6.2 The Bohr Model

The Bohr Model of the Hydrogen Atom

Niels Bohr proposed that electrons in hydrogen orbit the nucleus in fixed paths, jumping between energy levels by absorbing or emitting light. This explained why atomic spectra show discrete lines rather than a continuous rainbow of color. Though limited to hydrogen, the Bohr model introduced quantized energy levels and laid the groundwork for modern quantum mechanics.

Components of Bohr's Hydrogen Model

Bohr's model rests on a few core ideas. Electrons don't just float anywhere around the nucleus. Instead, they travel in fixed, circular paths called stationary states (or energy levels), labeled by the principal quantum number $n = 1, 2, 3, ...$ Each stationary state sits at a specific distance from the nucleus and has a specific energy. Electrons can only exist in these states, not in between them. This restriction is what "quantized" means.

• The ground state ($n = 1$) is the lowest energy level, closest to the nucleus
• Excited states ($n = 2, 3, ...$) are higher in energy and farther from the nucleus
• As $n$ approaches infinity, the electron is so far from the nucleus that it's essentially free. This corresponds to ionization

The radius of each stationary state grows with $n$:

$r = n^2 a_0$

where $a_0$ is the Bohr radius (0.529 Å). So the $n = 2$ orbit has a radius four times larger than $n = 1$, and $n = 3$ has nine times the radius.

Electron transitions happen when an electron jumps between energy levels:

• Moving from a lower level to a higher one requires absorbing a photon (excitation)
• Moving from a higher level to a lower one emits a photon (emission)

The photon's energy exactly equals the energy difference between the two levels:

$\Delta E = h\nu$

where $h$ is Planck's constant and $\nu$ is the frequency of the photon. That energy difference also determines the photon's wavelength, which is why different transitions produce different colors of light (visible, UV, or IR).

Energy Calculations with the Rydberg Equation

The Rydberg equation lets you calculate the wavelength of light absorbed or emitted during an electron transition in hydrogen:

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

• $\lambda$ = wavelength of the photon (in meters)
• $R_H$ = Rydberg constant for hydrogen ($1.097 \times 10^7 \, \text{m}^{-1}$)
• $n_1$ = lower energy level (smaller integer)
• $n_2$ = higher energy level (larger integer)

A key detail: $n_1$ is always the smaller quantum number and $n_2$ is the larger one. This keeps $\frac{1}{\lambda}$ positive.

Steps to calculate the energy of a transition:

1. Identify the two energy levels involved ($n_1$ and $n_2$, with $n_1 < n_2$)
2. Plug them into the Rydberg equation and solve for $\lambda$
3. Convert wavelength to energy using $E = \frac{hc}{\lambda}$, where $c = 2.998 \times 10^8 \, \text{m/s}$

The Rydberg equation predicts the wavelengths for several named spectral series in hydrogen:

• Lyman series (UV): transitions down to $n_1 = 1$
• Balmer series (visible): transitions down to $n_1 = 2$
• Paschen series (IR): transitions down to $n_1 = 3$

Bohr Model vs. Atomic Spectra

Atoms don't emit a continuous spectrum of light. Instead, they produce line spectra, with only specific wavelengths appearing. The Bohr model explains why: because energy levels are quantized, only certain energy differences are possible, so only certain wavelengths of light get emitted or absorbed.

Each element has a unique set of energy levels, which means each element produces a unique line spectrum. This is why spectroscopy can identify elements based on the light they emit or absorb.

For hydrogen specifically, the Bohr model's predictions match experiment remarkably well. The wavelengths calculated from the Rydberg equation line up with the observed lines in the Lyman, Balmer, and Paschen series. This agreement was strong evidence that Bohr's quantized energy levels were on the right track.

Limitations and Advancements Beyond the Bohr Model

The Bohr model works well for hydrogen (a one-electron atom), but it fails for multi-electron atoms. It can't accurately predict their spectra because electron-electron repulsions make the energy levels more complicated than a simple $n$-based formula can capture.

Other limitations worth knowing:

• It treats electrons as particles in fixed circular orbits, which doesn't reflect their actual wave-like behavior
• It can't explain molecular bonding or the varying intensities of spectral lines

Quantum mechanics replaced the Bohr model's fixed orbits with orbitals, which are probability distributions describing where an electron is likely to be found. You'll encounter this in the next sections of this unit. Still, the Bohr model's central insight, that electron energies are quantized, carries directly into the quantum mechanical model.