What Is Chemical Potential in Thermodynamics?

Chemical potential tells you how a system’s thermodynamic state changes when its amount of matter changes. It is especially useful for understanding diffusion, phase equilibrium, chemical reactions, and systems that can exchange particles with their surroundings.

The exact derivative depends on which variables are held fixed. In many practical settings, chemical potential is the change in Gibbs free energy per added mole at fixed temperature and pressure.

Definition and thermodynamic meaning

For a mixture with species amounts $N_1,N_2,\ldots$, the chemical potential of species $i$ is defined from the internal energy $U$ as

$$\mu_i=\left(\frac{\partial U}{\partial N_i}\right)_{S,V,N_{j\ne i}}.$$

This definition holds entropy $S$, volume $V$, and the amounts of all other species fixed. It says that $\mu_i$ is the marginal change in internal energy when you add a small amount of species $i$ under those constraints. The corresponding differential for a simple, uniform system is

$$dU=T\,dS-P\,dV+\sum_i\mu_i\,dN_i.$$

Other thermodynamic potentials give definitions with different constraints. For the Helmholtz free energy $F=U-TS$ and Gibbs free energy $G=U+PV-TS$,

$$\mu_i=\left(\frac{\partial F}{\partial N_i}\right)_{T,V,N_{j\ne i}} =\left(\frac{\partial G}{\partial N_i}\right)_{T,P,N_{j\ne i}}.$$

These definitions agree for the same equilibrium state; they simply use the potential suited to the conditions being considered. Chemical potential has units of energy per particle if $N_i$ counts particles, or energy per mole if amount is measured in moles.

Why it predicts matter transfer

Consider moving a small amount of a substance from region A to region B while temperature and pressure remain fixed. If $dn$ moles move, the Gibbs free-energy change is

$$dG=(\mu_B-\mu_A)\,dn.$$

If $\mu_A>\mu_B$, moving matter from A to B lowers $G$, so that transfer is thermodynamically favored. At equilibrium, the chemical potentials are equal across the regions, provided the substance can pass between them. The same principle applies to a substance in two phases: at phase equilibrium, its chemical potential is the same in each phase.

For a chemical reaction, the relevant balance is the sum of chemical potentials weighted by stoichiometric coefficients. If $\nu_i$ is positive for products and negative for reactants, equilibrium requires $\sum_i\nu_i\mu_i=0$. Thus chemical potential is not only about particles moving through space; it also determines which direction a reaction can proceed spontaneously under fixed temperature and pressure.

Worked example: an ideal gas

For an ideal gas at fixed temperature, the chemical potential varies with pressure as

$$\mu(T,p)=\mu^\circ(T)+RT\ln\left(\frac{p}{p^\circ}\right),$$

where $R$ is the gas constant and $p^\circ$ is a chosen standard pressure. The reference value $\mu^\circ(T)$ depends on the standard state; the pressure-dependent change does not.

At $T=298\,\mathrm{K}$, compare an ideal gas at $p_2=10p_1$. The change in molar chemical potential is

$$\begin{aligned} \mu(T,p_2)-\mu(T,p_1) &=RT\ln\left(\frac{p_2}{p_1}\right)\\ &=(8.314\,\mathrm{J\,mol^{-1}\,K^{-1}})(298\,\mathrm{K})\ln(10)\\ &\approx 5.71\times10^3\,\mathrm{J\,mol^{-1}}. \end{aligned}$$

So increasing the pressure tenfold at the same temperature raises the molar chemical potential by about $5.71\,\mathrm{kJ\,mol^{-1}}$. In an ideal-gas mixture, use the species’ partial pressure in this relation. This shows why pressure differences can drive gases to redistribute.

Connection to statistical mechanics

In statistical mechanics, chemical potential controls how particle number varies in a system that can exchange particles with a reservoir. In the grand canonical ensemble, states are weighted by a factor proportional to $\exp[-\beta(E-\mu N)]$, where $E$ is energy, $N$ is particle number, and $\beta=1/(k_BT)$. A larger $\mu$ makes states with more particles relatively more favorable. This is the statistical-mechanical counterpart of chemical potential’s role as the thermodynamic cost of changing particle number.

Common mistakes

• Calling it the energy of a particle: Chemical potential is a derivative of a thermodynamic potential, not generally the average energy carried by one particle.
• Leaving out what is held fixed: The derivative must specify constraints. For example, the Gibbs-free-energy definition holds temperature and pressure fixed.
• Mixing per-particle and per-mole units: A chemical potential quoted in joules per particle differs by Avogadro’s constant from one quoted in joules per mole.
• Treating the reference value as absolute: In formulas containing a logarithm, the standard state fixes the reference. Changes in chemical potential are often more directly meaningful than an isolated absolute value.