Bohr Model of the Atom

The Bohr Model of the Atom, proposed by Niels Bohr in 1913, marked a revolutionary step in understanding atomic structure. It successfully explained the stability of atoms and the discrete spectral lines of hydrogen, bridging the gap between classical physics and the emerging quantum theory. This post explores the historical background, postulates, mathematical derivations, applications, and limitations of the Bohr model in depth.

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1. Historical Background

Before Bohr, the atomic model underwent several transformations:

1.1 Thomson’s Plum Pudding Model (1904)

• Proposed by J.J. Thomson.
• Electrons embedded in a positively charged sphere.
• Could not explain atomic spectra or atomic stability.

1.2 Rutherford Model (1911)

• Gold foil experiment by Ernest Rutherford.
• Atom consists of a tiny, dense, positively charged nucleus.
• Electrons revolve around the nucleus.
• Limitations:
  • Classical physics predicted electrons would spiral into the nucleus.
  • Could not explain line spectra of hydrogen.

1.3 Need for Bohr’s Model

• Hydrogen emission spectrum shows discrete lines (Balmer series).
• Classical theory failed to explain why electrons do not radiate energy continuously while orbiting.
• Bohr introduced quantization to resolve these issues.

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2. Bohr’s Postulates

Bohr proposed three key postulates:

2.1 First Postulate: Quantized Orbits

• Electrons revolve around the nucleus in certain stable orbits without radiating energy.
• These orbits are called stationary states.

2.2 Second Postulate: Angular Momentum Quantization

• Electron’s angular momentum is quantized:

mevr=nh2π,n=1,2,3,…m_e v r = n \frac{h}{2\pi}, \quad n = 1, 2, 3, \dotsme​vr=n2πh​,n=1,2,3,…

Where:

• mem_eme​ = electron mass
• vvv = electron velocity
• rrr = orbit radius
• hhh = Planck’s constant
• nnn = principal quantum number

2.3 Third Postulate: Radiation Emission or Absorption

• Electrons emit or absorb energy only when jumping between stationary orbits.
• Energy of emitted/absorbed photon:

ΔE=Ei−Ef=hν\Delta E = E_i – E_f = h\nuΔE=Ei​−Ef​=hν

Where:

• EiE_iEi​ = initial orbit energy
• EfE_fEf​ = final orbit energy
• ν\nuν = frequency of radiation

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3. Mathematical Derivation of Bohr Model

3.1 Centripetal Force and Coulomb Attraction

Electrons revolve around the nucleus due to electrostatic force: ke2r2=mev2r\frac{k e^2}{r^2} = \frac{m_e v^2}{r}r2ke2​=rme​v2​

Where:

• k=8.99×109 Nm2/C2k = 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2k=8.99×109 Nm2/C2
• e=1.602×10−19e = 1.602 \times 10^{-19}e=1.602×10−19 C

From this, electron velocity: v=ke2merv = \sqrt{\frac{k e^2}{m_e r}}v=me​rke2​​

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3.2 Quantized Angular Momentum

mevr=nh2π⇒v=nh2πmerm_e v r = n \frac{h}{2\pi} \Rightarrow v = \frac{n h}{2 \pi m_e r}me​vr=n2πh​⇒v=2πme​rnh​

Equating centripetal and angular momentum conditions: ke2r2=mer(nh2πmer)2\frac{k e^2}{r^2} = \frac{m_e}{r} \left(\frac{n h}{2 \pi m_e r}\right)^2r2ke2​=rme​​(2πme​rnh​)2 rn=n2h24π2ke2mer_n = \frac{n^2 h^2}{4 \pi^2 k e^2 m_e}rn​=4π2ke2me​n2h2​

• This gives radius of the n-th orbit.
• Bohr radius (n=1):

a0=r1=0.529×10−10 ma_0 = r_1 = 0.529 \times 10^{-10} \text{ m}a0​=r1​=0.529×10−10 m

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3.3 Energy Levels of Electron

Total energy = kinetic + potential: E=K+U=12mev2−ke2rE = K + U = \frac{1}{2} m_e v^2 – \frac{k e^2}{r}E=K+U=21​me​v2−rke2​

From previous derivations: En=−k2mee42h2n2=−13.6 eVn2E_n = – \frac{k^2 m_e e^4}{2 h^2 n^2} = – \frac{13.6 \text{ eV}}{n^2}En​=−2h2n2k2me​e4​=−n213.6 eV​

• Negative sign indicates bound state.
• Energy levels are discrete.

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3.4 Spectral Lines of Hydrogen

When electron jumps from ni→nfn_i \to n_fni​→nf​: hν=Eni−Enf=13.6(1nf2−1ni2) eVh \nu = E_{n_i} – E_{n_f} = 13.6 \left(\frac{1}{n_f^2} – \frac{1}{n_i^2}\right) \text{ eV}hν=Eni​​−Enf​​=13.6(nf2​1​−ni2​1​) eV

• Explains Balmer, Lyman, Paschen series.
• Provides quantitative explanation of hydrogen spectrum.

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4. Bohr’s Model of Hydrogen Atom

• n=1: Ground state; lowest energy.
• n>1: Excited states; electrons can jump between levels.
• Ionization energy: 13.6 eV (energy to remove electron from n=1).
• Electron transitions produce line spectra observable experimentally.

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5. Radius of Orbits

• Radius increases with n2n^2n2:

rn=n2a0r_n = n^2 a_0rn​=n2a0​

• Implies electron farther from nucleus has higher energy.
• Orbits are quantized, not continuous.

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6. Velocity of Electron in Bohr Orbits

vn=ke2ℏ1n=2πke2h1nv_n = \frac{ke^2}{\hbar} \frac{1}{n} = \frac{2 \pi k e^2}{h} \frac{1}{n}vn​=ℏke2​n1​=h2πke2​n1​

• Electron velocity decreases with higher n.
• Velocity for n=1 hydrogen: v1≈2.19×106 m/sv_1 \approx 2.19 \times 10^6 \text{ m/s}v1​≈2.19×106 m/s.

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7. Bohr’s Model Predictions

1. Quantized Energy Levels: Explains discrete spectral lines.
2. Stability of Atoms: Electron orbits without radiating energy.
3. Ionization Energy: Matches experimental values for hydrogen.
4. Radius of Orbits: Predicted accurately for hydrogen.
5. Spectral Series: Balmer, Lyman, Paschen, Brackett explained.

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8. Limitations of Bohr Model

1. Only Works for Hydrogen-like Atoms: Fails for multi-electron atoms.
2. Does Not Explain Fine Structure: Spin-orbit coupling and relativistic effects ignored.
3. No Electron Spin: Quantum mechanical spin not included.
4. Cannot Explain Zeeman Effect: Interaction with magnetic fields not accounted.
5. Semi-Classical Approach: Combines classical orbits with quantum postulates; lacks wave mechanics.

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9. Extensions and Improvements

9.1 Sommerfeld Model (1916)

• Introduced elliptical orbits and relativistic corrections.
• Explained fine structure of spectral lines.

9.2 Quantum Mechanical Model

• Schrödinger wave equation replaced Bohr orbits with probability clouds.
• Bohr’s energy levels remain valid for hydrogen, but electron positions are not fixed orbits.

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10. Applications of Bohr Model

10.1 Atomic Spectra

• Explains emission and absorption lines of hydrogen.
• Basis for astronomy: identifying elements in stars.

10.2 Lasers

• Electron transitions between energy levels produce coherent light in lasers.

10.3 X-ray Spectroscopy

• Electron transitions in inner shells produce characteristic X-rays.
• Used in material analysis and medical imaging.

10.4 Atomic Clocks

• Energy transitions used to measure time accurately.

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11. Bohr Model Calculations: Examples

11.1 Energy Levels of Hydrogen

• n=1: E1=−13.6 eVE_1 = -13.6 \text{ eV}E1​=−13.6 eV
• n=2: E2=−3.4 eVE_2 = -3.4 \text{ eV}E2​=−3.4 eV
• n=3: E3=−1.51 eVE_3 = -1.51 \text{ eV}E3​=−1.51 eV

11.2 Wavelength of Transition

• Electron transition: ni=3→nf=2n_i = 3 \to n_f = 2ni​=3→nf​=2 (Balmer series):

1λ=RH(1nf2−1ni2)\frac{1}{\lambda} = R_H \left(\frac{1}{n_f^2} – \frac{1}{n_i^2}\right)λ1​=RH​(nf2​1​−ni2​1​) 1λ=1.097×107(122−132)\frac{1}{\lambda} = 1.097 \times 10^7 \left(\frac{1}{2^2} – \frac{1}{3^2}\right)λ1​=1.097×107(221​−321​)

λ≈656 nm\lambda \approx 656 \text{ nm}λ≈656 nm (red line of hydrogen spectrum)

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11.3 Radius of n=2 Orbit

r2=n2a0=22×0.529×10−10≈2.12×10−10 mr_2 = n^2 a_0 = 2^2 \times 0.529 \times 10^{-10} \approx 2.12 \times 10^{-10} \text{ m}r2​=n2a0​=22×0.529×10−10≈2.12×10−10 m

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11.4 Velocity of Electron in n=2 Orbit

v2=v1n=2.19×1062≈1.095×106 m/sv_2 = \frac{v_1}{n} = \frac{2.19 \times 10^6}{2} \approx 1.095 \times 10^6 \text{ m/s}v2​=nv1​​=22.19×106​≈1.095×106 m/s

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12. Significance of Bohr Model

1. Introduced quantization in atomic physics.
2. Explained stability of atoms.
3. Predicted hydrogen spectral lines accurately.
4. Provided basis for later quantum mechanics.
5. Paved way for laser, spectroscopy, and atomic physics research.

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13. Bohr Model and Modern Physics

While superseded by quantum mechanics, Bohr’s model remains:

• A pedagogical tool for understanding atomic structure.
• Accurate for hydrogen-like ions.
• Foundation for concepts of energy quantization and electron transitions.