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24.2: The Gibbs-Duhem Equation Relates Chemical Potential and Composition at Equilibrium

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    14504
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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.


    24.2: The Gibbs-Duhem Equation Relates Chemical Potential and Composition at Equilibrium is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.