= Chemical potential equality for particle exchange
{title2=$\mu_{{\rm m},A}=\mu_{{\rm m},B}$}
Consider two internally equilibrated reservoirs A and B at the same imposed <temperature> and <pressure>. They may have different compositions or represent different phases. Transfer a small positive <amount of substance> $\delta n$ of the same <chemical species> from A to B, holding the other inventories fixed. To first order, their total <Gibbs free energy> changes by
$$
\delta G_{\rm total}
=-\mu_{{\rm m},A}\delta n+\mu_{{\rm m},B}\delta n
=(\mu_{{\rm m},B}-\mu_{{\rm m},A})\delta n.
$$
Hence exchange toward lower <chemical potential> lowers <Gibbs free energy>. If transfer is allowed in both directions and the two reservoirs contain the exchangeable component, <thermodynamic equilibrium> requires equal <chemical potentials> for that component. See https://ocw.mit.edu/courses/5-60-thermodynamics-kinetics-spring-2008/e763a06aa3b5de09cd6bfea6280f219c_5_60_lecture14.pdf[MIT 5.60, Lecture 14, pages 1–2].
An impermeable partition can maintain unequal <chemical potentials>; equality is a condition on an allowed exchange. Two pure samples of the same <ideal gas> already have equal <chemical potentials> at the same <temperature> and <pressure>. Different compositions at a common <pressure> can instead give different <chemical potentials> for one component through its <thermodynamic activity>.
Equality concerns the same exchangeable component in different places, not all different <chemical species>. For a permitted <chemical reaction>, the corresponding interior <chemical equilibrium> criterion is $\sum_i\nu_i\mu_{{\rm m},i}=0$, with signed <stoichiometric vector> entries $\nu_i$. The existing <chemical equilibrium> and <constrained Gibbs minimization for chemical equilibrium> sections develop those composition constraints.
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