By the nineteenth century, substantial evidence supported the atomic hypothesis of matter. In 1897, J. J. Thomson discovered the electron , proving that atoms contain identical negatively charged constituents. Since atoms are electrically neutral, they must also contain a neutralizing positive charge.
Thomson proposed that the positive charge of the atom is uniformly distributed throughout its volume, with negatively charged electrons embedded in it like seeds in a watermelon. This model is historically known as the plum pudding model .
In 1911, H. Geiger and E. Marsden, under the direction of Ernst Rutherford, performed a landmark scattering experiment using \alpha-particles directed at a thin gold foil.
Historical Chronicle: Ernst Rutherford (1871 – 1937)
A collimated beam of 5.5 MeV \alpha-particles from a bismuth-214 source was directed at a thin gold foil of thickness 2.1 &× 10 -7 m . The scattered particles were detected using a rotatable zinc sulphide screen and microscope.
The trajectory of a scattered \alpha-particle depends on its impact parameter ($b$), which is the perpendicular distance of the initial velocity vector of the \alpha-particle from the center of the nucleus.
The electrostatic repulsive force is governed by Coulomb's Law:
$$F = \frac{1}{4\pi\epsilon_0} \frac{(2e)(Ze)}{r^2}$$Mathematical Problem Statement: In a Geiger-Marsden experiment, calculate the distance of closest approach to the nucleus of a 7.7 MeV \alpha-particle before it momentarily comes to rest and reverses its direction (for Gold, $Z = 79$).
Step-by-Step Algorithmic Solution Trace:
- Identify conservation of mechanical energy: Initial kinetic energy ($K$) equals final potential energy ($U$) at the distance of closest approach ($d$).$$K = 7.7 \text{ MeV} = 1.2 \times 10^{-12} \text{ J}$$
- Set up the potential energy equation:$$K = \frac{1}{4\pi\epsilon_0} \frac{2Ze^2}{d}$$
- Isolate the distance of closest approach ($d$):$$d = \frac{2Ze^2}{4\pi\epsilon_0 K}$$
- Substitute the values ($e = 1.6 \times 10^{-19} \text{ C}$, $Z = 79$, $\frac{1}{4\pi\epsilon_0} = 9.0 \times 10^9 \text{ N m}^2/\text{C}^2$):$$d = \frac{2 \times (9.0 \times 10^9) \times 79 \times (1.6 \times 10^{-19})^2}{1.2 \times 10^{-12}} \approx 3.0 \times 10^{-14} \text{ m} = 30 \text{ fm}$$
Final Evaluation Output: $d \approx 3.0 \times 10^{-14} \text{ m}$
Rutherford's nuclear model pictured the atom as a dynamically stable system where electrons revolve around the nucleus in circular orbits, balanced by electrostatic attraction.
The electrostatic force of attraction provides the necessary centripetal force for a stable orbit of radius $r$ and velocity $v$:
$$F_e = F_c \implies \frac{1}{4\pi\epsilon_0} \frac{e^2}{r^2} = \frac{mv^2}{r}$$This yields the relation for the orbit radius:
$$r = \frac{e^2}{4\pi\epsilon_0 mv^2}$$The total mechanical energy ($E$) of the electron is the sum of its kinetic energy ($K$) and potential energy ($U$):
$$K = \frac{1}{2}mv^2 = \frac{e^2}{8\pi\epsilon_0 r}$$$$U = -\frac{e^2}{4\pi\epsilon_0 r}$$$$E = K + U = -\frac{e^2}{8\pi\epsilon_0 r}$$The negative sign of the total energy indicates that the electron is bound to the nucleus.
According to classical electromagnetic theory, an accelerating charged particle (such as a revolving electron undergoing centripetal acceleration) must continuously emit electromagnetic radiation.
Niels Bohr combined classical mechanics with early quantum concepts to resolve the instability of Rutherford's model.
Historical Chronicle: Niels Henrik David Bohr (1885 – 1962)
By combining the quantization condition with orbital mechanics, we derive the radius of the $n$-th orbit ($r_n$):
$$r_n = \left(\frac{n^2}{m}\right) \left(\frac{h}{2\pi}\right)^2 \frac{4\pi\epsilon_0}{e^2} = \frac{n^2 h^2 \epsilon_0}{\pi m e^2}$$The total energy of the electron in the $n$-th stationary state is:
$$E_n = -\frac{m e^4}{8 \epsilon_0^2 n^2 h^2}$$$$E_n = -\frac{2.18 \times 10^{-18}}{n^2} \text{ J} = -\frac{13.6}{n^2} \text{ eV}$$In 1923, Louis de Broglie explained Bohr's quantization of angular momentum by introducing the wave nature of electrons.
An electron in a circular orbit behaves as a particle wave. For the wave to persist without destructive interference, it must form a standing wave. This requires the circumference of the orbit to be an integral multiple of the de Broglie wavelength ($\lambda$):
$$2\pi r_n = n\lambda \quad (n = 1, 2, 3, \dots)$$Substituting the de Broglie relation $\lambda = \frac{h}{p} = \frac{h}{m v_n}$:
$$2\pi r_n = n \frac{h}{m v_n}$$$$m v_n r_n = \frac{nh}{2\pi}$$This elegantly derives Bohr's second postulate from the wave-particle duality of matter.
Despite its success in explaining the hydrogen spectrum, the Bohr model has several critical limitations:
| Atomic Model | Core Structural Concept | Mathematical/Physical Formulation | Key Limitations |
|---|---|---|---|
| Thomson's Model | Plum pudding; uniform positive sphere with embedded electrons. | Electrostatically neutral sphere. | Failed to explain large-angle \alpha-particle scattering. |
| Rutherford's Model | Planetary model; tiny, massive positive nucleus with orbiting electrons. | $F_e = F_c = \frac{mv^2}{r}$ | Electromagnetically unstable; predicts continuous spectrum collapse. |
| Bohr's Model | Quantized stationary orbits; transitions emit discrete photons. | $m v r = \frac{nh}{2\pi}$, $E_n = -\frac{13.6}{n^2} \text{ eV}$ | Limited to single-electron systems; cannot predict line intensities. |