The Bohr model formulas
- Radius: rₙ = 0.529 × n² ÷ Z Å (52.9 pm for hydrogen's ground state)
- Energy: Eₙ = −13.6 × Z² ÷ n² eV = −2.18 × 10⁻¹⁸ × Z² ÷ n² J
- Speed: vₙ = 2.18 × 10⁶ × Z ÷ n m/s
- Energies: kinetic energy = −E, potential energy = 2E, so KE : PE : total = 1 : −2 : −1
Hydrogen, level by level
| n | Radius (Å) | Energy (eV) | Speed (m/s) |
|---|---|---|---|
| 1 | 0.529 | −13.6 | 2.18 × 10⁶ |
| 2 | 2.12 | −3.4 | 1.09 × 10⁶ |
| 3 | 4.76 | −1.51 | 0.727 × 10⁶ |
| 4 | 8.46 | −0.85 | 0.545 × 10⁶ |
| 5 | 13.2 | −0.544 | 0.436 × 10⁶ |
| 6 | 19 | −0.378 | 0.363 × 10⁶ |
Remember the patterns: radius grows as n², energy shrinks as 1/n², and speed falls as 1/n. Going from H to He⁺ halves the radius and multiplies the energy by 4.
For the wavelength of light given out when an electron drops between levels, use the hydrogen spectrum calculator.
Questions students ask
What is the radius of the first Bohr orbit?
0.529 Å, or 52.9 pm, for hydrogen. For He⁺ it's half that, 0.2645 Å, because the radius is divided by Z.
What is the energy of an electron in the second orbit of hydrogen?
−13.6 ÷ 4 = −3.4 eV.
Why is the energy negative?
Zero is the energy of a free electron far from the nucleus. A bound electron has less energy than that, so its energy is negative, and you must supply energy to remove it.
What is the ratio of radii of the 1st and 2nd orbits?
1 : 4, because the radius goes as n².