Bohr Model Calculator

Calculate Bohr orbital energy, radius, speed, angular momentum, frequency, and wavelength from atomic number and principal quantum number. Bohr Model Calculator helps students, educators, and researchers analyze hydrogen-like atomic structure and spectra.

Highlights

  • Calculate orbital energy and ionization energy for hydrogen-like atoms.
  • Determine Bohr orbit radius and electron speed from atomic number and energy level.
  • Compute quantized angular momentum for each principal quantum number.
  • Find orbital frequency and electron de Broglie wavelength.
  • Supports multiple units, including eV, joules, nanometers, and meters per second.

Table of Contents

  1. How Bohr Model Calculator Works?
  2. Practical Use Cases
  3. Accuracy and Limitations
  4. Quick Examples
  5. Frequently Asked Questions (FAQs)
  6. Cite This Calculator

How Bohr Model Calculator Works?

The Bohr model treats an electron in a hydrogen-like atom as occupying an allowed orbit identified by the principal quantum number n. The nucleus has atomic number Z, and the calculator accepts integer values of Z from 1 to 118 and n from 1 to 100. The model applies most directly to one-electron species such as hydrogen, He+, and Li2+.

The electron's total orbital energy is negative because the electron is bound to the nucleus. The energy level is calculated from:

En=−13.6Z2n2 eV

In joules, the equivalent expression is En=−mee4Z28ε2h2n2. The ionization energy is the positive energy required to move the electron from that orbit to zero energy:

Eion=−En=13.6Z2n2 eV

The orbit radius increases with n2 and decreases as nuclear charge increases:

rn=a0n2Z

Here, a0 is the Bohr radius. The electron speed is:

vn=αcZn

Bohr's angular-momentum condition gives:

L=nℏ

The orbital frequency is found from the speed and circumference of the orbit, while the de Broglie wavelength follows from the electron momentum:

f=vn2πrn,λ=hmevn

Calculated quantityBohr-model relationshipAvailable output units
Orbital energyEn∝−Z2/n2J, eV, kJ/mol
Ionization energyEion=−EnJ, eV, kJ/mol
Orbit radiusrn∝n2/Zm, cm, nm, Å
Electron speedvn∝Z/nm/s, km/s
Angular momentumL=nℏJ·s, eV·s
Orbital frequencyf=vn/2πrnHz, kHz, MHz, GHz
de Broglie wavelengthλ=h/mevnm, nm, Å, pm

Practical Use Cases

Physics students can use the calculator to examine how changing the atomic number or principal quantum number affects orbital energy, radius, speed, angular momentum, frequency, and electron wavelength in hydrogen-like systems.

Chemistry students can estimate energy levels and ionization energies for one-electron ions, helping connect atomic number and electron configuration to simple spectral and bonding concepts.

Educators can use the calculator to demonstrate quantized angular momentum, the dependence of energy on Z2/n2, and the increasing size of higher Bohr orbits through worked classroom examples.

Spectroscopy learners can compare the calculated orbital energies and electron wavelengths with the expected behavior of hydrogen-like atoms when interpreting idealized atomic spectra.

Researchers and technical users can generate quick first-order estimates for hydrogenic ions before applying more advanced quantum-mechanical, relativistic, or many-electron calculations.

Accuracy and Limitations

The calculator uses the nonrelativistic Bohr model with standard physical constants and is exact in its idealized form for hydrogen-like, one-electron species. It is not quantitatively reliable for neutral atoms or ions containing multiple electrons because electron–electron interactions, shielding, spin, orbital subshell structure, and electron correlation are omitted. It also excludes relativistic, fine-structure, Lamb-shift, hyperfine, and finite-nuclear-size effects, which become increasingly important for high atomic numbers. The allowed input ranges extend to Z = 118 and n = 100, but those numerical limits do not make the model physically accurate for every atom or highly excited state.

Quick Examples

Example #1: Hydrogen Ground State

Analyze the fundamental Bohr orbit of a hydrogen atom.

Example #2: Helium Ion Excited State

Calculate Bohr-model properties for a singly ionized helium atom in its second energy level.

Example #3: Lithium Ion Higher Orbit

Examine the orbital characteristics of a hydrogen-like lithium ion at the third principal energy level.

Frequently Asked Questions

The Bohr model describes electrons as occupying specific, quantized energy levels around a positively charged nucleus. Electrons can move between these levels by absorbing or emitting discrete amounts of energy.

The model works most accurately for hydrogen-like species, which have only one electron, such as hydrogen, He⁺, and Li²⁺. It is less accurate for atoms with multiple electrons because electron interactions are not fully represented.

The principal quantum number identifies an electron’s allowed energy level or orbit. Larger values of n generally correspond to higher energy, larger orbital radius, and lower electron speed.

For a hydrogen-like atom, increasing the atomic number increases the magnitude of the electron’s binding energy. Higher nuclear charge also produces smaller orbital radii and greater electron speeds.

An electron emits a photon when it moves to a lower energy level and absorbs a photon when it moves to a higher one. The photon’s frequency and wavelength are determined by the energy difference between the two levels.

Cite This Calculator