The Euler equation in thermodynamics are as followed:
$U=TS-PV+\mu N$
But The first law of thermodynamics states that
$dU=TdS-PdV+\mu dN$
But I think that The Euler equation can be written by
$dU=TdS+SdT-PdV-VdP+\mu dN+Nd\mu$
Then, $SdT-VdP+Nd\mu=0$
But I don't think that this is always correct.
Edit
I see that it's only correct if the system's homogeneous. Can you give me an example of a homogeneous system and nonhomogeneous system?
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2 Answers
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The Euler equation is a consequence of the extensive property of energy $U(\lambda S,\lambda V,\lambda N)= \lambda U(S,V,N)$. \begin{align}U(S,V,N)&=\Big(\frac{\partial{U}}{\partial{S}}\Big)_{N,V}S+\Big(\frac{\partial{U}}{\partial{V}}\Big)_{N,S}V+\Big(\frac{\partial{U}}{\partial{N}}\Big)_{S,V}N\\ &=TS-PV+\mu N\end{align}
And the first law of thermodynamics is just a statement which says that energy is conserved. $U(S,V,N)$ is a state function and therefore $dU(S,V,N)$ is an exact differential, it can be written in the form shown below.
\begin{align}dU(S,V,N)&= \Big(\frac{\partial{U}}{\partial{S}}\Big)_{N,V}dS+\Big(\frac{\partial{U}}{\partial{V}}\Big)_{N,S}dV+\Big(\frac{\partial{U}}{\partial{N}}\Big)_{S,V}dN\\&=TdS-PdV+\mu dN \end{align}
Both the Euler equation and the first law of thermodynamics are logical consequences of different properties of energy, $U(S,V,N)$ (extensivity and conservation of energy) and both are true.
As you concluded, this leads to $SdT-VdP+Nd\mu=0$, which is the Gibbs-Duhem equation. Your question is the proof of Gibbs-Duhem equation.
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This is the Gibbs–Duhem equation. It's related to extensiveness of energy, entropy, volume and number of particles. The assumption is that the edge effects are much smaller than the volume effects, in particular, when combining two systems the total energy is the sum of energies of the smaller systems, without "interaction terms".
