Quick answer In the Bohr model the hydrogen energy levels are Eₙ = −13.6/n² eV and the orbit radii are rₙ = n²·a₀ (a₀ = 0.529 Å = Bohr radius, n = principal quantum number). The emitted wavelength is λ[nm] = 1240/ΔE[eV]; the Balmer-α line (3→2) is 656 nm.
Change principal quantum number n to visualize energy levels, orbital radii, and emission spectra in real time. Explore Lyman, Balmer, and Paschen series transitions interactively.
Parameters
Results
E₁ (initial state)
-3.40 eV
E₂ (final state)
-0.85 eV
ΔE Released energy
2.55 eV
Wavelength λ
486 nm
Orbital radius r₁
0.848 nm
Orbital radius r₂
0.212 nm
Orbit Model
Energy Levels
Spectrum
Orbit
Arrow: n₁ → n₂ transition. Color corresponds to emitted wavelength.
Energy
Spectrum
Spectrum lines in the visible range (380-700 nm). The white line is the currently selected transition.
Series
💬 Conversation about the Bohr Model
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I heard that electrons in a hydrogen atom can only occupy specific "orbits" — why is there such a restriction?
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Bohr's 1913 condition was that electron angular momentum must be an integer multiple of $\hbar$: $mvr = n\hbar$. Combined with classical electromagnetism, this leads to discrete orbital radii $r_n = n^2 a_0$. Quantum mechanics later explained why this quantization occurs, but the model matched the experimental hydrogen spectrum remarkably well.
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The energy is $-13.6/n^2$ eV — why is it negative? What does negative energy mean?
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It is a matter of the reference point. The energy of an electron infinitely far from the nucleus is defined as zero. A bound electron can release work as it is attracted to the nucleus, so bound-state energies are negative. The value $-13.6$ eV at $n=1$ means that 13.6 eV is needed to remove the electron completely from hydrogen, or ionize it.
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We covered the Balmer series in class, but are there others like the Lyman series or Paschen series?
Yes. The final quantum number determines the spectral series. Transitions to $n=1$ form the Lyman series in ultraviolet, transitions to $n=2$ form the Balmer series in visible light, and transitions to $n=3$ form the Paschen series in near infrared. Dark Fraunhofer lines in the solar spectrum appear because hydrogen in the solar atmosphere absorbs specific wavelengths.
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So is the Bohr model perfect? Why do we use quantum mechanics now?
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It works for hydrogen, but not for multi-electron atoms such as helium and heavier elements. It also cannot explain electron spin, fine structure in spectral lines, or behavior in a magnetic field such as the Zeeman effect. Quantum mechanics with the Schrodinger equation uses probability density instead of fixed orbits and describes these effects accurately. The Bohr model is still useful as an introductory model for intuitively understanding energy levels.
Frequently Asked Questions
Increasing n expands the electron's orbital radius by a factor of n² and makes the energy level shallower at -13.6/n² eV. Additionally, the wavelength of light emitted during transitions between n values changes, allowing real-time observation of spectral series (Lyman, Balmer, etc.).
The Balmer series involves transitions from n ≥ 3 to n = 2, emitting light in the visible spectrum, while the Lyman series involves transitions from n ≥ 2 to n = 1, emitting light in the ultraviolet spectrum. You can compare the wavelengths of each series by changing the value of n in the simulator.
Because the reference point for zero energy is set at infinity (the state where the electron is completely separated from the nucleus), the energy of bound states is negative. The larger the absolute value of the negative number, the more strongly the electron is bound, with n = 1 being the most stable ground state.
No. The Bohr model assumes classical circular orbits for electrons and ignores wave-particle duality and the uncertainty principle. More accurate quantum mechanical models represent electrons using probability distributions (orbitals) rather than orbits, but the Bohr model is useful for basic understanding of energy levels.
What is Bohr Hydrogen Atom Model?
Bohr Hydrogen Atom Model is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Physical Model & Key Equations
The simulator is based on the governing equations behind Bohr Hydrogen Atom Model. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: The concepts behind Bohr Hydrogen Atom Model are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.
Set the initial quantum number n1 (ground state) using the n1Slider, typically starting at n=1 for hydrogen transitions from the ground state.
Adjust n2Slider to define the excited state quantum number; higher values (n=2,3,4...) reveal transitions in Lyman (UV), Balmer (visible), and Paschen (infrared) series.
Use the slider to set the maximum principal quantum number displayed in the energy level diagram, enabling visualization of multiple orbital radii and corresponding transition wavelengths.
Worked Example
For a Balmer-series transition in hydrogen: set n1=2 (lower level) and n2=4 (upper level). The simulator calculates the energy difference ΔE = 13.6 eV × (1/4 − 1/16) = 2.55 eV, yielding a photon wavelength of 487 nm (hydrogen-beta line, cyan emission). The orbital radius increases from a₀=0.053 nm at n=2 to r₄=0.212 nm at n=4, demonstrating the n² scaling relationship governing Bohr orbits.
Practical Notes
Lyman series (n1=1, n2≥2) produces UV photons above 91.2 nm; critical for stellar spectroscopy and plasma diagnostics in fusion reactors.
Balmer series (n1=2, n2≥3) generates visible wavelengths 656–365 nm; historically used for identifying hydrogen in nebulae and laboratory spectral analysis.
Paschen series (n1=3, n2≥4) emits infrared radiation (1875+ nm); relevant in thermal imaging and astrophysical observations of cool stellar atmospheres.
Larger nmax values reveal fine structure splitting and quantum defects neglected by the classical Bohr model, bridging toward quantum mechanical treatments.