Can someone tell me how I can derive Wien's law, i.e., [itex]\lambda_{max} T = constant[/itex] where [itex]\lambda_{max}[/itex] is the peak wavelength and [itex]T[/itex] is the absolute temperature of the black body, using the equation, [itex]P=\frac{U^{*}}{3}[/itex] where [itex]U^{*}[/itex] is the energy density. Note: I'm not looking for the derivation using Plank's formula. I'm looking for a purely thermodynamic derivation. Thanks in advance!!
There is a thermodynamic derivation of Wien's Law in Heat and Thermodynamics by Roberts and Miller. It invokes considering slow expansion of a cavity, Doppler shift of reflected radiation, and so on. It is long and complicated, Maybe slicker derivations exist. These day, most textbook writers don't bother with this sort of derivation, but derive it from Planck's law. But I know you don't want this.
[itex]P[/itex] is the Radiation Pressure. It relates to the first law of termodynamics definition of work, [itex]PdV[/itex]. Basically, I'm looking to derive Wien's law from the first law.