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.

A laser pointer commonly used in classroom demonstrations emits visible red electromagnetic radiation with a wavelength of $656.3\,\mathrm{nm}$. From these data, calculate:

1. a) The frequency associated with this electromagnetic radiation, expressed in SI units.
2. b) The energy carried by a single photon emitted by the laser device.

Data: speed of light in a vacuum, $c = 3.00 \times 10^8\,\mathrm{m\times s^{-1}}$; Planck constant, $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$.

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: $1\,\mathrm{eV} = 1,602 \times 10^{-19}\,\mathrm{J}$; $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; $c = 3.00 \times 10^8\,\mathrm{m\times s^{-1}}$.

In a discharge lamp containing monatomic hydrogen gas, a mass electronic transition takes place from the fourth quantised level ($n=4$) to the ground state ($n=1$).

1. a) Calculate the energy released by a single hydrogen atom in this de-excitation and determine the frequency of the emitted photon.
2. b) Calculate the total energy emitted, expressed in $\mathrm{kJ\times mol^{-1}}$, if one whole mole of hydrogen atoms undergoes this de-excitation simultaneously.

Data: Bohr formula for hydrogen: $E_n = -\frac{2.18 \times 10^{-18}}{n^2}\,\mathrm{J}$; Planck constant, $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; Avogadro constant, $N_A = 6,022 \times 10^{23}\,\mathrm{mol^{-1}}$.

In an optical spectrometer, an emission line of the hydrogen atom is recorded in the visible range (Balmer series, whose final level is $n_f = 2$) with an experimental wavelength of $\lambda = 434.1\,\mathrm{nm}$.

1. a) Calculate the energy of the emitted photon in joules and determine the principal quantum number of the initial orbital level $n_i$ from which the electron came, using the Bohr expression $E_n = -\frac{2.18 \times 10^{-18}}{n^2}\,\mathrm{J}$.
2. b) Justify rigorously, by means of the relevant energy balance, whether an incident photon of radiation with an energy of $3.00\,\mathrm{eV}$ could be absorbed by a hydrogen atom in the state $n=2$.

Data: $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; $c = 3.00 \times 10^8\,\mathrm{m\times s^{-1}}$; $1\,\mathrm{eV} = 1,602 \times 10^{-19}\,\mathrm{J}$.

A hydrogen discharge tube emits ultraviolet radiation corresponding to the Lyman series, in which all transitions end in the ground state $n_f = 1$. Two specific lines are considered: transition 1 from $n=2$ and transition 2 from $n=3$.

1. a) Calculate the energy of the photons emitted in each of the two transitions (in $\mathrm{J}$) and their associated wavelengths (in $\mathrm{nm}$).
2. b) Explain why the Bohr equations approximate the principal levels and spectral lines of the singly ionised helium ion $\mathrm{He}^+$ by including the factor $Z^2$ in the energy ($Z=2$) and neglecting fine structure and mass corrections, but are unable to calculate the emission frequencies of the neutral helium atom ($\mathrm{He}$).

Data: $E_n = -\frac{2.18 \times 10^{-18}}{n^2}\,\mathrm{J}$ (for H); $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; $c = 3.00 \times 10^8\,\mathrm{m\times s^{-1}}$.

The Bohr atomic model was a milestone in introducing the concept of quantisation into atomic structure, successfully explaining the lines of the hydrogen spectrum; however, it soon revealed insurmountable limitations.

1. a) Explain two specific shortcomings of the Bohr model: its inability to describe atomic systems with more than one electron, and the fine structure of spectra observed with high-resolution spectroscopes.
2. b) Explain why attributing to the electron a trajectory with simultaneously defined position and momentum is incompatible with Heisenberg's principle. Distinguish this limitation from the fact that a de Broglie standing wave can indeed provide an interpretation of Bohr's quantisation.

The ionisation energy of a gaseous chemical species is the minimum energy required to remove its outermost electron, taking it from its initial state to rest at infinity.

1. a) Using the Bohr model for hydrogen ($E_n = -\frac{2.18 \times 10^{-18}}{n^2}\,\mathrm{J}$), calculate the ionisation energy of a hydrogen atom in its ground state ($n=1$) in joules and the corresponding value for one mole of atoms in $\mathrm{kJ\times mol^{-1}}$.
2. b) If a gaseous sample of hydrogen atoms is previously excited in level $n=3$, determine the maximum wavelength (in $\mathrm{nm}$) that a beam of monochromatic light must have to photoionise the atoms from that excited level, and identify, with reasoning, the region of the electromagnetic spectrum it belongs to.

Data: $h = 6,626 \times 10^{-34}\,\mathrm{J\times s}$; $c = 3.00 \times 10^8\,\mathrm{m\times s^{-1}}$; $N_A = 6,022 \times 10^{23}\,\mathrm{mol^{-1}}$.