The atom is the basic building block of matter, yet understanding its internal structure required nearly a century of experiment and theory. This chapter traces the evolution of atomic models from Thomson's plum pudding model to Rutherford's nuclear model and finally to Bohr's quantum model of the hydrogen atom.
Rutherford's alpha particle scattering experiment revealed that the positive charge and nearly all the mass of an atom are concentrated in a tiny nucleus. This nuclear model, however, could not explain the stability of atoms or the observed line spectra. Classical physics predicted that an orbiting electron would radiate energy and spiral into the nucleus within a fraction of a second.
Niels Bohr resolved these problems in 1913 by applying quantum ideas: electrons occupy discrete orbits in which they do not radiate, and spectral lines are emitted when electrons jump between these orbits. Bohr's model successfully explained the hydrogen spectrum and laid the foundation for modern atomic physics.
J.J. Thomson proposed that an atom consists of a sphere of positive charge with electrons embedded in it, like plums in a pudding. This plum pudding model explained electrical neutrality but gave no structure to spectral lines and could not explain how electrons were arranged.
Ernest Rutherford tested the model by bombarding a thin gold foil with alpha particles. The results were startling: most alpha particles passed straight through, a few were deflected through large angles, and about one in 8000 bounced back. This showed that the atom is mostly empty space with a tiny, dense, positively charged nucleus at its centre.
From the scattering data, Rutherford concluded that the nucleus has a radius of about 10^-15 m (a few fermi), while the atomic radius is about 10^-10 m. The electrons revolve around the nucleus in orbits, held by the electrostatic attraction. The distance of closest approach and the impact parameter describe the trajectories of the alpha particles.
Rutherford's model had serious flaws. According to classical electromagnetic theory, an accelerated electron revolving around the nucleus would continuously radiate energy, lose speed, and spiral into the nucleus. The atom would collapse in about 10^-8 seconds, yet atoms are stable.
The model also predicted that the emitted radiation should span a continuous range of frequencies as the electron spirals inward. But atoms emit light only at discrete frequencies, forming line spectra. Both of these contradictions required a fundamentally new approach - the quantization of electron orbits.
Rutherford's model also could not explain the distribution of electrons around the nucleus, nor the atomic spectra of hydrogen. These failures motivated Bohr to introduce quantum conditions that could not be derived from classical physics.
When hydrogen gas is excited electrically, it emits light that, when passed through a prism, shows a discrete line spectrum rather than a continuous rainbow. These lines are grouped into series named after their discoverers.
The Lyman series lies in the ultraviolet, the Balmer series in the visible region, and the Paschen, Brackett, and Pfund series in the infrared. The wavelengths of all hydrogen lines fit a single empirical formula:
1/lambda = R (1/n1^2 - 1/n2^2)
where R = 1.097 x 10^7 m^-1 is the Rydberg constant, n1 and n2 are integers with n2 > n1. For the Lyman series n1 = 1, for the Balmer series n1 = 2, for the Paschen series n1 = 3, and so on. The line spectrum is unique to each element and is the fingerprint used in spectroscopy.
Bohr combined Rutherford's nuclear atom with the quantum ideas of Planck and Einstein through three postulates:
Using these postulates and the Coulomb force as the centripetal force, Bohr derived the radius of the nth orbit:
r_n = n^2 * (h^2 epsilon_0 / (pi m e^2)) = n^2 a0
where a0 = 0.53 angstrom is the Bohr radius. The energy of the nth orbit is:
E_n = -13.6 / n^2 eV
The energy is negative because the electron is bound to the nucleus. The ground state energy of hydrogen is -13.6 eV, and the energy levels converge to zero as n increases.
When an electron jumps from orbit n2 to n1, the emitted photon has wavelength given by the energy difference:
1/lambda = (13.6 eV / h c)(1/n1^2 - 1/n2^2)
This reproduces the Rydberg formula with the Rydberg constant:
R = m e^4 / (8 epsilon_0^2 h^3 c) = 1.097 x 10^7 m^-1
The spectral series are: - Lyman series (n1 = 1, ultraviolet): transitions to the ground state. - Balmer series (n1 = 2, visible): transitions to the first excited state. - Paschen series (n1 = 3, infrared). - Brackett series (n1 = 4) and Pfund series (n1 = 5).
Bohr's model explained the hydrogen spectrum with remarkable accuracy. The ionization energy of hydrogen, 13.6 eV, is the energy needed to remove the ground-state electron, matching experiment.
Bohr's quantization rule, that angular momentum is mvr = nh/2 pi, appeared arbitrary. In 1924, de Broglie provided a natural explanation: the electron orbit is allowed only if the de Broglie wave of the electron forms a standing wave around the orbit. This requires the circumference to be an integral multiple of the wavelength:
2 pi r = n lambda = n h / (m v)
Rearranging gives exactly Bohr's quantization condition m v r = n h/2 pi. The electron in a stable orbit behaves like a standing wave, and the integer n counts the number of complete waves around the orbit. This showed that Bohr's rule was a consequence of the wave nature of matter.
| Quantity | Formula | Value |
|---|---|---|
| Rydberg formula | 1/lambda = R(1/n1^2 - 1/n2^2) | R = 1.097 x 10^7 m^-1 |
| Bohr radius | a0 = h^2 epsilon_0/(pi m e^2) | 0.53 angstrom |
| Orbit radius | r_n = n^2 a0 | n^2 scaling |
| Energy levels | E_n = -13.6/n^2 eV | Ground state -13.6 eV |
| Angular momentum | m v r = n h/2 pi | Quantized |
| Photon energy | h f = E_n2 - E_n1 | Emission on transition |
| Series | n1 | Region |
|---|---|---|
| Lyman | 1 | Ultraviolet |
| Balmer | 2 | Visible |
| Paschen | 3 | Infrared |
| Brackett | 4 | Infrared |
| Pfund | 5 | Infrared |
This chapter traced the development of atomic models. Rutherford's scattering experiment revealed the atom's tiny, massive, positively charged nucleus surrounded by mostly empty space, but classical physics could not explain atomic stability or line spectra. Bohr's postulates - quantized angular momentum mvr = nh/2 pi, stationary orbits, and energy emitted in jumps - led to r_n = n^2 a0 and E_n = -13.6/n^2 eV, which reproduce the hydrogen spectrum and the Rydberg formula. De Broglie's standing wave condition explained Bohr's quantization naturally. The spectral series of hydrogen and the quantization of energy remain central ideas in modern physics and the basis for the study of nuclei and quantum mechanics.