1. Real Solutions
Lecture 7

2. Three Kinds of Behavior
Looking at the graph, we see 3 regions:
1. Ideal:
µi =µi˚ + RT ln Xi
2. Henry’s Law:
µi =µi˚ + RT ln hiXi
µi =µi˚ + RT ln hiXi + RT ln hi
Letting µ* = µ˚ + ln h
µi =µi* + RT ln Xi
µ* is chemical potential in ‘standard state’ of Henry’s Law behavior at Xi = 1.
3. Real Solutions
Need a way to deal with them.

3. Fugacities
We define fugacity to have the same relationship to chemical potential as the partial pressure of an ideal gas:
Where ƒ˚ is the ‘standard state’ fugacity. We are free to chose the standard state, but the standard state for µ˚ and ƒ˚ must be the same.
We can think of this as the ‘escaping tendency’ of the gas.
The second part of the definition is:
Fugacity and partial pressure are the same for an ideal gas.
We can imagine that at infinitesimal pressure any gas should behave ideally.

4. Fugacity Coefficient
We can express the relationship between pressure and fugacity as:
ƒ = ΦP
where Φ is the fugacity coefficient which will be a function of T and P.
For example, see fugacity coefficients for H2O and CO2 in Table 3.1.

5. Activities
Fugacities are useful for gases such as H2O and CO2, but we can extent the concept to calculate chemical potentials in real liquid and solid solutions.
Recalling:
We define the activity as:
Hence
Same equation as for an ideal solution, except that ai replaces Xi.
We have retained our ideal solution formulation and stuffed all non-ideal behavior into the activity.
Activity can be thought of as the effective concentration.

6. Activity Coefficients
We’ll express the relationship between activity and mole fraction as:
ai = λiXi
The activity coefficient is a function of temperature, pressure, and composition (including Xi).
For an ideal solution, ai = Xi and λi = 1.

7. Rational and Practical Activity Coefficients
The rational activity coefficient, λ, relates activity to mole fraction.
Although mole fraction is the natural thermodynamic concentration unit, other units, such as moles (of a solute) per kilogram or liter or solution are more commonly used (because they are easily measured).
In those units, we use the practical activity coefficient, γ.

8. Excess Functions
Comparing real and ideal solutions, we can express the difference as:
Gexcess = Greal – Gideal
Similarly for other thermodynamic functions, so that:
Gexcess = Hexcess – Tsexcess
Also
And

9. Water and Electolyte Solutions

10. Water Water is a familiar but very unusual compound.
Highest heat capacity (except ammonia)
Highest heat of evaporation
Highest surface tension
Maximum density at 4˚C
Negative Clayperon Slope
Best solvent
Its unusual properties relate to the polar nature of the molecule.

11. Solvation
The polar nature of the molecule allows it to electrostatically shield ions from each other (its high dielectric constant), hence dissolve ionic compounds (like salt).
Once is solution, it also insulates ions by surrounding them with a solvation shell.
First solvation shell usually 4 to 6 oriented water molecules (depending on ion charge) tightly bound to ion and marching in lock step with the ion.
Outer shell consists of additional loosely bound molecules.

12. Solvation Effects Enhances solubility
Electrostriction: water molecules in solvation shell more tightly packed, reducing volume of the solution.
Causes partial collapse of the H-bonded structure of water.
Non-ideal behavior

13. Some definitions and conventions
Concentrations
Molarity: M, moles of solute per liter
Molality: m, moles of solute per kg
Note that in dilute solutions these are effectively the same. pH
Water, of course dissociates to form H+ and OH–.
At 25˚C and 1 bar, 1 in 107 molecules will do so such that
aH+ × aOH– = 10-14
pH = -log aH+
Standard state convention
a˚ = m = 1 (mole/kg)
Most solutions are very non-ideal at 1 m, so this is a hypothetical standard state constructed by extrapolating Henry’s Law behavior to m = 1. Reference state (where measurements actually made) is infinite dilution.

14. Example: Standard Molar Volume of NaCl in H2O
Volume of the solution given by
Basically, we are assigning all the non-ideal behavior on NaCl.
Not true, of course, but that’s the convention.

15. How do deal with individual ions
We can’t simply add Na+ to a solution (positive ions would repel each other).
We can add NaCl. How do we partition thermodynamic parameters between Na+ and Cl–?
For a salt AB, the molarity is:
mA = νAmAB and mB = νBmAB
For a thermodynamic parameter Ψ (could be µ)
ΨAB = νAΨA + νBΨB
So for example for MgCl2:

16. Practical Approach to Electrolyte Activity Coefficients
Debye-Hückel and Davies

17. Debye-Hückel Extended Law
Assumptions
Complete dissociation
Ions are point charges
Solvent is structureless
Thermal energy exceeds electrostatic interaction energy
Debye-Hückel Extended Law
Where A and B are constants, z is ionic charge, å is effective ionic radius and I is ionic strength:

18. Debye-Hückel Limiting & Davies Laws
Limiting Law (for low ionic strength)
Davies Law:
Where b is a constant (≈0.3).
Assumption of complete dissociation one of main limiting factors of these approaches: ions more likely to associate and form ion pairs at higher concentrations.