Gibbs–Duhem equation

The Gibbs-Duhem equation in thermodynamics describes the relationship between changes in chemical potential for components in a thermodynamical system  :

$\sum _{i=1}^{I}N_{i}\mathrm {d} \mu _{i}=-S\mathrm {d} T+V\mathrm {d} p\,$

where $N_{i}\,$ is the number of moles of component $i\,$, $\mathrm {d} \mu _{i}\,$ the incremental 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. 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 Gibbs and Pierre Duhem.

Derivation

Deriving the Gibbs-Duhem equation from basic thermodynamic state equations is straightforward. The Gibbs free energy $G\,$ can be expanded locally in terms of the thermodynamic state as:

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

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

$\mathrm {d} G=V\mathrm {d} p-S\mathrm {d} T+\sum _{i=1}^{I}\mu _{i}\mathrm {d} N_{i}\,$

The chemical potential is just another name for the partial molar Gibbs free energy, and as such:

$G=\sum _{i=1}^{I}\mu _{i}N_{i}\,$

$\mathrm {d} G=\sum _{i=1}^{I}\mu _{i}\mathrm {d} N_{i}+\sum _{i=1}^{I}N_{i}\mathrm {d} \mu _{i}\,$

Subtracting yields the Gibbs-Duhem relation:

$\sum _{i=1}^{I}N_{i}\mathrm {d} \mu _{i}=-S\mathrm {d} T+V\mathrm {d} p\,$

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 $I\,$ different components, there will be $I+1\,$ independent parameters or "degrees of freedom". For example, a gas cylinder filled with nitrogen is at room temperature (298 K) and at 2500 psi, we can determine the gas density, 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 potential across a phase boundary be equal. 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. 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 $x_{1}+x_{2}=1\,$:

$x_{1}\left.{\frac {\mathrm {d} ln\gamma _{1}}{\mathrm {d} x_{1}}}\right|_{p,T}=x_{2}\left.{\frac {\mathrm {d} ln\gamma _{2}}{\mathrm {d} x_{2}}}\right|_{p,T}\,$

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.

External links

• A lecture from www.chem.neu.edu Link
• A lecture from www.chem.arizona.edu Link
• Encylcopedia Britannica entry link