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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)