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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?
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
• 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
k
kKeq
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:
nXkdt
Xd)(
)(
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 Rate Law
Integrated Rate Law Units for k
0 A=A0-kt mol/cm3 s
1 ln A=lnA0-kt s-1
2 cm3/mol s
kAdt
Ad
][
2][kA
dt
Ad
kdt
Ad
][
ktAA
0
11
• 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
• 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
kdt
Ad
][
First Order
• Rate is dependent on concentration of a reactant or product– Pyrite oxidation, sulfate reduction are examples
kAdt
Ad
][
First Order
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
kAdt
Ad
][ )(
0][
][ ktt eA
A ktA
A t 0][
][ln
)log(]log[]log[ 0kt
t eAA 0]log[434.0]log[ AktA t
Pseudo- 1nd Order• For a bimolecular reaction: A+B products
)])([]([]][[ 0022 xBxAkBAkdt
dx
If [B]0 is held constant, the equation above reduces to:
)0])([]([]][[ 0022 BxAkBAkdt
dx
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][
][ 22
OPFekdt
Fed
2nd Order• For a bimolecular reaction: A+B products
)])([]([]][[ 0022 xBxAkBAkdt
dx
tkBA
AB
BAxBA
xAB
BA 20
0
0000
00
00 ][][
][][ln
][][
1
)]([][
)]([][ln
][][
1
0
0002 ][
][log)][]([43.0
][
][log
A
BtBAk
B
A
t
t
[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
• Half-lives tougher to quantify if A≠B for 2nd order reaction kinetics – but if A=B:
02
21 ][1A
kt
If one reactant (B) is kept constant (pseudo-1st order rxns):
02
21 ][2lnA
kt
0
021 ][5.0
][ln12ln
A
A
kkt
3rd order Kinetics
• Ternary molecular reactions are more rare, but catalytic reactions do need a 3rd component…
)])([])([]([]][][[ 00023 xCxBxAkCBAkdt
dx
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:
• With reaction progress
• In summary there is a definite role that approach to equilibrium plays on overall forward reaction kinetics!
][][][
PkAkdt
Adrf
)])([]([ 00 xPxAkdt
dxf
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:
So that a plot of log K vs. 1/T is a straight line whose slope = -EA/2.303R
2303.2
log
RT
E
dT
kd A
Activation EnergyReaction ‘typical’ range of
EA (kcal/mol)
Physical adsorption 2 – 6
Aqueous diffusion <5
‘Biotic’ reactions 5 - 20
Mineral dissolution/precipitation 8- 36
Dissolution controlled by surface reaction
10 - 20
Isotopic exchange in solution 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