Quantum Chemistry/Example 10

Question Use Planck's radiation law to find the surface temperature of a star with a maximum intensity of EM radiation is emitted at 504 nm. Report your answer in Kelvin. Getting the correct function A blackbody material is defined by its ability to absorb ALL radiation that falls onto it.Quantum Hypothesis Used for Blackbody Radiation Law https://chem.libretexts.org/@go/page/210776 (accessed Nov 20, 2021). When the blackbody is at constant temperature, the distribution of its emission frequency can be determined by assuming its only direct relation is to temperature. Furthermore, the frequency of the electromagnetic (EM) radiation can also be measured in units of wavenumber . Planck's approach to defining was deriving for a closed form Harmonic Oscillator was based on Boltzmann's distribution.Rotations and Vibrations of Polyatomic Molecules http://dx.doi.org/10.1002/9783527618095.ch6 Molecular Physics 203–236 Weinheim, Germany Wiley-VCH Verlag GmbH 2021-11-202020-03-18 1.1: Blackbody Radiation Cannot Be Explained Classically https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/01%3A_The_Dawn_of_the_Quantum_Theory/1.01%3A_Blackbody_Radiation_Cannot_Be_Explained_Classically 2021-11-20 Chemistry LibreTexts en The resulting form of Planck's law can then be applied to the question now that all parameters are known except for the parameter in question. The model below shows the radiation intensity distribution (i.e., area of the curve) of EM with respect to frequency (in Hertz; Hz) and temperature (in Kelvins; K). Planck's law for a blackbody material predicts the behaviour of its quantitative properties (such as frequency, wavelength, and temperature) when the environment are in extreme conditions.Planck's Law of BlackBody Radiation Solving for the temperature The maximum value of the independent variable of any function can be found by equating the function's derivative to zero. Only is unknown and is with respect to the independent variable, . Thus, Planck's law will be derived with respect to . First, for simplicity, let . The first term consists only of non-zero constants so that leaves: Subbing back in gives: At max intensity, . Subbing in all known values and constants (in SI units) then rearranging to solve for T will determine the surface temperature of the Sun under these specific conditions. +Known ValuesBoltzmann constant Value, Dimensions, Symbol, & Facts Britannica https://www.britannica.com/science/Boltzmann-constant 2021-11-21 www.britannica.com enVariableValueUnits Therefore, the surface temperature of the star with a maximum blackbody radiation emission wavelength of 504 nm is just above 5709 Kelvins. References