Chapter 12 / 12.4 Bohr Model of the Hydrogen Atom

Bohr Model of the Hydrogen Atom 🚀

1 – Why did we need a new model?

The Rutherford picture looked like a tiny solar system, but a revolving charge should radiate energy, spiral inward, and crash into the nucleus 😲. That never happens, so something else must keep atoms stable.:contentReference[oaicite:0]{index=0}
Rutherford’s model also predicted a continuous rainbow of light while experiments showed sharp, bright lines. Atoms behave in steps, not smooth slides.:contentReference[oaicite:1]{index=1}

2 – Bohr’s three game-changing rules 🎯

Stationary states: An electron can zip around in special orbits without losing energy. Each orbit has a fixed total energy. These “parking spots” are called stationary states.:contentReference[oaicite:2]{index=2}
Quantised angular momentum: $$L = \frac{nh}{2\pi}\quad(n = 1,2,3,\dots)$$ Here $h$ is Planck’s constant. Only whole-number multiples work—no fractions allowed!:contentReference[oaicite:3]{index=3}
Photon rule: When the electron jumps from a higher level $E_i$ to a lower one $E_f$, it spits out one photon with $$h\nu = E_i – E_f$$ The photon’s frequency $\nu$ tells you the colour of the emitted light.:contentReference[oaicite:4]{index=4}

3 – The size of an orbit 📏

Using the angular-momentum rule, Bohr found the radius of the n^th orbit:

$$ r_n = \frac{\varepsilon_0\,n^{2}h^{2}}{\pi m e^{2}} \tag{12.7} $$

The smallest orbit ($n = 1$) has radius $a_0 = 5.3\times10^{-11}\,\text{m}$ and is called the Bohr radius.:contentReference[oaicite:5]{index=5}

4 – Energy of an orbit ⚡

$$ E_n = -\frac{m e^{4}}{8 \varepsilon_0^{2} h^{2}}\,\frac{1}{n^{2}} = -\frac{13.6\ \text{eV}}{n^{2}} \tag{12.8 & 12.10} $$

Negative sign ➡️ the electron is bound to the nucleus.
$E_1 = -13.6\ \text{eV}$ is the ground-state energy. Kicking the electron completely free needs $+13.6\ \text{eV}$ — the ionisation energy.:contentReference[oaicite:6]{index=6}

Excited levels (quick numbers)

n	Energy $E_n$ (eV)	Jump up from ground ✈️
2	−3.40	Needs 10.2 eV
3	−1.51	Needs 12.09 eV

Higher levels bunch closer together, so jumps get smaller as $n$ grows.:contentReference[oaicite:7]{index=7}

5 – Making the bright lines 🌈

When an electron drops from $n_i$ to $n_f$ ($n_f<n_i$), the photon frequency is set by Bohr’s photon rule above. Collections of these jumps create series of bright lines (Lyman, Balmer, etc.). Every element shows its own fingerprint pattern.:contentReference[oaicite:8]{index=8}

6 – Quick reality check (classical vs. quantum) 🧐

If classical physics ruled, an electron in the first orbit would whirl at $6.6\times10^{15}\ \text{Hz}$ and radiate itself into the nucleus almost instantly – clearly not what we see. Bohr’s quantised orbits rescue atomic stability.:contentReference[oaicite:9]{index=9}

7 – High-yield ideas for NEET 🏆

Ionisation energy of hydrogen: $13.6\ \text{eV}$.
Quantised angular momentum: $L = nh/2\pi$.
Energy formula: $E_n = -13.6/n^{2}\ \text{eV}$.
Bohr radius $a_0 = 5.3\times10^{-11}\ \text{m}$.
Photon energy rule: $h\nu = E_i – E_f$.

Keep practicing those orbit jumps ✨ – you’ve got this!