11.3 Bohr model of the atom

Structure of the Bohr Model

Niels Bohr proposed his atomic model in 1913 to solve a problem classical physics couldn't: why atoms emit light only at specific wavelengths. His model introduced the radical idea that electrons can only exist in certain discrete orbits around the nucleus, each with a fixed energy. This was the first atomic model to successfully merge classical orbital mechanics with the new concept of quantization.

Planetary Model Analogy

The Bohr model pictures the atom like a miniature solar system. A dense, positively charged nucleus (protons and neutrons) sits at the center, and negatively charged electrons orbit around it. Instead of gravity holding things together, it's the electrostatic (Coulomb) attraction between the positive nucleus and negative electrons that provides the centripetal force keeping electrons in orbit.

The analogy breaks down in one critical way: unlike planets, electrons can't orbit at just any distance. They're restricted to specific allowed orbits.

Quantized Energy Levels

Each allowed orbit corresponds to a specific energy, labeled by a quantum number $n$ (where $n = 1, 2, 3, \ldots$). The $n = 1$ orbit is closest to the nucleus and has the lowest energy (the ground state). Higher values of $n$ mean the electron is farther out and has more energy.

• Electrons don't gradually slide between orbits. They make quantum jumps, instantly transitioning from one energy level to another.
• The energy difference between levels is not uniform; the levels get closer together as $n$ increases.

Stationary States

Bohr called the allowed orbits stationary states because an electron in one of these orbits does not radiate energy, even though classical physics says an accelerating charge should. This was a deliberate break from classical electromagnetism.

• In a stationary state, the electrostatic attraction toward the nucleus exactly provides the centripetal force needed for circular motion.
• An electron only emits or absorbs energy when it transitions between stationary states, never while sitting in one.

Postulates of the Bohr Model

Bohr built his model on a few key assumptions that departed sharply from classical physics. These postulates are worth knowing precisely, since exam questions often ask you to state or apply them.

Circular Orbits

Bohr assumed electrons travel in perfectly circular paths around the nucleus. The electrostatic attraction between the nucleus (charge $+Ze$) and the electron (charge $-e$) supplies the centripetal force:

$\frac{ke^2 Z}{r^2} = \frac{m_e v^2}{r}$

This simplification makes the math manageable and works well for hydrogen, though real electron behavior is more complex.

Angular Momentum Quantization

This is the postulate that makes the model "quantum." Bohr proposed that the electron's orbital angular momentum can only take values that are integer multiples of the reduced Planck constant:

$L = n\hbar$

where $\hbar = \frac{h}{2\pi} \approx 1.055 \times 10^{-34} \text{ J·s}$ and $n = 1, 2, 3, \ldots$

This single condition is what restricts electrons to specific orbits and produces discrete energy levels. Without it, any orbit would be allowed, and you'd be back to classical physics.

Energy Emission and Absorption

When an electron drops from a higher energy level ($n_2$) to a lower one ($n_1$), it emits a photon. When it jumps up, it absorbs one. The photon's energy exactly equals the difference between the two levels:

$E_{\text{photon}} = E_{n_2} - E_{n_1} = h f$

where $f$ is the photon's frequency. This directly explains why atoms produce discrete spectral lines rather than a continuous rainbow of colors.

Electron Behavior

Allowed vs. Forbidden Orbits

Only orbits satisfying the angular momentum condition $L = n\hbar$ are permitted. Any orbit that doesn't meet this criterion is "forbidden," meaning an electron simply cannot exist there. There's no gradual transition; the electron is either in one allowed orbit or another, with nothing in between.

This quantization is why atomic spectra have sharp lines rather than broad, continuous bands.

Electron Transitions

Transitions between energy levels happen instantaneously. The electron doesn't pass through the space between orbits; it jumps directly. These jumps always involve a photon:

• Downward transition (higher $n$ → lower $n$): a photon is emitted
• Upward transition (lower $n$ → higher $n$): a photon is absorbed

The photon's wavelength depends on the energy gap, so each possible transition produces a specific color of light.

Emission Spectra

When atoms are excited (by heat, electricity, etc.), their electrons jump to higher levels and then fall back down, emitting photons at characteristic wavelengths. The resulting pattern of spectral lines is unique to each element.

• Hydrogen's visible spectrum (the Balmer series) shows lines at 656 nm (red), 486 nm (blue-green), 434 nm (violet), and 410 nm (violet).
• These spectral "fingerprints" are used in chemistry and astronomy to identify elements in unknown samples or distant stars.

Mathematical Foundations

These equations are the core quantitative tools of the Bohr model. You should be comfortable using each one.

Energy Level Equation

The energy of an electron in the $n$th orbit of a hydrogen-like atom is:

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

where $Z$ is the atomic number ($Z = 1$ for hydrogen).

• The negative sign means the electron is bound to the nucleus. You'd need to add energy to free it.
• As $n \to \infty$, $E_n \to 0$, which represents the ionization limit where the electron is no longer bound.
• The ground state of hydrogen ($n = 1$) has $E_1 = -13.6 \text{ eV}$.

Rydberg Formula

To find the wavelength of light emitted or absorbed during a transition:

$\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$

where $R_H \approx 1.097 \times 10^7 \text{ m}^{-1}$ is the Rydberg constant, $n_1$ is the lower level, and $n_2$ is the higher level.

Using this formula step by step:

1. Identify the two energy levels involved in the transition ($n_1 < n_2$).
2. Plug them into the formula and compute $\frac{1}{\lambda}$.
3. Take the reciprocal to get $\lambda$.

For example, the first Balmer line ($n_1 = 2$, $n_2 = 3$):

$\frac{1}{\lambda} = 1.097 \times 10^7 \left(\frac{1}{4} - \frac{1}{9}\right) = 1.097 \times 10^7 \times \frac{5}{36} \approx 1.524 \times 10^6 \text{ m}^{-1}$

$\lambda \approx 656 \text{ nm}$ (red light)

Bohr Radius

The radius of the smallest allowed orbit in hydrogen ($n = 1$) is the Bohr radius:

$a_0 = \frac{\hbar^2}{m_e k e^2} \approx 0.529 \text{ Å} = 0.0529 \text{ nm}$

The radius of the $n$th orbit scales as $r_n = n^2 a_0$, so higher energy levels are much farther from the nucleus. The $n = 2$ orbit is four times larger than $n = 1$, and $n = 3$ is nine times larger.

Limitations of the Bohr Model

The Bohr model works remarkably well for hydrogen, but it has serious shortcomings that motivated the development of full quantum mechanics.

Hydrogen-Like Atoms Only

The model accurately describes one-electron systems: hydrogen ($Z = 1$), singly ionized helium ($\text{He}^+$, $Z = 2$), doubly ionized lithium ($\text{Li}^{2+}$, $Z = 3$), and so on. Once you add a second electron, the model breaks down because it can't handle electron-electron repulsion.

It also can't account for fine structure (small energy splittings due to relativistic effects) or hyperfine structure (even smaller splittings from nuclear spin interactions).

Failure for Complex Atoms

For multi-electron atoms, the Bohr model:

• Can't predict the correct spectral lines
• Doesn't account for electron-electron repulsion or spin-orbit coupling
• Fails to explain why electrons fill orbitals the way they do
• Can't describe chemical bonding or molecular structure

Inconsistencies with Quantum Mechanics

The Bohr model assigns each electron a definite position and velocity at all times. This directly violates the Heisenberg uncertainty principle, which states you can't simultaneously know both with precision. The model also ignores the wave nature of electrons, which de Broglie and Schrödinger later showed is fundamental to understanding atomic structure.

These aren't minor issues. They're the reason the Bohr model was eventually replaced by the full quantum mechanical treatment using wavefunctions and probability distributions.

Historical Significance

Precursor to Quantum Mechanics

The Bohr model was the first to apply quantization directly to atomic structure. Even though it was eventually superseded, it introduced ideas that remain central to quantum mechanics: discrete energy levels, quantum numbers, and the concept that classical physics fails at the atomic scale. It set the stage for Heisenberg's matrix mechanics and Schrödinger's wave equation.

Explanation of Spectral Lines

Before Bohr, the Rydberg formula was purely empirical; it worked, but nobody knew why. Bohr's model gave it a physical explanation: spectral lines correspond to electron transitions between quantized energy levels. This was a major achievement and one of the strongest pieces of evidence that the model was on the right track.

Impact on Atomic Theory

Bohr's work challenged the classical assumption that energy is continuous and that electrons can orbit at any distance. By introducing stationary states and quantized angular momentum, he fundamentally changed how physicists think about atoms and paved the way for the quantum numbers ($n, l, m_l, m_s$) used in modern atomic theory.

Experimental Evidence

Hydrogen Spectrum

The Bohr model's greatest experimental success was predicting the wavelengths of hydrogen's spectral lines with high precision. It explained the known series:

• Lyman series: transitions to $n = 1$ (ultraviolet)
• Balmer series: transitions to $n = 2$ (visible light)
• Paschen series: transitions to $n = 3$ (infrared)

Each series matched the Rydberg formula, and the Bohr model explained why the formula worked.

Franck-Hertz Experiment

In 1914, James Franck and Gustav Hertz fired electrons at mercury atoms and measured the energy lost by the electrons. They found that electrons lost energy only in specific amounts (4.9 eV for mercury), not continuously. This provided direct, independent confirmation that atoms have discrete energy levels, just as Bohr predicted.

Atomic Emission Spectroscopy

Every element produces a unique set of spectral lines when excited. This technique is used routinely in chemistry, materials science, and astrophysics to identify elements. The underlying principle, that spectral lines arise from quantized electron transitions, comes directly from the Bohr model framework.

Applications and Extensions

Atomic Clocks

Modern atomic clocks use extremely precise electron transitions (typically in cesium or rubidium atoms) as their timekeeping reference. The concept that these transitions occur at exact, discrete frequencies traces back to Bohr's quantized energy levels. Cesium clocks are accurate to about 1 second in 300 million years and are essential for GPS and global communications.

Laser Technology

Lasers work by stimulating electrons to drop from a higher energy level to a lower one in a coordinated way, producing coherent light. The process depends on population inversion, where more atoms are in an excited state than the ground state. The idea that atoms have discrete energy levels with specific transition energies is foundational to laser design.

Quantum Computing Concepts

Quantum computing uses discrete quantum states (like the spin states of electrons or energy levels of atoms) as the basis for qubits. While quantum computing draws on the full machinery of quantum mechanics rather than the Bohr model specifically, the core idea that physical systems have discrete, quantized states connects directly to what Bohr first proposed.