2. Metastability and stability

3. Why do metastable phases form?
Ostwald’s Step Rule:
The first solid phase to precipitate is most soluble phase (i.e. the least stable, or metastable, phase)
Wilhelm Ostwald (1853 –1932)
Aragonite instead of calcite
SiO2. x(H2O) instead of quartz
FeS instead of pyrite
Ferrihydrite instead of hematite

4. Classical Nucleation Theory
The nucleus increases in Gibb’s free energy as it accretes
The size reaches a critical value
The free energy decreases with size until a negative value is reached (i.e. a more stable phase)
e.g. water at 0oC , critical radius is 8Å with ca. 90 H2O molecules

5. Classical Nucleation theory and the Ostwald Step Rule
G = Gbulk + Gsurface
The free energy required to make a nucleus is the sum of the free energy gained in making bonds plus the free energy required to make a surface.
Gsurface = 4r2
Where  is the interfacial energy or interfacial surface tension.
The interfacial free energy increases with decreasing solubility
Therefore the more soluble, least stable phase forms first because it has the lower interfacial energy.

7. 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

8. Nonequilibrium Equilibrium = DEATH for all organisms
Why? available metabolic energy:
DGR=DG0 + RTlnQ
Biogenic, atmospheric elements (C, N, P, S, O) are in nonequilibrium in natural waters
There are thousands of natural organic molecules and even more synthetic ones that are not thermodynamically stable in the presence of O2

9. Black Smokers
Life thrives here on the H2S and Fe2+ coming out of the vents
H2S and Fe2+ is derived from interaction of hot ( ºC) fluid interacting with basalts

10. What else affects disequilibrium?
Physical forces – gas rising, convection cells, particle settling, transport
Biological activity segregates redox species
Mineral reactions affect other reactions, perturbing redox equilibria
How long it lasts, the forces that maintain it  described by kinetics

11. Time Scales

12. 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  Fe SO H+
2 NaAlSi3O8 + 9 H2O + 2 H+  Al2Si2O5(OH)4 + 2 Na+ + 4 H4SiO4

13. 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

14. 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

15. 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

16. General Rate Laws Reaction order Rate Law Integrated Rate Law
Units for k
A=A0-kt
mol/cm3 s 1
ln A=lnA0-kt
s-1 2
cm3/mol s

17. Zeroth order: rate does not change with lower concentration
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
Graphs of different rates of reaction copy the chapter from Langmuir for them!

18. 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) = – 2598/T

19. First Order
Rate is dependent on concentration of a reactant or product
Pyrite oxidation, sulfate reduction are examples

20. First Order Find order 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

21. 1st-order Half-life
Time required for one-half of the initial reactant to react

22. 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

23. General Rate Laws Reaction order Rate Law Integrated Rate Law
Units for k
A=A0-kt
mol/cm3 s 1
ln A=lnA0-kt
s-1 2
cm3/mol s

24. 2nd Order For a bimolecular reaction: A+B  products
[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)

25. Pseudo- 1nd Order For a bimolecular reaction: A+B  products
If [A]0 or [B]0 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

26. 2nd order Half-life
Half-lives tougher to quantify if A≠B for 2nd order reaction kinetics – but if A=B:
If one reactant (B) is kept constant (pseudo-1st order rxns):

27. 3rd order Kinetics
Ternary molecular reactions are more rare, but catalytic reactions do need a 3rd component…

28. Zero order reaction NOT possible for elementary reactions
Common for overall processes – independent of any quantity measured
[A]0-[A]=kt

29. 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

30. Reversible Kinetics
Kinetics of reversible reactions requires a back-reaction term:
With reaction progress
In summary there is a definite role that approach to equilibrium plays on overall forward reaction kinetics!

31. 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

32. Consecutive Reactions
A  B  C
Reaction sequence when k1≈k2: k1 k2

33. Consecutive Reactions
A  B  C
Reaction sequence when k1≈k2: k1 k2

34. Secular Equilibrium*
Secular equilibrium is a kinetic steady-state  NOT thermodynamic equilibrium!
For our consecuative reaction: ABC, if kii>ki, then at some time t, [A] / [B] ratio remains constant