Thermal Radiation Spectrum Plotter
Thermal Radiation Spectrum Plotter visualizes blackbody radiance vs. wavelength or frequency at given T, using relations like (∂T/∂V)_S.
Thermal Radiation Overview
Planck’s law describes blackbody spectral radiance:
\\[
B(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/(\lambda k T)} – 1}
\\]
or in frequency:
\\[ B(\nu, T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/(k T)} – 1} \\]Key properties:
- Radiation pressure: \\( P = \frac{1}{3} u \\), where \\( u = \frac{4\sigma T^4}{c} \\)
- Stefan-Boltzmann constant: \\(\sigma = \frac{2\pi^5 k^4}{15 h^3 c^2}\\)
Maxwell relation for internal energy (\\(U\\)):
\\[ \left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V \\]For blackbody radiation: \\( P = \frac{1}{3} a T^4 \\), where \\( a = \frac{4\sigma}{c} \\)
Example Calculation
Example: Blackbody at \\(T = 5800 \, \text{K}, \lambda = 500 \, \text{nm}\\)
Differential: \\( dU = T dS – P dV \\)
Maxwell Relation: \\(\left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V\\)
Radiance: \\( B(\lambda, T) \approx 1.41 \times 10^7 \, \text{W/(m}^2 \cdot \text{sr} \cdot \text{nm)} \\)