Experimental

U

ionic-model prediction se zyada exothermic ⇒ extra covalent bonding (Fajans).

Oxygen ki 2nd electron affinity +ve hai ya –ve? Kyun?

Endothermic (+); pehle se negative O

^{-}

pe ek electron force karna energy cost karta hai.

Recall Feynman: 12-saal ke bachche ko explain karo

Soch ke dekho tiny magnetic balls: red (+) aur blue (–). Sodium se electron nikaal ke use "red" banana actually mehnat maangta hai. Toh phir namak khushi khushi kyun exist karta hai? Kyunki jab ek poore box mein red aur blue balls hote hain, woh ek super-tight grid mein snap ho jaate hain, aur bahut saari energy release hoti hai — jo tumne spend ki thi usse kaafi zyada. Born–Haber cycle ek bank statement jaisi hai: hum seedhe nahi dekh sakte ki snap-together mein kitni energy release hui, lekin hum final balance jaante hain (namak mein kitni energy hai) aur har doosra transaction, toh hum mystery amount nikaalte hain subtract karke. Kapustinskii ek cheat-sheet formula hai jo us snap energy ka andaza sirf balls ki size aur unke charges ki strength se lagata hai.

Connections

Coulomb's Law — lattice energy ke peeche physics engine
Hess's Law — Born–Haber cycle ko valid banata hai
Ionisation Energy & Electron Affinity — cycle inputs
Fajans' Rules — polarisation se covalent character
Ionic Radii — $r_{+} + r_{-}$ term
Enthalpy of Formation — loop ki closing quantity
Lattice Energy → Solubility — hydration vs lattice energy ke trends