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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 thatP(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.