Bohr ModelJEE Main

Energy levels and spectral lines — interactive Physics simulation for IIT-JEE.

Concept

Bohr proposed electrons orbit the nucleus only in quantised energy levels $E_n = -13.6\,Z^2/n^2$ eV. When an electron drops from a higher to a lower level, the energy difference leaves as a single photon, producing the sharp spectral lines we observe.

Key formula

E_n = -\frac{13.6\,Z^2}{n^2}\,\text{eV}, \quad \Delta E = 13.6\,Z^2\!\left(\frac1{n_f^2}-\frac1{n_i^2}\right), \quad r_n = \frac{0.529\,n^2}{Z}\,\text{Å}

Derivation

Quantising angular momentum ( $mvr = n\hbar$ ) and balancing Coulomb attraction against the centripetal need gives discrete radii $r_n \propto n^2/Z$ and energies $E_n \propto -Z^2/n^2$ .

A jump $n_i \to n_f$ releases $\Delta E = E_i - E_f$ . The photon wavelength follows from $\lambda = hc/\Delta E = 1240/\Delta E$ nm. Final level $n_f = 1,2,3$ name the Lyman, Balmer and Paschen series.

Scenarios to explore

Bohr Model — Energy levels and the hydrogen spectral series.
Lyman-alpha (2 → 1) — The biggest hydrogen jump from the first excited state: ultraviolet, 10.2 eV.
Paschen series (4 → 3) — Transitions ending at n = 3 fall in the infrared.
He⁺ 4 → 2 — A one-electron ion with Z = 2: energies scale by Z².

Real-world applications

Explaining the hydrogen emission spectrum (Balmer's red Hα at 656 nm).
Atomic absorption/emission spectroscopy of stars.
Lasers and discharge lamps.

JEE exam tips

Ground state of hydrogen: $E_1 = -13.6$ eV, $r_1 = 0.529$ Å. Memorise these.
Ionisation energy from level $n$ is $13.6\,Z^2/n^2$ eV.

Common mistakes

Forgetting the $Z^2$ scaling for He⁺, Li²⁺ and other one-electron ions.
Sign errors — bound energies are negative; the photon energy is the positive difference.
Confusing the series (defined by $n_f$ ) with the individual lines (set by $n_i$ ).

Exam traps to avoid

Largest wavelength in a series is the smallest-energy jump ( $n_f+1 \to n_f$ ).
The Bohr model works only for one-electron (hydrogen-like) systems.