Convert Bohr Radius to Planck Length
Convert bohr radii to planck lengths instantly. 1 bohr radius = 3.274095e+24 planck length — use the live calculator, the exact formula, a conversion table and worked examples. Also check the Planck Length to Bohr Radius converter for the reverse conversion.
Units explained
Bohr Radius
The Bohr radius is approximately 5.292×10⁻¹¹ meters — the most probable distance between the proton and electron in a hydrogen atom in its ground state.
Defined by Niels Bohr in his 1913 model of the hydrogen atom. The current CODATA 2018 value of 5.29177210903×10⁻¹¹ m is derived from the elementary charge, electron mass, and Planck's constant.
The Bohr radius is used throughout atomic physics, quantum chemistry, and computational chemistry as the natural length scale for electronic-structure calculations. The atomic unit (a.u.) of length equals the Bohr radius.
Defined by Bohr in 1913; current value via CODATA 2018.
Planck Length
The Planck length is approximately 1.616×10⁻³⁵ meters — the natural unit of length in Planck's system of natural units. It is constructed from the gravitational constant, the speed of light, and the reduced Planck constant.
Defined by Max Planck in 1899 as part of a natural unit system independent of any human or planetary reference. The current CODATA 2018 value is 1.616255×10⁻³⁵ m, derived from G, ℏ, and c.
The Planck length appears in quantum-gravity research and string theory as the scale at which classical notions of spacetime are expected to break down. It is not measurable directly with current technology.
Defined by Planck in 1899; value derived from fundamental constants, most recently updated by CODATA 2018.
Bohr Radius to Planck Length conversion formula
The relationship between bohr radii and planck lengths:
To convert bohr radii to planck lengths, multiply the value in bohr radii by 3.274095e+24. To reverse, multiply planck lengths by 3.054279e-25.
How to use this converter
Type a value into the calculator. The result in planck lengths updates as you type. Tap a quick value, copy the result with one click, or use the swap arrow to jump straight to the Planck Length to Bohr Radius converter for the reverse direction.
Step-by-step: convert bohr radii to planck lengths
- Write down the value in bohr radii (a₀).
- Multiply that value by the factor 3.274095e+24.
- The product is the equivalent value in planck lengths (ℓP).
- To reverse, multiply the planck length value by 3.054279e-25.
Worked examples
Example 1 — Convert 1 a₀ to ℓP:
1 × 3.274095e+24 = 3.274095e+24 ℓP
Example 2 — Convert 100 a₀ to ℓP:
100 × 3.274095e+24 = 3.274095e+26 ℓP
Real-world example — Molecular dimensions
The diameter of small molecular structures (around 2 bohr radii) is often converted into related sub-micron units when comparing measurements across different microscopy techniques or imaging modalities.
2 a₀ × 3.274095e+24 = 6.54819e+24 ℓP
Real-world example — Wavelengths across the spectrum
Optical and atomic-scale phenomena are routinely cross-converted between sub-micron units. A photon of wavelength 800 bohr radii can be re-expressed in planck lengths for direct comparison with another instrument's calibration data sheet.
800 a₀ × 3.274095e+24 = 2.619276e+27 ℓP
Bohr Radius to Planck Length conversion table
Standard reference values for converting bohr radii to planck lengths:
| Bohr Radius [a₀] | Planck Length [ℓP] |
|---|---|
| 0.01 | 3.274095e+22 |
| 0.1 | 3.274095e+23 |
| 1 | 3.274095e+24 |
| 2 | 6.54819e+24 |
| 3 | 9.822284e+24 |
| 4 | 1.309638e+25 |
| 5 | 1.637047e+25 |
| 10 | 3.274095e+25 |
| 20 | 6.54819e+25 |
| 30 | 9.822284e+25 |
| 40 | 1.309638e+26 |
| 50 | 1.637047e+26 |
| 100 | 3.274095e+26 |
| 500 | 1.637047e+27 |
| 1000 | 3.274095e+27 |
Frequently asked questions
How many planck lengths is 1 bohr radius?
How do I convert bohr radii to planck lengths?
How do I convert planck lengths back to bohr radii?
How many planck lengths is 100 bohr radii?
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Metric / SI (4 units)
Imperial / US Customary (4 units)
Sources & references
Conversion factor (1 a₀ = 3.274095e+24 ℓP) verified against the following authoritative sources:
- BIPM — The International System of Units (SI Brochure 9th ed.)
Official BIPM publication defining the seven SI base units (including the meter) and the rules for their use. The global authority on units of measurement.
- NIST — Guide to the SI
US National Institute of Standards and Technology reference covering the SI base and derived units with definitions and usage rules for US technical practice.
- NIST Special Publication 811 — Guide for the Use of the International System of Units
Detailed NIST guide covering exact conversion factors between SI and US customary units along with formatting and rounding conventions.
- NIST — Refinement of values for the yard and pound (Federal Register 1959)
The treaty (signed by US
- International Astronomical Union — System of Astronomical Constants
The IAU defines astronomical units including the AU (149597870700 m exactly) light-year and parsec used in astronomy and astrophysics.