Gibbs–Duhem equation
the infinitesimal increase in chemical potential for this component,
This equation shows that in thermodynamics intensive properties are not independent but related, making it a mathematical statement of the state postulate.
components have independent values for chemical potential and Gibbs' phase rule follows.
The Gibbs−Duhem equation applies to homogeneous thermodynamic systems.
[5] The equation is named after Josiah Willard Gibbs and Pierre Duhem.
The Gibbs–Duhem equation follows from assuming the system can be scaled in amount perfectly.
Gibbs derived the relationship based on the though experiment of varying the amount of substance starting from zero, keeping its nature and state the same.
are all of the extensive variables of system: entropy, volume, and particle numbers.
The internal energy is thus a first-order homogenous function.
Applying Euler's homogeneous function theorem, one finds the following relation: Taking the total differential, one finds From both sides one can subtract the fundamental thermodynamic relation, yielding the Gibbs–Duhem equation[6] 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.
independent parameters or "degrees of freedom".
For example, if we know a gas cylinder filled with pure nitrogen is at room temperature (298 K) and 25 MPa, we can determine the fluid density (258 kg/m3), enthalpy (272 kJ/kg), entropy (5.07 kJ/kg⋅K) or any other intensive thermodynamic variable.
[7] If instead the cylinder contains a nitrogen/oxygen mixture, we require an additional piece of information, usually the ratio of oxygen-to-nitrogen.
[8] Combining expressions for the Gibbs–Duhem equation in each phase and assuming systematic equilibrium (i.e. that the temperature and pressure is constant throughout the system), we recover the Gibbs' phase rule.
[9] At constant P (isobaric) and T (isothermal) it becomes: or, normalizing by total number of moles in the system
: 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.
Lawrence Stamper Darken 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
He has deduced the following relation[11] xi, amount (mole) fractions of components.
Making some rearrangements and dividing by (1 – x2)2 gives: or or The derivative with respect to one mole fraction x2 is taken at constant ratios of amounts (and therefore of mole fractions) of the other components of the solution representable in a diagram like ternary plot.
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
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:
