Bohr's Atomic Model

Introduction

In 1913, Niels Bohr proposed a revolutionary model of the atom that successfully explained the hydrogen spectrum. While later superseded by quantum mechanics, Bohr’s model remains essential for JEE and provides intuitive understanding of atomic structure.

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Interactive: Bohr Model Visualization

Click the buttons to see energy absorption and emission as electrons jump between orbits:

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Bohr’s Postulates

Postulate 1: Quantized Orbits

Electrons revolve around the nucleus in fixed circular orbits called stationary states or energy levels. While in these orbits, electrons do not radiate energy.

Postulate 2: Quantized Angular Momentum

The angular momentum of an electron is quantized:

$$\boxed{mvr = n\frac{h}{2\pi} = n\hbar}$$

where:

• $n$ = principal quantum number (1, 2, 3, …)
• $\hbar$ = reduced Planck’s constant = $\frac{h}{2\pi}$

Postulate 3: Energy Transitions

Energy is absorbed or emitted only when an electron jumps between orbits:

$$\boxed{\Delta E = E_2 - E_1 = h\nu}$$

• Absorption: electron jumps to higher orbit (n₁ → n₂, where n₂ > n₁)
• Emission: electron falls to lower orbit (n₂ → n₁)

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Interactive Demo: Bohr’s Model

Watch how electrons transition between energy levels. See absorption and emission in action:

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Derivation of Key Formulas

For a hydrogen-like atom with nuclear charge $Ze$:

Balancing Forces

Centripetal force = Electrostatic attraction:

$$\frac{mv^2}{r} = \frac{kZe^2}{r^2}$$

where $k = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9$ N·m²/C²

Combining with Angular Momentum

From $mvr = n\hbar$, we get $v = \frac{n\hbar}{mr}$

Substituting and solving:

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Results for Hydrogen-like Atoms

1. Radius of nth Orbit

$$\boxed{r_n = \frac{0.529 \times n^2}{Z} \text{ Å}}$$

or in SI units:

$$r_n = \frac{n^2 h^2 \epsilon_0}{\pi m Z e^2}$$

Key observations:

• Radius increases as $n^2$
• Radius decreases with increasing $Z$
• First Bohr radius ($a_0$) for H = 0.529 Å

2. Velocity in nth Orbit

$$\boxed{v_n = \frac{2.18 \times 10^6 \times Z}{n} \text{ m/s}}$$

Key observations:

• Velocity decreases as $\frac{1}{n}$
• Velocity increases with $Z$
• Electron in 1st orbit of H has velocity ≈ c/137

3. Energy of nth Orbit

$$\boxed{E_n = -\frac{13.6 \times Z^2}{n^2} \text{ eV}}$$

or in Joules:

$$E_n = -\frac{2.18 \times 10^{-18} \times Z^2}{n^2} \text{ J}$$

Key observations:

• Energy is negative (bound state)
• Energy increases (becomes less negative) with n
• Ground state energy of H = -13.6 eV
• At n = ∞, E = 0 (ionization)

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Energy Level Diagram

graph BT
    A[n=1: -13.6 eV] --> B[n=2: -3.4 eV]
    B --> C[n=3: -1.51 eV]
    C --> D[n=4: -0.85 eV]
    D --> E[n=∞: 0 eV]
    
    style A fill:#e74c3c
    style E fill:#2ecc71

n	Energy (eV)	Radius (Å)	Velocity (m/s)
1	-13.6	0.529	2.18 × 10⁶
2	-3.4	2.116	1.09 × 10⁶
3	-1.51	4.761	0.727 × 10⁶
4	-0.85	8.464	0.545 × 10⁶
∞	0	∞	0

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Important Relationships

Energy Ratios

For hydrogen-like atoms:

$$\frac{E_{n_1}}{E_{n_2}} = \frac{n_2^2}{n_1^2}$$

Orbital Period

Time for one revolution:

$$T_n = \frac{2\pi r_n}{v_n} \propto \frac{n^3}{Z^2}$$

Frequency of Revolution

$$f_n = \frac{v_n}{2\pi r_n} \propto \frac{Z^2}{n^3}$$

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Calculations

Example 1: Comparing Radii

Problem: Find the radius of the 3rd orbit of Li²⁺.

Solution: For Li²⁺: Z = 3, n = 3

$$r_3 = \frac{0.529 \times 3^2}{3} = \frac{0.529 \times 9}{3} = 1.587 \text{ Å}$$

Note: This equals the first Bohr radius of hydrogen times 3!

Example 2: Energy Difference

Problem: Calculate the energy required to excite an electron from n=2 to n=4 in hydrogen.

Solution:

$$\Delta E = E_4 - E_2 = -\frac{13.6}{16} - \left(-\frac{13.6}{4}\right)$$ $$\Delta E = -0.85 + 3.4 = 2.55 \text{ eV}$$

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Ionization Energy

Ionization energy is the energy needed to remove an electron from ground state to infinity:

$$IE = E_\infty - E_1 = 0 - (-13.6 Z^2) = 13.6 Z^2 \text{ eV}$$

For hydrogen: IE = 13.6 eV = 2.18 × 10⁻¹⁸ J

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Practice Problems

1. Calculate the velocity of electron in the 2nd orbit of He⁺.

2. In which orbit of hydrogen atom will the electron have the same energy as in the first orbit of Li²⁺?

3. Find the ratio of kinetic energies of electrons in the 1st and 3rd orbits of hydrogen.

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Related Topics

Within Atomic Structure

• Electromagnetic Radiation — Light as waves and photons
• Photoelectric Effect — Einstein’s particle theory of light
• Hydrogen Spectrum — Bohr model explains spectral lines!
• Quantum Mechanical Model — Modern replacement for Bohr’s model

Cross-Subject: Physics Connections

• Circular Motion — Electron orbits use same concepts!
• Electrostatics — Coulomb force between nucleus and electron
• Centripetal Force — What keeps electrons in orbit

Cross-Subject: Math Connections

• Sequences and Series — Energy level patterns (1/n²)

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