Solution
= Solution
Replicating a macroscopic equilibrium system scales $S,V,N$, and $E$ by the same factor, so $E(S,V,N)$ is homogeneous of degree one. Euler's theorem and the first law
$$
dE=T\,dS-p\,dV+\mu\,dN
$$
give
$$
\boxed{E=TS-pV+\mu N}.
$$
Differentiating this identity and comparing with the first law cancels the $T\,dS$, $-p\,dV$, and $\mu\,dN$ terms, leaving the <Gibbs-Duhem equation>
$$
S\,dT-V\,dp+N\,d\mu=0.
$$
Therefore
$$
\boxed{d\mu=\frac{V\,dp-S\,dT}{N}}.
$$