Thermodynamic Derivation of Wien's Law?

[Collisionman]

Can someone tell me how I can derive Wien's law, i.e.,

$\lambda_{max} T = constant$

where $\lambda_{max}$ is the peak wavelength and $T$ is the absolute temperature of the black body, using the equation,

$P=\frac{U^{*}}{3}$

where $U^{*}$ 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!

 

[mikeph]

what is P?

 

[Philip Wood]

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.

 

[Collisionman]

what is P?

$P$ is the Radiation Pressure. It relates to the first law of termodynamics definition of work, $PdV$. Basically, I'm looking to derive Wien's law from the first law.

 

[sardar]

The derivation of Wien's law using thermodynamics involves understanding the principles of thermodynamic equilibrium and the relationship between energy density and temperature in a black body.

Firstly, we must understand that a black body is a hypothetical object that absorbs all radiation incident upon it and emits radiation at all wavelengths. In thermodynamic equilibrium, the black body must radiate as much energy as it absorbs in order to maintain a constant temperature.

Using the equation provided, P=\frac{U^{*}}{3}, we can rewrite it as U^{*}=3P. This equation represents the energy density of the black body, where P is the power emitted per unit area.

Next, we can use the Stefan-Boltzmann law, which states that the power emitted per unit area by a black body is proportional to the fourth power of its temperature, to get P=\sigma T^4, where \sigma is the Stefan-Boltzmann constant.

Substituting this into our equation for energy density, we get U^{*}=3\sigma T^4.

Now, we can use the definition of peak wavelength, \lambda_{max}=\frac{b}{T} where b is a constant, to rewrite our equation as U^{*}=3\sigma b^4\lambda_{max}^{-4}.

Since we know that the energy density, U^{*}, is constant for a black body in thermodynamic equilibrium, we can set our two equations for U^{*} equal to each other and solve for \lambda_{max}.

This leads to the final result of \lambda_{max} T = constant, which is Wien's law.

In summary, the thermodynamic derivation of Wien's law involves understanding the principles of thermodynamic equilibrium, the Stefan-Boltzmann law, and the definition of peak wavelength. It shows that the peak wavelength of a black body's emission is inversely proportional to its temperature and is a result of maintaining thermodynamic equilibrium.