5: Bohr atom model

Bohr Atom Model: Explaining Atomic Stability and Spectra

Prior to Niels Bohr's revolutionary 1913 model, the classical planetary model of the atom faced two critical failures. Firstly, accelerating electrons (as required in circular orbits) should continuously radiate energy, leading to a catastrophic collapse into the nucleus – contradicting the observed stability of atoms. Secondly, classical physics predicted a continuous spectrum of emitted light as electrons spiraled inward, starkly contrasting the discrete line spectra (like the Balmer series for hydrogen) observed experimentally.

Bohr boldly introduced quantized atomic structure through two non-classical postulates:

1. Stationary States: Electrons orbit the nucleus only in specific, stable orbits ("stationary states") without radiating energy. These orbits satisfy a quantization condition for the electron's angular momentum: $L = m_e v r = n \hbar$, where $n$ is an integer (1, 2, 3, ...), $m_e$ is the electron mass, $v$ its speed, $r$ the orbit radius, and $\hbar = h/(2\pi)$ ($h$ is Planck's constant).
2. Quantum Jumps & Photon Emission: An electron can transition ("jump") from a higher-energy stationary state ($n_i$) to a lower-energy state ($n_f$). The energy difference ($\Delta E = E_i - E_f$) is emitted as a single photon with frequency $\nu$ given by $\Delta E = h\nu$.

Applying these postulates to the hydrogen atom (one electron orbiting a proton), Bohr derived quantized energy levels: $E_n = - \frac{m_e e^4}{8 \varepsilon_0^2 h^2} \cdot \frac{1}{n^2} = - \frac{13.6 \text{ eV}}{n^2}$ Here, $e$ is the electron charge, $\varepsilon_0$ is the vacuum permittivity, and $n$ is the principal quantum number ($n=1$ is the ground state). The negative sign indicates bound states.

Key Achievements:

• Stability Explained: Electrons in stationary states ($n \geq 1$) don't radiate, preventing collapse.
• Discrete Spectra Explained: The emitted photon frequency $\nu$ during a jump from $n_i$ to $n_f$ is $\nu = (E_i - E_f)/h$. This directly accounted for the empirical Rydberg formula for hydrogen's spectral lines: $1/\lambda = R_H (1/n_f^2 - 1/n_i^2)$, where $R_H$ is the Rydberg constant. Bohr's model provided the theoretical derivation for $R_H$.
• Ground State Radius: The smallest orbit ($n=1$) gave the Bohr radius: $a_0 = 4\pi\varepsilon_0 \hbar^2 / (m_e e^2) \approx 0.529 \text{ \AA}$, a fundamental atomic length scale.

Limitations: While groundbreaking for hydrogen, the Bohr model failed for multi-electron atoms. It couldn't explain:

• The relative intensities of spectral lines.
• The fine structure splitting of lines.
• Atoms with more than one electron.
• The fundamental reasons behind quantization (addressed later by wave mechanics).

Despite its limitations, Bohr's model was pivotal. It shattered classical notions, introduced quantization into atomic structure, successfully explained hydrogen's spectrum, and laid essential groundwork for the full development of quantum mechanics.