Aqueous Complexes
• Why do we care??1. Complexation of an ion also occuring in a
mineral increases solubility
2. Some elements occur as complexes more commonly than as free ions
3. Adsorption of elements greatly determined by the complex it resides in
4. Toxicity/ bioavailability of elements depends on the complexation
Defining Complexes
• Use equilibrium expressions:
G0R = -RT ln Keq
• cC + lHL CL + lH+
• Where B is just like Keq!
)reactants()( 000i
iii
iiR GnproductsGnG
lc
nc
i HLC
HCL
][][
][][
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
Outer Sphere Complexes• 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-, SO4
2-, and CO32-
H H
O
+ +
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
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 (UO2
2+), 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
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)
EN determines bonding – covalent as EN approaches 0 (more inner sphere), as EN > 1.7, more ionic and outer-sphere
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
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
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
Speciation• 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)2 CaF+ CaOH+
Ca(O-phth) CaH2SiO4 CaPO4-
CaB(OH)4+ CaH3SiO4
+ CaSO4
CaCH3COO+ CaHCO3+ CaHPO4
0
CaCO30
• Plus more species gases and minerals!!
Mass Action & Mass Balance
• mCa2+=mCa2++MCaCl+ + mCaCl20 + CaCL3- +
CaHCO3+ + CaCO3
0 + CaF+ + CaSO40 +
CaHSO4+ + CaOH+ +…
• Final equation to solve the problem sees the mass action for each complex substituted into the mass balance equation
lc
nc
i HLC
HCL
][][
][][
nxLmCamCa 22
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+, SO4
2- 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
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 = 0 = 1 / (1+ K[Cl-] + K[OH-])
• [PbCl+]=K[Pb2+][Cl-]– [Pb2+] / PbT = 0 [Pb2+] = 0PbT
– [PbCl+]=K 0PbT [Cl-]
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
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!
R(x)
BUT – what if there is more than one root????
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)
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
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
)(2
1 2ii zmI
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=1.824928x10601/2(T)-3/2, B=50.3 (T)-1/2
Where is the dielectric constant of water and is the density
IBa
IAz
i
ii
1log
2
IAzii2log
Iteration and activity example• Speciate a simple mix of Fe3+ and Cl-1. Starting analysis of Fe3+ and Cl-2. Calculate I3. Calculate i for each ion (Fe3+, Cl-, FeCl++)4. Calculate activity for each ion5. Recalculate I6. Recalculate i for each ion (Fe3+, Cl-, FeCl++)7. Recalculate activity for each ion8. Until the residual for these reduces…
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+ + FeCl2
0 + 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 +/-,
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!
aacc
aacc
zmzm
zmzmCBE
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.)
±KCl = [(K)(Cl)]1/2
Ksp= ±KCl2(mK+)(mCl-)
• MacInnes Convention K = Cl= ±KCl
– Measure other salts in KCl electrolyte and substitute ±KCl in for one ion to measure the other ion w.r.t. ±KCl and ±salt
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
)(2
1 2ii zmI
Mean Ion Activity Coefficients versus Ionic Strength
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=1.824928x10601/2(T)-3/2, B=50.3 (T)-1/2
Where is the dielectric constant of water and is the density
IBa
IAz
i
ii
1log
2
IAzii2log
Higher Ionic Strengths• Activity coefficients decrease to minimal
values around 1 - 10 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
• Adds a correction term to account for increase of i after certain ionic strength
• Truesdell-Jones (proposed by Huckel in 1925) is similar:
Extended Debye-Hückel
IIBa
IAzAz
i
ii 3.0
1log
22
bIIBa
IAz
i
ii
1log
2
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)
I
I
IAzi 3.0
1log 2
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
k
i jmIjiDz )(),,()log( 2
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
ijk
kjijki
jijii mmEmIDfyz ...)(ln 2
Setchenow Equationlog i=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
