Bohr Model Calculator

Enter the initial orbit (n₁), final orbit (n₂), and atomic number (Z) to compute electron transition properties using the Bohr Model Calculator. You'll get the energy difference (ΔE), photon frequency (ν), wavelength (λ), and orbital energy levels for hydrogen-like atoms. Works for H, He⁺, Li²⁺, and other one-electron ions. The calculator also identifies the spectral series (Lyman, Balmer, Paschen) and shows whether the transition emits or absorbs a photon.

Select a hydrogen-like ion or enter a custom atomic number.

Only used when 'Custom Z' is selected above.

The higher energy level the electron starts from (n ≥ 1).

The lower energy level the electron transitions to (n ≥ 1, must be < n₂ for emission).

Results

Energy Difference (ΔE)

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Energy Difference (ΔE)

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Photon Frequency (ν)

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Wavelength (λ)

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Wavenumber (1/λ)

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Energy at n₂ (initial level)

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Energy at n₁ (final level)

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Transition Type

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Spectral Series

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Energy Levels at n₁ and n₂ (eV)

Frequently Asked Questions

What is the Bohr model?

The Bohr model, proposed by Niels Bohr in 1913, describes the hydrogen atom as a nucleus surrounded by an electron orbiting in fixed, quantized circular paths. Each allowed orbit corresponds to a specific energy level given by Eₙ = −13.6 × Z²/n² eV. While superseded by quantum mechanics, it correctly predicts hydrogen's spectral lines.

What is a hydrogen-like atom or ion?

A hydrogen-like (or hydrogenic) atom is any ion that has only one electron, such as H, He⁺, Li²⁺, Be³⁺, or B⁴⁺. The Bohr model applies exactly to these systems because the single-electron simplification holds. The energy levels scale as Z², where Z is the atomic number.

How is the energy difference ΔE calculated?

The energy of level n is Eₙ = −13.6 × Z² / n² eV. The energy difference between two levels is ΔE = E_final − E_initial = 13.6 × Z² × (1/n₁² − 1/n₂²) eV. A positive ΔE means a photon is emitted (emission), while a negative value means a photon is absorbed.

How does the calculator find photon frequency and wavelength?

Once ΔE is known in joules, the photon frequency is ν = |ΔE| / h, where h = 6.626×10⁻³⁴ J·s (Planck's constant). The wavelength is λ = h·c / |ΔE|, where c = 3×10⁸ m/s. Wavelength is then converted to nanometres for practical use.

What are the Lyman, Balmer, and Paschen series?

These spectral series are named by the final orbit (n₁). Lyman series transitions end at n₁ = 1 and emit ultraviolet light. Balmer series transitions end at n₁ = 2 and produce visible light for hydrogen. Paschen series transitions end at n₁ = 3 and fall in the infrared region.

What is the Rydberg formula and how does it relate?

The Rydberg formula 1/λ = R∞ × Z² × (1/n₁² − 1/n₂²) directly gives the wavenumber of emitted or absorbed light, where R∞ ≈ 1.097×10⁷ m⁻¹ is the Rydberg constant. This is mathematically equivalent to ΔE = h·c/λ derived from the Bohr energy levels, providing the same result for wavelength.

Can the Bohr model be used for multi-electron atoms?

Strictly speaking, no. The Bohr model only works accurately for one-electron (hydrogen-like) ions. For multi-electron atoms, electron–electron repulsion and quantum mechanical effects make the simple Z²/n² energy formula inaccurate. Modern quantum mechanics (Schrödinger equation) is needed for those cases.

What does it mean if n₁ equals n₂?

If the initial and final quantum numbers are equal, there is no transition — the electron stays in the same orbit. ΔE = 0, meaning no photon is emitted or absorbed. A valid transition requires n₁ ≠ n₂; this calculator will flag that case and return zero values.

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