# Gibbs–Duhem equation
The **Gibbs–Duhem equation** is a thermodynamic identity that relates the intensive variables of a single phase, temperature, pressure, and the chemical potentials of its components, so that they cannot all change independently. In the form given by IUPAC, for a phase containing C components,
\[ S\,dT - V\,dp + \sum_{i=1}^{C} n_i\,d\mu_i = 0 \]
where S is entropy, V is volume, n_i is the amount of component i, and μ_i is its chemical potential[1](https://goldbook.iupac.org/terms/view/15329/html). The equation follows from the extensivity of thermodynamic potentials and is the reason a mixture's intensive state is described by one fewer independent variable than the raw count suggests[2](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/05%3A_Thermodynamic_Potentials_and_Equilibrium/5.01%3A_Thermodynamic_Potentials).
| Key fact | Detail |
|---|---|
| General form | \( S\,dT - V\,dp + \sum_i n_i\,d\mu_i = 0 \) for a phase with C components[1](https://goldbook.iupac.org/terms/view/15329/html) |
| Constant T and p | Reduces to \( \sum_i n_i\,d\mu_i = 0 \), or \( \sum_i x_i\,d\mu_i = 0 \) in mole fractions[3](https://www2.chem.umd.edu/thermobook/v10-print.pdf) |
| Binary form | \( x_A\,d\mu_A = -x_B\,d\mu_B \); the two chemical potentials cannot vary independently[4](https://mccullaghlab.github.io/physical_chemistry/thermodynamics/Thermodynamics_of_Solutions.html) |
| Variable count | Of the n + 2 intensive variables of a phase, only n + 1 are independently variable[5](https://cersonsky-lab.github.io/cbe710-notes/lecture_files/Lecture22.html) |
| Phase rule | One Gibbs–Duhem equation per phase gives F = C + 2 − P[6](https://serc.carleton.edu/research_education/equilibria/phaserule.html) |
| Practical use | Foundation of the thermodynamic consistency tests applied to vapor–liquid equilibrium data[7](https://trc.nist.gov/TDE/Help/TDE103b_v4_0/VLE-DataSets-ConsistencyTests/QualityFactor.htm) |
| Surface analog | The Gibbs adsorption equation is the corresponding relation for an interfacial phase[8](https://pubs.acs.org/doi/10.1021/acs.jpcb.0c05946) |
## Origin and derivation
The equation comes from the homogeneity of the extensive variables. Because the internal energy U is extensive in S, V, and the amounts n_i, its differential is
\[ dU = T\,dS - p\,dV + \sum_i \mu_i\,dn_i \]
and [Euler's theorem](https://www.edgepedia.org/eulers-theorem) for homogeneous functions gives the integrated form U = TS − pV + Σ μ_i n_i; for the Gibbs energy this reads G = Σ μ_i n_i[2](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/05%3A_Thermodynamic_Potentials_and_Equilibrium/5.01%3A_Thermodynamic_Potentials). Taking the total differential of G = Σ μ_i n_i by the product rule gives dG = Σ μ_i dn_i + Σ n_i dμ_i, while the fundamental relation gives dG = −S dT + V dp + Σ μ_i dn_i. Subtracting the common term leaves
\[ \sum_i n_i\,d\mu_i = -S\,dT + V\,dp \]
which is the Gibbs–Duhem equation in the energy representation[4](https://mccullaghlab.github.io/physical_chemistry/thermodynamics/Thermodynamics_of_Solutions.html). Gibbs himself derived it as equation 97 of his 1875–1878 paper; the same manipulation applied to his equation for a surface phase yields the Gibbs adsorption equation[8](https://pubs.acs.org/doi/10.1021/acs.jpcb.0c05946).
## What the equation constrains
**Intensive variables are linked.** At constant temperature and pressure the equation becomes Σ n_i dμ_i = 0, or equivalently Σ x_i dμ_i = 0 in mole fractions, showing that the chemical potentials of different species cannot be varied independently[3](https://www2.chem.umd.edu/thermobook/v10-print.pdf). More generally, a phase has n + 2 intensive variables (T, p, and one chemical potential per component), and the Gibbs–Duhem equation removes one degree of freedom, leaving n + 1 independently variable[5](https://cersonsky-lab.github.io/cbe710-notes/lecture_files/Lecture22.html).
**Binary mixtures.** For two components A and B at constant T and p,
\[ x_A\,d\mu_A = -x_B\,d\mu_B \]
so any change in one chemical potential forces an opposite change in the other[4](https://mccullaghlab.github.io/physical_chemistry/thermodynamics/Thermodynamics_of_Solutions.html). If μ_A is known as a function of composition, μ_B follows by integration. A standard example shows this works: if μ_A = μ_A* + RT ln x_A, then dμ_A = −(RT/x_A) dx_B, and substituting into the Gibbs–Duhem relation and integrating from the pure component gives μ_B = μ_B* + RT ln x_B[4](https://mccullaghlab.github.io/physical_chemistry/thermodynamics/Thermodynamics_of_Solutions.html).
**Any partial molar quantity.** The same structure holds for every extensive property X of a mixture: at constant T and p, Σ n_i dX_i = 0, where X_i is the partial molar quantity of species i. Applied to partial molar volumes in a binary mixture this gives n_A dV_A + n_B dV_B = 0, so changes in the two partial molar volumes are related as composition changes[3](https://www2.chem.umd.edu/thermobook/v10-print.pdf).
## Role in the phase rule
The Gibbs [phase rule](https://www.edgepedia.org/phase-rule) follows directly. A system with C components and P phases has C + 2 intensive variables per phase (T, p, and the C chemical potentials), and each phase contributes one equation of Gibbs–Duhem form. The variance is the difference, F = C + 2 − P, or P + F = C + 2 as Gibbs wrote it[6](https://serc.carleton.edu/research_education/equilibria/phaserule.html). The equation is also used in the same spirit to check thermodynamic consistency of activity-coefficient models for mixtures, a standard topic in chemical engineering thermodynamics[9](https://ocw.mit.edu/courses/10-40-chemical-engineering-thermodynamics-fall-2003/pages/readings/).
## Testing experimental data
**Vapor–liquid equilibrium.** A VLE data set must satisfy the constraint given by the Gibbs–Duhem equation, written for a molar thermodynamic property M in terms of its partial molar values and the changes in T, p, and liquid composition. When M is the excess Gibbs energy divided by RT, the constraint becomes a relation among activity coefficients, excess volume, and excess enthalpy[7](https://trc.nist.gov/TDE/Help/TDE103b_v4_0/VLE-DataSets-ConsistencyTests/QualityFactor.htm). NIST's Thermodynamics Research Center builds four consistency tests on these equations, combined with a pure-component test that checks the composition endpoints of the VLE curve against pure-component vapor pressures; each test yields a quality factor, and the factors combine into an overall quality factor for the data set[7](https://trc.nist.gov/TDE/Help/TDE103b_v4_0/VLE-DataSets-ConsistencyTests/QualityFactor.htm). A published algorithm of this kind applies to systems with at least three reported state variables (pressure, temperature, plus liquid and/or vapor composition) at subcritical conditions[10](https://www.nist.gov/publications/quality-assessment-algorithm-vapor-liquid-equilibrium-data). A two-step scheme combining liquid and vapor Gibbs–Duhem equations to check internal consistency of (P, y, x) at constant temperature, followed by a liquid-side test of activity coefficients, has been described as the most powerful of the approaches examined[11](https://pubs.acs.org/doi/pdf/10.1021/je9902692).
## Extensions and limits
**Other variables and fields.** The equation generalizes beyond the usual variables. When external fields are present, the Gibbs energy is redefined with extra terms, for example G_f = U − TS + pV − EP′ for dielectrics in an electric field, and the resulting generalized equation shows that pressure, temperature, and chemical potentials still cannot change independently[12](https://www.eng.uc.edu/~beaucag/Classes/AdvancedMaterialsThermodynamics/Homework%20Advanced%20Thermodynamics/Homework%202%20September%206/Extension_of_Gibbs%E2%80%93Duhem_equat.pdf). For elastically stressed, nonhydrostatically deformed crystals, several Gibbs–Duhem type equations exist because strain components can act either as intensive variables or as densities, depending on boundary conditions and geometry[13](https://www.sciencedirect.com/science/article/pii/0956715192901525).
**Where the standard form fails.** The classical relation assumes extensive, homogeneous systems in quasi-static equilibrium. In systems with long-range interactions, thermodynamic equilibrium states are spatially inhomogeneous and a Gibbs–Duhem relation is absent in the standard form; a generalized version instead relates the potential energy to temperature, pressure, and chemical potential[14](https://arxiv.org/abs/1302.4903). In nanoscale systems such as a compressible lattice gas crystal, the differentials of pressure, temperature, and chemical potential become independent, violating the standard Gibbs–Duhem equation, and an extended form treats the number of lattice sites M as an additional extensive variable with its own conjugate thermodynamic force[15](https://doi.org/10.1063/5.0283508). For interfaces, the surface analog is the Gibbs adsorption equation, which plays a role in surface-phase equilibria similar to that of the Gibbs–Duhem equations in bulk-phase equilibria and connects surface tension to adsorption[8](https://pubs.acs.org/doi/10.1021/acs.jpcb.0c05946) • [16](https://www.sciencedirect.com/science/article/pii/S0001868614000049).
## References
1. [IUPAC Gold Book: Gibbs–Duhem equation](https://goldbook.iupac.org/terms/view/15329/html)
2. [Thermodynamic Potentials, Physics LibreTexts (Nair)](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/05%3A_Thermodynamic_Potentials_and_Equilibrium/5.01%3A_Thermodynamic_Potentials)
3. [Thermodynamics and Chemistry, v10 (DeVoe), University of Maryland](https://www2.chem.umd.edu/thermobook/v10-print.pdf)
4. [Thermodynamics of Solutions, McCullagh Lab](https://mccullaghlab.github.io/physical_chemistry/thermodynamics/Thermodynamics_of_Solutions.html)
5. [Equilibrium in single-component systems, Advanced Thermodynamics for Chemical Engineers](https://cersonsky-lab.github.io/cbe710-notes/lecture_files/Lecture22.html)
6. [Gibbs' Phase Rule: Where it all Begins, SERC](https://serc.carleton.edu/research_education/equilibria/phaserule.html)
7. [VLE Consistency Tests and TDE Quality Factor, NIST TRC](https://trc.nist.gov/TDE/Help/TDE103b_v4_0/VLE-DataSets-ConsistencyTests/QualityFactor.htm)
8. [Gibbsian Surface Thermodynamics, J. Phys. Chem. B (Elliott)](https://pubs.acs.org/doi/10.1021/acs.jpcb.0c05946)
9. [Chemical Engineering Thermodynamics readings, MIT OpenCourseWare](https://ocw.mit.edu/courses/10-40-chemical-engineering-thermodynamics-fall-2003/pages/readings/)
10. [Quality Assessment Algorithm for Vapor-Liquid Equilibrium Data, NIST](https://www.nist.gov/publications/quality-assessment-algorithm-vapor-liquid-equilibrium-data)
11. [Consistency Tests for Binary VLE Data, J. Chem. Eng. Data](https://pubs.acs.org/doi/pdf/10.1021/je9902692)
12. [Extension of Gibbs–Duhem equation including influences of external fields](https://www.eng.uc.edu/~beaucag/Classes/AdvancedMaterialsThermodynamics/Homework%20Advanced%20Thermodynamics/Homework%202%20September%206/Extension_of_Gibbs%E2%80%93Duhem_equat.pdf)
13. [Gibbs-Duhem and Clausius-Clapeyron type equations for elastically stressed crystals, Acta Metallurgica](https://www.sciencedirect.com/science/article/pii/0956715192901525)
14. [Local thermodynamics and the generalized Gibbs-Duhem equation in systems with long-range interactions, arXiv](https://arxiv.org/abs/1302.4903)
15. [Thermodynamics of a compressible lattice gas crystal: Generalized Gibbs–Duhem equation and adsorption, J. Chem. Phys.](https://doi.org/10.1063/5.0283508)
16. [Gibbs adsorption equation for planar fluid–fluid interfaces: Invariant formalism, Advances in Colloid and Interface Science](https://www.sciencedirect.com/science/article/pii/S0001868614000049)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Thermodynamics › Laws, states, and potentials › Equilibrium and state functions › State variables and conjugate pairs › Chemical potential–particle number pair*
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