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*From*: "Jeffrey Schnick" <JSchnick@Anselm.Edu>*Date*: Sun, 17 Jun 2007 22:33:33 -0400

">>" <-> 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 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.

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.

**Follow-Ups**:**Re: [Phys-l] Photon Thermodynamics***From:*John Denker <jsd@av8n.com>

**References**:**[Phys-l] thermo differential and extensive/intensive variables***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*Stefan Jeglinski <jeglin@4pi.com>

**[Phys-l] Photon Thermodynamics***From:*"Jeffrey Schnick" <JSchnick@Anselm.Edu>

**Re: [Phys-l] Photon Thermodynamics***From:*John Denker <jsd@av8n.com>

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