Aquatic Chemical Kinetics
• Look at 3 levels of chemical change:
– Phenomenological or observational
• Measurement of reaction rates and interpretation of
data in terms of rate laws based on mass action
– Mechanistic
• Elucidation of reaction mechanisms = the
‘elementary’ steps describing parts of a reaction
sequence (or pathway)
– Statistical Mechanical
• Concerned with the details of mechanisms 
energetics of molecular approach, transition states,
and bond breaking/formation
How can you tell if any system is at
equilibrium?
• Beware of steady state (non-equilibrium)
conditions where proportions of reactants
are constant, but due to flux in-out and
relative rates of reaction!
Thermodynamic or kinetic
descriptions?
• When a reaction is reversible and the rate is fast
compared to residence time  thermodynamic
description
• When a reaction is irreversible, OR it’s reaction rate
is slower than the residence time  kinetic
description
• Partial Equilibrium  system where some reactions
fast, others are slow – sound familiar?
Time Scales
Reactions and Kinetics
• Elementary reactions are those that
represent the EXACT reaction, there are NO
steps between product and reactant in
between what is represented
• Overall Reactions represent the beginning
and final product, but do NOT include one or
more steps in between.
FeS2 + 7/2 O2 + H2O  Fe2+ + 2 SO42- + 2 H+
2 NaAlSi3O8 + 9 H2O + 2 H+  Al2Si2O5(OH)4 +
2 Na+ + 4 H4SiO4
Equilibrium and reversible kinetics
• For any reaction AT equilibrium, Keq is related to the
forward (k+) and reverse (k-) reaction rates
k
K eq 
k
• Example:
Fe2+ + H+ + 0.25 O2 = Fe3+ + 0.5 H2O
Log K=8.48, if k+=100 mol/min, then k-=3x10-7 mol/min
Extent of Reaction
• In it’s most general representation, we can
discuss a reaction rate as a function of the
extent of reaction:
Rate = dξ/Vdt
where ξ (small ‘chi’) is the extent of rxn, V is the
volume of the system and t is time
Normalized to concentration and stoichiometry:
rate = dni/viVdt = d[Ci]/vidt
where n is # moles, v is stoichiometric coefficient,
and C is molar concentration of species i
Rate Law
• For any reaction: X  Y + Z
• We can write the general rate law:
d(X )
n

 k(X )
dt
Rate = change in concentration
of X with time, t
Order of reaction
Rate Constant
Concentration of X
Reaction Order
• ONLY for elementary reactions is reaction
order tied to the reaction
• The molecularity of an elementary reaction
is determined by the number of reacting
species: mostly uni- or bi-molecular rxns
• Overall reactions need not have integral
reaction orders – fractional components
are common!
General Rate Laws
Reaction
order
0
1
2
Rate Law
d [ A]
 k
dt
Integrated
Rate Law Units for k
A=A0-kt
mol/cm3 s
ln A=lnA0-kt s-1
d [ A]
  kA
dt
3/mol s
cm
1
1
d [ A]
2

 kt
 kA
A A0
dt
• First step in
evaluating rate
data is to
graphically
interpret the
order of rxn
• Zeroth order:
rate does not change
with lower concentration
• First, second
orders:
Rate changes as a
function of
concentration
Zero Order
d [ A]
 k
dt
• Rate independent of the reactant or
product concentrations
• Dissolution of quartz is an example:
SiO2(qtz) + 2 H2O  H4SiO4(aq)
log k- (s-1) = 0.707 – 2598/T
First Order
d [ A]
  kA
dt
• Rate is dependent on concentration of a
reactant or product
– Pyrite oxidation, sulfate reduction are examples
First Order
d [ A]
  kA
dt
[ A]t
 e (  kt )
[ A]0
log[ A]t  log[ A]0  log( e  kt )
ln
[ A]t
  kt
[ A]0
log[ A]t  0.434kt  log[ A]0
Find rate constant from log[A]t vs t plot 
Slope=-0.434k
k = -(1/0.434)(slope) = -2.3(slope)
k is in units of: time-1
Pseudo- 1nd Order
• For a bimolecular reaction: A+B  products
dx
 k 2 [ A][ B]  k 2 ([ A]0  x)([ B]0  x)
dt
If [B]0 is held constant, the equation above reduces to:
dx
 k 2 [ A][ B]  k 2 ([ A]0  x)([ B]0  0)
dt
SO – as A changes B does not, reducing to a
constant in the reaction: plots as a first-order
reaction – USE this in lab to determine order of
reactions and rate constants of different reactions
Second Order
• Rate is dependent on two reactants or
products (bimolecular for elementary rxn):
• Fe2+ oxidation is an example:
Fe2+ + ¼ O2 + H+  Fe3+ + ½ H2O
2
d [ Fe ]
2
 k[ Fe ]PO2
dt
2nd Order
• For a bimolecular reaction: A+B  products
dx
 k 2 [ A][ B]  k 2 ([ A]0  x)([ B]0  x)
dt
[ B]0 ([ A]0  x)
[ B]0 [ A]
1
1
ln

ln
 k 2t
[ A]0  [ B]0 [ A]0 ([ B]0  x) [ A]0  [ B]0 [ A]0 [ B]
[ A]t
[ B]0
log
 0.43k 2 ([ A]0  [ B]0 )t  log
[ B]t
[ A]0
[A]0 and [B]0 are constant, so a plot of log [A]/[B] vs t
yields a straight line where slope = k2 (when A=B) or
= k2([A]0-[B]0)/2.3 (when A≠B)
Half-life
• Time required for one-half of the initial reactant to
react
t1 2
[ A]0
ln 2 1

 ln
k
k 0.5[ A]0
• Half-lives tougher to quantify if A≠B for 2nd order
reaction kinetics – but if A=B:
t1 2
1
 [ A]0
k2
If one reactant (B) is kept constant (pseudo-1st order rxns):
t1 2
ln 2

[ A]0
k2
3rd order Kinetics
• Ternary molecular reactions are more rare,
but catalytic reactions do need a 3rd
component…
dx
 k3 [ A][ B][C ]  k 2 ([ A]0  x)([ B]0  x)([C ]0  x)
dt
Reversible Reactions
• Preceeding only really accurate if
equilibrium is far off i.e, there is little
reaction in the opposite direction
– For A = B
– Rate forward can be: dA/dt = kf[A]
– Rate reverse can be: dB/dt = kr[B]
– At equilibrium: Rate forward = Rate reverse
kf[A] = kr[B]
Keq = [A] / [B] = kf / kr
Reversible Kinetics
• Kinetics of reversible reactions requires a
back-reaction term:
d [ A]
  k f [ A]  k r [ P ]
dt
• With reaction progress
dx
 k f ([ A]0  x)([ P]0  x)
dt
• In summary there is a definite role that
approach to equilibrium plays on overall
forward reaction kinetics!
T effect of reaction rates
• Arrhenius Expression:
k=AFexp(-EA/RT)
Where rate k is dependent on Temperature, the
‘A’ factor (independent of T) and the Activation
Energy, EA  differentating:
d log k
EA

2
dT
2.303RT
So that a plot of log K vs. 1/T is a straight line
whose slope = -EA/2.303R
Activation Energy
Physical adsorption
‘typical’ range of
EA (kcal/mol)
2–6
Aqueous diffusion
<5
‘Biotic’ reactions
5 - 20
Reaction
Mineral dissolution/precipitation 8- 36
Dissolution controlled by
surface reaction
Isotopic exchange in solution
10 - 20
18 - 48
Solid state diffusion in minerals 20 - 120
Pathways
• For an overall reaction, one or a few (for
more complex overall reactions)
elementary reactions can be rate limiting
Reaction of A to P  rate determined by slowest reaction in between
If more than 1 reaction possible at any intermediate point, the faster of
those 2 determines the pathway
Consecutive Reactions
A
B
C
k
k
1
2
Reaction sequence when k1≈k2:
d [ A]

 ki [ A]
dt
d [ B]
 ki [ A]  kii [ B ]
dt
d [C ]
 kii [ B]
dt