radius of the electrons orbit in a hydrogen atom
Published:
In Bohr’s atomic model, the radius of the electron’s orbit in a hydrogen atom (or hydrogen-like ion) is quantized — meaning it has discrete allowed values based on the principal quantum number $n$.
🧮 Formula for the Radius of Bohr Orbit:
\[r_n = n^2 \cdot r_1\]Where:
- $r_n$ = radius of the $n^\text{th}$ orbit
- $n$ = principal quantum number (1, 2, 3, …)
- $r_1$ = Bohr radius = $0.529 \times 10^{-10} \, \text{m}$ (or 0.529 Å)
So, the radius increases with the square of $n$:
🔢 Values of First Few Orbit Radii (for hydrogen):
| Orbit (n) | Radius $r_n$ (in meters) | Radius (in Ångströms, Å) |
|---|---|---|
| 1 | $0.529 \times 10^{-10} \, \text{m}$ | 0.529 Å |
| 2 | $2.116 \times 10^{-10} \, \text{m}$ | 2.116 Å |
| 3 | $4.761 \times 10^{-10} \, \text{m}$ | 4.761 Å |
| 4 | $8.464 \times 10^{-10} \, \text{m}$ | 8.464 Å |
⚛ General Formula (with constants):
\[r_n = \frac{n^2 h^2}{4\pi^2 m e^2 Z}\]Or simplified for hydrogen-like atoms:
\[r_n = \frac{n^2 \cdot a_0}{Z}\]Where:
- $a_0 = 0.529 \times 10^{-10} \, \text{m}$ (Bohr radius)
- $Z$ = atomic number (for hydrogen, $Z = 1$)
- $h$ = Planck’s constant
- $m$ = mass of the electron
- $e$ = charge of the electron
- $\pi$ = 3.14159…
