The photoelectric effect
When light of high enough frequency shines on a metal, electrons are emitted. Classical wave theory could not explain this. Einstein (1905) proposed that light comes in packets of energy called photons, each of energy E = hf (h = 6.63 × 10⁻³⁴ J s). One photon gives all its energy to one electron.
| Observation | Explanation |
|---|---|
| No emission below a threshold frequency | Each photon must have energy at least equal to φ |
| The maximum kinetic energy depends on frequency, not on brightness | E_k,max = hf − φ |
| Brighter light gives more electrons, not faster ones | More photons per second, each with the same energy |
| Emission is instant | One photon gives its energy to one electron immediately |
Choose the frequency of the light and the work function of the metal. Find the maximum kinetic energy of the emitted electrons (h = 4.14 × 10⁻¹⁵ eV s).
Energy levels and spectra
In the Bohr model of hydrogen, the electron has only certain allowed energies E_n = −13.6 eV ÷ n². When an electron falls from a higher to a lower level it emits a photon of energy equal to the difference. This explains the line emission spectrum. A continuous spectrum comes from a hot solid; an absorption spectrum has dark lines where a cool gas absorbs the same wavelengths it would emit.
De Broglie suggested that particles have wavelength λ = h ÷ p = h ÷ mv. The wave nature of electrons is shown by electron diffraction and is used in the electron microscope. X-rays are produced when fast electrons hit a target: a continuous spectrum with a minimum wavelength λ_min = hc ÷ eV and sharp characteristic lines from inner-shell transitions.
The nucleus: mass defect and binding energy
The mass of a nucleus is less than the total mass of its separate protons and neutrons. The difference is the mass defect Δm; the energy equivalent is the binding energy E = Δm c². The binding energy per nucleon is greatest near iron, which is why fusion of light nuclei and fission of heavy nuclei both release energy.
Radioactive decay law
A radioactive isotope has a half-life of 8.0 days. Find the decay constant and the fraction left after 20 days.
- Decay constant in per day.
- Fraction left.
Answer: λ = 0.087 per day; about 18% is left.
Key terms
- Photon
- A packet of electromagnetic energy E = hf.
- Work function
- The minimum energy needed to remove an electron from a metal surface.
- Binding energy
- The energy needed to separate a nucleus into its protons and neutrons.
- Decay constant
- The probability per unit time that a nucleus decays.
Summary
- Photoelectric effect: hf = φ + E_k,max; light behaves as photons.
- Atoms have discrete energy levels; transitions give line spectra.
- De Broglie: λ = h/mv shows matter has wave properties.
- Binding energy peaks at iron; N = N₀ e^(−λt); T½ = 0.693/λ.