I've been trying to derive the Stefan-Boltzmann law using thermodynamics, and have resorted to looking up the derivation in the feynman lectures and on wikipedia, and I'm confused by both. I think the wikipedia derivation is the best one to look at, it's here:
The line I'm not happy with is this one:
'Because the pressure is proportional to the internal energy density it depends only on the temperature and not on the volume.'
If the pressure doesn't depend on the volume (using temperature and volume as independent variables) then I have no problem in deriving the result, but I'm not sure why
the pressure doesn't depend on the volume.
The article claims that it's because the pressure is proportional to the internal energy density, but then isn't that also the case in an ideal gas? And in an ideal gas the pressure definitely does depend on the volume. In fact, the derivation of this law seems to rest on the fact that:
U = 3PV
But in an ideal gas:
U = (3/2)PV
And all the same arguments seem to apply, which would appear to suggest that for an ideal gas at constant volume, the pressure is proportional to T^(5/2), which I know is wrong.
Help on this would be much appreciated.