Gibbs-Duhem Relation/Entropic Equations of State

[mateomy]

I'm working through Callen (ch. 3.4) right now and I'm trying to follow his explanation of the Gibbs-Duhem relation:

Beginning with the ideal gas characterizations
<br /> PV = nRT<br />
<br /> U = cnRT<br />
there is some rearranging of terms and we are shown the relations
<br /> \frac{P}{T}=\frac{nR}{V}<br />
<br /> \frac{1}{T}=\frac{cNR}{U}<br />
and recognizing the further definitions of n/V \equiv v and N/U \equiv u.
Then Callen states 'from these two entropic equations of state we find the third equation of state'
<br /> \frac{\mu}{T} = \,function\, of\, u,\, v \,\,\,\,\,\,[1]<br />
by integrating the Gibbs-Duhem relation
<br /> d\left(\frac{\mu}{T}\right) = ud\left(\frac{1}{T}\right) + vd\left(\frac{P}{T}\right)<br />

I'm not seeing where he gets [1] from. Is that just a definition? I noticed a few sections before this he defined:
<br /> d\mu = -sdT + vdP<br />
I'm missing something but I don't know what. Just looking for some clarification, thanks.

 

[dextercioby]

He says that the integrated version [1] comes from the differential one called Gibbs-Duhem. So your question is actually where Gibbs-Duhem comes from and that should be answered by Callen.