24.2: The Gibbs-Duhem Equation

For a system at equilibrium, the Gibbs-Duhem equation must hold:

$\sum_i n_i d\mu_i = 0 \label{eq1}$

This relationship places a compositional constraint upon any changes in the chemical potential in a mixture at constant temperature and pressure for a given composition. This result is easily derived when one considers that $$\mu_i$$ represents the partial molar Gibbs function for component $$i$$. And as with other partial molar quantities,

$G_{tot} = \sum_i n_i \mu_i$

Taking the derivative of both sides yields

$dG_{tot} = \sum_i n_i d \mu_i + \sum_i \mu_i d n_i$

But $$dG$$ can also be expressed as

$dG = Vdp - sdT + \sum_i \mu_i d n_i$

Setting these two expressions equal to one another

$\sum_i n_i d \mu_i + \cancel{ \sum_i \mu_i d n_i } = Vdp - sdT + \cancel{ \sum_i \mu_i d n_i}$

And after canceling terms, one gets

$\sum_i n_i d \mu_i = Vdp - sdT \label{eq41}$

For a system at constant temperature and pressure

$Vdp - sdT = 0 \label{eq42}$

Substituting Equation \ref{eq42} into \ref{eq41} results in the Gibbs-Duhem equation (Equation \ref{eq1}). This expression relates how the chemical potential can change for a given composition while the system maintains equilibrium. So for a binary system, consisting of components $$A$$ and $$B$$ (the two most often studied compounds in all of chemistry)

$d\mu_B = -\dfrac{n_A}{n_B} d\mu_A$

Contributors

• Patrick E. Fleming (Department of Chemistry and Biochemistry; California State University, East Bay)