1. Gibbs-Duhem and the Chemical Potential of Ideal Solutions
Lecture 6

2. Review: Chemical Potential
The chemical potential is defined as partial molar Gibbs Free Energy:
such that
or, dividing each side by the total number of moles:
The chemical potential tells us how the Gibbs Free Energy will vary with the number of moles, ni, of component i holding temperature, pressure, and the number of moles of all other components constant.
For a pure substance, the chemical potential is equal to its molar Gibbs Free Energy (also the molar Helmholz Free Energy):

3. Gibbs-Duhem Equation
The Free Energy of a system (or phase) is the sum of chemical potentials of its components:
Differentiating:
Equating with the earlier equation:
We can rearrange this as the Gibbs-Duhem Equation:

4. Interpreting Gibbs-Duhem
In a closed system at equilibrium, net changes in chemical potential will occur only as a result of changes in temperature or pressure. At constant temperature and pressure, there can be no net change in chemical potential at equilibrium:
This equation further tells us that the chemical potentials do not vary independently, but change in a related way.

5. Final point about chemical potential:
In spontaneous processes, components or species are distributed between phases so as to minimize the chemical potential of all components.

6. Thermodynamics of Ideal Solutions

7. Chemical Potential in Ideal Solutions
In terms of partial molar quantities
For an ideal gas:
Integrating from P˚ to P:
Where P˚ is the pressure of pure substance in its ‘standard state’ and µ˚ is the chemical potential of component i in that state. In that case, P/P˚ = Xi and:

8. This equation tells us that in an ideal solution:
the chemical potential of component i is always less than that of pure i, since X is by definition less than 1 and hence ln X is negative.
the chemical potential of component i increases linearly with the log of its concentration in the solution.

9. Volume and enthalpy changes of solutions
Water–alcohol is an example of a non-ideal solution. The volume is expressed as:
∆Vmixing term may be negative, as in rum ‘n coke.
Similarly, mix nitric acid and water and the solution gets hot – that thermal energy is the enthapy of mixing. Enthalpy of solutions is expressed as:
The ∆H term is positive in the nitric acid case.
For ideal solutions, however, ∆Vmixing = ∆Hmixing = 0

10. Entropy changes of solution
What about entropy and free energy changes of ideal solutions?
Even in ideal solutions, there is an entropy change (increase) because we have increased the randomness of the system.
The entropy change of ideal solution is:
Note similarity to configurational entropy.
Note negative sign. How will entropy change?
The total (molar) entropy of an ideal solution is then:

11. What does this tell us about how Gibbs Free Energy will change in a ideal solution?
∆Gr = ∆Hr – T∆Sr

12. Free Energy Change of Solution
∆Gmixing = ∆Hmixing - T∆Smixing
For an ideal solution, ∆Hmixing = 0
And
Because the log term is always negative, ideal solutions have lower free energy than a mixture of their pure constituents and ∆G decreases with increasing T. This is why things are usually more soluble at higher T.
Total Free Energy of an ideal solution is:

13. Total Free Energy of an Ideal Solution
Recalling that and
We can substitute
Into the above and obtain:

14. Total Free Energy of an Ideal Solution
The Free Energy of a solution is simply the sum of the chemical potentials of the components times their mole fractions. (Note that I have shortened the subscript from ideal solution to simply ideal).
Since
This equation is entirely equivalent to the one we previously derived:
The first term on the right is the sum of the contribution of the chemical potentials of the pure components.
The second term on the right is the decrease in free energy that comes from the increase in entropy.
The second term on the right will always be negative because the mole fractions are by definition less than 1, hence their logs are always negative.
Equivalently:

15. The change due to solution is simply the second term on the right:
Let’s now see what happens to Free Energy when we dissolve component 2 in component 1 to create an ideal solution at various temperatures
Recall:
The change due to solution is simply the second term on the right:

16. Free Energy of Mixing in an Ideal Solution