chemicalpotential

Chemical potential is a thermodynamic property that measures the change in a system's energy when the amount of a substance is varied. For a component i in a mixture, the chemical potential μ_i is the partial molar Gibbs free energy, defined as μ_i = (∂G/∂n_i)_{T,P,n_j≠i}. It can also be expressed as the partial derivative of other thermodynamic potentials with their natural variables, such as μ_i = (∂U/∂n_i)_{S,V,n_j≠i} or μ_i = (∂A/∂n_i)_{T,V,n_j≠i}. In practice, μ_i represents the energy change when an infinitesimal amount of species i is added to the system at fixed temperature, pressure, and the amounts of the other components.

In a single-component system, μ is the molar Gibbs free energy of that species. In mixtures, μ_i depends on composition due to molecular interactions. For ideal mixtures, μ_i = μ_i^0 + RT ln a_i, where a_i is the activity. For an ideal gas, μ_i = μ_i^0 + RT ln(p_i/p^0). Non-ideality alters the activity coefficients embedded in a_i.

Equilibrium and transport: at phase equilibrium, μ_i is equal for the same species across all coexisting phases. Gradients in chemical potential drive mass transfer and diffusion, with flow from regions of higher μ to lower μ. In open systems, μ acts as a generalized driving force for reactions and phase changes.

In statistical mechanics, μ is the Lagrange multiplier for particle number in the grand canonical ensemble, governing exchange of particles with a reservoir and linking to distribution functions in quantum statistics.