Calculate electron orbital radius, energy levels, and velocity for hydrogen-like atoms using the quantum-mechanical Bohr model.
ISO 8601 • Quantum Mechanics • 2024
Must be an integer greater than or equal to 1.
Use for hydrogen-like systems with one electron.
Energy (Eₙ)
-13.60
eV
Orbital Radius (rₙ)
0.529
Å
Electron Velocity (vₙ)
2.18e+6
m/s
The Bohr model, proposed by Niels Bohr in 1913, describes electrons as occupying discrete circular orbits around a nucleus, each corresponding to a specific quantized energy level. Unlike classical mechanics (where orbiting charges should radiate energy and collapse), the Bohr model postulates that electrons in certain "allowed" orbits do not radiate. Transitions between orbits occur when electrons absorb or emit photons with energy matching the energy difference. This revolutionary model successfully explained the hydrogen atom's observed spectral lines through the formula E = 13.6 eV × (Z²/n²), where n is the principal quantum number and Z is the nuclear charge. The model correctly predicted hydrogen's Lyman, Balmer, and Paschen series and established the foundation for quantum mechanics. Although modern quantum mechanics reveals that electrons do not orbit in classical sense but inhabit probabilistic wavefunctions (orbitals), the Bohr model remains remarkably accurate for hydrogen-like ions (single-electron systems) and provides intuitive insight into atomic structure and energy quantization.
The Bohr model's key insight is quantization: angular momentum is restricted to L = nℏ, where ℏ is the reduced Planck constant. This constraint emerges from the wave nature of electrons—only orbits where an integer number of de Broglie wavelengths fit around the circumference are stable. For hydrogen (Z=1), the ground state (n=1) has energy -13.6 eV and orbital radius 0.53 Å (Bohr radius). Higher levels have increasingly negative energies (becoming less bound) and larger radii; an electron at n=∞ has zero energy and is ionized. The binding energy of state n is 13.6 × Z²/n² eV—the energy required to ionize from that level. Historical impact: the model bridged classical and quantum physics, inspired quantum mechanics development, and directly supported Planck's quantization hypothesis. Modern refinements account for relativistic effects, fine structure, hyperfine structure, and multi-electron atoms, yet Bohr's core principles remain pedagogically vital and computationally accurate for simple systems.
Input Principal Quantum Number (n): Enter the energy level (n=1, 2, 3, ...). Ground state is n=1 (lowest energy, most stable). Excited states are n≥2. Larger n means electron is further from nucleus and less tightly bound. For hydrogen, measurable transitions occur between low n values.
Input Atomic Number (Z): Enter the nuclear charge (number of protons). Z=1 for hydrogen, Z=2 for He⁺ ion, Z=3 for Li²⁺ ion, etc. These formulas apply ONLY to hydrogen-like systems with exactly one electron. Multi-electron atoms require more complex models (Hartree-Fock, quantum field theory).
Apply Energy Formula: Calculate Eₙ = -13.6 × (Z²/n²) eV. Negative values indicate bound states. Magnitude increases with Z (more protons means stronger attraction). For He⁺ (Z=2) in ground state (n=1): E = -13.6 × 4 = -54.4 eV.
Compute Orbital Radius: Use rₙ = 0.529 × (n²/Z) Ångströms. This is the Bohr radius (a₀ = 0.529 Å) scaled by n²/Z. Ground state hydrogen (n=1, Z=1): r = 0.529 Å. He⁺ is half this size due to stronger nuclear pull. Excited states are much larger (n=3 is 9× larger than n=1).
Calculate Electron Velocity: Apply vₙ = (2.18×10⁶ × Z/n) m/s. Speed decreases with n (outer electrons move slower). Ground state hydrogen electron: v = 2.18×10⁶ m/s ≈ 0.73% light speed. Higher Z gives faster velocities (stronger pull accelerates electron).
The Bohr model assumes: (1) Circular orbits (no ellipses), (2) Quantized angular momentum L = nℏ, (3) Coulomb attraction provides centripetal force, (4) Zero orbital angular momentum radiation (postulate, not derived), (5) Single electron (hydrogen-like). Violations of these assumptions complicate real atoms. For multi-electron atoms, electron-electron repulsion breaks this simple picture.
Scenario: Calculate orbital properties for a He⁺ ion (Z=2) with its single electron in the n=2 excited state.
Interpretation: He⁺ in n=2 state has the same orbital velocity as ground-state hydrogen (2.18×10⁶ m/s ≈ 0.73%c). Orbital radius matches He⁺'s n=1 being twice hydrogen's n=1 (scaled by 1/Z). Energy is -13.6 eV—He⁺ is more tightly bound than hydrogen due to Z=2. Ionization energy from n=2 is 13.6 eV.
Negative energy indicates the electron is bound to the nucleus. Zero energy represents the ionized limit (electron infinitely far away). Negative energies mean work is required to remove the electron. For hydrogen: E₁ = -13.6 eV means 13.6 eV ionization energy.
No, not well. These formulas apply only to hydrogen-like ions with ONE electron (H, He⁺, Li²⁺, etc.). Multi-electron atoms have electron-electron repulsion that breaks this model. Quantum mechanics and numerical solutions are needed for accurate predictions.
As n increases: energy becomes less negative (less bound), orbital radius grows (proportional to n²), electron velocity decreases (proportional to 1/n). Electron spends more time far from nucleus, is easier to ionize, and responds weakly to external fields.
No. n must be a positive integer (1, 2, 3, ...). n=0 would give infinite negative energy and zero radius—physically impossible. The ground state is n=1 with lowest possible energy. This quantization constraint is central to quantum mechanics.
When an electron jumps between levels (e.g., n=2 to n=1), it emits or absorbs a photon with energy ΔE = |Eₙ - Eₙ'|. Wavelength follows: E = hc/λ. Hydrogen's Balmer series (visible lines) comes from transitions TO n=2 from higher levels.
Yes, for pedagogical clarity and practical calculations involving hydrogen-like systems. Modern quantum mechanics (Schrödinger equation) provides more complete description including orbital shapes, but Bohr model often gives correct energies and gives intuitive understanding of quantization.
It emerges from balancing Coulomb attraction and centripetal acceleration for the ground state: a₀ = 4πε₀ℏ²/(meₑ²) ≈ 0.529 Å. This characteristic size scale of atoms comes from fundamental constants. Historically, it matched spectroscopic data perfectly.
Ground state (n=1): v = 2.18×10⁶ m/s ≈ 0.73% of light speed. Fast enough to be relativistic for heavy nuclei, but non-relativistic for light atoms. This supports Bohr's self-consistency: centripetal acceleration doesn't radiate (postulate), and speeds are modest enough for non-relativistic mechanics.
The Bohr model is historically important and remains useful for simple hydrogen-like systems, even though modern quantum mechanics provides a more complete description of atoms.
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