Exercises: Spectra and the Bohr model

The analysis of light emitted or absorbed by substances is a fundamental tool for understanding the electron shell of the atom.

1. a) Explain the essential physical difference between a continuous spectrum and a discontinuous or line spectrum.
2. b) Justify why the atomic emission spectrum of a low-pressure gas consists of discrete lines, and explain how these lines correspond to those observed in its absorption spectrum.

In the atomic model proposed by Bohr for the hydrogen atom, the energy associated with each electronic level depends on the principal quantum number $n$ through the expression:

$$E_n = -\frac{2.18 \times 10^{-18}}{n^2}\,\mathrm{J}$$

1. a) Calculate the energy of the electron in the ground state ($n=1$) and in the first excited state ($n=2$).
2. b) Explain the physical meaning of the negative sign assigned to these energy levels and what the state in which $n \rightarrow \infty$ represents.

The following diagram shows the first three energy levels of the hydrogen atom and two electronic transitions labelled A and B:

. Using the diagram and the data provided:

1. a) Explain whether transitions A and B correspond to processes of energy absorption or emission.
2. b) Calculate the energy (in $\mathrm{eV}$) and the wavelength (in $\mathrm{nm}$) of the photon involved in transition A ($3 \rightarrow 2$), and state, with reasoning, which of the two transitions (A or B) emits radiation of shorter wavelength.

Data: $E_1 = -13.6\,\mathrm{eV}$; $E_2 = -3.4\,\mathrm{eV}$; $E_3 = -1.51\,\mathrm{eV}$; conversion factor: $1\,\mathrm{eV} = 1,602 \times 10^{-19}\,\mathrm{J}$; $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; $c = 3.00 \times 10^8\,\mathrm{m\times s^{-1}}$.