1. Aqueous Complexes Why do we care??
Complexation of an ion also occuring in a mineral increases solubility
Some elements occur as complexes more commonly than as free ions
Adsorption of elements greatly determined by the complex it resides in
Toxicity/ bioavailability of elements depends on the complexation

2. Defining Complexes Use equilibrium expressions: DG0R = -RT ln Keq
cC + lHL  CL + lH+
Where B is just like Keq!

3. Closer look at complexation
Stability of complexes generally increases with increasing charge or decreasing radius ratio (i.e. factors increasing bond strength)
Cations forming strong complexes with certain ligands also tend to form minerals with low solubilities
Complexation tends to increase mineral solubility that contain the species being complexed
More salinity = more multinuclear complexes

4. Outer Sphere Complexesd+ ..
Water’s polar nature is key:
Cations are usually surrounded by H2O’s
Outer-sphere complexes (aka ion pairs) – Cation complexed with an anion BUT the anion does NOT displace a water:
Ca(H2O)6SO40
Long-range electrostatic interaction
Commonly involve mono and di-valent cations and anions like Cl-, HCO3-, SO42-, and CO32-
Draw the Ca(H2O)6 ion on the board, then put in SO4…

5. Inner Sphere Complexes
Inner-sphere complexes – ligand does displace the water
M(H2O)n + L-  ML(H2O)n-1 + H2O
n for any complex is based on Pauling’s first rule (radius ratio, close packed structures)
Cations get more inner-sphere as charge increases and radius decreases  scales as Ionic potential, I=z/r

6. Ionization Potential
z/r (charge/radius) also relates to a surface charge density on a cation ‘surface’
With increasing IP, charge density repulses H+ on H2O and forms oxycations (UO22+), hydroxycations (Fe(H2O)5OH2+), and hydroxyanions (Fe(OH)4-)
This effectively displaces the equilibrium distribution as a function of pH when comparing cations of varying IP

7. Electronegativities The power of an atom or ion to attract electrons
High EN (>2) = Lewis bases (nonmetals and ligands; e- donor)
Low EN (<2) = Lewis acids (metal cations; e- acceptor)
DEN determines bonding – covalent as DEN approaches 0 (more inner sphere), as DEN > 1.7, more ionic and outer-sphere
Lewis acid = e-pair-acceptor; base= e-pair donor

8. HSAB
Classification of cations and ligands as hard or soft acids and bases
Soft  species electron cloud is polarizable (deformable, soft) which prefers to participate in covalent bonding
Hard  low polarizability, e- cloud is rigid and prefers ionic bonding
Hard-hard = ionic (outer sphere)
Soft-soft = covalent (inner sphere)
Opposite  Weak bonds, rare complexes

9. Schwarzenbach Classification
Considers the electronic structure of individual cations divided into 3 classes:
Class A  noble gas configurations (highest orbital level filled) spherical symmetry and low polarizablity – hard spheres (Na+, Al3+, Ca2+)
Class B  electron configurations Ni0, Pd0, Pt0, highly polarizable – soft spheres (Ag+, Zn2+, Cd2+, Hg2+, Sn4+)
Class C  Transition metals with 0-10 e- in the d shell, intermediate polarizability
Polarizability – ease to which the cation’s e- cloud is deformed (ease to which the e- easily move in response to an attractive/repulsive force (presumably electrostatic)

10. Toxicity
Toxicity of a particular contaminant is partly based on complexation reactions  Hg2+ for instance is a soft acid, forming strong bonds with sulfur sites in amino acids like methionine and cysteine, breaking down enzyme function

11. Speciation Plus more species  gases and minerals!!
Any element exists in a solution, solid, or gas as 1 to n ions, molecules, or solids
Example: Ca2+ can exist in solution as:
Ca CaCl CaNO3+
Ca(H3SiO4) CaF CaOH+
Ca(O-phth) CaH2SiO CaPO4-
CaB(OH) CaH3SiO CaSO4
CaCH3COO CaHCO CaHPO40
CaCO30
Plus more species  gases and minerals!!

12. Mass Action & Mass Balance
mCa2+=mCa2++MCaCl+ + mCaCl20 + CaCL3- + CaHCO3+ + CaCO30 + CaF+ + CaSO40 + CaHSO4+ + CaOH+ +…
Final equation to solve the problem sees the mass action for each complex substituted into the mass balance equation

13. Coupling mass action and mass balance  governing equations
Start with a set of basis species
Mass balance for each of those basis species (includes all complexes of one basis species with other possible basis species – Cd2+ with Cl-, OH+, SO42- for example)
Using mass action for each complex in each mass balance – get an equation using only basis species to determine activity of each basis species – each secondary species then calculated based on the solution for the basis

14. Example: Pb2+, Cl-, OH- basis
PbT=[Pb2+]+[PbCl+]+[PbOH+]
Pb2+ + Cl- = PbCl+ K= [PbCl+] / [Pb2+][Cl-]
Pb2+ + OH- = PbCl+ K= [PbOH+] / [Pb2+][OH-]
[PbCl+]=K[Pb2+][Cl-] ; [PbOH+]=K[[Pb2+][OH-]
PbT=[Pb2+]+ K[Pb2+][Cl-] + K[Pb2+][OH-]
PbT=[Pb2+](1+ K[Cl-] + K[OH-])
[Pb2+] / PbT = a0 = 1 / (1+ K[Cl-] + K[OH-])
[PbCl+]=K[Pb2+][Cl-]
[Pb2+] / PbT = a0  [Pb2+] = a0PbT
[PbCl+]=K a0PbT [Cl-]

15. Non-linearity
Unknown variables (species activities and activity coefficients) are products raised to reaction coefficients
Multiple basis species – multiple equations need to be solved simulaneously
Set of values that satisfies a set of equations is called a root
Iterative procedures guess at the root value and tries to improve it incrementally until it satisfies the equations to a desired accuracy

16. Newton’s Method Newton’s method – for a function f(x)=a
An initial guess (x0) will yield a residual (R(x)), which is the amount that guess is still ‘off’
Subsequent guesses ideally improve, resulting in a smaller residual – keep going to the root!
Start at
R(x)
BUT – what if there is more than one root????

17. Newton - Raphson Multi-dimensional counterpart to Newton’s method
Used for the multiple governing equation for each basis species
Results in a matrix of functions where the residuals are recalculated iteratively to a small number (epsilon value in GWB, default=5e-11), the matrix, called the Jacobian matrix is n x n (where n are the number of basis species)

18. Uniqueness
Any set of equations that has more than one possible root can become a non-unique situation
There are several geochemical examples where 2 roots are physically realistic

19. Ionic Strength
Dealing with coulombic interaction of selected ions to each other in a matrix (solution) of many ions
Ionic strength is a measure of how many of those ions are in the matrix which affect how selected ions interact
Ionic strength (I):
Where m is the molality of species i and z is the charge of species i

20. Debye-Hückel
Assumes ions interact coulombically, ion size does not vary with ionic strength, and ions of same sign do not interact
A, B often presented as a constant, but:
A= x106r01/2(T)-3/2, B=50.3 (T)-1/2
Where  is the dielectric constant of water and r is the density

21. Iteration and activity example
Speciate a simple mix of Fe3+ and Cl-
Starting analysis of Fe3+ and Cl-
Calculate I
Calculate gi for each ion (Fe3+, Cl-, FeCl++)
Calculate activity for each ion
Recalculate I
Recalculate gi for each ion (Fe3+, Cl-, FeCl++)
Recalculate activity for each ion
Until the residual for these reduces…

22. Geochemical Models
Step 1: Defining the problem  Define basis species, used to then distribute between all species for that element or group
Al3+ = Al3+ + Al(OH)2+ + Al(OH)2+ + Al(OH)30 + Al(NO3)2- +… OR Fe2+ = Fe2+(H2O)6 + FeCl+ + FeCl20 + FeCl3- + FeNO3+ + FeHCO3+ + …)
Step 2 – Calculate the distribution of species
Step 3 – Calculate mineral and gas equilibria, find S.I.
THEN many models continue with a reaction  titration (T, +/- anything), mineral +/-, gas +/-,

23. Charge Balance
Principle of electroneutrality  For any solution, the total charge of positively charged ions will equal the total charge of negatively charged ions.
Net charge for any solution must = 0
Charge Balance Error (CBE)
Tells you how far off the analyses are (greater than 5% is not good, greater than 10% is terrible…)
Models adjust concentration of an anion or cation to make the charges balance before each iteration!

24. Activity Coefficients
No direct way to measure the effect of a single ion in solution (charge balance)
Mean Ion Activity Coefficients – determined for a salt (KCl, MgSO4, etc.)
g±KCl = [(gK)(gCl)]1/2
Ksp= g±KCl2(mK+)(mCl-)
MacInnes Convention  gK = gCl= g±KCl
Measure other salts in KCl electrolyte and substitute g±KCl in for one ion to measure the other ion w.r.t. g±KCl and g±salt
Direct measurement of g can be made from solubility measurments as well as freezing point lowering, boiling point elevation, water-vapor pressure, osmotic pressure, transport properties (including conductivity and diffusion)

25. Ionic Strength
Dealing with coulombic interaction of selected ions to each other in a matrix (solution) of many ions
Ionic strength is a measure of how many of those ions are in the matrix which affect how selected ions interact
Ionic strength (I):
Where m is the molality of species i and z is the charge of species i

26. Mean Ion Activity Coefficients versus Ionic Strength

27. Debye-Hückel
Assumes ions interact coulombically, ion size does not vary with ionic strength, and ions of same sign do not interact
A, B often presented as a constant, but:
A= x106r01/2(T)-3/2, B=50.3 (T)-1/2
Where  is the dielectric constant of water and r is the density

28. Higher Ionic Strengths
Activity coefficients decrease to minimal values around m, then increase
the fraction of water molecules surrounding ions in hydration spheres becomes significant
Activity and dielectric constant of water decreases  in a 5 M NaCl solution, ~1/2 of the H2O is complexed, decreasing the activity to 0.8
Ion pairing increases, increasing the activity effects

29. Extended Debye-Hückel
Adds a correction term to account for increase of gi after certain ionic strength
Truesdell-Jones (proposed by Huckel in 1925) is similar:

30. Davies Equation
Lacks ion size parameter –only really accurate for monovalent ions
Often used for Ocean waters, working range up to 0.7 M (avg ocean water I)

31. Specific Ion Interaction theory
Ion and electrolyte-specific approach for activity coefficients
Where z is charge, i, m(j) is the molality of major electrolyte ion j (of opposite charge to i). Interaction parameters, (i,j,I) describes interaction of ion and electrolyte ion
Limited data for these interactions and assumes there is no interaction with neutral species

32. Pitzer Model
At ionic strengths above 2-3.5, get +/+, -/- and ternary complexes
Terms above describe binary term, fy describes interaction between same or opposite sign, terms to do this are called binary virial coefficients
Ternary terms and virial coefficients refine this for the activity coefficient

33. Setchenow Equation log gi=KiI
For molecular species (uncharged) such as dissolved gases, weak acids, and organic species
Ki is determined for a number of important molecules, generally they are low, below 0.2  activity coefficients are higher, meaning mi values must decline if a reaction is at equilibrium  “salting out” effect