Bohr Model

Developed by Danish physicist Niels Bohr in 1913, the Bohr Model marked a revolutionary transition from classical physics to quantum mechanics. It modified Ernest Rutherford’s nuclear model by incorporating Max Planck’s quantum theory to explain why atoms remain stable and emit discrete spectral lines.

Core Postulates

• Quantized Stationary Orbits: Electrons in an atom do not revolve around the nucleus in arbitrary paths. Instead, they rotate only in specific, non-radiating circular paths known as stationary orbits or energy shells. While moving within these defined shells, an electron does not lose or radiate energy, maintaining atomic stability.
• Angular Momentum Quantization: An electron can only revolve in orbits where its orbital angular momentum (L) is an integral multiple of h/2π. This condition is mathematically expressed as:
  L = mvr = nh/2π
  where:
  • m = mass of the electron
  • v = velocity of the electron in the orbit
  • r = radius of the orbit
  • h = Planck’s constant (6.626 × 10^-34 J⋯)
  • n = Principal Quantum Number, representing the shell number (n = 1, 2, 3, …, or designated as K, L, M, N, … shells).
• Energy Transitions and Photons: Energy is neither emitted nor absorbed as long as the electron remains in its designated stationary orbit. Energy changes occur only when an electron jumps from one energy level to another:
  • Absorption: An electron absorbs a photon of precise energy to jump from a lower energy orbit (ground state) to a higher energy orbit (excited state).
  • Emission: An electron emits a photon of precise energy when it falls from a higher energy orbit to a lower energy orbit. The energy of the emitted or absorbed photon matches the exact difference between the two states:
    Δ E = E_higher – E_lower = hν = hc/λ
    where ν is the frequency and λ is the wavelength of the radiation, and c is the speed of light.

Mathematical Formulations for Hydrogen-like Atoms

Bohr’s equations apply precisely to single-electron systems (hydrogen-like species) such as Hydrogen (₁^1H), singly ionized Helium (He^+), doubly ionized Lithium (Li²⁺), and triply ionized Beryllium (Be³⁺).

Radius of Orbits

By balancing the electrostatic force of attraction between the nucleus and the electron with the centripetal force, and applying the angular momentum quantization rule, the radius (r_n) of the n^th orbit is given by:

• Z = Atomic number of the element.
• Bohr Radius (a₀): The radius of the first orbit of a hydrogen atom (n = 1, Z = 1), which equals 0.529 Å (0.529 × 10^-10 m).
• Proportionality: The radius is directly proportional to the square of the shell number (r ∝ n²) and inversely proportional to the atomic number (r ∝ 1/Z).

Velocity of Electrons

The velocity (v_n) of an electron moving within the n^th orbit is formulated as:

• Proportionality: The velocity is directly proportional to the atomic number (v ∝ Z) and inversely proportional to the principal quantum number (v ∝ 1/n). An electron moves fastest in the innermost shell (n = 1).

Total Energy of the Electron

The total mechanical energy (E_n) of an electron is the sum of its Kinetic Energy (KE) and Potential Energy (PE).

• Kinetic Energy: KE = +1/2 Z e²/4πϵ₀ r
• Potential Energy: PE = -Z e²/4πϵ₀ r (The negative sign indicates a bound state due to attractive electrostatic forces).
• Total Energy Expression:
  E_n = KE + PE = -13.6 × Z²/n² eV/atom
• Relationship Between Energies:
  KE = -E_n and PE = 2 · E_n
• Significance of Negative Energy: The negative sign signifies that the electron is bound to the nucleus. Energy must be supplied to remove the electron from the atom. As n → ∞, E → 0, indicating that the electron is completely free from nuclear attraction.

Explanation of the Hydrogen Spectral Series

When a high voltage is applied across a tube filled with hydrogen gas, it emits light. When passed through a prism, this light separates into distinct, sharp spectral lines. The Bohr model successfully predicted these lines using transitions between quantized energy states.

The Rydberg Formula

The wavenumber (ν), which is the reciprocal of wavelength (1/λ), for any electronic transition between an initial higher shell (n₂) and a final lower shell (n₁) is given by:

Where R_H is the Rydberg Constant ≈ 1.097 × 10⁷ m^-1.

Summary of Spectral Lines

Depending on the final shell (n₁) where the returning electron lands, the hydrogen emission spectrum is categorized into five distinct series of lines.

Series	Lower Level (n1​)	Higher Level (n2​)	Spectral Region	Key Characteristics
Lyman	1	2, 3, 4, …	Ultraviolet (UV)	Highest energy transitions; invisible to the naked eye.
Balmer	2	3, 4, 5, …	Visible Light	Only series with lines visible directly to human eyes (H_α, H_β, H_γ, H_δ).
Paschen	3	4, 5, 6, …	Near-Infrared (IR)	Low energy heat radiation transitions.
Brackett	4	5, 6, 7, …	Mid-Infrared (IR)	Infrared region line group.
Pfund	5	6, 7, 8, …	Far-Infrared (IR)	Very low energy transitions.

Limitations of the Bohr Model

While the Bohr model successfully explained single-electron systems, it encountered fundamental limitations that eventually led to the development of modern quantum mechanics.

Multi-Electron Systems

The model cannot calculate energy levels or explain spectral patterns for atoms containing more than one electron, such as Helium (Z = 2, two electrons), Lithium (Z = 3, three electrons), or heavier elements. It completely neglects electron-electron electrostatic repulsions.

Fine Structure and Spectral Splitting

High-resolution spectroscopes revealed that individual spectral lines are actually composed of multiple, closely spaced fine lines. Bohr’s model could not account for this fine structure. It also failed to explain the splitting of spectral lines under external fields:

• Zeeman Effect: The splitting of spectral lines when the light-emitting source is placed inside a strong magnetic field.
• Stark Effect: The splitting of spectral lines under the influence of an external electric field.

Violation of Fundamental Quantum Principles

• Wave-Particle Duality: Bohr treated the electron purely as a localized material particle traveling along a precise geometric trajectory (circular orbit), which contradicts Louis de Broglie’s thesis that moving electrons display wave-like properties.
• Heisenberg Uncertainty Principle: The Bohr model assumes that both the exact position (radius of the orbit) and the exact velocity (momentum) of an electron can be calculated simultaneously. This directly violates Werner Heisenberg’s principle (Δ x · Δ p ≥ h/4π).

Comparative Matrix: Atomic Models

Parameter	Thomson Model	Rutherford Model	Bohr Model	Quantum Mechanical Model
Electron Distribution	Embedded uniformly	Revolving in arbitrary paths	Confined to quantized fixed orbits	Found inside high-probability 3D clouds (orbitals)
Nucleus Concept	Absent	Present (Dense central positive core)	Present	Present
Atomic Stability	Assumed statically	Unexplained (Electromagnetic collapse)	Explained via non-radiating orbits	Explained using wave equations and probability
Spectrum Form	Not addressed	Continuous spectrum predicted	Discrete line spectrum explained	Detailed fine structure spectrum explained

Last Modified: May 28, 2026