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Photons and energy levels

Waves, sound and light · level 8, lesson 1 of 6

A question to start

Heat a little salt in a flame and it glows a bright yellow-orange. Not a mix of colours: one very particular yellow. Why does each element glow with its own colours?

The idea

Light comes in tiny packets of energy called photons. A photon of frequency \(f\) carries energy

\[E = hf = \frac{hc}{\lambda},\]

where \(h = 6.63\times10^{-34}\) J s is Planck's constant. Atomic energies are small, so we often use the electronvolt: \(1 \text{ eV} = 1.60\times10^{-19}\) J.

The electrons in an atom can only have certain energies: energy levels. When an electron drops from a higher level to a lower one, the atom gives out one photon carrying exactly the difference:

\[hf = E_\text{upper} - E_\text{lower}.\]

Each element has its own set of levels, so it gives out its own set of wavelengths. For hydrogen, the levels are

\[E_n = -\frac{13.6 \text{ eV}}{n^2}, \quad n = 1, 2, 3, \dots\]
n = 1: −13.6 eVn = 2: −3.4 eVn = 3: −1.51 eVn = 4656 nm

Worked examples

1. A red photon. \(\lambda = 656\) nm: \(E = 6.63\times10^{-34} \times 3.00\times10^8/656\times10^{-9} = 3.03\times10^{-19}\) J \(= 1.89\) eV.

2. Hydrogen, 3 to 2. \(E_3 - E_2 = -1.51 - (-3.40) = 1.89\) eV, the same: this jump gives the red line at 656 nm.

3. Sodium's yellow. At 589 nm, \(E = 2.11\) eV. Every sodium atom gives out the same yellow, so a street lamp full of sodium glows yellow.

Try it

  1. Find the energy, in eV, of a photon of violet light at 400 nm.
  2. What wavelength does a photon of 2.55 eV have?
  3. Find the energy hydrogen gives out when an electron drops from \(n = 4\) to \(n = 2\).
  4. Why does each element give out only certain wavelengths?
Answers
  1. \(hc/\lambda = 4.97\times10^{-19}\) J \(= 3.1\) eV.
  2. \(\lambda = hc/E = 1.99\times10^{-25}/(2.55 \times 1.60\times10^{-19}) = 487\) nm.
  3. \(-0.85 - (-3.40) = 2.55\) eV.
  4. Its electrons can only have certain energy levels, so only certain jumps, and photon energies, are possible.

Common mistake

"A brighter light has more energetic photons." Brightness is the number of photons. The energy of each one is set by its frequency alone.

Where it leads