1. Rutherford's Model and Atomic Spectra

Ernest Rutherford's gold foil experiment led to the nuclear model of the atom, where a small, dense, positively charged nucleus is at the center, with electrons orbiting it. However, this classical model predicted that orbiting electrons should continuously radiate energy and spiral into the nucleus, which contradicts the stability of atoms and the existence of discrete atomic spectra. Atomic spectra are unique patterns of light emitted or absorbed by atoms, consisting of specific wavelengths, indicating that electron energies are quantized.

2. Bohr Model of Hydrogen Atom

Niels Bohr proposed a model for the hydrogen atom that successfully explained atomic spectra. He postulated that electrons orbit the nucleus in specific, stable orbits or energy levels without radiating energy. Electrons can jump between these orbits by absorbing or emitting photons of energy equal to the difference between the energy levels ($E = h\nu$). The energy levels for hydrogen are quantized and given by $E_n = -\frac{13.6 \, \text{eV}}{n^2}$, where $n$ is the principal quantum number. This model provided a quantum explanation for spectral lines.

3. Line Spectra and De Broglie's Explanation

The characteristic line spectra observed for elements arise from electrons transitioning between discrete energy levels. De Broglie's hypothesis of matter waves provided a deeper insight into Bohr's postulates. Bohr's third postulate, that the angular momentum of an electron in its orbit is quantized ($mvr = n\frac{h}{2\pi}$), can be justified by assuming that the electron orbits are standing waves, where the circumference of the orbit is an integer multiple of the electron's de Broglie wavelength ($2\pi r = n\lambda$).

4. Additional: Limitations of Bohr Model

While the Bohr model successfully explained the hydrogen spectrum, it had several limitations. It could not explain the spectra of atoms with more than one electron, the fine structure of spectral lines (splitting of lines into closely spaced components), the Zeeman effect (splitting of spectral lines in a magnetic field), or the relative intensities of spectral lines. These limitations paved the way for the development of more sophisticated quantum mechanics, introducing concepts like quantum numbers and electron spin.