30.3 Bohr’s Theory of the Hydrogen Atom

Bohr's Theory of the Hydrogen Atom

Bohr's theory of the hydrogen atom explained why atoms emit and absorb light only at specific wavelengths. By introducing the idea of quantized energy levels, it bridged the gap between classical physics and the quantum mechanics that would follow. While the model only works precisely for hydrogen, it remains one of the most important stepping stones in atomic physics.

Significance of Atomic Spectra

Every element produces a unique pattern of light wavelengths when its atoms emit or absorb energy. For hydrogen, these patterns fall into named series based on the energy level the electron drops to:

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

An atom's absorption spectrum contains the same wavelengths as its emission spectrum, just appearing as dark lines against a continuous background instead of bright lines against a dark background.

Classical electromagnetic theory predicted that an orbiting electron should radiate energy continuously, producing a smooth, continuous spectrum. That prediction didn't match what experiments showed. Bohr set out to explain why the spectrum was discrete rather than continuous.

Key Principles of Bohr's Model

Bohr built his model on a few core postulates:

Stationary states. Electrons orbit the nucleus in specific circular paths, each with a fixed radius and energy. These are the only orbits allowed. While in a stationary state, the electron does not radiate energy, which was a deliberate break from classical physics.

Quantized angular momentum. The angular momentum of an electron in orbit can only take values that are integer multiples of $\frac{h}{2\pi}$:

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

where $n = 1, 2, 3, ...$ is the principal quantum number, $m$ is the electron mass, $v$ is its orbital velocity, and $r$ is the orbit radius. This quantization condition is what restricts the electron to discrete energy levels.

Photon emission and absorption. An electron can jump between energy levels by absorbing or emitting a photon whose energy exactly matches the difference between the two levels:

$\Delta E = hf$

where $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J·s}$) and $f$ is the photon's frequency. Absorbing a photon moves the electron to a higher level; emitting one drops it to a lower level.

Coulomb attraction. The electrostatic force between the positive nucleus and the negative electron provides the centripetal force that keeps the electron in its circular orbit.

Energy-Level Diagrams for Hydrogen

Energy-level diagrams are a visual way to represent the allowed states of an atom. Here's how to read one:

• Each horizontal line represents a stationary state, labeled by its quantum number $n$.
• The lowest line, $n = 1$, is the ground state with the most negative energy.
• Higher levels ($n = 2, 3, 4, ...$) have progressively higher (less negative) energies, and the spacing between levels gets smaller as $n$ increases.
• Downward arrows represent photon emission; upward arrows represent photon absorption.

The energy of each level in hydrogen is given by:

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

So the ground state has $E_1 = -13.6 \text{ eV}$, the second level has $E_2 = -3.4 \text{ eV}$, and so on. As $n$ approaches infinity, the energy approaches zero, which corresponds to a free electron.

The ionization energy of hydrogen is 13.6 eV. That's the energy needed to remove the electron from the ground state entirely.

To find the wavelength of a photon emitted or absorbed during a transition, use the Rydberg formula:

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

where $R = 1.097 \times 10^7 \text{ m}^{-1}$ is the Rydberg constant, $n_1$ is the lower level, and $n_2$ is the higher level. This formula gives a positive result when $n_1 < n_2$.

Bohr's Model vs. Earlier Theories

Thomson's "plum pudding" model proposed that electrons were embedded throughout a uniform sphere of positive charge. Rutherford's gold foil experiment disproved this by showing that most of the atom's mass and positive charge is concentrated in a tiny, dense nucleus.

Rutherford's nuclear model correctly placed electrons orbiting a central nucleus, but it had a fatal flaw: classical electromagnetism says an accelerating charge (and a circular orbit is acceleration) should continuously radiate energy. The electron would spiral into the nucleus in a fraction of a second. Bohr solved this by simply postulating that electrons in stationary states don't radiate, and that only transitions between states involve energy exchange.

Successes and Limitations of Bohr's Theory

Successes:

1. Explained why hydrogen's emission spectrum is discrete rather than continuous.
2. Predicted the wavelengths of hydrogen's spectral lines with remarkable accuracy using the Rydberg formula.
3. Introduced quantized energy levels, a concept that carries directly into modern quantum mechanics.
4. Correctly calculated hydrogen's ionization energy as 13.6 eV.

Limitations:

1. Failed to accurately predict spectra for atoms with more than one electron, because it doesn't account for electron-electron interactions.
2. Treated electrons purely as particles in fixed orbits, ignoring their wave-like nature.
3. Could not explain fine structure (small splittings in spectral lines due to relativistic and spin effects) or the Zeeman effect (splitting in a magnetic field) without additional modifications.

Quantum Mechanical Developments

Bohr's model was a critical bridge, but several advances replaced it with a more complete picture:

• de Broglie's hypothesis proposed that electrons have a wavelength given by $\lambda = \frac{h}{mv}$. This wave-particle duality explained why angular momentum is quantized: only orbits whose circumference fits a whole number of electron wavelengths are stable.
• The Schrödinger equation replaced Bohr's fixed orbits with probability distributions (orbitals) that describe where an electron is likely to be found. This approach works for multi-electron atoms, not just hydrogen.
• Atomic number ($Z$) specifies the number of protons in the nucleus. For a neutral atom, it also equals the number of electrons. Bohr's energy formula can be extended to hydrogen-like ions (one electron) by including $Z$: $E_n = \frac{-13.6 \, Z^2}{n^2} \text{ eV}$.