REACTION KINETICS
Pierre Glynn, March 2003
(w/ notes from Alex Blum)
General Concepts
 Transport vs. Reaction Control
 Elementary vs. Overall Reactions
• Detailed Balancing
• Microscopic Reversability
 Temperature Dependance
 Transition State Theory
 Michaelis-Menten Kinetics
 Surface Area
Silicate mineral dissolution kinetics & weathering
Transport vs. Reaction Control
a) Transport control
b) Surface reaction control
c) Mixed Transport and surface-reaction control
Transport limitations:
• diffusion in solution
• solid-state diffusion
Reaction limitations:
• surface reaction control
• surface characteristics
crystal defects
impurities
crystal morphology
Elementary vs. overall reactions
Reactions are the result of molecular collisions; & almost
invariably depend on the collision of no more than 2 molecular
species at a time.
Overall reactions, such as:
2 KAlSi3O8 + 2H + + 9H 2O 
Al 2Si 2O5 (OH) 4 + 4H 4SiO4 + 2K +
do not reveal the sequential, and possibly parallel, sets of
molecular interactions, i.e. elementary reactions, that are
actually involved.
Example of fast reactions (only 1 elementary step):
Ag + + Cl-  AgCl(s)
CO2,aq + OH -  HCO-3
Example of a two step reaction:
O3  O2 + O
O + O3  2 O2
_____________
Overall reaction:
2 O3  3 O2
Determining a rate law requires knowledge of the rate-limiting
elementary reaction (usually only one). Allows accounting for the
stoichiometry and the reaction order. If this is not possible (eg.
for an overall reaction), rate laws are determined experimentally.
Principle of Detailed Balancing:
The net rate of a reaction is the difference between the forward
and the backward microscopic rates (eg. microscopic dissolution
vs. precipitation).
Principle of Microscopic Reversability:
The forward and the backward reactions have the same
molecular mechanism.
Activation Energies and Temperature Dependence
Reaction rates are often exponentially dependent on
temperature, and are also highly depend on the activation
energy EA required for a molecular reaction.
 EA 
Rate  A exp 

RT


or log Rate 
1
T
The preexponential factor A may depend on pH, solution
chemistry, surface characteristics, and many other factors
including temperature.
Activation Energy (EA)
1) Reaction rates are exponentially dependent on EA
2) EA depends on the direction of a reaction
3) Catalysis lowers the EA required for a reaction (note
activated complex)
Exothermic reaction
Endothermic reaction
EA
from http://www.ucdsb.on.ca/tiss/stretton/chem2/rate03
Transition State Theory
Applies statistical mechanics to individual elementary reactions.
Meaningless for overall reactions.
Focuses on the activated complex, the molecular configuration
present at the top of the energy barrier (actually a saddle point)
between reactants and products in an elementary reaction.
Assumes this complex is a true chemical species and assumes
that the initial reactants are always at equilibrium with the
complex.
Transition State Theory
Predicts that the rate is proportional to the number of
activated complexes and to their rate of decomposition.
Applies near equilibrium.
TST is somewhat similar to the idea that the rate is
proportional to the deviation from equilibrium (or the degree
of supersaturation or undersaturation).

Q
 nG  
Rate  kdiss 1  exp 

G

RT
ln

RT
K eq



 G 0 
kdiss
 K eq  exp 

k precip
RT


(From Burch et al., 1993)
G = -8 kcal/mol => log Q/K = -3.7
=> Sat = 0.02%
Surface speciation kinetic model
1) Fast reversible adsorption reaction to form a
surface species
2) Irreversible dissolution reaction at that surface
species
Surface complexation
theory allows guessing the
form of the “activated
complex”.
In this case (dissolution of
Amelia albite; Blum, 1994,
GCA), the dissolution rate
is proportional to the
degree of protonation, or
deprotonation of the
surface
-4.0
-6.5
Michaelis-Menten kinetics
1) Based on enzyme kinetics
k1
k2
E + S  ES  EP  E + P
k1
2) Similar to TST theory; based on the idea of an
“activated complex”, or an “enzyme-substrate”
(ES) compound, whose concentration controls
the rate of reaction
3) Assumes that the concentration of ES is at
steady-state (d(ES)/dt = 0)
1) Measurements needed are:
•
•
•
the total amount of enzyme, ET = E + ES
the concentration of substrate, S
the measured steady state velocity: V = k2 (ES)
2) The maximal velocity is measured, Vmax = k2 (ET), using the
highest substrate concentration
3) The Michaelis-Menten constant, KM = (k-1 + k2)/k1, is simply
the substrate concentration that gives a reaction velocity
half of Vmax. Also, for a slow reaction, k2 << k-1, KM = k-1/k1
= Keq
4) The Michaelis-Menten equation is:
V  Vmax
(S )
KM  (S )
Surface Area
• Critical to rate calculations and predictions
• Geometric area estimation (often requires
averaging or stochastic theory)
• BET measurements, usually w/ N2 (4Å
compared to 3Å for H2O)
• Surface roughness (BET/Geometric):
– SR = 5 - 12 for fresh ground silicate
– SR = 300 - 2000 for deeply weathered
natural silicates
Feldspar dissolution kinetics
and composition
%An
Silicate dissolution kinetics
and pH
Albite
pH
K-feldspar
pH
Silicate weathering reactions
Conclusions: silicate weathering
1) Natural weathering rates in soils are 102 to 105
slower than exp. rates
2) Weathering rates in aquifers depend on (water
chem)/(res. time)/(extent of react)??
3) Accumulation of solutes retards dissolution rates??
4) Discrepancies betw. natural and exp. rates are
consistent w/ models of solute accumulation in
pores
Silicate weathering & the C cycle
Exercise K1
1) Enter the 2 waters from the Norman, OK, landfill into a
PHREEQC input file. Assume a temperature of 16.6 C for
both. Units are mg/L. Do not enter DOC. Which water is
contaminated? How did you specify redox conditions?
Description
pH
MLSNPD -6 7.01
MLS38-6
6.78
Ca
Mg
166 52.6
514 232
Na
91.8
606
K
Alk.
as HCO3
2.5
626
14.4
2642
Cl
S(6)
181 113.6
1027
0.1
Br
Si
Fe(2)
Mn
0.79 18.60 0.13 1.94
7.54 35.90 19.30 0.90
Sr
Ba
1.06 0.14
9.87 7.12
N(5) N(-3) DOC
as NO3as NH4
0.05
2.1
2.9
3.48
15.0 159
2) (Ulrich et al., 2003) used field and lab techniques to identify
the biogeochemical factors affecting the rate of sulfate
reduction in the leachate contaminated water. They obtained a
Michaelis-Menten type relationship with KM & Vmax values of 80
and 0.83 mM SO4/day, resp.
Norman landfill
Norman landfill
Exercise K1 (cont)
3) Enter the Rates keyword in PHREEQC to describe the
Michaelis-Menten type rate law for SO4 reduction using the
constants determined by Ulrich et al. (2003). Notice that in the
screen shot, the KM & Vmax constants have been converted into
units of moles SO4 per second. Call the rate law
“Sulfate_reduction”.
4) Use the SELECTED_OUTPUT keyword to have PHREEQC
output total concentrations on S(6), S(-2), C(4), and info on the
kinetics of “Sulfate_reduction”.
Michaelis-Menten kinetics for sulfate reduction
(constants from Ulrich et al., ES&T, 2003)
Exercise K1 (cont)
5) Write the reaction for SO4 reduction with CH2O oxidation. How
many moles of CH2O are used per ml of SO4 reduced? Use
the KINETICS keyword to enter the CH2O formula, it’s stoich.
coeff., and the total amount added (hint: use the DOC value).
6) Specify output steps at 1 second, 0.1, 1, 10, 20, 40, 60, 80 and
100 years.
7) Graph the decay of SO4, and the increase in S(-2) and C(4) w/
time.
8) How long does it take to oxidize the dissolved organic carbon?
9) What are the final concentrations of SO4, S(-2), and TDIC?
Exercise K1 (cont)
10) Examine the SO4 concentration in the initial water and
compare it to the Michaelis-Menten constant. Which one is
significantly smaller? What does that say about the order of
the rate law applicable at the beginning of the SO4 reduction
process?
11) Compare the Michaelis Menten constant with the SO4
concentration obtained after 100 years of reaction. What is the
order of the rate law then?
Exercise K1 (part 2)
10) Ulrich et al. claim that barite dissolution provides the source of
most of the S(6) in the Norman waters. In a new simulation,
use the background water from near the Norman landfill.
Maintain equilibrium with barite. Use the rate law and the
kinetics keywords previously defined in part 1, but provide a
input of organic carbon up to 10x greater.
11)Specify output steps at 1 second, 0.1, 1, 10, 20, 40, 60, 80 and
100 years.
12)Graph the decay of SO4, and the increase in S(-2) and C(4),
and Ba w/ time.
13)How do the values compare with those observed in the
leachate-contaminated water?
Exercise K2
1) Enter the 2 Sierra Spring waters from Garrels & Christ into a
PHREEQC input file. Assume a temperature of 25 C for both.
Units are mmol/L.
Description
Ephemeral Spring
Perennial Spring
pH
6.2
6.8
Ca
0.078
0.26
Mg
0.029
0.071
Na
0.134
0.259
K
0.028
0.04
Alk.
0.328
0.895
Cl
0.014
0.03
S(6)
0.01
0.025
Si
0.273
0.41
2) Using the most dilute water, add the RATE laws and
suggested KINETICS blocks that are present in the
phreeqc.dat file, to simulate the kinetic reaction of the water
with albite and K-spar.
3) Ignoring other possible reactions, how long would it take to
obtain a water with Si, Na and K concentrations similar to
those in the “Perennial Spring”?
Exercise K2 (part 2)
4) Examine the rate and kinetic blocks for K-spar and albite
dissolution. Write out, and explain the rate laws and equations
used.
Exercise K3
1) Examine the rate and kinetic blocks for Pyrite dissolution
present in the phreeqc.dat file. Write out, and explain the rate
laws and equations used.
2) Simulate the reaction of pyrite (as stipulated in the phreeqc.dat
example) with a O2-equilibrated recharge water such as the
OK recharge water (used in sorption exercise S3). How long
does it take to react away the oxygen dissolved in one unit
volume of water? Given the initial concentration of pyrite
specified, how many volumes of recharge water would it take
to consume all the pyrite?
Exercise K4
1) Examine the rate and kinetic blocks for Organic carbon
reaction, using Monod kinetics, present in the phreeqc.dat file.
Write out, and explain the rate laws and equations used.
2) Simulate the kinetic reaction specified using a water
equilibrated with atmospheric O2 and and a log pCO2 of –1.5.
What is the initial organic carbon concentration specified?
How long does it take for the reactants to disappear?
3) Repeat the problem using the background water from the
Norman landfill
4) Repeat the problem using the contaminated water from the
Norman landfill