About

The Bohr model, proposed in 1913, remains the foundational framework for computing discrete energy levels in hydrogen-like atoms. It predicts orbital radii as r_n = n² ⋅ a₀ ÷ Z, where a₀ = 0.529177 Å is the Bohr radius and Z is the atomic number. Energy quantization follows E_n = −13.6 ⋅ Z² ÷ n² eV. Applying this model to multi-electron atoms requires approximation: nuclear shielding reduces the effective charge experienced by outer electrons. This tool computes exact Bohr parameters for hydrogen-like systems and provides shell-filling visualization for all 118 elements. Errors in energy level calculations propagate directly into spectral line predictions, producing incorrect wavelength values that invalidate experimental comparisons.

The model breaks down for heavy atoms where relativistic corrections become significant (roughly Z > 40). It also cannot account for fine structure, Lamb shift, or spin-orbit coupling. For hydrogen and hydrogen-like ions, accuracy is within 0.01% of experimental values. For multi-electron atoms, treat results as pedagogical approximations. Pro tip: compare calculated Lyman-series wavelengths against NIST Atomic Spectra Database values to gauge model validity for your element.

Formulas

The Bohr model quantizes electron orbits via the principal quantum number n. All properties derive from three constants: the Bohr radius a₀ = 0.529177 Å, the Rydberg energy 13.6 eV, and the fine-structure constant α = 1/137.036.

Orbital radius for shell n:

r_n = n² ⋅ a₀Z

Energy of level n:

E_n = −13.6 ⋅ Z²n² eV

Electron velocity in orbit n:

v_n = Z ⋅ α ⋅ cn

Photon wavelength for transition n_i → n_f:

1λ = R_∞ ⋅ Z² ⋅ (1n_f² − 1n_i²)

De Broglie wavelength of orbiting electron:

λ_dB = 2π ⋅ r_nn

Where Z = atomic number, n = principal quantum number (1, 2, 3, …), a₀ = Bohr radius (5.29177 × 10⁻¹¹ m), R_∞ = Rydberg constant (1.0974 × 10⁷ m⁻¹), α = fine-structure constant, c = speed of light (2.998 × 10⁸ m/s).

Reference Data

Element	Z	Ground State Config	E₁ (eV)	r₁ (Å)	Ionization Energy (eV)	Series Limit λ (nm)
Hydrogen	1	1s¹	−13.60	0.529	13.60	91.2
Helium	2	1s²	−54.42	0.265	24.59	22.8
Lithium	3	1s²2s¹	−122.4	0.176	5.39	10.1
Beryllium	4	1s²2s²	−217.6	0.132	9.32	5.7
Boron	5	1s²2s²2p¹	−340.0	0.106	8.30	3.6
Carbon	6	1s²2s²2p²	−489.6	0.088	11.26	2.5
Nitrogen	7	1s²2s²2p³	−666.4	0.076	14.53	1.86
Oxygen	8	1s²2s²2p⁴	−870.4	0.066	13.62	1.42
Fluorine	9	1s²2s²2p⁵	−1101.6	0.059	17.42	1.13
Neon	10	1s²2s²2p⁶	−1360.0	0.053	21.56	0.91
Sodium	11	[Ne]3s¹	−1645.6	0.048	5.14	0.75
Magnesium	12	[Ne]3s²	−1958.4	0.044	7.65	0.63
Aluminum	13	[Ne]3s²3p¹	−2298.4	0.041	5.99	0.54
Silicon	14	[Ne]3s²3p²	−2665.6	0.038	8.15	0.47
Phosphorus	15	[Ne]3s²3p³	−3060.0	0.035	10.49	0.41
Sulfur	16	[Ne]3s²3p⁴	−3481.6	0.033	10.36	0.36
Chlorine	17	[Ne]3s²3p⁵	−3930.4	0.031	12.97	0.31
Argon	18	[Ne]3s²3p⁶	−4406.4	0.029	15.76	0.28
Potassium	19	[Ar]4s¹	−4909.6	0.028	4.34	0.25
Calcium	20	[Ar]4s²	−5440.0	0.026	6.11	0.23
Iron	26	[Ar]3d⁶4s²	−9193.6	0.020	7.90	0.13
Copper	29	[Ar]3d¹⁰4s¹	−11438.4	0.018	7.73	0.11
Silver	47	[Kr]4d¹⁰5s¹	−30046.4	0.011	7.58	0.041
Gold	79	[Xe]4f¹⁴5d¹⁰6s¹	−84886.4	0.0067	9.23	0.015
Uranium	92	[Rn]5f³6d¹7s²	−115059.2	0.0058	6.19	0.011

Frequently Asked Questions

At high atomic numbers, inner-shell electrons reach velocities approaching a significant fraction of the speed of light. The Bohr model uses non-relativistic mechanics, so it cannot account for relativistic mass increase. For gold (Z = 79), the 1s electron velocity is approximately 58% of c, producing a ~20% error in orbital radius versus Dirac equation predictions. This relativistic contraction of inner shells is responsible for gold's color and mercury's liquid state at room temperature.

The Bohr energy formula E_n = −13.6 × Z²/n² is exact only for one-electron systems (H, He⁺, Li²⁺). For multi-electron atoms, this calculator applies the unscreened nuclear charge Z, which overestimates binding energies for outer shells. The electron configuration display uses the Aufbau filling order (1s, 2s, 2p, 3s, ...) with correct subshell capacities. Shell populations shown in the visualization reflect actual electron distribution across energy levels.

For hydrogen (Z = 1): Lyman series (n_f = 1) produces ultraviolet lines from 121.6 nm down to 91.2 nm. Balmer series (n_f = 2) produces visible lines from 656.3 nm (red) to 364.6 nm (violet). Paschen series (n_f = 3) lies in near-infrared at 1875 nm downward. Brackett (n_f = 4) and Pfund (n_f = 5) series fall in mid-infrared. For higher Z, all wavelengths scale as 1/Z², shifting the entire spectrum toward shorter wavelengths.

The Bohr radius for hydrogen (0.529 Å) agrees well with the most probable radius from quantum mechanical wavefunctions (also 0.529 Å for the 1s orbital). However, for multi-electron atoms, the outermost Bohr orbit r_n = n² × a₀/Z significantly underestimates atomic radii because it ignores electron-electron repulsion and screening. Experimental covalent radii for carbon (0.77 Å) and iron (1.32 Å) are much larger than their Bohr n=1 radii of 0.088 Å and 0.020 Å respectively.

For hydrogen and hydrogen-like ions (single-electron species), the calculated wavelengths match experimental values to within 0.01%. The Balmer series lines at 656.3, 486.1, 434.0, and 410.2 nm are diagnostic for hydrogen in stellar spectra. For neutral multi-electron atoms, the Bohr model does not produce correct spectral lines because electron-electron interactions split and shift energy levels. Use NIST Atomic Spectra Database for precise multi-electron wavelengths.

Negative energy indicates a bound state. The zero-energy reference is an electron at rest infinitely far from the nucleus. E₁ = −13.6 eV for hydrogen means 13.6 eV of energy must be supplied to remove the electron (ionization). As n increases, E_n approaches zero: the electron becomes less tightly bound. At n = ∞, the electron is free. The energy difference between two levels equals the photon energy emitted or absorbed during a transition.