24.2: The Gibbs-Duhem Equation Relates Chemical Potential and Composition at Equilibrium
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[ G_\text{tot} = \sum_i n_i \mu_i \nonumber \]
Taking the derivative of both sides yields:
\[ dG_\text{tot} = \sum_i n_i d \mu_i + \sum_i \mu_i d n_i \nonumber \]
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:
\[V~dp - s~dT = 0 \label{eq42} \]
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 \(B\):
\[ n_Bd\mu_B + n_Ad\mu_A = 0 \nonumber \]
Rearranging:
\[ d\mu_B = -\dfrac{n_A}{n_B} d\mu_A \nonumber \]
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 \(G\):
\[ \begin{align*} dG &= d(\mu_An_A) + d(\mu_Bn_B) \\[4pt] &= n_Ad\mu_A+\mu_Adn_A + n_Bd\mu_B+\mu_Bdn_B \end{align*} \]
However, if we simply write out a change in \(G\) due to the number of moles we have:
\[dG = \mu_A dn_A +\mu_B dn_B \nonumber \]
Consequently the other terms must add up to zero:
\[0 = n_Ad\mu_A+ n_Bd\mu_B \nonumber \]
\[d\mu_A= - \dfrac{n_B}{n_A}d\mu_B \nonumber \]
\[d\mu_A= - \dfrac{\chi_B}{\chi_A}d\mu_B \nonumber \]
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:
\[d\mu_A = - \dfrac{\chi_B}{\chi_A} d\mu_B \nonumber \]
As:
\[d\mu_B = 0 + \dfrac{RT}{\chi_B} \nonumber \]
with \(x_B\) being the only active variable at constant temperature, we get:
\[d\mu_A = - \dfrac{\chi_B}{\chi_A} \dfrac{RT}{\chi_B} = \dfrac{RT}{\chi_A} \nonumber \]
If we now wish to find \(\mu_A\) we need to integrate \(d\mu_A\), e.g. form pure 1 to \(\chi_A\). This produces:
\[\mu_A = \mu^*_A + RT \ln \chi_A \nonumber \]
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.