Q3(b). Partial molar Gibbs energy (chemical potential)

1. Definition. The partial molar Gibbs energy of component i is the change in the Gibbs energy of a mixture per mole of i added, at constant T, p and amounts of all other components: μᵢ=(∂ G/∂ nᵢ)_T,p,n_(j≠ i). It is the slope of G against nᵢ, and is called the chemical potential.
2. SI unit: G in J, nᵢ in mol: J mol⁻¹
3. Intensive. It is the ratio of two extensive quantities. If the mixture is doubled at the same composition, G and nᵢ both double and μᵢ is unchanged, so it does not depend on the size of the system. Hence G=Σᵢ nᵢμᵢ: the extensive G is built from intensive μᵢ.
4. Factors: (i) temperature, (∂μᵢ/∂ T)_(p,n)=- Sᵢ; (ii) pressure, (∂μᵢ/∂ p)_(T,n)= Vᵢ; (iii) composition, μᵢ=μᵢ°+RTln aᵢ (RTln xᵢ for an ideal solution, RTln(pᵢ/p°) for a perfect gas); (iv) the nature of i and of its neighbours, which fixes μᵢ° and the activity coefficient.
5. Gibbs–Duhem: at constant T and p, Σᵢ nᵢ dμᵢ=0: the chemical potentials in a mixture cannot change independently.
6. Meaning: matter flows spontaneously from high μᵢ to low μᵢ; at equilibrium μᵢ is the same in every phase.

Application. equal chemical potential in every phase is the basis of the phase rule, Raoult's law and the colligative properties; ΔᵣG=Σνᵢμᵢ=0 gives the law of mass action.