Bohr model of hydrogen — postulates, radius rₙ = 0.529 n² - Z Å, energy Eₙ = −13.6 Z² - n² eV

1.2.7 · Chemistry › Atomic Structure (Classical)

The three postulates (WHAT Bohr assumed)

YEH KYUN? Postulate 3 atom ko classical collapse se bachata hai (ek accelerating charge ko radiate karna chahiye — Bohr sirf allowed orbits mein ise forbid kar deta hai). Postulate 2 woh quantisation seed hai jo energies ko discrete banata hai → discrete spectral lines.

Deriving the radius from scratch (HOW)

Step 1 — Force balance. Coulomb attraction = centripetal force.

$\frac{1}{4 π ε _{0}} \frac{( Z e ) ( e )}{r ^{2}} = \frac{m v ^{2}}{r}$

Yeh step kyun? Electron fly off nahi karta, toh inward electric pull mass×centripetal-acceleration ke barabar honi chahiye jo ise circling mein rakhe. Nuclear charge $+ Z e$ hai, electron $- e$ hai.

Simplify karo (ek $r$ cancel karo): $\frac{kZe^2}{r} = mv^2, \qquad k=\frac{1}{4\pi\varepsilon_0} \tag{1}$

Step 2 — Quantisation. Postulate 2 se, $v = \frac{nh}{2 π m r}$ .

Yeh step kyun? Hamare paas do unknowns $v$ aur $r$ hain; quantisation condition woh doosri equation deti hai jo humein chahiye.

Step 3 — Substitute $v$ ko (1) mein: $\frac{k Z e ^{2}}{r} = m (\frac{nh}{2 π m r})^{2} = \frac{n ^{2} h ^{2}}{4 π ^{2} m r ^{2}}$

Dono sides ko $r^{2}$ se multiply karo aur $k Z e^{2}$ se divide karo:

$r_{n} = \frac{n ^{2} h ^{2}}{4 π ^{2} mk Z e ^{2}} = \frac{ε _{0} n ^{2} h ^{2}}{π m Z e ^{2}}$

Step 4 — Numbers plug karo ( $n = 1, Z = 1$ ke liye): $n^{2} / Z$ ko chhod kar baaki sab ek constants ka bundle hai jo $0.529 \times 1 0^{- 10}$ m evaluate karta hai.

Deriving the energy (HOW)

Step 1 — Total energy = kinetic + potential. $E = K E \frac{1}{2} m v^{2} + P E (- \frac{k Z e ^{2}}{r})$

Yeh step kyun? PE negative hai kyunki attraction energy ko lower karta hai (bound = ek well mein trapped).

Step 2 — (1) use karo: $m v^{2} = \frac{k Z e ^{2}}{r}$ , toh $K E = \frac{k Z e ^{2}}{2 r}$ .

$E = \frac{kZe^2}{2r} - \frac{kZe^2}{r} = -\frac{kZe^2}{2r} \tag{2}$

Yeh step kyun? Notice karo $E = - K E = \frac{1}{2} P E$ — yeh inverse-square force ki ek pehchaan hai (virial theorem). Total energy negative hai → electron bound hai.

Step 3 — $r_{n}$ insert karo: $E_{n} = - \frac{k Z e ^{2}}{2} \cdot \frac{4 π ^{2} mk Z e ^{2}}{n ^{2} h ^{2}} = - \frac{2 π ^{2} m k ^{2} Z ^{2} e ^{4}}{n ^{2} h ^{2}}$

Constants ka bundle $13.6$ eV evaluate karta hai.

Figure — Bohr model of hydrogen — postulates, radius rₙ = 0.529 n² - Z Å, energy Eₙ = −13.6 Z² - n² eV

Spectral lines (WHY hydrogen has a line spectrum)

$n_{2} \to n_{1}$ jump ( $n_{2} > n_{1}$ ke saath) ek photon emit karta hai: $Δ E = E_{n_{2}} - E_{n_{1}} = 13.6 Z^{2} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}) eV$ $\overset{ν}{ˉ} = \frac{1}{λ} = R_{H} Z^{2} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}), R_{H} = 1.097 \times 1 0^{7} m^{- 1}$

Kyunki $n$ discrete hai, sirf certain $Δ E$ exist karte hain → sharp lines (Lyman $n_{1} = 1$ , Balmer $n_{1} = 2$ , etc.).

Worked examples

Common mistakes (Steel-man → fix)

Flashcards

Bohr model mein centripetal force kaun si force provide karti hai?

Nucleus (+Ze) aur electron (−e) ke beech Electrostatic (Coulomb) attraction.

Bohr ki quantisation condition batao.

m v r = nh /2 π = n ℏ

(angular momentum quantised hai).

Bohr orbit radius ka formula?

r_{n} = 0.529 (n^{2} / Z)

Å.

Bohr energy levels ka formula?

E_{n} = - 13.6 (Z^{2} / n^{2})

eV.

Total energy negative kyun hoti hai?

Electron bound hai; PE (negative) KE se zyada hai, deta hai

E = - K E = \frac{1}{2} P E

Radius n aur Z ke saath kaise scale karta hai?

r_{n} \propto n^{2}

aur

\propto 1/ Z

Ground-state hydrogen ki ionisation energy?

13.6 eV (electron ko

n = 1

n = \infty

tak le jaane ki energy).

Bohr model kin species ke liye kaam karta hai?

One-electron (hydrogen-like) species: H, He⁺, Li²⁺, Be³⁺.

Line spectra ke liye Rydberg formula?

\overset{ν}{ˉ} = R_{H} Z^{2} (1/ n_{1}^{2} - 1/ n_{2}^{2})

R_{H} = 1.097 \times 1 0^{7}

m⁻¹.

H ke

n = 1

orbit mein kya special hai?

Iska radius Bohr radius

a_{0} = 0.529

Å hai; ground state,

E = - 13.6

eV.

Recall Feynman: explain to a 12-year-old

Ek ball ko string par imagine karo jo circle mein ghoom rahi hai — string nucleus ki electric pull jaisi hai jo electron ko ghoomte rehne deti hai. Ab imagine karo ki electron sirf certain "magic circles" par hi ride kar sakta hai, jaise sirf staircase ke specific steps par khade hone ki ijazat ho, beech mein kabhi nahi. Ek step par woh calm rehta hai aur koi light nahi deta. Jab woh ek step NEECHE jump karta hai, toh ek exact colour ki light ki flash fenk deta hai; UPAR jaane ke liye use woh exact colour nighalni padti hai. Isliye hydrogen kuch sharp colours mein chamakta hai — har colour do steps ke beech ek particular jump hai.

Connections

Rutherford model — Bohr iske stability problem ko fix karta hai.
Hydrogen spectrum & Rydberg formula — direct application.
Photoelectric effect & Planck's quantisation — same $E = h ν$ quantum idea.
de Broglie wavelength — standing-wave picture Bohr ka $m v r = n ℏ$ derive karta hai.
Quantum mechanical model of atom — orbits ki jagah orbitals aate hain; explain karta hai kahan Bohr fail karta hai.
Ionisation energy — hydrogen-like ions ke liye $∣ E_{1} ∣$ ke barabar hai.