Aqueous Complexes

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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

iiR GnproductsGnG

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-

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

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

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!

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

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

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!

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

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

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

ii 3.0

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)

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

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

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

Aqueous Complexes

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