1. Activity Coefficients; Equilibrium Constants
Lecture 8

2. How do deal with individual ions in aqueous solution?
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
as usual ν is the stoichiometric coefficient
For some thermodynamic parameter Ψ (e.g., µ)
ΨAB = νAΨA + νBΨB
So for example for MgCl2:

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

4. Debeye-Hückel Theory
In an electrolyte solution, each ion exerts an electrostatic force on every other ion. These forces will decrease with the square of distance between ions.
The forces between ions will be reduced by the presence of water molecules, due to their dielectric nature.
As total solute concentration increases, the mean distance between ions will decrease. Thus electrostatic interaction, and consequently activity, should depend on the total ionic concentration.
The extent of electrostatic interaction will also depend on the charge of the ions involved: the force between Ca2+ and Mg2+ ions will be greater at the same distance than between Na+ and K+ ions.
In the Debye-Hückel Theory, a given ion is considered to be surrounded by an atmosphere or cloud of oppositely charged ions (this atmosphere is distinct from, and unrelated to, the solvation shell). If it were not for the thermal motion of the ions, the structure would be analogous to that of a crystal lattice, though looser. Thermal motion inhibits a fixed structure.
The density of charge in this ion atmosphere increases with the square root of the ionic concentrations, but increases with the square of the charges on those ions. The dielectric effect of intervening water molecules will tend to reduce the interaction between ions.

5. 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:

6. 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.

7. Activities in Solid Solutions
Many minerals are also solutions.
How do we compute activities of components in the solutions?
For example, if we substitute Fe2+ for Mg2+ in forsterite, what are the activities of Fe2+ and Mg2+?
Olivine Structure:
Fe and Mg ions are located between unlinked silica tetrahedra

8. Mixing-On-Site Model
Many crystalline solids, for example olivine, can be treated as ideal solutions. A simple ideal solution model is the mixing on site model, which considers the substitution of species in any site individually. In this model, the activity of an individual species is calculated as:
ai,ideal = (Xi)ν
where X is the mole fraction of the ith atom and ν is the number of sites per formula unit on which mixing takes place (the stoichiometric coefficient). For example, ν=2 in the Fe-Mg exchange in olivine, (Mg,Fe)2SiO4. Olivine has two (nearly identical) sites that can be occupied by Mg and Fe. We could treat them separately, then the activity of Mg would be, in effect, the sum of its activity in the two sites:
For example, if Mg were distributed randomly between the two sites and the total mole fraction were 0.5, the activity would be = 0.5.
While olivine has two sites that can be occupied by Mg and Fe (M1 and M2), they are effectively equivalent. So we could simplify things by choosing (Mg,Fe)Si1/2O2 as the formula unit (we must then choose all other thermodynamic parameters to be 1/2 those of (Mg,Fe)2SiO4). In this case, the activity of Mg2+ is simply its mole fraction.

9. Activities of Phase Components
Now suppose we chose of component to be not an ion, but a pure phase, such as pyrope, Mg3Al2Si3O12, in garnet, whose general formula is X3Y2Si3O12.
Suppose the garnet of interest has the composition:
The chemical potential of pyrope in garnet contains mixing contributions from both Mg in the cubic site and Al in the octahedral site. The activity of pyrope is thus given by:
For example:

10. Free Energy of Solution in Ideal Solid Solutions
∆G = ∆H – T∆S
But ∆H = 0 in the ideal case so
∆Gideal mixing = -T∆Sideal mixing
where ∆Sideal mixing is simply the configurational entropy:
In the pyrope example, the chemical potential of pyrope in garnet would be:

11. The Equilibrium Constant
Consider the reaction
aA + bB = cC + dD
The Free Energy change of reaction is:
∆G = cµC + dµD – aµA – bµB
At equilibrium:
Expanding the µ term: or
We define the right term as the equilibrium constant:

12. Free Energy and the Equilibrium Constant
Since:
then
and
Note of caution: our thermodynamic parameters are additive, but because of the exponential relation between the equilibrium constant and free energy, equilibrium constants are multiplicative.

13. Manipulating Equilibrium Constants
Suppose we want to know the equilibrium constant for a reaction that can be written as the sum of two reactions,
e.g., we can sum
to yield
The equilibrium constant of the net reaction would be the product of the equilibrium constants of the individual reactions.
For this reason and because equilibrium constants can be very large or very small numbers, it is often convenient to work with logs of equilibrium constants:
pK = - log K
(we can then sum the pK’s).

14. Apparent Equilibrium Constants and Distribution Coefficient
In practice, other kinds of equilibrium constants are used based on concentrations rather than activities.
Distribution Coefficient
Apparent Equilibrium Constant

15. Other Forms
A ‘solubility constant’ is an equilibrium constant. For example:
Since the activity of NaCl in halite = 1, then
Henry’s Law constants for describing solubility of gases in solution (e.g., CO2 in water).
Since Pi = hiXi

16. Law of Mass Action Important to remember our equation
describes the equilibrium condition. At non-equilibrium conditions the r.h.s. is called the reaction quotient, Q.
Written for the reaction H2CO3 = HCO3- + H+
We can see that addition of H+ will drive the reaction to the left.
“Changing the concentration of one species in a reaction in a system at equilibrium will cause a reaction in a direction that minimizes that change”.

17. Le Chatelier’s Principle
We can generalize this to pressure and temperature:
dG = VdP - SdT
An increase in pressure will drive a reaction in a direction such as to decrease volume
An increase in temperature will drive a reaction in a direction such as to increase entropy.
“When perturbed, a system reacts to minimize the effects of perturbation.”

18. Temperature and Pressure Dependence
Since ∆G˚ = ∆H˚ - T∆S˚ and ∆G˚ = -RT ln K then
Temperature and pressure dependencies found by taking derivatives of this equation with respect to T:
(this is known as the van Hoff equation).
For pressure, since

19. Oxidation and Reduction
“Redox” refers to processes in which atoms gain or lose electrons, e.g., oxidation of iron:
Fe2+ ⟶ Fe3+

20. Valence and Redox
We define valence as the charge an atom acquires when it is dissolved in solution.
Conventions
Valence of all elements in pure form is 0.
Sum of valences much equal actual charge of species
Valence of hydrogen is +1 except in metal hydrides when it is -1
Valence of O is -2 except in peroxides when it is -1.
Elements generally function as either electron donors or acceptors.
Metals in 0 valence state are electron donors (become positively charged)
Oxygen is the most common electron acceptor (hence the term oxidation)
Redox
A reduced state can be thought of as one is which the availability of electrons is high
An oxidized state is one in which the availability of electrons is low.
Really idiotic pneumonic:
LEO the Lion Says GRR:
Loss Equals Oxidation
Gain Refers to Reduction