Geochemical Kinetics

1
Geochemical 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

2
Time Scales
3
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

4
Extent of Reaction

• In its most general representation, we can
  discuss a reaction rate as a function of the
  extent of reaction
• Rate d?/Vdt
• where ? (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 dCi/vidt
• where n is moles, v is stoichiometric
  coefficient, and C is molar concentration of
  species i

5
Rate Law

• For any reaction X ? Y Z
• We can write the general rate law

Rate change in concentration of X with time, t
Order of reaction
Rate Constant
Concentration of X
6
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, even
  zero is possible

7
General Rate Laws
Reaction order Rate Law Integrated Rate Law Units for k
0 AA0-kt mol/cm3 s
1 ln AlnA0-kt s-1
2 cm3/mol s
8

• 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

9
Zero Order

• 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

10
First Order

• Rate is dependent on concentration of a reactant
  or product
• Pyrite oxidation, sulfate reduction are examples

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First Order

• Find order from logAt vs t plot ?
• Slope-0.434k
• ?k -(1/0.434)(slope) -2.3(slope)
• k is in units of time-1

12
1st-order Half-life

• Time required for one-half of the initial
  reactant to react

13
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

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General Rate Laws
Reaction order Rate Law Integrated Rate Law Units for k
0 AA0-kt mol/cm3 s
1 ln AlnA0-kt s-1
2 cm3/mol s
15
2nd Order

• For a bimolecular reaction AB ? products

A0 and B0 are constant, so a plot of log
A/B vs t yields a straight line where slope
k2 (when AB) or k2(A0-B0)/2.3 (when A?B)
16
Pseudo- 1nd Order

• For a bimolecular reaction AB ? products

If A0 or B0 are held constant, the equation
above reduces to
SO as A changes B does not, reducing to a
constant in the reaction plots as a first-order
reaction
17
2nd order Half-life

• Half-lives tougher to quantify if A?B for 2nd
  order reaction kinetics but if AB

If one reactant (B) is kept constant (pseudo-1st
order rxns)
18
3rd order Kinetics

• Ternary molecular reactions are more rare, but
  catalytic reactions do need a 3rd component

19
Zero order reaction

• NOT possible for elementary reactions
• Common for overall processes independent of any
  quantity measured
• A0-Akt

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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
21
Initial Rate, first order rxn example

• For the example below, lets determine the order
  of reaction A B ? C
• Next, lets solve the appropriate rate law for k

Run Initial A (A0) Initial B (B0) Initial Rate (v0)
1 1.00 M 1.00 M 1.25 x 10-2 M/s
2 1.00 M 2.00 M 2.5 x 10-2 M/s
3 2.00 M 2.00 M 2.5 x 10-2 M/s
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Rate Limiting Reactions

• For an overall reaction, one or a few (for more
  complex overall reactions) elementary reactions
  will 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
23
Activation Energy, EA

• Energy required for two atoms or molecules to
  react

24
Transition State Theory

• The activation energy corresponds to the energy
  of a complex intermediate between product and
  reactant, an activated complex
• A B ? C ? AB

It can be derived that EA RT DHC
25
Collision Theory

• collision theory is based on kinetic theory and
  supposes that particles must collide with both
  the correct orientation and with sufficient
  kinetic energy if the reactants are to be
  converted into products.
• The minimum kinetic energy required in a
  collision by reactant molecules to form product
  is called the activation energy, Ea.
• The proportion of reactant molecules that collide
  with a kinetic energy that is at least equal to
  the activation energy increases rapidly as the
  temperature increases.

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T dependence on k

• Svante Arrhenius, in 1889, defined the
  relationship between the rate constant, k, the
  activation energy, EA, and temperature in
  Kelvins
• or
• Where A is a constant called the frequency
  factor, and eEA/RT is the Boltzmann factor,
  fraction of atoms that aquire the energy to clear
  the activation energy

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Arrhenius Equation

• y mx b
• Plot values of k at different temperatures log
  k vs 1/T ? slope is EA/2.303R to get activation
  energy, EA

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Activation Energy

• EA can be used as a general indicator of a
  reaction mechanism or process (rate-limiting)

Reaction / Process EA range
Ion exchange gt20
Biochemical reactions 5-20
Mineral dissolution / precipitation 8-36
Mineral dissolution via surface rxn 10-20
Physical adsorption 2-6
Aqueous diffusion lt5
Solid-state diffusion in minerals 20-120
Isotopic exchange in solution 18-48