Covalent bonding — bond length, bond energy, bond order

2.3.5 · Chemistry › Chemical Bonding

WHAT are we describing?

WHY does a covalent bond exist at all? (Derivation from first principles)

Do H atoms ko infinity se ek doosre ki taraf laao. Total energy $E$ ko separation $r$ ke saath track karo.

Step 1 — Kaunsi forces kaam karti hain?

Nucleus–nucleus repulsion: $\propto + \frac{1}{r}$ (jab $r \to 0$ toh blast ho jaata hai).
Electron–electron repulsion: yeh bhi positive hai.
Nucleus–electron attraction: negative hai, aur — crucially — electron density nuclei ke beech pile up ho jaati hai, isliye har nucleus shared cloud ki taraf attract hota hai.

Density beech mein kyun pile up hoti hai? Kyunki jab atomic orbitals in phase overlap karte hain (wavefunctions ki constructive interference), toh internuclear region mein probability $∣ ψ ∣^{2}$ badh jaati hai. Woh extra negative charge do positive nuclei ke beech hi unhe bind karta hai.

Step 2 — Energy curve banao. Bade $r$ par: attraction dominate karta hai → energy girती hai jab atoms paas aate hain. Chhote $r$ par: nuclear repulsion $\propto 1/ r$ jeet jaati hai → energy upar shoot karti hai. Kahin beech mein, $E$ ek minimum par pahunchti hai. Ek convenient model hai Morse potential:

$E (r) = D_{e} (1 - e^{- a (r - r_{e})})^{2} - D_{e}$

Step 3 — Do quantities read off karo.

Minimum $r = r_{e}$ par baithti hai: $\frac{d E}{d r} = 0$ set karo.

$\frac{d E}{d r} = 2 D_{e} (1 - e^{- a (r - r_{e})}) \cdot a e^{- a (r - r_{e})}$

Yeh zero hoga jab $1 - e^{- a (r - r_{e})} = 0 \Rightarrow r = r_{e}$ . Toh bond length exactly well minimum hai. ✔

Well ki depth bond energy hai. $r = r_{e}$ par, $E = - D_{e}$ ; $r \to \infty$ par, $E \to 0$ . Toh well se escape karne ki energy hai

$B E = E (\infty) - E (r_{e}) = 0 - (- D_{e}) = D_{e} .$

Figure — Covalent bonding — bond length, bond energy, bond order

The master trend

Carbon–carbon evidence (yeh numbers yaad karo):

Bond	BO	Length (pm)	Energy (kJ mol $^{- 1}$ )
C–C	1	154	348
C=C	2	134	614
C≡C	3	120	839

MO theory se bond order (diatomics ke liye):

$BO = \frac{N _{b} - N _{a}}{2}$

jahan $N_{b}$ = bonding MOs mein electrons, $N_{a}$ = antibonding MOs mein electrons. Example O $_{2}$ : $N_{b} = 10, N_{a} = 6 \Rightarrow$ BO $= 2$ . Fractional orders yahan bhi aate hain (e.g. O $_{2}^{+}$ ka BO $2.5$ hai).

Worked examples

Common mistakes (Steel-man → Fix)

Estimating reaction enthalpy (the 80/20 payoff)

Flashcards

Bond length kya hai?

Equilibrium internuclear distance — potential-energy curve ke minimum par separation.

Bond energy kya hai?

Gas phase mein ek mole bond break karne ke liye required energy (homolytically); hamesha positive.

Bond order length aur energy ko kaise affect karta hai?

Higher bond order → shorter bond length aur higher bond energy.

C=C bond exactly C–C se twice as strong kyun nahi hai?

Doosra bond ek π bond hai (weaker side-on overlap) pehle σ bond se, isliye har extra bond kam energy add karta hai.

Bond order ke liye MO formula?

BO = (N_bonding − N_antibonding)/2.

MO theory se O2 ka bond order?

(10 − 6)/2 = 2.

NO⁺ NO se stronger/shorter-bonded kyun hai?

Ek antibonding electron remove karna BO ko 2.5 se 3 tak raise karta hai.

Bond energies se reaction enthalpy ka formula?

ΔH = Σ(bonds broken) − Σ(bonds formed).

Benzene C–C length 139 pm (single aur double ke beech) kyun hai?

Delocalisation effective bond order ~1.5 deta hai.

Nuclei ke beech electron density atoms ko kyun bind karti hai?

In-phase orbital overlap do positive nuclei ke beech negative charge pile karta hai, dono ko andar kheenchta hai.

Recall Feynman: 12-year-old ko explain karo

Do magnets imagine karo jo ek doosre ko dhakelte hain (nuclei). Ab ek chewy sticky glue ka blob (shared electrons) bilkul unke beech rakho. Glue dono ko pakad leta hai aur andar kheenchta hai jab tak dhakka aur khichaav exactly balance nahi ho jaate — woh gap bond length hai. Unhe alag kheenchne ke liye tumhe glue scrape off karni padegi, aur jo effort lagta hai woh bond energy hai. AUR ZYADA glue daalo (double, triple bond = higher bond order) aur magnets aur paas baithenge aur alag karna aur mushkil hoga. Simple!

Connections

Molecular Orbital Theory — jahan bond order = (Nb − Na)/2 aata hai
VSEPR and Molecular Geometry — bond angles bond lengths ko complement karte hain
Resonance and Delocalisation — fractional bond orders (benzene, NO)
Hess's Law and Enthalpy — bond energies ΔH calculations mein jaati hain
Electronegativity and Bond Polarity — size/polarity bhi length affect karte hain
Sigma and Pi Bonds — kyun σ > π strength mein