Gibbs–Duhem equation

The Gibbs–Duhem equation follows from assuming the system can be scaled in amount perfectly. Gibbs derived the relationship based on the thought experiment of varying the amount of substance starting from zero, keeping its nature and state the same.

Mathematically, this means the internal energy $U$ scales with its extensive variables as follows:{{cite book | first = Herbert B. | last = Callen | author-link = Herbert Callen | year = 1985 | title = Thermodynamics and an Introduction to Thermostatistics | edition = 2nd | publisher = John Wiley & Sons | location = New York | isbn = 978-0-471-86256-7 | pages=61-62 }}

$$U(\backslash lambda\; S,\; \backslash lambda\; V,\; \backslash lambda\; N\_1,\; \backslash lambda\; N\_2,\; \backslash ldots)\; =\; \backslash lambda\; U\; (S,\; V,\; N\_1,\; N\_2,\; \backslash ldots)$$

where $S,\; V,\; N\_1,\; N\_2,\; \backslash ldots$ 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:

$$U\; =\; TS\; -\; pV\; +\; \backslash sum\_\{i=1\}^I\; \backslash mu\_i\; N\_i$$

Taking the total differential, one finds

$$\backslash mathrm\{d\}U\; =\; T\backslash mathrm\{d\}S\; +\; S\backslash mathrm\{d\}T\; -\; p\backslash mathrm\{d\}V\; -\; V\; \backslash mathrm\{d\}p\; +\; \backslash sum\_\{i=1\}^I\; \backslash mu\_i\; \backslash mathrm\{d\}\; N\_i\; +\; \backslash sum\_\{i=1\}^I\; N\_i\; \backslash mathrm\{d\}\; \backslash mu\_i$$

From both sides one can subtract the fundamental thermodynamic relation,

$$\backslash mathrm\{d\}U\; =\; T\backslash mathrm\{d\}S\; -\; p\backslash mathrm\{d\}V\; +\; \backslash sum\_\{i=1\}^I\; \backslash mu\_i\; \backslash mathrm\{d\}\; N\_i$$

yielding the Gibbs–Duhem equation

$$0\; =S\backslash mathrm\{d\}T\; -\; V\; \backslash mathrm\{d\}p\; +\; \backslash sum\_\{i=1\}^I\; N\_i\; \backslash mathrm\{d\}\; \backslash mu\_i\; .$$

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.Calculated using REFPROP: NIST Standard Reference Database 23, Version 8.0 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.Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 710 {{ISBN|0-471-07681-3}} 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.The Properties of Gases and Liquids, 5th Edition Poling, Prausnitz and O'Connell, p. 8.13, {{ISBN|0-07-011682-2}} At constant P (isobaric) and T (isothermal) it becomes:

$$0=\; N\_1\; \backslash mathrm\{d\}\backslash mu\_1\; +\; N\_2\; \backslash mathrm\{d\}\backslash mu\_2$$

or, normalizing by total number of moles in the system $N\_1\; +\; N\_2,$ substituting in the definition of activity coefficient $\backslash gamma$ and using the identity $x\_1\; +\; x\_2\; =\; 1$:{{ cite book | last = Stølen | first = Svein | title = Chemical Thermodynamics of Materials: Macroscopic and Microscopic Aspects | last2 = Grande | first2 = Tor | last3 = Allan | first3 = Neil L. | date = 2008 | publisher = Wiley |isbn = 978-0-471-49230-6 | edition = Transferred to digital print |location = Chichester | page = 79}}

$$0\; =\; x\_1\; \backslash mathrm\{d\}\backslash ln(\backslash gamma\_1)\; +\; x\_2\; \backslash mathrm\{d\}\backslash ln(\backslash gamma\_2)$$

This equation is instrumental in the calculation of thermodynamically consistent and thus more accurate expressions for the vapor pressure of a binary mixture from limited experimental data. One can develop this further to the Duhem–Margules equation which relates to vapor pressures directly.

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 $\backslash bar\; \{G\_2\}$ of only one component (here component 2) at all compositions. He has deduced the following relation{{Cite journal | last1 = Darken | first1 = L. S | title = Application of the Gibbs-Duhem Equation to Ternary and Multicomponent Systems | doi = 10.1021/ja01163a030 | journal = Journal of the American Chemical Society | volume = 72 | issue = 7 | pages = 2909–2914 | year = 1950 }}

$$\backslash bar\{G\_2\}\; =\; G\; +\; (1-x\_2)\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\_2\}\backslash right)\_\{\backslash frac\{x\_1\}\{x\_3\}\}$$

x_i, amount (mole) fractions of components.

Making some rearrangements and dividing by (1 – x₂)² gives:

$$\backslash frac\{G\}\{(1-x\_2)^2\}\; +\; \backslash frac\{1\}\{1-x\_2\}\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\_2\}\backslash right)\_\{\backslash frac\{x\_1\}\{x\_3\}\}\; =\; \backslash frac\{\backslash bar\{G\_2\}\}\{(1-x\_2)^2\}$$

or

$$\backslash left(\backslash mathfrak\{d\}\; \backslash frac\{G\}\{\backslash frac\{1\; -\; x\_2\}\{\backslash mathfrak\{d\}\; x\_2\}\}\backslash right)\_\{\backslash frac\{x\_1\}\{x\_3\}\}\; =\; \backslash frac\{\backslash bar\{G\_2\}\}\{(1\; -\; x\_2)^2\}$$

or

$$\backslash left(\backslash frac\; \{\backslash frac\{\backslash partial\; G\}\{1-x\_2\}\}\{\backslash partial\; x\_2\}\backslash right)\_\{\backslash frac\{x\_1\}\{x\_3\}\}\; =\; \backslash frac\{\backslash bar\{G\_2\}\}\{(1\; -\; x\_2)^2\}$$ as formatting variant

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.

The last equality can be integrated from $x\_2\; =\; 1$ to $x\_2$ gives:

$$G\; -\; (1\; -\; x\_2)\; \backslash lim\_\{x\_2\backslash to\; 1\}\; \backslash frac\{G\}\{1\; -\; x\_2\}\; =\; (1\; -\; x\_2)\; \backslash int\_\{1\}^\{x\_2\}\backslash frac\{\backslash bar\{G\_2\}\}\{(1\; -\; x\_2)^2\}\; dx\_2$$

Applying LHopital's rule gives:

$$\backslash lim\_\{x\_2\backslash to\; 1\}\; \backslash frac\{G\}\{1\; -\; x\_2\}\; =\; \backslash lim\_\{x\_2\backslash to\; 1\}\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\_2\}\backslash right)\_\{\backslash frac\{x\_1\}\{x\_3\}\}\; .$$

This becomes further:

$$\backslash lim\_\{x\_2\backslash to\; 1\}\; \backslash frac\{G\}\{1\; -\; x\_2\}\; =\; -\backslash lim\_\{x\_2\backslash to\; 1\}\; \backslash frac\; \{\backslash bar\{G\_2\}\; -\; G\}\{1\; -\; x\_2\}.$$

Express the mole fractions of component 1 and 3 as functions of component 2 mole fraction and binary mole ratios:

$$x\_1\; =\; \backslash frac\{1-x\_2\}\{1+\backslash frac\{x\_3\}\{x\_1\}\}$$

$$x\_3\; =\; \backslash frac\{1-x\_2\}\{1+\backslash frac\{x\_1\}\{x\_3\}\}$$

and the sum of partial molar quantities

$$G=\backslash sum\; \_\{i=1\}^3\; x\_i\; \backslash bar\{G\_i\},$$

gives

$$G=\; x\_1\; (\backslash bar\; \{G\_1\})\_\{x\_2\; =1\}\; +\; x\_3\; (\backslash bar\; \{G\_3\})\_\{x\_2\; =1\}\; +\; (1\; -\; x\_2)\; \backslash int\_\{1\}^\{x\_2\}\backslash frac\{\backslash bar\{G\_2\}\}\{(1\; -\; x\_2)^2\}\; dx\_2$$

$(\backslash bar\{G\_1\})\_\{x\_2\; =1\}$ and $(\backslash bar\{G\_3\})\_\{x\_2\; =1\}$ 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

$$(\backslash bar\; \{G\_1\})\_\{x\_2\; =1\}\; =\; -\; \backslash left(\backslash int\_\{1\}^\{0\}\backslash frac\{\backslash bar\{G\_2\}\}\{(1\; -\; x\_2)^2\}\; dx\_2\; \backslash right)\_\{x\_3=0\}$$

and

$$(\backslash bar\; \{G\_3\})\_\{x\_2\; =1\}\; =\; -\; \backslash left(\backslash int\_\{1\}^\{0\}\backslash frac\{\backslash bar\{G\_2\}\}\{(1\; -\; x\_2)^2\}\; dx\_2\; \backslash right)\_\{x\_1=0\}$$

The final expression is given by substitution of these constants into the previous equation:

$$G=\; (1\; -\; x\_2)\; \backslash left(\backslash int\_\{1\}^\{x\_2\}\backslash frac\{\backslash bar\{G\_2\}\}\{(1\; -\; x\_2)^2\}\; dx\_2\; \backslash right)\_\{\backslash frac\{x\_1\}\{x\_3\}\}\; -\; x\_1\; \backslash left(\backslash int\_\{1\}^\{0\}\backslash frac\{\backslash bar\{G\_2\}\}\{(1\; -\; x\_2)^2\}\; dx\_2\; \backslash right)\_\{x\_3=0\}\; -\; x\_3\; \backslash left(\backslash int\_\{1\}^\{0\}\backslash frac\{\backslash bar\{G\_2\}\}\{(1\; -\; x\_2)^2\}\; dx\_2\; \backslash right)\_\{x\_1=0\}$$

• Margules activity model
• Darken's equations
• Gibbs–Helmholtz equation

{{reflist}}

• [https://pubs.acs.org/doi/abs/10.1021/j100833a006 J. Phys. Chem. Gokcen 1960]
• [https://web.archive.org/web/20070322035057/http://www.chem.neu.edu/Courses/1382Budil/PartialMolarQuantities.htm A lecture from www.chem.neu.edu]
• [https://web.archive.org/web/20070707224025/http://www.chem.arizona.edu/~salzmanr/480a/480ants/opensys/opensys.html A lecture from www.chem.arizona.edu]
• [https://www.britannica.com/science/Gibbs-Duhem-equation Encyclopædia Britannica entry]

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Category:Chemical thermodynamics

Category:Thermodynamic equations

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