Bohr Model Calculator

Calculate electron energy levels, orbital radius, and velocity for hydrogen-like atoms. Visualise electron orbits and photon transitions.

Key formulas (Bohr model):

Energy: En = –13.6 eV · Z² / n²

Radius: rn = a₀ · n² / Z   (a₀ = 0.529 Å)

Velocity: vn = (Z / n) · 2.18×10⁶ m/s

Transition (photon wavelength): 1/λ = R∞·Z²·(1/nf² – 1/ni²)

Number of protons (hydrogen-like ions: He⁺ Z=2, Li²⁺ Z=3, ...)
Energy level (n ≥ 1)
Photon transition (optional)
For emission, ni > nf; for absorption, ni < nf.
Calculating...

Understanding the Bohr Model

Proposed by Niels Bohr in 1913, the Bohr model was the first successful quantum description of the hydrogen atom. It combines classical circular orbits with quantisation of angular momentum.

Bohr's postulates:

  1. Electrons move in circular orbits under Coulomb attraction; only certain orbits are allowed.
  2. Angular momentum is quantised: L = n·ħ, where ħ = h/(2π) and n = 1,2,3,…
  3. Electrons emit/absorb photons only when jumping between orbits, with photon energy ΔE = hν.

Derivation of Key Formulas

From the balance of Coulomb force and centripetal force:
\(\frac{1}{4\pi\epsilon_0} \frac{Ze^2}{r^2} = \frac{mv^2}{r}\)

Quantisation of angular momentum: \(mvr = n\hbar\)

Solving these gives:

  • Radius: \(r_n = \frac{4\pi\epsilon_0 \hbar^2}{m e^2} \frac{n^2}{Z} = a_0 \frac{n^2}{Z}\) where \(a_0 = 0.529\) Å (Bohr radius).
  • Velocity: \(v_n = \frac{Z e^2}{4\pi\epsilon_0 \hbar n} = \frac{Z}{n} \alpha c\) with fine‑structure constant α ≈ 1/137.
  • Energy: \(E_n = -\frac{m e^4 Z^2}{2(4\pi\epsilon_0)^2 \hbar^2 n^2} = -13.6\,\text{eV} \frac{Z^2}{n^2}\).

Rydberg Formula for Transitions

The wavelength of emitted/absorbed light is given by:

\(\frac{1}{\lambda} = R_\infty Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)\)

where \(R_\infty = 1.097373 \times 10^7 \,\text{m}^{-1}\) (Rydberg constant). The photon energy is \(E_{\text{photon}} = |E_{n_i} - E_{n_f}|\).

Limitations of the Bohr Model

  • Works only for single‑electron systems (hydrogen, He⁺, Li²⁺, …).
  • Cannot explain fine structure (splitting of spectral lines).
  • Does not incorporate wave nature of electrons (de Broglie).
  • Violates the Heisenberg uncertainty principle for n=0 (would be at nucleus).
  • Replaced by the full quantum mechanical Schrödinger equation.

Real‑World Applications

  • Astronomy: Identifying elements in stars via spectral lines (Balmer series for hydrogen).
  • Lasers: Understanding population inversion and photon emission.
  • Quantum mechanics education: First stepping stone to modern atomic theory.
  • X‑ray spectra: Moseley's law for K‑alpha lines (similar Z² dependence).

Calculator Features:

  • Calculates energy (eV), radius (Å), velocity (m/s) for any hydrogen‑like ion.
  • Provides transition wavelength, photon energy, and wave number.
  • Visual diagram of allowed orbits (n = 1 to 5) with electron on selected level.
  • Includes Rydberg constant and fundamental constants.

Frequently Asked Questions

Negative energy means the electron is bound to the nucleus. Zero energy corresponds to an electron at rest infinitely far away. The more negative, the stronger the binding.

α ≈ 1/137 is a fundamental constant characterising the strength of electromagnetic interaction. In the Bohr model, v₁ = αc for hydrogen, so the electron speed is about 1/137 of light speed.

No, the Bohr model only works for one‑electron systems. For multi‑electron atoms, electron‑electron repulsion changes the energies significantly.

These are sets of spectral lines for hydrogen: Lyman (nf = 1, ultraviolet), Balmer (nf = 2, visible), Paschen (nf = 3, infrared). Our calculator shows any transition.