This set of equations forms the cornerstone of the Phase Rule; they show how the chemical potential for any given *end member* (i.e. phase component) varies as a function of the other "thermodynamic potentials",(i.e., T, P and the chemical potentials of the other end-members for the phase).
For example, the Gibbs-Duhem equation for pure liquid water is:
du = -SdT + VdP eq. 1
where u = chemical potential, and S,V = molar entropy and molar volume of liquid water, respectively (note that both vary with T and P);
for a two component phase phase such as liquid NaCl - H2O, the G-D equation is:
N1du1 + N2du2 = -SdT + VdP eq.2
where u1 and N1 are the chemical potential and molefraction of component 1 (NaCl) in the solution; H2O (component 2) is defined similarly.
Equation 1 says that the chemical potential of water in the liquid phase decreases with increasing temperature and increases with increasing pressure. Equation 2. suggests that, at a constant P and T, the chemical potential for NaCl decreases with increasing chemical potential of water.