Bohr Model Calculator

The Bohr model predicts discrete energy levels in hydrogen. Electrons transition between levels, emitting or absorbing photons at specific wavelengths that match observed spectral lines exactly.

⚛️ Quantum Physics📐 Eⁿ = -13.6/n² eV🌈 Spectroscopy

Initial Level n₁

Final Level n₂

Emission: n₁ > n₂ | Absorption: n₁ < n₂

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Energy Level	E_n	E_n = −13.6/n² eV	eV
Transition Energy	ΔE	13.6(1/n₁² − 1/n₂²)	eV
Wavelength	λ	hc/ΔE	nm
Ground State	n=1	E = −13.6 eV	eV
Ionization	ΔE	13.6 eV from n=1	eV

Step-by-Step Examples

Example 1

H-alpha Line

n1=3 to n2=2 (Balmer series).

dE = 13.6*(1/4 - 1/9) = 13.6*0.1389 = 1.89 eV
lambda = 1240/1.89 = 656 nm

✓ 656 nm — red H-alpha line

Example 2

Lyman Alpha

n1=2 to n2=1.

dE = 13.6*(1 - 0.25) = 10.2 eV
lambda = 1240/10.2 = 121.6 nm

✓ 121.6 nm — UV Lyman-alpha line

Example 3

Ionization from n=2

Energy to ionize hydrogen from n=2.

E2 = -13.6/4 = -3.4 eV
Ionization energy = 3.4 eV from n=2

✓ 3.4 eV needed to ionize from first excited state

Real-World Applications

🔭

Stellar Spectroscopy

Hydrogen spectral lines identify composition and temperature of stars. H-alpha at 656 nm is most prominent.

🌀

Quasar Redshift

Lyman-alpha forest in quasar spectra traces hydrogen clouds at various redshifts.

🧪

Atomic Physics

Bohr model was first successful quantum theory. Predicts all hydrogen wavelengths to 0.01% accuracy.

💡

Lasers

Population inversion between specific levels creates laser emission at Bohr-predicted wavelengths.

Common Mistakes to Avoid

⚠️

n is a positive integer only

n = 1, 2, 3... No fractional levels. Energy is quantized.

⚠️

Sign convention

E_n is negative (bound state). Energy of emitted photon = |E_n1 - E_n2| where n1 > n2.

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Bohr model works only for hydrogen

Works for hydrogen-like ions (He+, Li2+) with charge Z: E_n = -13.6*Z^2/n^2 eV. Fails for multi-electron atoms.

Frequently Asked Questions

Why are hydrogen energy levels discrete? ▾

Bohr postulated that only orbits with angular momentum L = n*h-bar are allowed. This quantization gives discrete energies E_n = -13.6/n^2 eV.

What is the Lyman series? ▾

Transitions to n=1 (ground state). All lines are in UV (91.2-121.6 nm). Lyman-alpha (n=2 to 1) at 121.6 nm is strongest.

What is the Balmer series? ▾

Transitions to n=2. First 4 lines are visible: H-alpha 656 nm (red), H-beta 486 nm (blue-green), H-gamma 434 nm (violet), H-delta 410 nm.

Where does the Bohr model fail? ▾

Cannot handle multi-electron atoms, spin, fine structure, or Zeeman effect. Quantum mechanics (Schrodinger equation) is needed for these.

What is the ionization energy of hydrogen? ▾

13.6 eV from ground state (n=1 to n=infinity). Corresponds to photon wavelength of 91.2 nm (far UV).

Related Physics Calculators

Formula Explorer connections

Interpretation: This relationship connects quantized energy, wavelength, probability, nuclear mass or radioactive change. Assumption: Use the correct particle, quantum state, nuclide and energy units. Idealized potentials, nonrelativistic motion or single decay channels may be assumed.

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