World Library  
Flag as Inappropriate
Email this Article

Gibbs–Duhem equation

Article Id: WHEBN0008891344
Reproduction Date:

Title: Gibbs–Duhem equation  
Author: World Heritage Encyclopedia
Language: English
Subject: Josiah Willard Gibbs, Pierre Duhem, Thermodynamic equations, Double layer forces, Chemical thermodynamics
Publisher: World Heritage Encyclopedia

Gibbs–Duhem equation

In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamical system:[1]

\sum_{i=1}^I N_i\,\mathrm{d}\mu_i = - S\,\mathrm{d}T + V\,\mathrm{d}p \,

where N_i\, is the number of moles of component i\,, \mathrm{d}\mu_i\, the infinitesimal increase in chemical potential for this component, S\, the entropy, T\, the absolute temperature, V\, volume and p\, the pressure. It shows that in thermodynamics intensive properties are not independent but related, making it a mathematical statement of the state postulate. When pressure and temperature are variable, only I-1\, of I\, components have independent values for chemical potential and Gibbs' phase rule follows. The law is named after Josiah Willard Gibbs and Pierre Duhem.

The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena.[2]


Deriving the Gibbs–Duhem equation from basic thermodynamic state equations is straightforward.[3] The total differential of the Gibbs free energy G\, in terms of its natural variables is

\mathrm{d}G =\left. \frac{\partial G}{\partial p}\right | _{T,N}\,\mathrm{d}p +\left. \frac{\partial G}{\partial T}\right | _{p,N}\,\mathrm{d}T +\sum_{i=1}^I \left. \frac{\partial G}{\partial N_i}\right | _{p,T,N_{j \neq i}}\,\mathrm{d}N_i \,.

With the substitution of two of the Maxwell relations and the definition of chemical potential, this is transformed into:[4]

\mathrm{d}G =V \,\mathrm{d}p-S \,\mathrm{d}T +\sum_{i=1}^I \mu_i \, \mathrm{d}N_i \,

As shown in the Gibbs free energy article, the chemical potential is just another name for the partial molar (or just partial, depending on the units of N) Gibbs free energy, thus

G = \sum_{i=1}^I \mu_i N_i \,.

The total differential of this expression is[4]

\mathrm{d}G = \sum_{i=1}^I \mu_i \, \mathrm{d}N_i + \sum_{i=1}^I N_i \,\mathrm{d}\mu_i \,

By subtracting the two expressions for the total differential of the Gibbs free energy gives the Gibbs–Duhem relation:[4]

\sum_{i=1}^I N_i\,\mathrm{d}\mu_i = - S\,\mathrm{d}T + V\,\mathrm{d}p \,


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 I\, different components, there will be I+1\, 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.[5] 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.[6] 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.

One particularly useful expression arises when considering binary solutions.[7] At constant P (isobaric) and T (isothermal) it becomes:

0= N_1\,\mathrm{d}\mu_1 + N_2\,\mathrm{d}\mu_2 \,

or, normalizing by total number of moles in the system N_1 + N_2 \,, substituting in the definition of activity coefficient \gamma\ and using the identity x_1 + x_2 = 1\,:

x_1 \left. \frac{\partial \ln \gamma_1}{\partial x_1} \right |_{p,T} =x_2 \left. \frac{\partial \ln \gamma_2}{\partial x_2} \right |_{p,T} \, [8]

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.

See also


  1. ^ A to Z of Thermodynamics Pierre Perrot ISBN 0-19-856556-9
  2. ^ Stephenson, J. (1974). "Fluctuations in Particle Number in a Grand Canonical Ensemble of Small Systems". American Journal of Physics 42 (6): 478–471.  
  3. ^ Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 538 ISBN 0-471-07681-3
  4. ^ a b c Salzman, William R. (2001-08-21). "Open Systems". Chemical Thermodynamics. University of Arizona. Archived from the original on 2007-07-07. Retrieved 2007-10-11. 
  5. ^ Calculated using REFPROP: NIST Standard Reference Database 23, Version 8.0
  6. ^ Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 710 ISBN 0-471-07681-3
  7. ^ The Properties of Gases and Liquids, 5th Edition Poling, Prausnitz and O'Connell, p. 8.13, ISBN 0-07-011682-2
  8. ^ Engineering and Chemical Thermodynamics, 1st Edition Milo D. Koretsky, p. 335, ISBN 978-0-471-38586-8

External links

  • A lecture from
  • A lecture from
  • Encyclopædia Britannica entry
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Hawaii eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.