Unit 08a : Advanced Hydrogeology

Aqueous Geochemistry

Aqueous Systems

• In addition to water, mass exists in the subsurface as:– Separate gas phases (eg soil CO2)– Separate non-aqueous liquid phases (eg

crude oil)– Separate solid phases (eg minerals

forming the pm)– Mass dissolved in water (solutes eg Na+,

Cl-)

Chemical System in Groundwater

• Ions, molecules and solid particles in water are not only transported.

• Reactions can occur that redistribute mass among various ion species or between the solid, liquid and gas phases.

• The chemical system in groundwater comprises a gas phase, an aqueous phase and a (large) number of solid phases

Solutions

• A solution is a homogeneous mixture where all particles exist as individual molecules or ions. This is the definition of a solution.

• There are homogeneous mixtures where the particle size is much larger than individual molecules and the particle size is so small that the mixture never settles out.

• Terms such as colloid, sol, and gel are used to identify these mixtures.

Concentration Scales

• Mass per unit volume (g/L, mg/L, g/L) is the most commonly used scale for concentration

• Mass per unit mass (ppm, ppb, mg/kg, g/kg) is also widely used

• For dilute solutions, the numbers are the same but in general:mg/kg = mg/L / solution density (kg/L)

Molarity

• Molar concentration (M) defines the number of moles of a species per litre of solution (mol.L-1)

• One mole is the formula weight of a substance expressed in grams.

Molarity Example

• Na2SO4 has a formula weight of 142 g

• A one litre solution containing 14.2 g of Na2SO4 has a molarity of 0.1 M (mol.L-1)

• Na2SO4 dissociates in water:

Na2SO4 = 2Na+ + SO42-

• The molar concentrations of Na+ and SO4

2- are 0.2 M and 0.1 M respectively

Seawater Molarity• Seawater contains roughly 31,000 ppm of NaCl and

has a density of 1028 kg.m-3. What is the molarity of sodium chloride in sea water?

• M = (mc/FW) *where mc is mass concentration in g/kg; is in kg/m3; and FW is in g.

• Formula weight of NaCl is 58.45• 31 g is about 0.530 moles• Seawater molarity = 0.530 * 1.028 = 0.545 M (mol.L-1)

Molality

• Molality (m) defines the number of moles of solute in a kilogram of solvent (mol.kg-1)

• For dilute aqueous solutions at temperatures from around 0 to 40oC, molarity and molality are similar because one litre of water has a mass of approximately one kilogram.

Molality Example

• Na2SO4 has a formula weight of 142 g• One kilogram of solution containing 0.0142 kg

of Na2SO4 contains 0.9858 kg of water.• The solution has a molality of 0.101 m

(mol.kg-1)• Na2SO4

dissociates in water:

Na2SO4 = 2Na+ + SO42-

• The molal concentrations of Na+ and SO42-

are 0.202 m and 0.101 m respectively

Seawater Molality• Seawater contains roughly 3.1% of NaCl. What

is the molality of sodium chloride in sea water?

m = (mc/FW)/(1 – TDS) where mc is mass concentration in g/kg; TDS is in kg/kg and FW is in g.

• Formula weight of NaCl is 58.45• 31 g is about 0.530 moles• Average seawater TDS is 35,500 mg/kg (ppm)• m = (31/58.45)/ (1- 0.0355) = 0.550 mol.kg-1

Molar and Molal

• The molarity definition is based on the volume of the solution. This makes molarity a temperature-dependent definition.

• The molality definition does not have a volume in it and so is independent of any temperature changes.

• The difference is IMPORTANT for concentrated solutions such as brines.

Brine Example

• Saturated brine has a TDS of about 319 g/L• Saturated brine has an average density of

1.203 at 15oC• The concentration of saturated brine is

therefore 265 g/kg or 319 g/L• The molality m = (265/58.45)/(1-0.319)) is

about 6.7 m (mol.kg-1)• The molarity M = (265/58.45)*1.203 is about

5.5 M (mol.L-1)

Equivalents

• Concentrations can be expressed in equivalent units to incorporate ionic charge

meq/L = mg/L / (FW / charge)• Expressed in equivalent units, the

number of cations and anions in dilute aqueous solutions should approximately balance

Partial Pressures

• Concentrations of gases are expressed as partial pressures.

• The partial pressure of a gas in a mixture is the pressure that would be exerted by the gas if it occupied the volume alone.

• Atmospheric CO2 has a partial pressure of 10-3.5atm or about 32 Pa.

Mole Fractions

• In solutions, the fundamental concentration unit in is the mole fraction Xi; in which for j components, the ith mole fraction is

•Xi = ni/(n1 + n2 + ...nj),

•where the number of moles n of a component is equal to the mass of the component divided by its molecular weight.

Mole Fractions of Unity

• In an aqueous solution, the mole fraction of water, the solvent, is always near unity.

• In solids that are nearly pure phases, e.g., limestone, the mole fraction of the dominant component, e.g., calcite, will be near unity.

• In general, only the solutes in a liquid solution and gas components in a gas phase will have mole fractions that are significantly different from unity.

Structure of Water

• Covalent bonds between H and O

• 105o angle H-O-H

• Water molecule is polar

• Hydrogen bonds join molecules– tetrahedral structure

• Polar molecules bind to charged species to “hydrate” ions in solution

105o

-

+

+

Chemical Equilibrium

• The state of chemical equilibrium for a closed system is that of maximum thermodynamic stability

• No chemical energy is available to redistribute mass between reactants and products

• Away from equilibrium, chemical energy drives the system towards equilibrium through reactions

Kinetic Concepts

• Compositions of solutions in equilibrium with solid phase minerals and gases are readily calculated.

• Equilibrium calculations provide no information about either the time to reach equilibrium or the reaction pathway.

• Kinetic concepts introduce rates and reaction paths into the analysis of aqueous solutions.

Reaction Rates

After Langmuir and Mahoney, 1984

Mineral Recrystallization

Solute-Solute

Hydrolysis of multivalent ions (polymerization)

Adsorption-Desorption

Mineral-Water Equilibria

Secs Mins Hrs Days Months Years Centuries My

Gas-Water

Solute-Water

Reaction Rate Half-Life

Relative Reaction Rates

• An equilibrium reaction is “fast” if it takes place at a significantly greater rate than the transport processes that redistribute mass.

• An equilibrium reaction is “slow” if it takes place at a significantly smaller rate than the transport processes that redistribute mass.

• “Slow” reactions in groundwater require a kinetic description because the flow system can remove products and reactants before reactions can proceed to equilibrium.

Partial Equilibrium• Reaction rates for most important reactions are

relatively fast. Redox reactions are often relatively slow because they are mediated by micro-organisms. Radioactive decay reactions and isotopic fractionation are extremely variable.

• This explains the success of equilibrium methods in modelling many aspects of groundwater chemistry.

• Groundwater is best thought of as a partial equilibrium system with only a few reactions requiring a kinetic approach.

Equilibrium Model

• Consider a reaction where reactants A and B react to produce products C and D with a,b,c and d being the respective number of moles involved.

aA + bB = cC + dD• For dilute solutions the law of mass action describes

the equilibrium mass distribution

K = (C)c(D)d

(A)a(B)b

where K is the equilibrium constant and (A),(B),(C), and (D) are the molal (or molar) concentrations

Activity

• In non-dilute solutions, ions interact electrostatically with each other. These interactions are modelled by using activity coefficients () to adjust molal (or molar) concentrations to effective concentrations

[A] = a(A)

• Activities are usually smaller for multivalent ions than for those with a single charge

• The law of mass action can now be written:

K = c(C)c d(D)d = [C]c[D]d

a(A)a b(B)b [A]a[B]b

Debye-Hückel Equation

• The simplest model to predict ion ion activity coefficients is the Debye-Hückel equation:

log i = - Azi2(I)0.5

where A is a constant, zi is the ion charge, and I is the ionic strength of the solution given by:

I = 0.5 Mizi2

where (Mi) is the molar concentration of the ith species

• The equation is valid and useful for dilute solutions where I < 0.005 M (TDS < 250 mg/L)

Extended Debye-Hückel Equation

• The extended Debye-Hückel equation is used to increase the solution strength for which estimates of can be made:

log i = - Azi2(I)0.5

1 + Bai(I)0.5

where B is a further constant, ai is the ionic radius

• This equation extends the estimates to solutions where I < 0.1 M (or TDS of about 5000 mg/L)

More Activity Coefficient Models

• The Davis equation further extends the ionic strength range to about 1 M (roughly 50,000 mg/L) using empirical curve fitting techniques

• The Pitzer equation is a much more sophisticated ion interaction model that has been used in very high strength solutions up to 20 M

Monovalent Ions

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.001 0.01 0.1 1 10

Ionic Strength

Act

ivity

Coe

ffici

ent

Debye-Huckel

Extended

Davis

Pitzer

Divalent Ions

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.001 0.01 0.1 1 10

Ionic Strength

Act

ivit

y C

oef

fici

ent

Debye-Huckel

Extended

Davis

Pitzer

Activity and Ionic Charge

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.001 0.01 0.1 1 10

Ionic Strength

Act

ivity

Coe

ffici

ent

Debye-Huckel

Extended

Davis

Pitzer

Monovalent

Divalent

Non-Equilibrium

• Viewing groundwater as a partial equilibrium system implies that some reactions may not be equilibrated.

• Dissolution-precipitation reactions are certainly in the non-equilibrium category.

• Departures from equilibrium can be detected by observing the ion activity product (IAP) relative to the equilibrium constant (K) where

IAP = [C]c[D]d = products [A]a[B]b reactants

Dissolution-Precipitation

aA + bB = cC + dD• If IAP<K (IAP/K<1) then the reaction is

proceeding from left to right.• If IAP>K (IAP/K>1) then the reaction is

proceeding from right to left.• If the reaction is one of mineral dissolution

and precipitation– IAP/K<1 the system in undersaturated and is

moving towards saturation by dissolution– IAP/K>1 the system is supersaturated and is

moving towards saturation by precipitation

Saturation Index

• Saturation index is defined as:

SI = log(IAP/K)

• When a mineral is in equilibrium with the aqueous solution SI = 0

• For undersaturation, SI < 0

• For supersaturation, SI > 0

Calcite• The equilibrium constant for the calcite dissolution reaction is K =

4.90 x 10-9 log(K) = -8.31• Given the activity coefficients of 0.57 for Ca2+ and 0.56 for CO3

2- and molar concentrations of 3.74 x 10-4 and 5.50 x 10-5 respectively, calculate IAP/K.

• Reaction: CaCO3 = Ca2+ + CO32-

IAP = [Ca2+][CO32-] = 0.57x3.37x10-4x0.56x5.50x10-5

[CaCO3] 1.0 = 6.56 x 10-9 and log(IAP) = -8.18

{IAP/K}calcite = 6.56/4.90 = 1.34

log{IAP/K}calcite = 8.31 - 8.18 = 0.13• The solution is slightly oversaturated wrt calcite.

Dolomite• The equilibrium constant for the calcite dissolution reaction is K =

2.70 x 10-17 and log(K) = -16.57• Given activity coefficients of 0.57, 0.59 and 0.56 for Ca2+, Mg2+ and

CO32- and molar concentrations of 3.74 x 10-4, 8.11 x 10-5 and 5.50 x

10-5 respectively, calculate IAP/K.• Reaction: CaMg(CO3)2 = Ca2+ + Mg2+ + 2 CO3

2-

• Assume the effective concentration of the solid dolomite phase is unity

log[Ca2+] = -3.67 log[Mg2+] = -4.32 log[CO32-] = -4.51

log(IAP)=log([Ca2+][Mg2+][CO32-]2)= -3.67-4.32-9.02= -16.31

log{IAP/K}dolomite = 16.57 – 17.01 = -0.44• The solution is undersaturated wrt dolomite.

Kinetic Reactions• Reactions that are “slow” by comparison with

groundwater transport rates require a kinetic modelk1

aA + bB = cC + dDk2

where k1 and k2 are the rate constants for the forward (L to R) and reverse (R to L) reactions

• Each constituent has a reaction rate:

rA = dA/dt; rB = dB/dt; rc = dC/dt; rD = dD/dt;• Stoichiometry requires that:

-rA/a = -rB/b = rC/c = rD/d

Rate Laws

• Each consituent has a rate law of the form:

rA = -k1(A)n1(B)n2 + k2(C)m1(D)m2

where n1, n2, m1 and m2 are empirical or stoichiometric constants

• If the original reaction is a single step (elementary) reaction then n1=a, n2=b, m1=c and m2=d

Irreversible Decay

14C = 14N + e

d(14C)/dt = -k1(14C) + k2(14N)(e)

• Here there is only a forward reaction and k2 for the reverse reaction is effectively zero

d(14C)/dt = -k1(14C)

• k1 is the decay constant for radiocarbon

Elementary Reactions

Fe3+ + SO42- = FeSO4

+

d(Fe3+)/dt = -k1(Fe3+)(SO42-) + k2(FeSO4

+)

• The reaction rate depends not only on how fast ferric iron and sulphate are being consumed in the forward reaction but also on the rate of dissociation of the FeSO4

+

ion.