Atomic Models: Thorson, Rutherford, and Bohr

Thorson Model of the Atom

The Thorson model represents an early theoretical framework regarding atomic structure, proposing how internal atomic constituents are arranged within a defined spatial volume to maintain net electrical neutrality in matter. Under this conceptual framework, positive electrical charge is conceived as a continuous, diffuse sphere or medium, within which discrete negative charges are embedded. This structural configuration was designed to maintain electrostatic equilibrium under non-dynamic conditions.

The electrostatic forces governing interactions within the Thorson atomic framework provided an initial mathematical basis for calculating atomic radii and understanding atomic interactions prior to high-energy collision experiments. However, despite its historical role in establishing models of charged subatomic constituents, the Thorson framework encountered severe theoretical limitations. Specifically, it could not account for large-angle particle deflection patterns observed in scattering experiments, which eventually necessitated a centralized, dense nuclear model.

Rutherford Model of the Atom

The Rutherford model of the atom established a nuclear framework based on high-velocity alpha particle scattering experiments through thin metallic foils. Empirical measurements revealed that while the vast majority of energetic particles passed through target foils with minimal deflection, a small fraction experienced deflection angles exceeding $90^\times$ degrees. This observed scattering distribution demonstrated that positive charge and virtually the entire mass of an atom are concentrated within an extraordinarily small, dense central region known as the nucleus.

In the Rutherford atomic model, negatively charged electrons orbit the dense central nucleus in planetary trajectories, bound by electrostatic attractive forces governed by Coulomb's law. The magnitude of the attractive electrostatic force $F_e$ acting between a central nucleus of charge $Z e$ and an electron of charge $e$ separated by orbital distance $r$ is expressed as:

$F_e = \frac{1}{4 \pi \epsilon_0} \frac{Z e^2}{r^2}$

This central Coulomb force supplies the necessary centripetal acceleration required for continuous circular orbital motion of an electron with mass $m_e$ and orbital velocity $v$:

$\frac{m_e v^2}{r} = \frac{1}{4 \pi \epsilon_0} \frac{Z e^2}{r^2}$

Despite its success in describing central alpha particle deflection dynamics, the Rutherford model possessed a critical failure under classical electrodynamics. According to classical Maxwellian electromagnetic theory, any accelerated charged particle continuously radiates energy in the form of electromagnetic waves. Because an electron in circular motion undergoes continuous centripetal acceleration, it should continuously lose orbital kinetic energy and spiral inward into the nucleus within approximately $10^{-10}\,s$. Additionally, this continuous orbital decay would produce a continuous emission spectrum, directly contradicting the discrete spectral lines observed experimentally from atomic gases.

Bohr Model of the Atom

The Bohr model resolved the dynamic instability inherent in classical nuclear models by incorporating quantum concepts into electronic motion. Formulated specifically for single-electron hydrogenic systems, the model postulated that electrons orbit the central nucleus only along specific, non-radiating trajectories designated as stationary states. An electron occupying a stationary orbit experiences no energy loss despite undergoing centripetal acceleration.

Bohr postulated that the orbital angular momentum $L$ of an electron is quantized, restricted to integer multiples of Dirac's constant $\hbar = \frac{h}{2 \pi}$, where $h$ is Planck's constant. For an electron of mass $m_e$ orbiting at radius $r$ with tangential velocity $v$, this quantization rule is defined as:

$L = m_e v r = n \hbar = \frac{n h}{2 \pi}$

In this equation, the principal quantum number $n$ takes discrete integer values $n \in \{1, 2, 3, \dots\}$. Combining the angular momentum quantization condition with classical electrostatic orbital equilibrium allows the orbital radius $r_n$ for a given quantum state $n$ in a hydrogen atom ($Z = 1$) to be calculated as:

$r_n = \frac{4 \pi \epsilon_0 n^2 \hbar^2}{m_e e^2} = n^2 a_0$

where $a_0 \approx 5.29 \times 10^{-11}\,m$ denotes the fundamental Bohr radius. The corresponding total mechanical energy $E_n$ associated with the $n$-th stationary orbit is quantized according to:

$E_n = -\frac{m_e e^4}{32 \pi^2 \epsilon_0^2 \hbar^2 n^2} = -\frac{13.6\,eV}{n^2}$

Electromagnetic radiation is emitted or absorbed exclusively when an electron undergoes a quantum transition between two distinct stationary states characterized by initial and final principal quantum numbers $n_i$ and $n_f$. The frequency $\nu$ and wavelength $\lambda$ of the involved photon are dictated by the energy difference $\Delta E$ between the states, governed by the Planck-Einstein relation:

$\Delta E = E_i - E_f = h \nu = \frac{h c}{\lambda}$

This framework accurately reproduced the Rydberg formula for the hydrogen spectrum:

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

where $R_H \approx 1.097 \times 10^7\,m^{-1}$ represents the Rydberg constant. While successful for single-electron systems, the Bohr model could not account for multi-electron atomic spectra, fine spectral splitting, or the full quantum mechanical wave properties of matter.