Updated: 24 October 2013 CEE697K Lecture #11 1 Print version CEE 697K ENVIRONMENTAL REACTION KINETICS Lecture #11 Kinetic Theory: Encounter Model, Transition State Model & Ionic Strength Effects Brezonik, pp. 130-158 Introduction David A. Reckhow
Structure of Water 2 S&M: Fig. 1.3 sp 3 hybridization 2 bonding and 2 non-bonding orbitals Dipolar Character S&M: Fig. 1.4 Origin of Water’s Unusual properties High melting and boiling point High heat of vaporization Expands upon freezing High surface tension Excellent polar solvent B: Fig 1.2 CEE697K Lecture #11 David A. Reckhow
Water’s intermolecular structure 3 Dominated by Hydrogen Bonds Fig. 1.5a Pg. 8 Ice Open tetrahedral structure Water Flickering cluster model 100 ps lifetime 0.1 ps molecular vibration Avg cluster size 65 molecules @ 0ºC Fig. 1.5b 12 molecules @ 100ºC Pg. 8 CEE697K Lecture #11 David A. Reckhow
Solutes in Water 4 Great solvent for ionic or ionizable substances Ion-dipole bonds improves stability Energy increases with charge of ion and S&M: Fig. 1.6 decreases with size Solvent hole model As solute-water bonding strengthens compared to water-water bonding, B: Fig 1.4 solubility goes up Hydrophilic solute Weak solute-water bonds reduce solubility Hydrophobic solutes CEE697K Lecture #11 David A. Reckhow
Activation Energy 5 Activation Energy must always be positive Unlike ∆ H, which may be positive or negative Differing reaction rates Activated Complex Activated Complex E a E a Energy Energy reactants reactants ∆ = ∆ = ∆ ∆ E H E H f f products products Reaction Coordinate Reaction Coordinate CEE697K Lecture #11 David A. Reckhow
Encounter Theory I 6 Uncharged Solutes Nature of diffusion in water Encounter within a solvent cage Random diffusion occurs through elementary jumps of λ = distance Molecular Molecular 2 r diameter radius Average time between jumps λ 2 λ For a continuous medium: 2 = 2 τ = or D τ More 2 D For a semi-crystalline structure: appropriate λ 6 λ 2 = 2 τ = or D τ for water 6 D For water, D ~ 1x10 -5 cm 2 s -1 , and λ = 4x10 -8 cm, so τ ~ 2.5x10 -11 s If time between vibrations is ~ 1.5x10 -13 s, then the average water molecule vibrates 150 times (2.5x10 -11 /1.5x10 -13 ) in its solvent cage before jumping to the next one. CEE697K Lecture #11 David A. Reckhow
Encounter Theory II 7 Probability of Encounter If A and B are the same size as water They will have 12 nearest neighbors Probability that “A” will encounter “B” in a solvent cage of 12 neighbors is: Proportional to the mole fraction of “B” = P 6 X With each new jump, “A’ has 6 new neighbors B A B n Where: = # molecules of “B” per cm 3 B X B 1 # molecules of solvent per cm 3 γλ 3 Geometric packing factor Molecular volume (cm 3 ) CEE697K Lecture #11 David A. Reckhow
Encounter Theory III 8 And combining the rate of movement with the probability of encountering “B”, we get an expression for the rate of [ ] encounter with “B” = 1 6 D P τ λ 2 A B AB Then substituting in for the probability γλ 3 6 D ( 6 n ) = 1 B τ λ 2 AB = γλ 36 n D B For water, γ =0.74, and the effective diffusion coefficient, D AB = D A + D B , and λ =r AB , the sum of the molecular radii Then we get: = 1 25 r D n τ AB AB B AB # of encounters/sec for each molecule of “A ” CEE697K Lecture #11 David A. Reckhow
Encounter Theory IV 9 Now the total # of encounters between “A” and “B” per cm 3 per second is: n = A 25 r D n n τ AB AB A B AB In terms of moles of encounters (encounter frequency) this becomes: 3 3 cm cm 1000 n 1000 = = L A L 25 Z r D n n τ e , AB AB AB A B molecules molecules N N o Mole o Mole = 25 r D [ A ] n AB AB B n B =[B]/N 0 /1000 L/cm 3 cm 2 s -1 M -1 s -1 cm #/Mole − = 2 Z 2 . 5 x 10 r D N [ A ][ B ] e , AB AB AB 0 CEE697K Lecture #11 David A. Reckhow
Encounter Theory V 10 Frequency Factor − = 2 Z 2 . 5 x 10 r D N [ A ][ B ] e , AB AB AB 0 A When E a = 0, k=A Activated Complex − E a = E a / RT k Ae Energy reactants products Reaction Coordinate CEE697K Lecture #11 David A. Reckhow
Transition State Theory I 11 Consider the simple bimolecular reaction + → k A B C Even though it is an elementary reaction, we can break it down into two steps ≠ + ⇔ ≠ → k A B AB C “Activated Complex” ≠ Where the first “equilibrium” is: ≠ = [ AB ] K ≠ = [ A ][ B ] ≠ [ ] [ ][ ] AB K A B Activated So the forward rate is: Complex E a [ ] d C = ≠ ≠ = ≠ ≠ k [ AB ] k K [ A ][ B ] Energy reactants dt k products CEE697K Lecture #11 David A. Reckhow Reaction Coordinate
Transition State Theory II 12 Now the transition state is just one bond vibration away from conversion to products Frequency of vibration (s -1 ) E vib = ν Planks Law: h vibrational Planck’s constant (6.62 x 10 -27 ergs∙s) energy Bond energy must be in the thermal region: Temperature (ºK) E bond ≈ Bond kT energy Boltzman constant (1.3807×10 − 16 ergs ºK -1 ) So equating, we get: ν = kT ν = h kT h And since conversion occurs on the next vibration: kT ≠ ≠ ≠ = = ≠ = ν and k k K K k h CEE697K Lecture #11 David A. Reckhow
Transition State Theory III 13 Now from basic thermodynamics: ∆ G o ∆ = − − G o = RT ln K or RT K e And also ∆ = ∆ − ∆ G o H T S ∆ S e ∆ − So: H = R RT K e ∆ ≠ ∆ ≠ kT − And combining: S H = R RT k e e h Recall: ∆ = ∆ − ∆ ≈ ∆ E H P V H − ∆ ≠ kT E S a = And substituting back in: R RT k e e h A CEE697K Lecture #11 David A. Reckhow
Activation Energy 14 Activation Energy must always be positive Unlike ∆ H, which may be positive or negative Differing reaction rates Activated Complex Activated Complex E a E a Energy Energy reactants reactants ∆ = ∆ = ∆ ∆ E H E H f f products products Reaction Coordinate Reaction Coordinate CEE697K Lecture #11 David A. Reckhow
Temperature Effects 15 Arrhenius Equation ( ) − k T T E d ln k E = = 2 2 1 a a ln = − E a / RT 2 k Ae RT dT k RT T 1 1 2 Log A ( ) − ∆ K T T H = 2 2 1 0 Log k ln E a /2.3R K RT T 1 1 2 Analogous to Van’t Hoff Equation for Equilibria 1/T CEE697K Lecture #11 David A. Reckhow
Ionic Strength Effects 16 Ion-ion Reactions Based on activated complex theory [ ] d C ≠ = ≠ ≠ = ≠ ≠ + ⇔ ≠ → k k [ AB ] k K [ A ][ B ] A B AB C dt So let’s look at the equilibrium constant ≠ γ ≠ γ γ [ AB ] { AB } ≠ = ≠ ≠ ≠ = = or A B [ AB ] K [ A ][ B ] AB K γ γ γ { A }{ B } [ A ] [ B ] ≠ AB A B Which means: γ γ d [ C ] kT ≠ = A B K [ A ][ B ] γ dt h ≠ AB o (for I=0) K 2 CEE697K Lecture #11 David A. Reckhow
Reactions with charged ions 17 Using the Debye-Huckel Equation − γ i = 2 0 . 5 log 0 . 55 z i I I<0.005 { } ( ) = + − − + + 2 o 2 2 0 . 5 log k log k 0 . 51 z 0 . 51 z 0 . 51 z z I 2 2 A B A B = + o 0 . 5 log k 1 . 02 z z I + + 2 2 z 2 z z z 2 A B A A B B Using the Guntelberg Approximation 2 0 . 5 0 . 55 z i I − γ = log + 0 . 5 i ( 1 I ) I<0.01 0 . 5 I = + o log k log k 1 . 02 z z + 2 2 A B 0 . 5 ( 1 I ) CEE697K Lecture #11 David A. Reckhow
I corrections (cont.) 18 Neutral species − γ i = log b i I { } I = + + − o log k log k b b b ≠ 2 2 A B AB Some case studies: CEE697K Lecture #11 David A. Reckhow
Case Study: TCP Note: both TCP and TCAC refer to the 1,1,1-trichloropropanone 19 Observed loss of 1,1,1- trichloropropanone in distribution systems Lab studies show that chloroform is the product Logically presumed to be a simple hydrolysis Reckhow & Singer, 1985 “Mechanisms of Organic Halide Formation During Fulvic Acid Chlorination and Implications with Respect to Preozonation”, In Jolley et al., Water Chlorination; CEE697K Lecture #11 David A. Reckhow Chemistry, Environmental Impact and Health Effect, Volume 5, Lewis.
TCP (cont.) 20 Ionic strength effects = − − ln k H 4 . 81 1 . 4 I = − − log k H 2 . 08 0 . 6 I Rate with chlorine Increases greatly High intercept [ ] = + k 0 . 024 32 HOCl T T CEE697K Lecture #11 David A. Reckhow
Disagreement with prior study 21 Gurol & Suffet showed 10x higher rate constants Phosphate? CEE697K Lecture #11 David A. Reckhow
Putting it together 22 CEE697K Lecture #11 David A. Reckhow
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