Chemical equilibrium

A thermodynamic system is in chemical equilibrium when there is no spontaneous tendency for its composition to change through matter transfer or chemical reaction, given the allowed exchanges with its environment and its constraints.

Two common cases are:

1. Particle exchange (diffusive equilibrium). If two systems can exchange particles through a permeable boundary, chemical equilibrium requires equality of the chemical potential for each exchanged species:

   $$\mu_i^{(1)} = \mu_i^{(2)} \quad \text{for each species } i.$$

2. Chemical reactions. For a reaction with stoichiometric coefficients $\nu_i$, chemical equilibrium requires that the reaction has no net driving force. In terms of chemical potentials this can be written as

   $$\sum_i \nu_i \mu_i = 0,$$

   with the sum over all species participating in the reaction.

Chemical equilibrium is a necessary component of thermodynamic equilibrium , alongside mechanical and thermal equilibrium.

Physical interpretation

Chemical potential measures the “escaping tendency” of particles (or, more generally, how the system’s free energy changes with composition). If two regions have different chemical potentials for a species, matter transfer or reaction progress can increase total entropy (or decrease the appropriate free-energy potential), so a net change occurs until the chemical potentials satisfy the equilibrium condition.

In practice, chemical equilibrium is what makes composition stable: once reached, the system may still exchange energy as heat or perform work (depending on constraints), but it has no net tendency to change its particle content or reaction extent.

Key relations and thermodynamic potentials

Entropy maximization with particle exchange. For two subsystems that can exchange particles (but not volume) with fixed totals, maximizing total entropy implies

Using the identity from equilibrium thermodynamics

this yields $\mu_1/T_1 = \mu_2/T_2$. When the subsystems are also in thermal equilibrium so that $T_1=T_2$, it reduces to $\mu_1=\mu_2$.

Equivalent definitions of chemical potential. For a single-component system,

linking $\mu$ to derivatives of internal energy $U$, Helmholtz free energy $F$, and Gibbs free energy $G$ in the natural variables of each potential.

Open systems and reservoirs. A system that can exchange particles with a reservoir is an open system ; at fixed $(T,V,\mu)$ the equilibrium state minimizes the grand potential .