Wiens displacement law
Wien's displacement law could also be derived from the same model. Wilhelm Carl Wien (18641928) noticed that the shape of the blackbody radiation spectrum bore a remarkable resemblance to the distribution of speeds of the molecules in a gas. This, he thought, may not be a coincidence since if the molecules of the blackbody are thermally agitated, then their velocities and accelerations may be related to the molecular velocities of a gas which is also thermally agitated. The radiation which results from the vibrating charges might well display similar characteristics to the corresponding distribution of molecular energies. On this basis Wien was able to show that his displacement law, XmT = constant, is a special case of a more general law which states that at corresponding wavelengths the energy density of cavity radiation varies as:
Yx = T 5f (IT ) (Wien's spectral distribution law) (11.1)
where f (XT) is an unknown function. If we knew this function, we would know the shape of the curve. The peak of the curve depends on the temperature, and is proportional to T5. As we saw before, the area under the curve is proportional to T4.
Plotting MX/T5 against XT, we should get a curve which is unique in the sense that each point on the curve corresponds to a given value of XT. If we choose different temperatures, i.e. different values of T, corresponding values of X such that X1T1 = X2T2 will give the same point on the curve. Wien's displacement law is a special case for the point at the apex of the graph.
Each point on the x axis of the graph below corresponds to a value of XT. The maximum occurs at XT = 2.9 x 103 mK.
Wien's reasoning was also based on principles established in the 19th century. It is remarkable that the powerful methods of classical thermodynamics could go so far as to predict not only the StefanBoltzmann law but also the existence of a function f (XT), as verified by the experimental results illustrated in the graph.
At that point all that is possible had been done on the basis only of pure thermodynamic reasoning. This had now to be supplemented with a more detailed model to derive a mathematical expression which gives the shape of the function.
— experiment
Figure 11.5 Experimental verification of Wien's displacement law. A set of measurements at one temperature determines f (XT). Measured points at other temperatures fall on the same curve.
Wien's law predicts the same graph for all temperatures, but says nothing about the shape of the graph.
— experiment
Figure 11.5 Experimental verification of Wien's displacement law. A set of measurements at one temperature determines f (XT). Measured points at other temperatures fall on the same curve.
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