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

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

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 d+ .. 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…

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 (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

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

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 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)

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 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)2 CaF+ CaOH+ Ca(O-phth) CaH2SiO4 CaPO4- CaB(OH)4+ CaH3SiO4+ CaSO4 CaCH3COO+ CaHCO3+ CaHPO40 CaCO30 Plus more species  gases and minerals!!

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

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

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-]

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! Start at 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

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=1.824928x106r01/2(T)-3/2, B=50.3 (T)-1/2 Where  is the dielectric constant of water and r is the density

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…

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 +/-,

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!

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)

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

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=1.824928x106r01/2(T)-3/2, B=50.3 (T)-1/2 Where  is the dielectric constant of water and r is the density

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

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:

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)

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

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

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