Explanation of Line Spectrum of Hydrogen

The Line Spectrum of Hydrogen: A Quantitative Explanation

Bohr's model provides a complete quantitative explanation for the line spectrum of hydrogen. The key is the relationship between electron transitions between orbits and the absorption or emission of radiation.

When an electron jumps from a lower energy orbit (smaller principal quantum number $n$) to a higher energy orbit (larger $n$), the atom absorbs energy. Conversely, when an electron falls from a higher orbit to a lower orbit, the atom emits energy. The energy change involved in any such transition is the difference between the energies of the two orbits involved.

The Energy Gap Between Orbits

The energy of an electron in the $n$th orbit of hydrogen is given by:

where $R_H = 2.18 \times 10^{-18} \text{ J}$ is the Rydberg constant for hydrogen.

For a transition from an initial orbit $n_i$ to a final orbit $n_f$, the energy change $\Delta E$ is:

Substituting the energy expression:

This is equation (2.17) in the textbook. In terms of the numerical value of $R_H$:

From Energy to Frequency and Wavenumber

The frequency $\nu$ of the photon absorbed or emitted is obtained from Planck's relation $E = h\nu$:

Evaluating the constant:

This is equation (2.19). Note that for emission, $n_i > n_f$, so the term in parentheses is positive and gives the magnitude of the frequency.

The wavenumber $\bar{\nu}$ (the reciprocal of wavelength, measured in m$^{-1}$) is related to frequency by $\bar{\nu} = \nu/c$, where $c = 3.0 \times 10^8 \text{ m s}^{-1}$:

This is equation (2.21). The constant $1.09677 \times 10^7 \text{ m}^{-1}$ is the Rydberg constant expressed in wavenumber units.