Geochemical Rate Models
––Online materials––
This online material offers two kinds of support for Geochemical Rate Models. First,
example problems are provided to give instructors ideas for exercises that will enhance a
model-building course. The answers for each problem are given in the problem document
and the accompanying spreadsheet. Second, spreadsheet models that were used to
construct figures are given in order to save the reader the time needed to recreate those
calculations. These spreadsheet models show the governing equations and data sources.
These spreadsheets can be easily modified for use in research projects or to construct
additional teaching problems.
The materials are listed below by chapter. Figures that are used to illustrate an Example are
grouped under that example.
J.D. Rimstidt
Chapter 1 Geochemical models
Problem 1.1. Carbon content of deciduous forests–teaches estimation
Problem 1.2. Hair weight–teaches estimation
Chapter 2 Modeling tools
Example 2.4. Finding the rate of Fe(III) reaction with pyrite
Figure 2.1. Fit of the PRF10 data to polynomials
Example 2.5. Kaolinite to illite transformation under diagenetic conditions
Figure 2.3. Arrhenius plot used to extrapolate rate constants for the kaolinite to illite
transformation reaction
Example 2.6 (Figure 2.5). Rate of reaction of Fe3+ with pyrite in a batch reactor
Figure 2.5. (a) Concentration of Fe3+ versus time from experiment BR5 (b) Numerical
derivatives of the concentration of Fe 3+
Figure 2.2. The relative error in a statistically estimated value decreases as the reciprocal of
the square root of the number of data
Problem 2.1. Feldspar weathering reactions–teaches balancing chemical equations
Problem 2.2. Iron sulfide oxidation reactions–teaches balancing chemical equations
Problem 2.3: Surface area to volume constant for a cube–teaches dimensional analysis
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Problem 2.4. Furlong per fortnight in SI units–teaches units conversion
Problem 2.5. mg/L to molal conversion–teaches units conversion
Problem 2.6. CO2 hydration/dehydration rate constants–teaches calculating mean and
standard error
Problem 2.7. Forward propagation of error for jarosite dissolution rate–teaches error
propagation
Problem 2.8. Covellite dissolution rate equation–teaches linear regression of rate versus
temperature to Arrhenius equation
Problem 2.9. Numerical differentiation of pyrite rate data–teaches numerical differentiation
(with accompanying spreadsheet)
Chapter 3 Rate equations
Example 3.2. Dissolved oxygen consumption by reaction with pyrite
Figure 3.1. Fraction of DO remaining as a function of time for 1 kg of air saturated
solution reacting with 1 m2 of pyrite surface
Example 3.3. Oxidation of hydrogen sulfide
Figure 3.2. Half-life of H2S in air-saturated solution as a function of pH at 25°C
Example 3.4 Hydration of CO2
Figure 3.4. Graph of mB/mBe versus time
Example 3.5. Quartz equilibration times
Figure 3.5. Graph showing the concentration versus time values for 1 m2 of quartz
surface dissolving into 1 kg of water
Figure 3.6. t1/2 for the quartz precipitation–dissolution reaction as a function of
temperature
Figure 3.3. Reaction rate normalized to the far-from equilibrium rate versus chemical
potential driving the reaction
Problem 3.1. Fe(II) reaction with chromate and dissolved oxygen–teaches calculating
reaction half life (with accompanying spreadsheet)
Problem 3.2. CO2 hydration/dehydration reaction–teaches graphing of integrated rate
equations (with accompanying spreadsheet)
Problem 3.3. Quartz reaction rates–teaches graphing rate equations (with two
accompanying spreadsheets)
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Chapter 4 Chemical reactors
Example 4.2. Hot spring flow rate
Figure 4.3. Concentration of tracer in a hot spring as a function of time
Example 4.3. Intitial rate of scorodite dissolution using the polynomial and chord method
Figure 4.4. Concentration of arsenic released by scorodite dissolution versus time fi t to
a second-order polynomial
Figure 4.5. (a) Concentration of arsenic versus time graph with chords connecting the
concentration at zero time and the concentration at the time of each sampling (b) A
graph of the slope of the chords versus time
Example 4.5. Rate of Fe3+ consumption by reaction with pyrite in a PFR
Figure 4.6. Slope of the chords versus time for the Fe3+ reaction with pyrite in a plug
flow reactor experiment
Example 4.6. Fluorapatite dissolution as a function of pH
Figure 4.7. Logarithm of the Ca2+ release flux versus pH for fluorapatite dissolution
Example 4.7. Ca2+ flux from dissolving wollastonite in a non-steady state MFR
Figure 4.8. (a) Ca concentration in the MFR effluent solution as a function of time (b)
Rate of Ca release from dissolving wollastonite as a function of time
Problem 4.1. PDQ contamination–teaches tracer kinetics
Problem 4.2. The big spill–teaches tracer kinetics (with one accompanying spreadsheet)
Problem 4.3. Rate of reaction of Fe(III) with pyrite–teaches initial polynomial and chord
initial rate methods (with two accompanying spreadsheets)
Problem 4.4. Quartz dissolution in acidic HF–teaches multiple linear regression and the loglog method (with one accompanying spreadsheet)
Problem 4.5. Non-steady state wollastonite dissolution flux–teaches extraction of nonsteady state rates from MFR experiments (with two accompanying spreadsheets)
Chapter 5 Molecular kinetics
Example 5.2. Finding the activation energy for the ferric thiosulfate reaction using the
Arrhenius equation
Figure 5.4. Arrhenius plot for the ferric thiosulfate reaction
Example 5.3. Testing the postulated stoichiometry of the ferric thiosulfate transition state
using rate versus ionic strength data
Figure 5.6. The graph of log k versus √ I for the ferric thiosulfate reaction
Example 5.4. The activation volume for Mn(II) catalyzed SO2 oxidation
Figure 5.7. The effect of pressure on the Mn-catalyzed rate of oxidation of dissolved SO2
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Problem 5.1. Activation energy of ants–teaches calculating activation energy from rates
(with two accompanying spreadsheets)
Problem 5.2. CO2 hydration/dehydration rate–teaches extrapolation of rates to higher
temperature and pressure
Problem 5.3. Sulfur dioxide oxidation rate–teaches Brönsted-Bjerrum model (ionic strength
effect on rates)
Chapter 6 Surface kinetics
Example 6.3 (Figure 6.1). A/M ratio for packed beds
Figure 6.1. A/M ratio for packed beds filled with a solution with a density of 1.0 g/cm3
as a function of grain diameter and bed porosity
Example 6.4. Oxidation rate of UO2
Figure 6.3. Graph of the uranium dissolution flux from oxidizing UO2 as a function of
bicarbonate and dissolved oxygen concentrations
Example 6.6. Forsterite dissolution flux determined using the shrinking particle model
Figure 6.4. (a) Fraction of forsterite reacted away as a function of time (b) The reacted
fraction data fit to the shrinking particle model
Figure 6.2. Langmuir sorption isotherms contoured for values of Kb
Figure 6.5. Dimensionless diameter, surface area, and volume of a dissolving spherical
particle as a function of dimensionless time
Figure 6.7. (a) A quartz dissolution flux at 300°C, J, as a function of the affinity, A , driving
the reaction (b) Quartz dissolution flux, J/J+ , as a function of the linearized affinity, 1 – eA/RT
Problem 6.1. Water monolayers on mineral surfaces–teaches unit analysis and estimation of
surface site density
Problem 6.2. Serpentinite formation–teaches layer thickness growth rate model
Problem 6.3. Specific surface as a function of grain diameter–teaches geometric surface area
model (with one accompanying spreadsheet)
Problem 6.4. Pyrite grain lifetime–teaches particle lifetime calculation based on shrinking
particle model
Problem 6.5. Pyrite oxidation rate from shrinking particle model–teaches rate data analysis
using the shrinking particle model and fitting rate constants to the Arrhenius equation
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Chapter 7 Diffusion and advection
Example 7.1. Diffusion coefficient of silica as a function of temperature
Figure 7.3. Diffusion coefficient of H4SiO4 as a function of temperature estimated using
the Stokes–Einstein equation
Example 7.2. Gypsum (CaSO4•2H2O) dissolution in an unstirred solution
Figure 7.4. Concentration of Ca2+ or SO42− as a function of distance and elapsed time
from a gypsum surface
Example 7.4 (Figure 7.7). Gypsum dissolution as a function of fluid velocity
Figure 7.6. Dissolution flux of gypsum into pure water flowing past a flat plate
Example 7.5. Gypsum dissolution flux as a function of temperature
Figure 7.7. Comparison of the CaSO4 fluxes predicted for reaction limited dissolution
Example 7.6. Iron oxyhydroxide coating on pyrite for AMD mitigation
Figure 7.9. (a) Thickness of an iron oxyhydroxide coating on pyrite undergoing
oxidation in a solution buffered to a near-neutral pH (b) Rate of oxygen consumption by
the pyrite oxidation reaction as a coating of iron oxyhydroxide grows in thickness
Example 7.7. Development of a limonite pseudomorph after pyrite
Figure 7.11. Fraction of pyrite converted to goethite as a function of time
Problem 7.1. Laminar versus turbulent flow in a fracture–teaches calculation and
interpretation of Reynolds number (with one accompanying spreadsheet)
Problem 7.2. Diffusion coefficient of CO2(aq) as a function of temperature–teaches
temperature extrapolation of diffusion coefficient using Stokes-Einstein equation (with one
accompanying spreadsheet)
Problem 7.3. Gypsum dissolution in a packed bed–teaches use of Nernst film model for
diffusion limited dissolution (with one accompanying spreadsheet)
Problem 7.4. Pseudomorphic replacement of anglesite by cerussite–teaches finding
diffusion coefficient in a coating using the shrinking core model (with two accompanying
spreadsheets)
Chapter 8 Quasi-kinetics
Example 8.1. Equilibration of dissolved silica with quartz in a fracture
Figure 8.1. Map of DaI versus Pe that shows that the Local Equilibrium Assumption
(LEA) is valid for DaI >~10 and Pe > ~100
Example 8.2. Perturbation of the global water cycle
Figure 8.4. Change in the water content of the Earth’s atmosphere in response to a
sudden 10% decrease in the evaporation rate
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Example 8.3. Rate of growth of quartz veins
Figure 8.5. Growth rate of a layer of quartz on the wall of a fracture with a 1 cm aperture
predicted by the phase transfer model
Example 8.4 (Figure 8.6). Reaction of potassium feldspar with rainwater
Figure 8.6. Reaction path model of solution composition as potassium feldspar is
titrated into rainwater
Example 8.6. Trapping toxic trace elements from solutions using a mixed flow reactor
Figure 8.7. Fraction of trace element removed from solution as a function of the fraction
of major element precipitated in a mixed flow reactor
Example 8.7. Trace elements in calcite cements
Figure 8.8. Fraction of trace element removed from solution as a function of the fraction
of major element precipitated in a batch reactor
Example 8.8. Zn2+ migration in lateritic soil
Figure 8.9. (a) Breakthrough curve for three Br− tracer experiments showing that the
inert tracer breakthrough occurs at one pore volume (b) Breakthrough curve for the
Zn2+ experiment
Problem 8.1. Short term carbon cycle–teaches box models (with one accompanying
spreadsheet)
Problem 8.2. Radium capture from solution–teaches distribution coefficient model for batch
reactor (with one accompanying spreadsheet)
Problem 8.3. Effect of geothermal gradient on quartz vein formation (with one
accompanying spreadsheet)
Problem 8.4. Chrysotile dissolution reaction path–teaches reaction path modeling (with one
accompanying spreadsheet)
Problem 8.5. Weathering of the Sierra Nevada batholith–teach mass balance modeling (with
one accompanying spreadsheet)
Chapter 9 Accretion and transformation kinetics
Example 9.1. Silica scaling
Figure 9.1. Silica monomer concentration as a function of time due to monomer
incorporation into polymers
Example 9.2. Barite nucleation rate
Figure 9.3. (a) Free energy of formation of BaSO4 clusters as a function of the number of
BaSO4 molecules in the cluster (b) Number of BaSO4 molecules in the critical cluster as a
function of log S (c) Nucleation rate of BaSO4 as a function of log S
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Example 9.3. Aggregation time for silica clusters
Figure 9.5. Logarithm of experimentally determined stability ratios for colloidal silica as
a function of the logarithm of the ionic strength
Figure 9.6. Logarithm of the time needed for one half of the silica clusters to aggregate
Example 9.4. Schwertmannite to goethite transformation
Figure 9.8. (a) TTT diagram contoured in the extent of transformation of
schwertmannite to goethite (b) When a reciprocal temperature axis is used the contours
are straight lines
Figure 9.2. (a) Correlation between the interfacial free energy of ionic solids and the
logarithm of their molar solubility (b) Solubility product of gypsum and barite clusters
relative to the solubility product of the bulk solid as a function of cluster diameter
Figure 9.4. Time needed for half the clusters to aggregate (a) as a function of cluster
concentration and (b) as a function of temperature. (c) Arrhenius plot showing log ks as a
function of 1/T for diffusion limited cluster aggregation
Figure 9.7. (a) Examples of the effect of changing k and n on the extent of the transformation
versus time pattern predicted by the classical Avrami equation (b) Examples of the effect of
changing k and n on the extent of the transformation versus time pattern predicted by the
modified Avrami equation
Problem 9.1. Ferrihydrite to goethite TTT diagram–teaches use of modified Avrami
equation and construction of TTT diagram (with one accompanying spreadsheet)
Problem 9.2. Gypsum nucleation rate–teaches calculation of nucleation rate as a function of
degree of saturation (with one accompanying spreadsheet)
Problem 9.3. Aragonite to calcite transformation–teaches fitting transformation rate data to
the Avrami equation and the construction of a TTT diagram from the fitted equation (with
one accompanying spreadsheet)
Chapter 10 Pattern formation
No examples, figures, or problems
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