= Solution
The <Gibbs free energy> is $G=U-TS+pV$. For an extensive thermodynamic system, scaling the amount of matter at fixed pressure, temperature and composition scales $G$ by the same factor, hence $G(T,p,N)=N\mu(T,p)$. Euler's relation $U=TS-pV+\mu N$ gives the same conclusion. This uses the ordinary thermodynamic-limit extensivity assumption, rather than a universal claim about finite-size or long-range-interacting systems.
The fundamental thermodynamic identity gives $dG=-S\,dT+V\,dp+\mu\,dN$. Comparing with $d(N\mu)$ at fixed $N$ yields
$$
\boxed{\left(\frac{\partial\mu}{\partial p}\right)_T=\frac VN.}
$$
Thus the <chemical potential> changes with pressure at a rate equal to the volume per particle.
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