Deriving the Gibbs–Duhem equation from the fundamental thermodynamic equation is straightforward. The total differential of the extensive Gibbs free energy in terms of its natural variables is Since the Gibbs free energy is the Legendre transformation of the internal energy, the derivatives can be replaced by its definitions transforming the above equation into: The chemical potential is simply another name for the partial molar Gibbs free energy. Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition, i.e. The total differential of this expression is Combining the two expressions for the total differential of the Gibbs free energy gives which simplifies to the Gibbs–Duhem relation:
Alternative derivation
Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. Extensivity implies that where denotes all extensive variables of the internal energy. The internal energy is thus a first-order homogenous function. Applying Euler's homogeneous function theorem, one finds the following relation when taking only volume, number of particles, and entropy as extensive variables: Taking the total differential, one finds Finally, one can equate this expression to the definition of to find the Gibbs-Duhem equation
Applications
By normalizing the above equation by the extent of a system, such as the total number of moles, the Gibbs–Duhem equation provides a relationship between the intensive variables of the system. For a simple system with different components, there will be independent parameters or "degrees of freedom". For example, if we know a gas cylinder filled with pure nitrogen is at room temperature and 25 MPa, we can determine the fluid density, enthalpy, entropy or any other intensive thermodynamic variable. If instead the cylinder contains a nitrogen/oxygen mixture, we require an additional piece of information, usually the ratio of oxygen-to-nitrogen. If multiple phases of matter are present, the chemical potentials across a phase boundary are equal. Combining expressions for the Gibbs–Duhem equation in each phase and assuming systematic equilibrium, we recover the Gibbs' phase rule. One particularly useful expression arises when considering binary solutions. At constant P and T it becomes: or, normalizing by total number of moles in the system substituting in the definition of activity coefficient and using the identity : This equation is instrumental in the calculation of thermodynamically consistent and thus more accurate expressions for the vapor pressure of a fluid mixture from limited experimental data.
Ternary and multicomponent solutions and mixtures
has shown that the Gibbs-Duhem equation can be applied to the determination of chemical potentials of components from a multicomponent system from experimental data regarding the chemical potential of only one component at all compositions. He has deduced the following relation xi, amount fractions of components. Making some rearrangements and dividing by 2 gives: or or The derivative with respect to one mole fraction x2 is taken at constant ratios of amounts of the other components of the solution representable in a diagram like ternary plot. The last equality can be integrated from to gives: Applying LHopital's rule gives: This becomes further: Express the mole fractions of component 1 and 3 as functions of component 2 mole fraction and binary mole ratios: and the sum of partial molar quantities gives and are constants which can be determined from the binary systems 1_2 and 2_3. These constants can be obtained from the previous equality by putting the complementary mole fraction x3 = 0 for x1 and vice versa. Thus and The final expression is given by substitution of these constants into the previous equation:
