Chemical potential

Definition and physical interpretation

The chemical potential $\mu$ is an intensive thermodynamic variable conjugate to the particle number $N$. It quantifies how the system’s energy (or free energy) changes when one adds or removes matter, holding appropriate other variables fixed.

Physically, gradients or mismatches in $\mu$ drive particle transfer: when two subsystems can exchange particles through a permeable boundary, net flow occurs in the direction that lowers the appropriate thermodynamic potential, and equilibrium is reached when chemical potentials match (the condition for chemical equilibrium ).

Thermodynamic definitions

For a simple compressible single-component system with fundamental relation $U=U(S,V,N)$,

and therefore

Using thermodynamic potentials obtained by a Legendre transform gives equivalent and often more practical characterizations:

• From the Helmholtz free energy $F(T,V,N)=U-TS$,

  $$\mu=\left(\frac{\partial F}{\partial N}\right)_{T,V}.$$

• From the Gibbs free energy $G(T,p,N)=U-TS+pV$,

  $$\mu=\left(\frac{\partial G}{\partial N}\right)_{T,p}.$$

These identities make clear which laboratory controls (fixed $T,V$ or fixed $T,p$) correspond to the “cost of adding a particle” in different experimental situations.

Exchange, equilibrium, and the grand potential

When particle exchange with a reservoir is allowed (an open system ), it is natural to work with the grand potential $\Omega = U - TS - \mu N$. At fixed temperature , volume, and $\mu$, equilibrium corresponds to minimizing $\Omega$ (subject to constraints). This is the thermodynamic underpinning of the grand canonical ensemble description in statistical mechanics.

Key relations and constraints

• Euler and Gibbs–Duhem structure: For systems satisfying the extensivity postulate and standard homogeneity assumptions, $\mu$ participates in the Euler relation and is not independent of the other intensive variables. Infinitesimal constraints among intensives are captured by the Gibbs–Duhem relation .

• Connection to number density: For homogeneous matter, $\mu$ is often viewed as a function of $(T,n)$ where $n$ is the number density . This emphasizes that $\mu$ controls composition at fixed temperature, much like pressure controls mechanical equilibrium.

• Sign and “chemical work” bookkeeping: The term $\mu\,dN$ in the fundamental differential is the standard accounting of energy change due to matter exchange; conventions for separating this from other work-like contributions are summarized by the chemical work convention .