de Broglie’s Insight into Bohr’s “Magic” Rule ✨
Bohr boldly claimed that an electron circling the nucleus can have only certain values of angular momentum: \(L_n = n\dfrac{h}{2\pi}\) with \(n = 1,2,3,\dots\). Why these special values? Louis de Broglie’s wave idea makes the mystery vanish! :contentReference[oaicite:0]{index=0}
1️⃣ How Standing Waves Make the Rule
- Think of the electron not just as a tiny ball but as a wave that wraps around its orbit. 🌊
- The wave must “join up with itself” neatly. That happens only when the orbit’s circumference fits an exact number of wavelengths:
\(2\pi r_n = n\lambda\) ( \(n = 1,2,3…\) ). :contentReference[oaicite:1]{index=1} - de Broglie gave every particle a wavelength: \(\lambda = \dfrac{h}{p} = \dfrac{h}{mv_n}\) when the speed is far below light speed. :contentReference[oaicite:2]{index=2}
- Plugging that λ into the circumference condition gives \(m v_n r_n = n\dfrac{h}{2\pi}\) – exactly Bohr’s quantisation rule! 🎉 :contentReference[oaicite:3]{index=3}
- Only these resonant orbits survive; other waves cancel themselves out. Electrons “choose” these stable tracks, so atoms emit or absorb light at specific colours. 🌈
2️⃣ Must-Know Equations 💡
- Standing-wave fit: \(2\pi r_n = n\lambda\)
- de Broglie relation: \(\lambda = \dfrac{h}{p} = \dfrac{h}{mv_n}\)
- Quantised angular momentum: \(L_n = m v_n r_n = n\dfrac{h}{2\pi}\)
- Photon energy for a jump: \(h\nu_{if} = E_{n_i} – E_{n_f}\) (gives the bright lines seen in spectra)
3️⃣ What This Means for the Atom 🚀
Because only whole-number standing waves persist, electrons occupy discrete orbits. Every leap between orbits releases (or absorbs) a photon whose colour matches the energy gap. That’s why atomic spectra look like neat bar-codes, not rainbows.
4️⃣ Where Bohr’s Picture Falls Short 🤔
- Works great for hydrogenic atoms – one electron circling a nucleus \(+Ze\) (examples: H, He+, Li2+). :contentReference[oaicite:4]{index=4}
- Fails for atoms with two or more electrons: Bohr ignored the push-and-pull between electrons.
- Predicts the positions of spectral lines but not their brightness; experiments show some lines shine brighter than others, and Bohr can’t explain why. :contentReference[oaicite:5]{index=5}
- Quantum mechanics steps in with a fuller, more flexible toolkit for complex atoms. 🧮
🎯 High-Yield NEET Nuggets
- Standing-wave fit \(2\pi r_n = n\lambda\) directly leads to Bohr’s quantised angular momentum.
- de Broglie wavelength \(\lambda = \dfrac{h}{p}\) links a particle’s motion to its wave nature.
- Core quantisation formula \(L_n = n\dfrac{h}{2\pi}\) underpins energy-level calculations.
- Bohr model ≈ hydrogenic atoms only; multi-electron atoms need full quantum mechanics.
- Emission and absorption spectra arise from electrons jumping between these allowed orbits.
