24.2: The Gibbs-Duhem Equation Relates Chemical Potential and Composition at Equilibrium
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At equilibrium, there is no change in chemical potential for the system:
\[\sum_i n_i d\mu_i = 0 \label{eq1}\]
This is the Gibbs-Duhem relationship and it 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:
\[ G_text{tot} = \sum_i n_i \mu_i\]
Taking the derivative of both sides yields:
\[ dG_text{tot} = \sum_i n_i d \mu_i + \sum_i \mu_i d n_i \]
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.
Gibbs-Duhem for Binary Systems
For a binary system consisting of components two components, \(A\) and \(B\):
\[ n_Bd\mu_B + n_Ad\mu_A = 0 \]
Rearranging:
\[ d\mu_B = -\dfrac{n_A}{n_B} d\mu_A\]
Consider a Gibbs free energy that only includes \(μ_n\) conjugate variables as we obtained it from our scaling experiment at \(T\) and \(P\) constant:
\[G = \mu_An_A + \mu_Bn_B \nonumber \]
Consider a change in \(G\):
\[dG = d(\mu_An_A) + d(\mu_Bn_B) \nonumber \]
\[dG = n_Ad\mu_A+\mu_Adn_A + n_Bd\mu_B+\mu_Bdn_B \nonumber \]
However, if we simply write out a change in \(G\) due to the number of moles we have:
\[dG = \mu_Adn_A +\mu_Bdn_B \nonumber \]
Consequently the other terms must add up to zero:
\[0 = n_Ad\mu_A+ n_Bd\mu_B \nonumber \]
\[d\mu_A= - \dfrac{n_B}{n_A}d\mu_B \nonumber \]
\[d\mu_A= - \dfrac{x_B}{x_A}d\mu_B \nonumber \]
In the last step we have simply divided both denominator and numerator by the total number of moles. This expression is the Gibbs-Duhem equation for a 2-component system. It relates the change in one thermodynamic potential (\(d\mu_A\)) to the other (\(d\mu_B\)).
The Gibbs-Duhem equation relates the change in one thermodynamic potential (\(d\mu_A\)) to the other (\(d\mu_B\)).
Gibbs-Duhem in the Ideal Case
In the ideal case we have:
\[\mu_B = \mu^*_B + RT \ln x_B \nonumber \]
Gibbs-Duhem gives:
\[d\mu_A = - \dfrac{x_B}{x_A} d\mu_B \nonumber \]
As:
\[d\mu_B = 0 + \dfrac{RT}{x_B} \nonumber \]
with \(x_B\) being the only active variable at constant temperature, we get:
\[d\mu_A = - \dfrac{x_B}{x_A} \dfrac{RT}{x_B} = \dfrac{RT}{x_A} \nonumber \]
If we now wish to find \(\mu_A\) we need to integrate \(d\mu_A\), e.g. form pure 1 to \(x_A\). This produces:
\[\mu_A = \mu^*_A + RT \ln x_A \nonumber \]
This demonstrates that Raoult's law can only hold over the whole range for one component if it also holds for the other over the whole range.