Bohr Radius Calculator
Find the radius of an electron orbit in the Bohr model from the quantum number and atomic number.
⚛️ What is the Bohr Radius Calculator?
The Bohr radius calculator finds the radius of an electron's orbit in the Bohr model of the atom. Niels Bohr's 1913 model pictured electrons circling the nucleus in fixed, quantised orbits, and the radius of each is set by a simple rule: r equals n squared divided by Z, times the Bohr radius a-nought. Give it the shell number and the atomic number and it returns the orbit size.
It is a core teaching tool in chemistry and physics. Students use it to see how atoms are sized, why higher shells sit much farther out, and how a heavier nucleus pulls its electrons inward. Working out that the hydrogen ground state has a radius of about 53 picometres, while the n = 3 shell reaches nearly 480 picometres, makes the structure of the atom concrete. It also anchors the idea of the Bohr radius as a fundamental constant, the natural yardstick for atomic distances.
Two dependencies drive the result. The radius grows with the square of the principal quantum number n, so the second shell is four times the ground-state size and the third is nine times. It shrinks in proportion to the atomic number Z, because more protons mean a stronger pull. The formula is exact only for hydrogen-like systems, those with a single electron such as He+ or Li2+; for atoms with many electrons it is an instructive approximation, since electrons shield one another from the nucleus.
This calculator is useful because it turns two whole numbers into an orbit radius expressed every way you might need, in metres, nanometres, picometres, and as a multiple of the Bohr radius, with the working shown so the n-squared-over-Z scaling is clear.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Hydrogen ground state
Example 2 - Hydrogen second shell
Example 3 - Singly ionised helium (He+)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Bohr radius?
The Bohr radius, written a-nought and equal to about 5.29 x 10^-11 metres (0.529 angstroms), is the radius of the electron's ground-state orbit in a hydrogen atom according to the Bohr model. It sets the natural size scale of atoms and appears throughout atomic physics as a fundamental length.
How do you calculate the Bohr model orbit radius?
Use r = n squared divided by Z, times the Bohr radius a-nought. Here n is the principal quantum number (the shell) and Z is the atomic number. For hydrogen's ground state (n = 1, Z = 1) the radius is a-nought, about 5.29 x 10^-11 m. For n = 2 it is four times larger.
How does the orbit radius change with n?
It grows as n squared. The n = 1 orbit has radius a-nought, n = 2 is 4 a-nought, n = 3 is 9 a-nought, and so on. So electrons in higher shells orbit much farther from the nucleus, and the spacing between shells increases rapidly with n.
How does atomic number affect the Bohr radius?
The radius is inversely proportional to Z, the number of protons. A larger nuclear charge pulls the electron closer, so a hydrogen-like ion with Z = 2 (singly ionised helium) has orbits half the size of hydrogen's for the same n. The formula r = n squared / Z times a-nought captures this.
What is the Bohr radius in angstroms or picometres?
The Bohr radius is about 0.529 angstroms, or 52.9 picometres, or 0.0529 nanometres. These smaller units are convenient at the atomic scale. This calculator shows the orbit radius in metres, nanometres, and picometres so you can use whichever suits your work.
Is the Bohr model still accurate?
The Bohr model correctly predicts hydrogen's energy levels and orbit sizes and is a superb teaching tool, but it is a simplification. Modern quantum mechanics replaces fixed orbits with probability clouds (orbitals). The Bohr radius survives as the most probable distance of the electron in the true hydrogen ground state.
What is a hydrogen-like atom?
A hydrogen-like atom or ion has a single electron orbiting a nucleus of charge Z, such as He+, Li2+, or Be3+. The Bohr formula r = n squared / Z times a-nought applies exactly to these one-electron systems. For multi-electron atoms it is only a rough guide, because electrons shield one another.
Why is the ground-state radius the smallest?
Because n = 1 is the lowest allowed orbit in the Bohr model, and radius grows with n squared. The electron cannot spiral closer, which is one of the model's key ideas: quantisation sets a minimum orbit and prevents the atom from collapsing. Larger n means a larger, higher-energy orbit.