Bohr model of the atom Knowledge-Based Study

Study notes

1. The Crisis of Classical Physics

   In 1911, Ernest Rutherford proposed a planetary model of the atom in which negatively charged electrons orbit a dense, positively charged nucleus. While this model explained alpha particle scattering, it contradicted classical electrodynamics. According to Maxwell's theory, an accelerating electric charge continuously radiates electromagnetic energy. Because an orbiting electron undergoes centripetal acceleration, it should steadily lose energy and spiral into the nucleus within a fraction of a second. Furthermore, continuous radiation would produce a continuous spectrum, contradicting the discrete line spectra observed experimentally for atomic hydrogen.

   Bohr's Fundamental Postulates

   In 1913, Niels Bohr resolved these contradictions by introducing quantum ideas into the planetary model through three core postulates:

   • Stationary States: Electrons exist in specific, non-radiating circular orbits known as stationary states. While in these orbits, an electron maintains constant energy and does not emit electromagnetic radiation despite its centripetal acceleration.
   • Quantization of Angular Momentum: The orbital angular momentum $L$ of an electron is quantized and can only assume integral multiples of the reduced Planck constant: $$L = m_e v r = n \hbar = \frac{n h}{2\pi}$$ where $n = 1, 2, 3, \dots$ is the principal quantum number, $m_e$ is the electron mass, $v$ is orbital speed, and $r$ is the orbital radius.
   • Frequency Condition (Transitions): Radiation is emitted or absorbed only when an electron transitions between two stationary states. The energy of the emitted or absorbed photon equals the difference in energy between the initial state $E_i$ and the final state $E_f$: $$\Delta E = |E_i - E_f| = h f = \frac{h c}{\lambda}$$

   Derivation of Quantized Radii

   Consider a hydrogen-like ion with atomic number $Z$ and a single electron of mass $m_e$ and charge $e$. The electrostatic Coulomb force provides the necessary centripetal force for circular motion:

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

   where $k_e = \frac{1}{4\pi \varepsilon_0}$ is Coulomb's constant. Rearranging this relation yields:

   $$m_e v^2 r = k_e Z e^2$$

   Using Bohr's quantization condition, the orbital velocity can be expressed as:

   $$v = \frac{n \hbar}{m_e r}$$

   Substituting this expression for $v$ into the force equation gives:

   $$m_e \left(\frac{n \hbar}{m_e r}\right)^2 r = k_e Z e^2 \implies r_n = \frac{n^2 \hbar^2}{m_e k_e Z e^2}$$

   The Bohr Radius

   For neutral hydrogen ($Z = 1$) in its ground state ($n = 1$), this expression defines the Bohr radius $a_0$:

   $$a_0 = \frac{\hbar^2}{m_e k_e e^2} \approx 5.29 \times 10^{-11} \text{ m} = 0.0529 \text{ nm}$$

   Thus, the radius of the $n$-th orbit scales quadratically with $n$:

   $$r_n = \frac{n^2}{Z} a_0$$

   Orbital Velocity

   Substituting $r_n$ back into the quantization condition reveals the velocity of the electron in the $n$-th orbit:

   $$v_n = \frac{k_e Z e^2}{n \hbar} = \left(\frac{Z}{n}\right) \alpha c$$

   where $\alpha \approx \frac{1}{137}$ is the fine-structure constant and $c$ is the speed of light. As $n$ increases, the electron moves farther from the nucleus and its orbital speed decreases inversely with $n$.

   Total Energy of Stationary States

   The total mechanical energy $E$ of the electron is the sum of its kinetic energy $K$ and electrical potential energy $U$:

   $$K = \frac{1}{2} m_e v^2 = \frac{k_e Z e^2}{2 r}$$$$U = -\frac{k_e Z e^2}{r}$$$$E = K + U = -\frac{k_e Z e^2}{2 r}$$

   Notice that $E = -K = \frac{1}{2} U$, consistent with the virial theorem. Substituting the expression for $r_n$ yields the quantized total energy:

   $$E_n = -\frac{m_e k_e^2 Z^2 e^4}{2 n^2 \hbar^2}$$

   For atomic hydrogen ($Z = 1$), substituting known physical constants results in:

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

   The negative sign signifies a bound system; as $n \to \infty$, $E \to 0$, corresponding to an ionized, free electron at rest.

   The Rydberg Formula and Spectral Series

   When an electron drops from a higher level $n_i$ to a lower level $n_f$, a photon is emitted with energy:

   $$h f = E_i - E_f = 13.6 \text{ eV} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$

   Expressing this relationship in terms of inverse wavelength (wavenumber $\frac{1}{\lambda}$) produces the empirical Rydberg formula:

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

   where $R_H \approx 1.097 \times 10^7 \text{ m}^{-1}$ is the Rydberg constant.

   • Lyman Series ($n_f = 1$): Transitions down to the ground state; falls in the ultraviolet region.
   • Balmer Series ($n_f = 2$): Transitions down to the first excited state; falls largely in the visible spectrum.
   • Paschen Series ($n_f = 3$): Transitions down to the second excited state; falls in the infrared region.
   • Brackett ($n_f = 4$) and Pfund ($n_f = 5$) Series: Transitions falling further into the infrared region.

   Successes of the Model

   The Bohr model represented a breakthrough in modern physics by bridging classical mechanics and early quantum theory:

   • It provided a quantitative explanation for the stability of atoms.
   • It accurately calculated the Rydberg constant from fundamental constants ($m_e, e, h, \varepsilon_0$).
   • It correctly predicted the absorption and emission spectra of one-electron systems such as $\text{H}$, $\text{He}^+$, $\text{Li}^{2+}$, and $\text{Be}^{3+}$.

   Limitations of the Model

   Despite its remarkable success with hydrogenic systems, the model has significant conceptual and empirical limitations:

   • Multi-Electron Atoms: It fails to predict the energy levels and spectra of atoms with two or more electrons because it ignores electron-electron electrostatic repulsion and quantum exchange effects.
   • Spectral Line Intensities: It cannot predict transition probabilities or explain why certain spectral lines are brighter than others.
   • Fine and Hyperfine Structure: High-resolution spectroscopy reveals that individual spectral lines are split into closely spaced multiplets, an effect requiring relativistic corrections and electron spin.
   • Zeeman and Stark Effects: It cannot fully account for the splitting of spectral lines in external magnetic or electric fields without ad hoc modifications.
   • Violation of Quantum Mechanics: It assumes precise simultaneously known values for electron position and momentum in fixed orbits, violating the Heisenberg Uncertainty Principle formulated later.

   Ultimately, the Bohr model served as an indispensable stepping stone toward modern wave mechanics and the full quantum-mechanical treatment of the Schrödinger equation.