the Gibbs-Duhem equation
/ gibz doo-EM /
You might expect temperature, pressure and chemical potential to be three knobs you can turn independently. The Gibbs-Duhem equation says no: because thermodynamic potentials scale with system size (they are extensive), the intensive variables are secretly tied together. Fix any two and the third is determined; you cannot change T, p and mu all at will.
For a single-component system the relation reads S dT - V dp + N dmu = 0, or per particle dmu = -s dT + v dp, where s = S/N and v = V/N are the molar entropy and volume. It follows from Euler's relation G = mu N (a consequence of G being first-order homogeneous in the extensive variables): differentiate to get dG = mu dN + N dmu, compare with dG = -S dT + V dp + mu dN, and the two mu dN cancel to leave the constraint. For several species it generalizes to sum N_i dmu_i = -S dT + V dp.
The equation is why intensive variables are not all independent — the root of the Gibbs phase rule — and it is a practical tool: in a binary mixture, measuring how one component's chemical potential (or activity) varies fixes the other's. Caveat: it constrains only the intensive variables among themselves; it says nothing quantitative until you also supply an equation of state.
For a two-component liquid at constant T and p, the Gibbs-Duhem relation reduces to N_1 dmu_1 + N_2 dmu_2 = 0. So if adding more of component 1 raises its chemical potential, component 2's chemical potential must fall in proportion — you can obtain the second from measurements of the first alone.
In a binary mixture the two chemical potentials cannot vary independently.
The Gibbs-Duhem equation is a direct consequence of extensivity (Euler's homogeneous-function theorem); it is what forbids all intensive variables from being independent and underlies the phase rule.