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# Re: [Phys-l] Photon Thermodynamics

">>" <-> Jeff Schnick
">" <-> John Denker

He takes the partial derivative of U=3PV (comes
from equation 39.17) holding T constant and obtains:
dU/dV (const T) = 3P
I was expecting:
dU/dV (const T) = 3P + 3V dP/dV (const T)

That expectation is correct.

Why is the second term missing? Its absence would suggest that
P(T,V)
is actually only a function of T, P(T).

For photons under black-body conditions, P(T,V) is independent
of V. You might have surmised as much from a scaling argument:
We know P is intensive and T is intensive, so if there are no
other variables involved, you can't have an extensive variable
on one side of the equation and not on the other.

Remember: The idea that XXX is an intensive property is just
a particularly simple scaling property: It means that XXX
scales like the size of the system to the zeroth power.

Thanks for your informative response. I perceive the scaling argument
to be elegant and powerful. At present, however, I don't follow it in
this case. I assume the equation to which you refer when you write "you
can't have an extensive variable on one side of the equation and not the
other" is:

(1) dU/dV (const T) = 3P + 3V dP/dV (const T)

On the left is the ratio of a change in an extensive variable to a
change in an extensive variable. That ratio is an intensive variable.
The first term on the right is clearly an intensive variable. The
second term on the right is an extensive variable times the ratio of a
change in an intensive variable to a change in an extensive variable.
The ratio is a reciprocal extensive variable. The product of the
extensive variable 3V times that reciprocal extensive variable is thus
an intensive variable. As such, every term in equation (1) scales like
the zeroth power of the size of the system and we can't use the scaling
argument to establish the fact that the second term on the right must
vanish.