In the previous module, we introduced the chemical potential as the quantity that describes how the Gibbs function changes when the composition of a system changes.
An important question naturally follows:
Can the chemical potentials of the components of a mixture change independently?
The answer is no.
The chemical potentials of the components of a mixture are linked by a fundamental thermodynamic relationship known as the Gibbs-Duhem equation.
For a multicomponent system, the Gibbs function can be written as the sum of the contributions from the individual components:
\[ \boxed{ G = \sum_i n_i\mu_i } \]
where \(n_i\) is the number of moles of component \(i\) and \(\mu_i\) is its chemical potential.
Differentiating this expression yields
\[ dG = \sum_i \mu_i\,dn_i + \sum_i n_i\,d\mu_i \]
Earlier, we derived the general differential form of the Gibbs function:
\[ dG = V\,dp - S\,dT + \sum_i \mu_i\,dn_i \]
Since both expressions describe the same quantity, they must be equal:
\[ \sum_i \mu_i\,dn_i + \sum_i n_i\,d\mu_i = V\,dp - S\,dT + \sum_i \mu_i\,dn_i \]
The terms involving \(dn_i\) cancel:
\[ \sum_i n_i\,d\mu_i = V\,dp - S\,dT \]
This relationship is sometimes called the generalized Gibbs-Duhem equation.
Most mixtures are studied at constant temperature and pressure.
Under these conditions,
\[ dT=0 \qquad dp=0 \]
and the previous expression simplifies to
\[ \boxed{ \sum_i n_i\,d\mu_i = 0 } \]
This is the Gibbs-Duhem equation.
For a mixture containing only two components, A and B, the Gibbs-Duhem equation becomes
\[ \boxed{ n_A\,d\mu_A + n_B\,d\mu_B = 0 } \]
Rearranging gives
\[ \boxed{ d\mu_B = - \frac{n_A}{n_B} d\mu_A } \]
This result shows that if the chemical potential of one component changes, the chemical potential of the other component must also change.
The two chemical potentials cannot vary independently.
The Gibbs-Duhem equation is a consequence of the fact that the composition of a mixture is constrained.
If the mole fraction of one component increases, the mole fractions of the other components must adjust to compensate.
As a result, the chemical potentials are linked together by the thermodynamic requirement that the total Gibbs function remain consistent.
The Gibbs-Duhem equation therefore tells us that a mixture possesses fewer independent variables than might initially be expected.
The Gibbs-Duhem equation is one of the most important relationships in the thermodynamics of mixtures because it allows information about one component to be used to determine information about the others.
It plays a central role in:
Big picture: The Gibbs-Duhem equation shows that the chemical potentials of the components of a mixture are not independent. A change in the chemical potential of one component requires compensating changes in the others, reflecting the fact that all components contribute to a single thermodynamic system.
A binary mixture contains
\[ n_A = 2.00\ \mathrm{mol} \]
and
\[ n_B = 3.00\ \mathrm{mol}. \]
At constant temperature and pressure, the chemical potential of component A changes by
\[ d\mu_A = -15.0\ \mathrm{J\,mol^{-1}}. \]
Calculate the corresponding change in the chemical potential of component B.
For a binary mixture, the Gibbs-Duhem equation is
\[ n_A\,d\mu_A + n_B\,d\mu_B = 0 \]
Rearranging to solve for \(d\mu_B\):
\[ d\mu_B = - \frac{n_A}{n_B} d\mu_A \]
Substituting the values:
\[ d\mu_B = - \frac {2.00\ \mathrm{mol}} {3.00\ \mathrm{mol}} \left( -15.0\ \mathrm{J\,mol^{-1}} \right) \]
\[ d\mu_B = +10.0\ \mathrm{J\,mol^{-1}} \]
\[ \boxed{ d\mu_B = +10.0\ \mathrm{J\,mol^{-1}} } \]
Interpretation: The chemical potential of A decreases, but the chemical potential of B increases. The Gibbs-Duhem equation requires these changes to compensate one another so that the thermodynamic description of the mixture remains internally consistent.
This example illustrates the central message of the Gibbs-Duhem equation: the chemical potentials of the components of a mixture are not independent. A change in one chemical potential necessarily produces changes in the others.
| Question 1 |
|---|
| A binary mixture contains \(n_A = 2.00\ \mathrm{mol}\) and \(n_B = 5.00\ \mathrm{mol}\). If \[ d\mu_A = -20.0\ \mathrm{J\,mol^{-1}} \] at constant temperature and pressure, what is \(d\mu_B\)? |
Show AnswerUsing \[ n_A d\mu_A+n_B d\mu_B=0 \] gives \[ d\mu_B = -\frac{n_A}{n_B}d\mu_A = -\frac{2.00}{5.00}(-20.0) = +8.00\ \mathrm{J\,mol^{-1}}. \] |
| Question 2 |
|---|
| Which statement is the most important consequence of the Gibbs-Duhem equation? |
Show AnswerThe Gibbs-Duhem equation \[ \sum_i n_i d\mu_i = 0 \] links the chemical potentials of all components in a mixture. A change in one chemical potential requires compensating changes in the others. |
| Question 3 |
|---|
| A binary mixture contains equal amounts of A and B: \[ n_A=n_B. \] If \[ d\mu_A = +12.0\ \mathrm{J\,mol^{-1}}, \] what must be true of \(d\mu_B\)? |
Show AnswerSince \[ n_A=n_B, \] the Gibbs-Duhem equation becomes \[ d\mu_A+d\mu_B=0. \] Therefore, \[ d\mu_B=-d\mu_A=-12.0\ \mathrm{J\,mol^{-1}}. \] |
Big picture: The Gibbs-Duhem equation provides a fundamental constraint on mixtures by linking the chemical potentials of all components in the system. It demonstrates that the thermodynamic properties of a mixture are interconnected and that no component can be considered completely independently of the others.