= Chemical potential
{title2=$\mu$}
{wiki}
The <chemical potential> measures how a <thermodynamic potential> changes when matter is added under specified constraints. It plays for matter exchange the role that <temperature> plays for <heat> exchange: a difference supplies a direction, and equality characterizes <thermodynamic equilibrium> when the corresponding exchange is allowed. This connects <diffusion>, <phase coexistence> and <chemical equilibrium> through one quantity.
For a single component, the particle convention is
$$
\mu_{\rm particle}=\left(\frac{\partial U}{\partial N}\right)_{S,V},
$$
where $U$ is <internal energy>, $N$ counts particles, $S$ is <entropy> and $V$ is <volume (thermodynamics)>. This <partial derivative> holds <entropy> and <volume (thermodynamics)> fixed; it measures an incremental change rather than the average <energy> per particle. See https://davidtong.org/pdfs/teaching/statistical-physics/statphys1.pdf[David Tong, Statistical Physics, section 1.4].
In <chemistry> the molar convention instead uses <amount of substance> $n$, measured in <moles>:
$$
\mu_{\rm m}=\left(\frac{\partial G}{\partial n}\right)_{T,p},
\qquad n=\frac{N}{N_{\rm A}},
\qquad \mu_{\rm m}=N_{\rm A}\mu_{\rm particle}.
$$
Here $G$ is <Gibbs free energy>, $T$ is <temperature>, $p$ is <pressure> and $N_{\rm A}$ is the <Avogadro constant>. The units are $\mathrm J$ per particle versus $\mathrm{J\,mol^{-1}}$. For several <chemical species>, define each <chemical potential> by varying only its own particle count or <amount of substance>, with all other components' counts or amounts fixed. Many texts use $\mu$ for either convention, so the units and the variable being differentiated must identify it.
For a <simple compressible system> with variable component inventories, the fundamental differentials are
$$
dU=T\,dS-p\,dV+\sum_i\mu_{{\rm particle},i}\,dN_i,
\qquad
dG=-S\,dT+V\,dp+\sum_i\mu_{{\rm m},i}\,dn_i.
$$
They describe changes between states in <thermodynamic equilibrium> with only <pressure>-<volume (thermodynamics)> <work>. If every component inventory is fixed, the sums vanish. The <Gibbs free energy> form follows by differentiating $G=U-TS+pV$ and using $dN_i=N_{\rm A}\,dn_i$; this is why the two definitions describe the same <chemical potential> in different units.
A <chemical potential> can be negative. Holding <entropy> fixed while adding matter need not increase <internal energy>; Tong explains this explicitly for a classical <ideal gas> in https://davidtong.org/pdfs/teaching/statistical-physics/statphys2.pdf[Statistical Physics, section 2.2.4]. Its absolute sign also depends on the reference <energy>: replacing $U$ by $U+\epsilon_0N$ shifts $\mu_{\rm particle}$ by $\epsilon_0$. A common reference shift cancels from differences for the same <chemical species>, which are what determine exchange. Thus “energy cost” means a signed increment, not an always positive amount that must be supplied.
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