7.4: The Gibbs-Duhem Equation
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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,
\[ G_{tot} = \sum_i n_i \mu_i \nonumber \]
Taking the derivative of both sides yields
\[ dG_{tot} = \sum_i n_i d \mu_i + \sum_i \mu_i d n_i \nonumber \]
But \(dG\) can also be expressed as
\[dG = Vdp - sdT + \sum_i \mu_i d n_i \nonumber \]
Setting these two expressions equal to one another
\[ \sum_i n_i d \mu_i + \cancel{ \sum_i \mu_i d n_i } = Vdp - sdT + \cancel{ \sum_i \mu_i d n_i} \nonumber \]
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)
\[ d\mu_B = -\dfrac{n_A}{n_B} d\mu_A \nonumber \]