(i) use of Wien’s displacement law 1 T max \ m to estimate the peak…

The idea

Wien’s displacement law states that the wavelength of peak emission λmax is inversely proportional to the surface temperature T: λmax T = 2.9 × 10⁻³ m K. To estimate a star’s surface temperature, measure the wavelength at which its continuous spectrum is most intense, then rearrange to T = 2.9 × 10⁻³ / λmax. For example, if λmax = 5.8 × 10⁻⁷ m, then T = 2.9 × 10⁻³ / 5.8 × 10⁻⁷ ≈ 5000 K. Hotter stars peak at shorter wavelengths (blue), cooler stars at longer wavelengths (red). The constant has units of metre kelvin, so λmax must be in metres and T in kelvin.

What earns the marks

• Wien’s displacement law states λmax T = 2.9 × 10⁻³ m K.
• λmax is the wavelength of peak intensity in the continuous spectrum.
• Surface temperature is estimated using T = 2.9 × 10⁻³ / λmax.
• Hotter stars have shorter peak wavelengths; cooler stars have longer peak wavelengths.
• The constant 2.9 × 10⁻³ has units of metre kelvin, so λmax must be in metres and T in kelvin.

Common mistakes and how to put them right

• Using λmax in nanometres or angstroms without converting to metres before substituting into the equation.

• Inverting the relationship and multiplying λmax by the constant instead of dividing.

• Confusing Wien’s displacement law with the Stefan–Boltzmann law, which relates power to T⁴ rather than peak wavelength to T.

• Forgetting that the constant has units m K, so the temperature obtained is in kelvin, not degrees Celsius.

Exam tips

• Always convert λmax to metres before substituting into T = 2.9 × 10⁻³ / λmax.
• Check that your answer is a sensible stellar temperature, typically between about 2000 K and 40000 K.
• Use the direction of the relationship as a check: a shorter peak wavelength must give a higher temperature.