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# [Phys-l] thermo differential and extensive/intensive variables

• From: Stefan Jeglinski <jeglin@4pi.com>
• Date: Fri, 15 Jun 2007 00:40:48 -0400

I'm looking for a better discussion than I've found regarding thermodynamic functions. In particular, in reading various treatments, it would appear that E = E(S,V) and E = E(V,T) are valid, but E = E(S,T) is not. I was thinking originally that there was some rule about mixing intensive and extensive variables, but the 2nd of the above 3 functional forms seems to kill that idea.

The first of the 3 leads to the most recognizable differential form [1]:

dE = (dE/dS)dS + (dE/dV)dV = TdS - PdV

The second is less obvious:

dE = (dE/dT)dT + (dE/dV)dV = CvdT + (?)dV

Through several manipulations, the dE/dV can be cast as T*(dP/dT) - P, or

dE = CvdT + T*(dP/dT) dV - PdV

which leads to (using the top equation):

TdS = CvdT + T*(dP/dT) dV

But why can't I use the 3rd functional form, E(S,T)? It would seem no more or less valid than E(V,T) from an intensive/extensive standpoint, but I'd get

dE = (dE/dS)dS + (dE/dT)dT = TdS + CvdT

which is clearly at odds with the above.

Stefan Jeglinski

[1] Interpret the parenthetical quantities as garden variety partial derivatives with the appropriate variable held constant.