Bohr model calculator
Hydrogen's ground state sits at −13.6 eV and one Bohr radius; the next level sits at −3.4 eV and four Bohr radii. Enter any positive whole quantum number to get its energy and orbit radius. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- N
- 1
- Energy ev
- -13.6
- Binding energy ev
- 13.6
- Radius m
- 0.0000000001
- Radius angstrom
- 0.5291772109
The Bohr model of the hydrogen atom assigns a discrete energy and a circular orbital radius to each principal quantum number n. Ground state n equals one yields minus 13.6 electronvolts and one Bohr radius; n equals two yields minus 3.4 electronvolts and four Bohr radii.
One whole number, two textbook formulas
In 1913 Niels Bohr pictured the hydrogen electron as a point charge on a circular orbit whose angular momentum can only be a whole multiple of ħ. That single rule, with the nuclear charge fixed at one, gives the two formulas every introductory course uses: the energy is −13.6 electronvolts divided by n squared, and the radius is n squared times the Bohr radius. So the ground state sits at −13.6 eV on one Bohr radius, the first excited state at −3.4 eV on four Bohr radii, and the third level at −13.6/9 eV on nine. As n grows, the energy climbs toward zero, the ionization limit.
What the model does not cover
This is the hydrogen Bohr atom and nothing more. Ions with a larger nuclear charge, atoms with several electrons, fine structure, the Lamb shift and reduced-mass corrections beyond the classroom 13.6 eV constant are out of scope, and so are photon wavelengths between two levels. The calculator also refuses zero, negative and fractional quantum numbers, so it never invents an orbit that does not exist. That makes it a dependable check for homework and for auto-graded exam questions.
What the Bohr model computes from the principal quantum number
Niels Bohr’s 1913 model treats the hydrogen electron as a point charge in a circular orbit whose angular momentum is an integer multiple of ħ. That single quantization condition, with nuclear charge Z fixed at one, yields two closed formulas every introductory chemistry and physics course uses.
Bound-state energy is E_n equals minus 13.6 electronvolts divided by n squared, so the ground state n equals one sits at minus 13.6 electronvolts, the first excited state n equals two sits at minus 3.4 electronvolts, n equals three sits at minus 13.6 over nine electronvolts, and the ionization limit as n tends to infinity is zero.
Radius therefore grows as n squared: the n equals two orbit is four Bohr radii, the n equals three orbit is nine Bohr radii, and Rydberg atoms with large n become macroscopic by atomic standards. Hydrogen-like ions with Z greater than one are also out of scope: this landing is the hydrogen Bohr atom only.
What you can do with it
Classroom and homework checks
Let students submit a principal quantum number n and verify hydrogen energy in electronvolts and orbital radius against textbook Bohr-model methods.
Exam and LMS auto-grading
Mark Bohr-model items with a deterministic oracle that rejects non-integer or non-positive n and never invents a fractional orbit.
FAQ
How do I calculate hydrogen Bohr energy levels from n?
Send a positive integer principal quantum number n.
What happens if n is not a positive integer?
Zero, negative, and non-integer values are not allowed principal quantum numbers and are not billed.
How does the orbit radius grow with n?
With the square of n, so the orbit for n equal to 3 is already nine Bohr radii. One Bohr radius is 0.529 ångströms, or 5.29177210903 × 10⁻¹¹ metres.
How much energy does it take to ionize hydrogen from the ground state?
13.6 electronvolts. The ground state sits at −13.6 eV and the ionization limit, reached as n grows without bound, is zero; the result also reports the binding energy of the level as a positive number.
Can I use it for helium ions or atoms with several electrons?
No. It covers the hydrogen Bohr atom only, with the nuclear charge fixed at one. Hydrogen-like ions, multi-electron atoms, fine structure, the Lamb shift and photon wavelengths between levels are outside its scope.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/phys/bohr-model \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":1}'const res = await fetch("https://api.kit.forhosting.com/phys/bohr-model", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/bohr-model",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/bohr-model", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/bohr-model", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"n": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.bohr_model",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_n | 10000 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This Bohr model calculator takes that positive integer and returns the bound-state energy in electronvolts together with the orbital radius in metres and angstroms, using the classroom hydrogen formulas E_n equals minus 13.6 electronvolts over n squared and r_n equals n squared times the Bohr radius a₀. Zero, negative, and non-integer quantum numbers are rejected with invalid_input because they are not allowed principal quantum numbers.
What the Bohr model computes from the principal quantum number
Orbital radius is r_n equals n squared times the Bohr radius a₀, where a₀ is 5.29177210903 times ten to the minus eleven metres, or 0.529177210903 angstroms. This endpoint evaluates both formulas for one positive integer n and always returns energy in electronvolts as the primary spectroscopic quantity, plus radius in metres and angstroms so a notebook can quote either convention. It does not treat multi-electron atoms, fine structure, Lamb shift, reduced-mass corrections beyond the classroom 13.6 electronvolt constant, or photon wavelengths between two levels; those are different search intents.
Input contract, response shape, and error behaviour
The failure is not billed. Pair the numbers with your own energy-level diagram if you need a sketch; this capability solves the two closed forms only and does not emit plots, Balmer-series wavelengths, or step-by-step prose beyond the formula field. POST a JSON body with one required field: n, the principal quantum number, which must be a positive integer from one through ten thousand. Aliases principal_quantum_number, quantum_number, and principal_n are accepted and mean the same thing. JSON numbers that are exact integers, and decimal-free numeric strings such as "2", are valid; values such as zero, minus one, 1.5, empty strings, and non-numeric text fail with invalid_input naming n. On success the body echoes n; energy_ev, the signed bound-state energy in electronvolts; binding_energy_ev, the positive ionization energy from that level, equal to 13.6 over n squared electronvolts; radius_m and radius_angstrom for the orbital radius; the constants rydberg_energy_ev equals 13.6 and bohr_radius_m; formula as a short ASCII identity; and unit_energy plus unit_radius set to eV and m. Classic checks: n equals one returns energy_ev minus 13.6 and radius one Bohr radius; n equals two returns energy_ev minus 3.4 and radius four Bohr radii.
Where a Bohr model calculator earns its keep
Classrooms and tutoring platforms use a Bohr model calculator so learners can check hydrogen energies and radii for arbitrary n without mixing up the n-squared factor in the denominator of energy versus the numerator of radius. Exam-prep generators and LMS plugins need a deterministic oracle that marks wrong a submitted decimal or a zero quantum number instead of silently returning a non-physical orbit. Failed validation does not consume balance. Spectroscopy primers and first-year lab notebooks often tabulate E_n in electronvolts next to r_n in angstroms when introducing line spectra; calling a tiny parse API avoids pulling a full quantum-chemistry stack for two high-school formulas. Keep n inside the published ceiling of ten thousand, pin the rounded energy_ev in tests, and you get reproducible hydrogen Bohr numbers for homework, grading pipelines, and edge workers without standing up a symbolic algebra system of your own.
Integration use cases
Lab notebooks and spectroscopy primers
Tabulate E_n in eV and r_n in metres or angstroms for hydrogen levels without importing a quantum-chemistry library.
Technical questions
What orbital radius does the Bohr model give?
r_n = n² a₀, with a₀ = 5.29177210903×10⁻¹¹ m (about 0.529 Å). The response includes radius_m and radius_angstrom.
What energy does n = 1 and n = 2 return?
Ground state n = 1 returns −13.6 eV and one Bohr radius. The first excited state n = 2 returns −3.4 eV and four Bohr radii.
What does the Bohr model calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.