Electron Configurations of Atoms and Ions

Lessons

Senior 4 · Chemistry

Electron Configurations of Atoms and Ions

5 lessons

Lesson 1 of 5

Bohr's Model, Energy Levels and Atomic Spectra

25 min

Every gas glows its own colour

Fill a glass tube with hydrogen at low pressure, apply a few thousand volts, and it glows pink. Nitrogen in the same tube glows violet and mercury vapour a bluish white. The colour is a signature: it does not depend on the shape of the tube or the voltage, only on which element is inside. Pass the light through a prism and the signature becomes sharper still. Instead of a rainbow you see a few separate bright lines. In 1913 Niels Bohr used those lines to argue that the electron in an atom can have only certain energies. This unit begins with his idea, because it is the first step to the s, p, d and f configurations you will use for the rest of A-level chemistry.

Definition

Electromagnetic radiation: energy that travels as a wave at the speed of light, $c = 3.00 \times 10^{8}\,\mathrm{m\,s^{-1}}$ . Its wavelength $\lambda$ and frequency $f$ are linked by $c = f\lambda$ .

Photon: a single packet (quantum) of electromagnetic energy. Its energy is $E = hf = \frac{hc}{\lambda}$ where $h = 6.626 \times 10^{-34}\,\mathrm{J\,s}$ is the Planck constant. The shorter the wavelength, the more energy each photon carries.

Table 1.1: Parts of the electromagnetic spectrum used in this unit

Region	Wavelength	Photon energy	Hydrogen series seen here
ultraviolet	about 10 nm to 400 nm	high	Lyman
visible: violet to red	about 400 nm to 700 nm	medium	Balmer
infrared	about 700 nm to 1 mm	low	Paschen, Brackett, Pfund

$1\,\mathrm{nm} = 10^{-9}\,\mathrm{m}$ . Visible light is a narrow band in the middle.

Worked example 1: the energy of a photon

Sodium street lamps give out yellow light of wavelength 589 nm. Calculate (a) its frequency, (b) the energy of one photon, (c) the energy of one mole of these photons, in $\mathrm{kJ\,mol^{-1}}$ . ( $N_A = 6.022 \times 10^{23}\,\mathrm{mol^{-1}}$ )

Solution. Convert first: $\lambda = 589 \times 10^{-9}\,\mathrm{m}$ .

(a) $f = \frac{c}{\lambda} = \frac{3.00 \times 10^{8}}{589 \times 10^{-9}} = 5.09 \times 10^{14}\,\mathrm{Hz}$

(b) $E = hf = 6.626 \times 10^{-34} \times 5.09 \times 10^{14} = 3.37 \times 10^{-19}\,\mathrm{J}$

(c) $3.37 \times 10^{-19} \times 6.022 \times 10^{23} = 2.03 \times 10^{5}\,\mathrm{J\,mol^{-1}} = 203\,\mathrm{kJ\,mol^{-1}}$

Answer: $5.09 \times 10^{14}\,\mathrm{Hz}$ ; $3.37 \times 10^{-19}\,\mathrm{J}$ ; $203\,\mathrm{kJ\,mol^{-1}}$ , similar in size to the energy of a chemical bond.

Three kinds of spectrum

A spectroscope spreads light out by wavelength with a prism or a diffraction grating, so you can see which wavelengths are present.

A continuous spectrum contains every wavelength in a range, merging into one another. A hot solid, such as the filament of an old lamp, gives one.
An emission (line) spectrum is a set of separate bright lines on a dark background. A hot or electrically excited gas of atoms gives one, and each element has its own pattern.
An absorption spectrum is a continuous spectrum crossed by dark lines. It forms when white light passes through a cooler gas: the atoms remove exactly the wavelengths they would emit if excited.

The spectrum of sunlight, spread into strips from violet (top) to red (bottom). The dark Fraunhofer lines are an absorption spectrum: atoms in the Sun's outer layers absorb light of particular wavelengths. Image: NASA, public domain, via Wikimedia Commons

Bohr's model of the hydrogen atom (1913)

Rutherford's nuclear atom had a problem. An electron circling a nucleus should, by the physics of the time, radiate energy continuously and spiral into the nucleus, giving a continuous spectrum. Atoms are stable, and they give line spectra. Bohr proposed four ideas:

The electron moves round the nucleus only in certain circular orbits, each with a fixed energy. These are the energy levels, numbered $n = 1, 2, 3, \dots$ outwards.
While it stays in one orbit, the electron does not radiate energy.
The electron can move to a higher level by absorbing exactly the energy difference between the levels, and falls to a lower level by emitting that energy as one photon: $\Delta E = E_{\text{upper}} - E_{\text{lower}} = hf$
Because only certain energies are allowed, only certain photon energies, and so certain wavelengths, can be emitted or absorbed. That is why the spectrum is made of lines.

Definition

Quantised: able to take only certain fixed values, not any value. In Bohr's model the energy of the electron is quantised.

Ground state: the lowest energy level an electron can occupy (for hydrogen, $n = 1$ ). Excited state: any higher level, reached by absorbing energy from heat, an electric discharge or light.

The energy levels of hydrogen

Bohr calculated the energy of level $n$ in a hydrogen atom:

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

The energies are negative because the zero of energy is chosen as the electron completely removed from the atom ( $n = \infty$ ). A bound electron has less energy than a free one, and work must be done to pull it away. As $n$ increases, the levels become less negative and closer together, crowding towards zero.

Worked example 2: a line in the hydrogen spectrum

Calculate the wavelength of the photon emitted when the electron in a hydrogen atom falls from $n = 3$ to $n = 2$ . In which region of the spectrum is it?

Solution.

\Delta E = E_3 - E_2 = 2.18 \times 10^{-18}\left(\frac{1}{2^2} - \frac{1}{3^2}\right) = 2.18 \times 10^{-18} \times \frac{5}{36} = 3.03 \times 10^{-19}\,\mathrm{J}

\lambda = \frac{hc}{\Delta E} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^{8}}{3.03 \times 10^{-19}} = 6.56 \times 10^{-7}\,\mathrm{m} = 656\,\mathrm{nm}

Answer: 656 nm (the measured value is 656.3 nm): the red line of the hydrogen spectrum, in the visible region.

What Bohr's model achieved, and where it fails

Achievements	Limitations
Explains why atoms give line spectra, not continuous ones	Works only for hydrogen and one-electron ions such as $\mathrm{He^{+}}$ and $\mathrm{Li^{2+}}$
Predicts the wavelengths of all the hydrogen lines	Cannot explain why some lines are close pairs, or why lines split in a magnetic field
Gives the ionisation energy of hydrogen correctly	Treats the electron as a particle on a fixed path; electrons also behave as waves, and their exact paths cannot be known
Introduces energy levels and quantised energy	Cannot explain chemical bonding or the shapes of molecules

The modern model keeps Bohr's energy levels (shells) but replaces orbits with orbitals, which Lesson 3 describes.

Warning

Common mistakes. (1) A line is produced when an electron falls to a lower level, not when it rises; rising absorbs energy. (2) The photon's energy is the difference between two levels, never the energy of one level. (3) Convert nm to m ( $\times 10^{-9}$ ) before using $E = \frac{hc}{\lambda}$ .

Safety. Never look at the Sun through a spectroscope, prism or any lens: it can blind you in seconds. Discharge tubes run at thousands of volts; only your teacher connects and switches them on.

Tip

Think of energy levels as the rungs of a ladder that get closer together as you climb. The electron can stand only on a rung, never between them. Each jump down gives one photon, and the bigger the jump, the bluer the light.

Ben's Chem Videos: Emission Spectra and the Bohr Model (6 min). Watch how each line in an emission spectrum is matched to one jump of the electron between two energy levels.

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Practice

Calculate the energy of a photon of wavelength 410 nm, in joules.
A photon has energy $4.09 \times 10^{-19}\,\mathrm{J}$ . Calculate its wavelength in nm and name its colour.
Distinguish between an emission spectrum and an absorption spectrum of the same element. What do they have in common?
State the four ideas of Bohr's model in your own words.
Calculate the energy, in J, needed to move a hydrogen electron from $n = 1$ to $n = 2$ .
Calculate the wavelength of the photon emitted when a hydrogen electron falls from $n = 4$ to $n = 2$ . Is it visible?
Explain why Bohr's model could not predict the spectrum of helium atoms.

Show answersHide answers

$E = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^{8}}{410 \times 10^{-9}} = 4.85 \times 10^{-19}\,\mathrm{J}$ .
$\lambda = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^{8}}{4.09 \times 10^{-19}} = 4.86 \times 10^{-7}\,\mathrm{m} = 486\,\mathrm{nm}$ : blue-green (cyan).
Emission: bright lines on a dark background, from an excited gas. Absorption: dark lines on a continuous spectrum, from white light passing through cool gas. The lines are at the same wavelengths, because both come from the same energy gaps.
Fixed orbits (energy levels) only; no radiation while in an orbit; energy is absorbed or emitted only in jumps between levels, $\Delta E = hf$ ; so only certain wavelengths appear, as lines.
$\Delta E = 2.18 \times 10^{-18}\left(1 - \frac{1}{4}\right) = 1.64 \times 10^{-18}\,\mathrm{J}$ .
$\Delta E = 2.18 \times 10^{-18}\left(\frac{1}{4} - \frac{1}{16}\right) = 4.09 \times 10^{-19}\,\mathrm{J}$ ; $\lambda = 486\,\mathrm{nm}$ . Yes: blue-green.
Helium has two electrons, which repel each other. Bohr's model only handles a single electron attracted by the nucleus, so its energy formula does not apply.

Quick check

Which change makes a hydrogen atom give out a photon?

Key points

$c = f\lambda$ and $E = hf = \frac{hc}{\lambda}$ : shorter wavelength, more energy per photon.
Hot solids give continuous spectra; excited gases give emission line spectra; cool gases in white light give absorption spectra with dark lines at the same wavelengths.
Bohr: the electron has only fixed energy levels; a jump between two levels absorbs or emits one photon with $\Delta E = hf$ .
For hydrogen, $E_n = -\frac{2.18 \times 10^{-18}}{n^2}\,\mathrm{J}$ ; the model fails for atoms with more than one electron.

Sources and credits

Image: Gase-in-Entladungsroehren.jpg by Alchemist-hp (pse-mendelejew.de), via Wikimedia Commons, licensed under CC BY-SA 3.0 DE.
Image: FraunhoferLinesDiagram.jpg by NASA, via Wikimedia Commons, in the public domain as a work of NASA.
Physical constants are the CODATA values published by NIST.
Video: Emission Spectra and the Bohr Model, Ben's Chem Videos, embedded from YouTube.
The text, diagrams, tables, worked examples and Practice items were written and drawn by Ganzaa.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1Every gas glows its own colour

2Three kinds of spectrum

3Bohr's model of the hydrogen atom (1913)

4The energy levels of hydrogen

5What Bohr's model achieved, and where it fails

Every gas glows its own colour

Three kinds of spectrum

Bohr's model of the hydrogen atom (1913)

The energy levels of hydrogen

What Bohr's model achieved, and where it fails