Chemical potential

Remarks

Equivalently, with the Gibbs free energy $G(T, p, N_1, \dots, N_r)$, one has $\mu_i = (\partial G / \partial N_i)_{T, p, N_{j \ne i}}$. The fundamental relation then reads $dU = T\,dS - p\,dV + \sum_i \mu_i\,dN_i$. For one species, $G = \mu N$, so $\mu$ is the Gibbs free energy per particle or per mole.

Example: for $n$ moles of an ideal gas at temperature $T$ and pressure $p$, $G = n\big(\mu^\circ(T) + RT \ln(p/p^\circ)\big)$, where $p^\circ$ is a reference pressure and $R$ the gas constant. Differentiating in $n$ at fixed $T$ and $p$ gives $\mu = \mu^\circ(T) + RT \ln(p/p^\circ)$.

Non-example: the derivative $(\partial U / \partial N_i)_{T, V}$ at fixed temperature instead of fixed entropy is in general not $\mu_i$. For a monatomic ideal gas, $U = \tfrac{3}{2} N k T$, so this derivative is $\tfrac{3}{2} kT$, which does not depend on the pressure, while $\mu$ does.

The chemical potential is intensive, and its unit is energy per particle or energy per mole.