Calculate the energy level, orbital radius, and electron velocity of a hydrogen-like atom in the Bohr model, and the transition energy between levels.
Energy Level
Principal quantum number. 1 = ground state, 2 = first excited state, etc.
Z = 1 for hydrogen, Z = 2 for He⁺, Z = 3 for Li²⁺.
Energy levels n=1–6 for Z=1
Level
Eₙ (eV)
rₙ (Å)
Orbital Radius & Velocity
Principal quantum number. 1 = ground state, 2 = first excited state, etc.
Z = 1 for hydrogen, Z = 2 for He⁺, Z = 3 for Li²⁺.
Transition
The electron's starting energy level.
The electron's ending energy level. Higher than n1 = absorption; lower = emission.
Z = 1 for hydrogen, Z = 2 for He⁺, Z = 3 for Li²⁺.
Result
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Enter values above to compute.
4 min read3 steps7 terms3 examples6 FAQsEₙ = −13.6 · Z² / n² eV
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Walk-through
How to Use This Calculator
3 steps▸
1
Pick the tab that matches your question
Use the Energy Level tab when you want Eₙ for a given level and element. Use the Radius & Velocity tab for the electron's orbital size and speed at that level. Use the Transition tab when you want the energy and wavelength of a photon absorbed or emitted as an electron moves between two levels.
2
Set n and the atomic number Z
n is the principal quantum number (1 = ground state, 2 = first excited state, and so on) — enter a whole number of 1 or higher. Z is the atomic number of the hydrogen-like ion (Z=1 for hydrogen, Z=2 for He⁺, Z=3 for Li²⁺); the Bohr model is exact only for one-electron systems.
3
Read the result and the energy ladder
The result card shows the headline value for the active tab, plus a plain-language interpretation. On the Energy Level tab, the table below the inputs lists Eₙ and rₙ for n=1 through 6 at the same Z, so you can see how the levels compare at a glance.
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Reference
Formula & Methodology
3 formulas▸
Energy of Level n
Eₙ = −13.6 · Z² / n² eV
The energy of an electron in level n of a hydrogen-like ion, in electron-volts. The −13.6 eV is the Rydberg energy (the ground-state binding energy of hydrogen, Z=1, n=1). Energy is negative because the electron is bound; it approaches zero as n→∞, which is the point of ionization. Example: hydrogen ground state (Z=1, n=1) gives Eₙ = −13.6 eV exactly, matching the measured hydrogen ionization energy.
Orbital Radius & Velocity
rₙ = 0.529 · n² / Z Å; vₙ = 2.188×10⁶ · Z / n m/s
The Bohr radius rₙ (in ångströms) is the classical orbit radius at level n, scaled from the Bohr radius constant a₀ = 0.529 Å. It grows with the square of n and shrinks as Z increases (a more charged nucleus pulls the electron in tighter). The orbital velocity vₙ comes from α·c·Z/n, where α is the fine-structure constant and c is the speed of light — at n=1, Z=1 this gives about 2,188,000 m/s, roughly 0.73% of the speed of light.
The energy exchanged when an electron moves from level n1 to level n2. A positive ΔE (n2 > n1) means the electron absorbs a photon to jump up; a negative ΔE (n2 < n1) means it emits a photon falling down — the calculator reports the magnitude and labels the direction. The wavelength uses the standard hc ≈ 1240 eV·nm shortcut. Example: n1=2 → n2=1 (Z=1) gives ΔE = 10.2 eV emitted, λ ≈ 121.6 nm — the hydrogen Lyman-alpha line.
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Glossary
Key Terms Explained
7 terms▸
Bohr model ↗An early quantum model of the atom (Niels Bohr, 1913) in which electrons orbit the nucleus only in fixed, quantized energy levels, rather than any arbitrary orbit, and jump between levels by absorbing or emitting a photon.
Energy level ↗One of the discrete, quantized energy states an electron can occupy in an atom, labeled by the principal quantum number n = 1, 2, 3, … Lower n means a more tightly bound, lower-energy state.
Orbital radius ↗The distance from the nucleus to the electron's orbit in the Bohr model at a given energy level n, which scales with n² and shrinks as the nuclear charge Z increases.
Bohr radius ↗The orbital radius of the electron in hydrogen's ground state (n=1, Z=1), a physical constant equal to about 0.529 Å (5.29×10⁻¹¹ m), often written a₀.
Ionization energy ↗The energy required to completely remove an electron from a given level to n = ∞ (a free, unbound electron). For level n it equals the magnitude of that level's (negative) energy, 13.6·Z²/n² eV.
Transition ↗The jump of an electron from one energy level to another, accompanied by absorption (moving to a higher level) or emission (moving to a lower level) of a photon whose energy exactly matches the level difference.
Hydrogen-like atom ↗Any atom or ion with exactly one electron orbiting a nucleus of charge Z, such as H (Z=1), He⁺ (Z=2), or Li²⁺ (Z=3) — the only systems the simple Bohr model solves exactly.
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Scenarios
Real-World Examples
3 worked examples▸
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Hydrogen Ground State
The textbook baseline value
Energy level n 1Atomic number Z 1
Eₙ = −13.6·1²/1² = −13.6 eV — the defining ground-state energy of hydrogen, and the number every other Bohr-model result is measured against. It also equals hydrogen's ionization energy exactly, since removing this electron to n=∞ requires exactly 13.6 eV.
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Ground-State Orbit Size
How big is a hydrogen atom, really?
Energy level n 1Atomic number Z 1
rₙ = 0.529·1²/1 = 0.529 Å — the Bohr radius a₀, the classic textbook size of a hydrogen atom's electron orbit — with an orbital velocity of about 2,188,000 m/s, roughly 0.73% of light speed, matching the standard reference values used across chemistry and physics textbooks.
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Lyman-Alpha Emission
An electron falls from n=2 to n=1
Initial level n1 2Final level n2 1Atomic number Z 1
ΔE = 13.6·(1/1² − 1/2²) = 10.2 eV emitted, giving λ = 1240/10.2 ≈ 121.6 nm — the hydrogen Lyman-alpha line, in the ultraviolet, one of the most studied spectral lines in astrophysics and laboratory spectroscopy.
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Reference
Cite This Calculator
APA & MLA▸
Use either format to cite this calculator in a paper, report, or resource list.
The Bohr model was the first quantum picture of the atom: instead of a continuum of possible orbits, an electron can only occupy a fixed ladder of energy levels, and it moves between them by absorbing or emitting a photon of exactly the right energy. This calculator applies the model to hydrogen and hydrogen-like ions to compute energy, orbital size, speed, and transition energies.
Why energy is quantized and negative
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In the Bohr model, an electron bound to a nucleus can only exist at specific, discrete energy levels indexed by the principal quantum number n = 1, 2, 3, … The energy of each level, Eₙ = −13.6·Z²/n² eV, is negative because the electron is bound — it takes positive energy input to pull it away from the nucleus. As n increases, Eₙ climbs toward zero, and at n = ∞ the electron is completely free (ionized). The magnitude of Eₙ at any level is therefore also that level's ionization energy: the energy needed to remove the electron entirely from that starting point.
How orbital radius and velocity scale
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The orbital radius rₙ = 0.529·n²/Z Å grows with the square of n, meaning higher-energy orbits are dramatically larger — the n=2 orbit is four times the radius of n=1. A larger nuclear charge Z pulls the electron into a tighter orbit, shrinking rₙ proportionally. The orbital velocity vₙ = 2.188×10⁶·Z/n m/s comes from balancing the Coulomb attraction toward the nucleus against the centripetal force needed to keep the electron in a circular path — it scales up with Z (a stronger pull requires more speed to maintain orbit) and down with n (higher, more distant orbits move more slowly).
Reading a transition: absorption vs. emission
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The Transition tab computes the energy exchanged when an electron moves from an initial level n1 to a final level n2. If n2 is higher than n1, the electron has gained energy — it absorbed a photon to make the jump. If n2 is lower than n1, the electron has released energy — it emitted a photon falling to a more tightly bound state. The calculator always reports the magnitude of that energy difference alongside a clear absorption/emission label, plus the corresponding photon wavelength using λ(nm) = 1240/ΔE(eV).
Where the simple Bohr model breaks down
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The Bohr model is exact only for hydrogen-like systems — a single electron orbiting a nucleus of charge Z, such as hydrogen itself, He⁺, or Li²⁺. It ignores electron spin, relativistic corrections, and the electron-electron repulsion (shielding) present in any atom with more than one electron, so it cannot be applied directly to neutral helium or heavier elements. Modern quantum mechanics — the Schrödinger equation and its relativistic extension, the Dirac equation — is needed to model multi-electron atoms accurately, though the Bohr model's energy-level formula remains a remarkably good first approximation for one-electron systems.
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Questions
Frequently Asked Questions
6 questions▸
What is the formula for energy levels in the Bohr model?+
Eₙ = −13.6 · Z² / n² electron-volts, where n is the principal quantum number (n=1,2,3,…) and Z is the atomic number of the hydrogen-like ion. The result is negative because the electron is bound to the nucleus; the magnitude equals the ionization energy from that level.
What is hydrogen's ground-state energy?+
For hydrogen (Z=1) in its lowest level (n=1), Eₙ = −13.6 eV exactly — this is the Rydberg energy and also hydrogen's ionization energy, one of the most-cited reference values in atomic physics.
What is the formula for the Bohr orbital radius?+
rₙ = 0.529 · n² / Z ångströms, where 0.529 Å is the Bohr radius a₀ (hydrogen's ground-state orbit size). Radius grows with the square of n and shrinks as the atomic number Z increases.
What is an electron transition, and how do I know if it's absorption or emission?+
A transition is an electron jumping between energy levels. On the Transition tab, moving to a higher level (n2 greater than n1) is absorption; falling to a lower level (n2 less than n1) is emission. The calculator reads your n1 and n2 order and labels the direction automatically.
What happens as n approaches infinity?+
As n→∞, Eₙ approaches zero — the electron becomes unbound (ionized) and free of the nucleus. The energy needed to reach that point starting from level n is the ionization energy, 13.6·Z²/n² eV.
What units does this calculator use?+
Energy is in electron-volts (eV), orbital radius in ångströms (Å), orbital velocity in meters per second (m/s), and transition wavelength in nanometers (nm) — the standard units used across atomic physics and chemistry textbooks.
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