2.3.14
Ek electron jo proton se bound hai, woh aise hai jaise ek ball ek kuen mein phans gayi ho. Woh sirf kuch khaas shelves (energy levels) par hi baithh sakta hai, kabhi beech mein nahi. Sabse gehri shelf (n = 1 E = 0 mushkil se pakda hota hai. Number − 13.6 eV sabse neechi shelf kitni gehri hai yeh batata hai. Baaki sab 1/ n 2
Definition Hydrogen energy levelsEk hydrogen atom (ek proton + ek electron) ke paas sirf yeh energies ho sakti hain
E n = − n 2 13.6 eV , n = 1 , 2 , 3 , … n principal quantum number hai. Energy negative hai kyunki electron bound hai — usse free karne ke liye aapko energy add karni padegi. Use ground state se free karne ke liye jo energy chahiye woh ionization energy = 13.6 eV
Negative kyun? Hum E = 0 kam energy rakhti hai, isliye woh negative hai. Zyada negative = zyada tightly bound.
Hume sirf teen physical ideas chahiye. Dekho kaise 1/ n 2
Constant 8 ε 0 2 h 2 m e 4 = 13.6 eV Rydberg energy ) purely m , e , ε 0 , h 13.6
Worked example 1) Ground state energy aur ionizationn = 1 E 1 = − 13.6/ 1 2 = − 13.6 eV Kyun? Yeh sabse gehri shelf hai. Ionize karne ke liye (electron ko E = 0 Δ E = 0 − ( − 13.6 ) = 13.6 eV Isliye hydrogen ki ionization energy exactly 13.6
n = 2 n = 3 E 2 = − 13.6/4 = − 3.40 eV E 3 = − 13.6/9 = − 1.51 eV Yeh step kyun? Bas n 2 1 → 2 10.2 2 → 3 1.89 0
n = 3 → n = 2 Δ E = 13.6 ( 4 1 − 9 1 ) = 13.6 × 36 5 = 1.89 eV λ = Δ E h c = 1.89 eV 1240 eV⋅nm ≈ 656 nm red light (visible Balmer line!).
Yeh step kyun? Handy constant h c = 1240 eV⋅nm
Worked example 4) Forecast-then-VerifyForecast: "Kaun sa transition sabse energetic photon deta hai: 2 → 1 3 → 2 Δ E 2 → 1 bahut bade bottom gap ko span karta hai → 10.2 3 → 2 → 1.89 Verify: 10.2 > 1.89 2 → 1 n = 1
Common mistake "Energy positive honi chahiye — ek electron move kar raha hai with KE."Kyun sahi lagta hai: kinetic energy sach mein positive hoti hai. Pakad: total energy = KE + PE, aur negative PE, KE ko outweigh karta hai (E = − KE Fix: hamesha potential energy include karo E ( ∞ ) = 0
E n = − 13.6 n 2 Kyun sahi lagta hai: upar ka n Pakad: upar ka n kamzor binding = energy 0 1/ n 2 Fix: yeh − 13.6/ n 2 n
Common mistake "Photon wavelength sirfE n i Kyun sahi lagta hai: tumhare paas ek level number hai. Pakad: photon do levels ke difference se aata hai. Fix: hamesha Δ E = E n i − E n f
− 13.6 Pakad: yeh hydrogen (Z = 1 E n = − 13.6 Z 2 / n 2 × 4 Fix: hidden Z 2
Recall Feynman: 12-saal ke bacche ko explain karoSocho atom ke andar ek seedhi hai. Electron step 1, step 2, step 3 par khada ho sakta hai… lekin steps ke beech mein kabhi nahi . Step 1 sabse neecha, sabse gehra basement hai; upar ke steps ek doosre ke paas squeeze ho jaate hain ground floor ke paas (matlab "escape ho gaya"). Upar chadne ke liye electron ko exactly sahi maatra mein light nigalni padti hai; neeche girne ke liye woh ek exact color ki light ugalta hai. Number 13.6
"Negative, divide, kabhi multiply mat karo." E n = − 13.6/ n 2 "In-to-1 = brightest hit" — n = 1 "Bottom par bada gap, top par baby steps."
What is the hydrogen energy level formula? E n = − 13.6/ n 2 n = 1 , 2 , 3 , …
Hydrogen energy levels negative kyun hote hain? Reference E = 0
Hydrogen ki ground state se ionization energy kya hai? 13.6 − 13.6 0
E n (1) Coulomb force = centripetal force; (2) angular momentum quantized: m v r = n ℏ
Total energy − 2 1 − Force balance m v 2 = k e 2 / r = 2 1 k e 2 / r = − k e 2 / r E = − 2 1 k e 2 / r = −
Orbit radius n r n = a 0 n 2 a 0 = 0.529
n = 2 − 3.40
Transition n i → n f Δ E = 13.6 ( 1/ n f 2 − 1/ n i 2 )
3 → 2 ≈ 656 Δ E = 1.89 λ = h c /Δ E
eV→nm conversion ke liye handy constant? h c = 1240
One-electron ion of charge Z E n = − 13.6 Z 2 / n 2
n Kyunki 1/ n 2 E = 0
Coulomb force = centripetal
m v squared = e^2 / 4 pi eps0 r
Bohr quantization m v r = n h-bar
Ionization energy = 13.6 eV
Principal quantum number n
