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Energy Levels in Atoms and Line Spectra

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Energy Levels of Atoms

1. Electron shells

2. Transitions between electron shells

3. Forum

↓ Show theory [1] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:energy_levels_in_atoms_and_line_spectra/4547/

Example 1.

The figure below shows the visible part of the emission spectrum from hydrogen gas in a laboratory on the Earth. The numbers indicate the wavelength, in , represented by each line.

Explain how the emission spectrum provides evidence for the existence of discrete energy levels for the electron in a hydrogen atom.
The figure below shows five of the energy levels in the hydrogen atom. The wavelengths of radiation shown in the figure above relate to transitions to the level in the figure below.

Show that the energy level X is .

Example 2.

Explain what is meant by a photon.
An emission spectrum is seen as a series of differently coloured lines on a black background. Suggest how this observation provides evidence for discrete electron energy levels in atoms.

State what is meant by a photon.
Describe the appearance of a visible line emission spectrum, as seen using a diffraction grating.
The lowest electron energy levels in an isolated hydrogen atom are shown in the figure below.

(not to scale)
- An electron is initially at the energy level . State the total number of different wavelengths that may be emitted as the electron de-excites (loses energy).
- Photons resulting from electron de-excitation from the energy level are incident on the surface of a sample of platinum. Platinum has a work function energy of . Determine
  - the maximum kinetic energy, in , of a photoelectron emitted from the surface of the platinum,
  - the wavelength of the photon producing the photoelectron in (ii) part 1.

The emission spectrum of atomic hydrogen consists of a number of discrete wavelengths. Explain how this observation leads to an understanding that there are discrete electron energy levels in atoms.
Some electron energy levels in atomic hydrogen are illustrated in the figure below.

The longest wavelength produced as a result of electron transitions between two of the energy levels shown in the figure above is .
- On the figure above
  - draw, and mark with the letter L, the transition giving rise to the wavelength of ,
  - draw, and mark with the letter S, the transition giving rise to the shortest wavelength.
- Calculate the wavelength for the transition you have shown in (i) part 2.
Photon energies in the visible spectrum vary between approximately and . Determine the energies, in , of photons in the visible spectrum that are produced by transitions between the energy levels shown in the figure above.

Some of the electron energy bands in a solid are illustrated in the figure below.

In isolated atoms, electron energy levels have discrete values. Suggest why, in a solid, there are energy bands, rather than discrete energy levels.
A light-dependent resistor (LDR) consists of an intrinsic semiconductor. Use band theory to explain the dependence on light intensity of the resistance of the LD when it is at constant temperature.

Some of the electron energy bands in a semiconductor material at the absolute zero of temperature are shown in the figure below.

Use band theory to explain why, as the temperature of the semiconductor material rises, the electrical resistance of the sample of material decreases.

A beam of light consists of a continuous range of wavelengths from to . The light passes through a cloud of cool gas, as shown in Figure I below.

The spectrum of the light emerging from the cloud of cool gas is viewed using a diffraction grating. Explain why this spectrum contains a number of dark lines.
Some of the electron energy levels of the atoms in the cloud of gas are represented in Figure II below.

II (not to scale)
- Light of wavelength has a photon energy of . Calculate the photon energy, in , of light of wavelength .
- Use data from (i) and your answer in (i) to show, on figure II, the changes in energy levels giving rise to the dark lines in (a).

Four electron energy levels in an isolated atom are shown in the figure below.

For the emission spectrum associated with these energy levels,

on the figure above, mark with an arrow the transition that gives rise to the shortest wavelength,
show that the wavelength of the transition in (a) is .

Some of the electron energy bands in a solid are illustrated in the figure below.

The width of the forbidden band and the number of charge carriers occupying each band depends on the nature of the solid. Use band theory to explain why the resistance of a sample of a metal at room temperature changes with increasing temperature.

10.

White light passes through a cloud of cool, low-pressure gas, as illustrated in the figure below.

For light that has passed through the gas, its continuous spectrum is seen to contain a number of darker lines.
Use the concept of discrete electron energy levels to explain the existence of these darker lines.
The uppermost electron energy bands in a solid are illustrated in the figure below.

Use band theory to explain the dependence on light intensity of the resistance of a light-dependent resistor (LDR).

11.

White light is incident on a cloud of cool hydrogen gas, as illustrated in Figure I below.

The spectrum of the light emerging from the gas cloud is found to contain a number of dark lines.

Explain why these dark lines occur.
Some electron energy levels in a hydrogen atom are illustrated in Figure II below.

II

One dark line is observed at a wavelength of .
- Calculate the energy, in , of a photon of light of wavelength .
- On figure II above, draw an arrow to indicate the energy change that gives rise to this dark line.

12.

Figure I below shows the lowest four energy levels of an electron in an isolated atom.

Figure II below shows the lines in the emission spectrum of the atom that correspond to the transitions of the electron from to and from to .

Explain, with reference to photons, why there is a single frequency of electromagnetic radiation that corresponds to each of these transitions.
- On Figure II, draw a line that corresponds to the transition of the electron from to . Label this line A.
- On Figure II, draw a line that corresponds to the transition of the electron from to . Label this line B.
The frequency of radiation represented by line A is . The frequency of radiation represented by line B is . The energy of the ground state () is . Determine an expression, in terms of , , and the Planck constant , for the energy of the energy level .

13.

Explain how the line spectrum of hydrogen provides evidence for the existence of discrete electron energy levels in atoms.
Some electron energy levels in atomic hydrogen are illustrated in the figure below.

Two possible electron transitions, A and B, giving rise to an emission spectrum, are shown. These electron transitions cause light of wavelengths and to be emitted.
- On the figure above, draw an arrow to show a third possible transition.
- Calculate the wavelength of the emitted light for the transition in (i).

14.

The figure below shows part of the emission spectrum of visible radiation emitted by hydrogen gas in a star in a distant galaxy.

The galaxy is moving away from the Earth at a speed of .

- Explain how the positions of the lines in the emission spectrum seen by an observer on the Earth differ from the positions shown in the figure above.
- On the figure above, draw the three lines in possible positions in the spectrum seen by the observer.
The lines in the figure above correspond to electron transitions down to the energy level . One of the lines represents emitted radiation of wavelength .
- Calculate the energy of a photon of this radiation.
- Determine the energy, in , of the energy level from which the electron transition originates to cause the emission of this radiation.
- Determine the wavelength, in , of this radiation as detected by the observer on the Earth.
A value for the Hubble constant is . Determine the distance of the galaxy from the Earth.

15.

State what is meant by a photon.
Light in a beam has a continuous spectrum that lies within the visible region. The photons of light have energies ranging from to . The beam passes through some hydrogen gas. It then passes through a diffraction grating, and an absorption spectrum is observed.
- All of the light absorbed by the hydrogen is re-emitted. Explain why dark lines are still observed in the absorption spectrum.
- Some of the energy levels of an electron in a hydrogen atom are illustrated in the figure below.
  
  (not to scale)
  
  The dark lines in the absorption spectrum are the result of electron transitions between energy levels. On the figure, draw arrows to show the initial electron transitions between energy levels that could give rise to dark lines in the absorption spectrum.
- Calculate the shortest wavelength of the light in the beam.

16.

Four electron energy levels in an atom are shown in the figure below.

(not to scale)

An emission spectrum is associated with the electron transitions between these energy levels. For this spectrum,

state the number of lines,
calculate the minimum wavelength.

17.

State what is meant by a photon.
Calculate the energy, in , of a photon of light of wavelength .
The outermost electron energy bands of a semiconductor material are illustrated in the figure below.

The width of the forbidden band is . Explain why, when photons of light, each of energy , are incident on the semiconductor material, its resistance decreases.

18.

Explain how a line emission spectrum leads to an understanding of the existence of discrete electron energy levels in atoms.
Some of the lines of the emission spectrum of atomic hydrogen are shown in the figure below.

The photon energies associated with some of these lines are shown in the table below.

Complete the table above by calculating the photon energy for a wavelength of .

19.

A beam of white light passes through a cloud of cool gas. The spectrum of the transmitted light is viewed and contains a number of dark lines. Explain why these dark lines occur.
Some energy levels for the electron in an isolated hydrogen atom are illustrated in the figure below.

The table below shows the wavelengths of photons that are emitted in the transitions to from the other energy levels shown in the figure above.

The energy associated with the energy level is . Calculate the energy, in , of energy level .

20.

- Briefly describe the concept of a photon.
- Explain how lines in the emission spectrum of gases at low pressure provide evidence for discrete electron energy levels in atoms.
Three electron energy levels in atomic hydrogen are represented in the figure below.

The wavelengths of the spectral lines produced by electron transitions between these three energy levels are , , and .
- On the figure above, draw arrows to show the electron transitions between the energy levels that would give rise to these wavelengths. Label each arrow with the wavelength of the emitted photon.
- Calculate the maximum change in energy of an electron when making transitions between these levels.