Bring two H atoms from infinity toward each other. Track total energy E vs separation r.
Step 1 — What forces act?
Nucleus–nucleus repulsion: ∝+r1 (blows up as r→0).
Electron–electron repulsion: also positive.
Nucleus–electron attraction: negative, and — crucially — electron density piles up between the nuclei, so each nucleus is attracted toward the shared cloud.
Why does density pile up in the middle? Because when atomic orbitals overlap in phase (constructive interference of the wavefunctions), the probability ∣ψ∣2 in the internuclear region increases. That extra negative charge between two positive nuclei is what binds them.
Step 2 — Build the energy curve.
At large r: attraction dominates → energy falls as atoms approach.
At small r: nuclear repulsion ∝1/rwins → energy shoots up.
Somewhere between, E hits a minimum. A convenient model is the Morse potential:
E(r)=De(1−e−a(r−re))2−De
Step 3 — Read off the two quantities.
The minimum sits at r=re: set drdE=0.
drdE=2De(1−e−a(r−re))⋅ae−a(r−re)
This is zero when 1−e−a(r−re)=0⇒r=re. So the bond length is exactly the well minimum. ✔
The depth of the well is the bond energy. At r=re, E=−De; at r→∞, E→0. So the energy to escape the well is
where Nb = electrons in bonding MOs, Na = electrons in antibonding MOs. Example O2: Nb=10,Na=6⇒ BO =2. Fractional orders arise here too (e.g. O2+ has BO 2.5).
The equilibrium internuclear distance — the separation at the minimum of the potential-energy curve.
What is bond energy?
Energy required to break one mole of that bond in the gas phase (homolytically); always positive.
How does bond order affect length and energy?
Higher bond order → shorter bond length and higher bond energy.
Why is a C=C bond not exactly twice as strong as C–C?
The second bond is a π bond (weaker side-on overlap) than the first σ bond, so each extra bond adds less energy.
MO formula for bond order?
BO = (N_bonding − N_antibonding)/2.
Bond order of O2 by MO theory?
(10 − 6)/2 = 2.
Why is NO⁺ stronger/shorter-bonded than NO?
Removing an antibonding electron raises BO from 2.5 to 3.
Formula for reaction enthalpy from bond energies?
ΔH = Σ(bonds broken) − Σ(bonds formed).
Why is benzene C–C length 139 pm (between single and double)?
Delocalisation gives an effective bond order of ~1.5.
Why does electron density between nuclei bind the atoms?
In-phase orbital overlap piles negative charge between the two positive nuclei, pulling both inward.
Recall Feynman: explain to a 12-year-old
Imagine two magnets that push each other away (the nuclei). Now put a blob of sticky glue (shared electrons) right between them. The glue grabs both and pulls them together until the pushing-away exactly balances the pulling-in — that gap is the bond length. To yank them apart you must scrape off the glue, and the effort you spend is the bond energy. Add MORE glue (double, triple bond = higher bond order) and the magnets sit closer and are harder to pull apart. Simple!
Dekho, covalent bond ka matlab hai do nuclei ke beech electrons ka sharing. Dono nuclei positive hote hain, isliye ek doosre ko push karte hain. Par jab electrons beech mein aa jaate hain (in-phase orbital overlap se density beech mein badh jaati hai), to woh dono nuclei ko andar ki taraf kheech lete hain. Jahan yeh push aur pull barabar ho jaate hain, wahi distance hai bond length (re) — potential energy curve ka lowest point.
Ab us gadde (well) ki gehrai hi bond energy hai — yaani atoms ko alag karne ke liye kitni energy chahiye. Ise hamesha positive likhte hain kyunki breaking mein energy lagti hai. Ek zabardast rule yaad rakho: bond order badhne pe length ghatti hai aur energy badhti hai. C–C, C=C, C≡C dekho — 154→134→120 pm length, aur 348→614→839 kJ/mol energy. Lekin energy exactly double-triple nahi hoti, kyunki pehla σ bond sabse strong hota hai, extra π bonds thoda kam add karte hain.
MO theory se bond order nikaalna simple hai: BO=(Nb−Na)/2. O2 ke liye (10−6)/2=2. Aur agar antibonding electron nikaal do (jaise NO se NO+), to BO badh jaata hai — bond aur strong, aur chhota. Yeh counterintuitive lagta hai par logic clear hai: antibonding hataao, net bonding badho.
Exam ka 80/20: reaction enthalpy nikaalne ka formula — ΔH=Σ(broken)−Σ(formed). Bas bonds break ki energy add karo, bonds form ki energy minus karo. Isse H2+Cl2→2HCl jaise sawaal minute mein ho jaate hain. Yeh teen concepts (length, energy, order) hamesha saath chalte hain — ek yaad rakho, teeno samajh jaoge.
