1

MODELING

• Solution-Phase Nonlinear Dynamics

• Heterogeneous Dynamics

2

Brief History of Chemical NLD

• Chemical waves and oscillations rejected– Violate Second Law– Difficult to reproduce

• Prigogine– No Second Law violation far from equilibrium

• BZ reaction– Easily reproduced

• Doubts allayed─new theories, experiments

3

Mechanism and Model

• Mechanism─elementary steps at molecular level

• Model─simplified mechanism– Abstract: Lotka-Volterra– Derived: Oregonator, Brusselator– Empirical: Rate Law

4

Calculating Rate Constants

• Diffusion-controlled limits

• Marcus theory of electron transfer

• Eigen theory of proton transfer

• Eigen theory of metal complex formation

5

Lotka-Volterra Model

A + X 2X, k1

X + Y 2Y, k2

Y , k3

dX/dt = k1AX - k2XY

dY/dt = k2XY - k3Y

6

Lotka-Volterra:Predator-Prey

0

3

6

9

12

15

18

0 2 4 6 8

Time

Po

pu

lati

on

7

Lotka-Volterra:Increase in Predator Efficiency

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8

Time

Po

pu

lati

on

8

Empirical Model:Iodate-Sulfite-Ferrocyanide

A) IO3- + 8I- + 6H+ 3I3

- + 3H2O

B) I3- + HSO3

- + H2O 3I- + HSO4- + 2H+

C) I3- + 2Fe(CN)6

4- 3I- + 2Fe(CN)63-

D) IO3- + 3HSO3

- I- + 3HSO4-

E) H+ + SO32- = HSO3

-

9

Derived Model-1:Epstein-Orbán-Edblom (EOE)

• Concentrations in large excess: iodate, iodide,

ferrocyanide, sulfate

– Incorporate into rate equation

• Four variables: A = [SO32-], X =[HSO3

-],Y = [H+],

Z = [I2]

• Reciprocal residence time, k0, varied

• Inflow variables: Y0, A0

10

Derived EOE Model-2

Number Reaction Rate Eqn Rate ConstantN1 A+Y X vN1 = kN1AY 5e10 M-1s-1

-N1 X A+Y v-N1 = k-N1X 8.1e3 s-1

N2 X Y vN2 = kN2X 6e-2 s-1

N3 2Y Z vN3 = kN3Y2 7.5e4 M-1s-1

N4 Z+X 3Y vN4 = kN4ZX 2.3e9 M-1s-1

N5 Z vN5 = kN5Z 3e1 s-1

A = [SO32-], X =[HSO3

-],Y = [H+], Z = [I2]

11

EOE: Oscillations in a Flow Reactor

0

2

4

6

8

10

0 10 20 30 40

Time (min)

pH

12

EOE: Oscillations in a Flow Reactor; Close-up

0

2

4

6

8

14.25 14.3 14.35 14.4 14.45

time (min)

pH

13

EOE: Changing k2N from 6.0e-2 to

6.5e-2 M-1s-1

14

Photoresponse of the Chlorine Dioxide-Iodine-Malonic Acid Reaction in a Flow Reactor

• First experimental demonstration of Turing structures

• Inflow solution A– 0.04 M H2SO4 with 0.0022 M I2

• Inflow solution B– varying amounts of– malonic acid (w/wo) starch with chlorine dioxide

15

CDIMA: No Light

(1) I2 + MA H+ + I- + IMA

(2) ClO2 + I- ½I2 + ClO2-

(3) ClO2- + 4I- + 4H+ Cl- + 2I2 + 2H2O

(4) I- + I2 + Starch StarchI3-

(5) StarchI3- I- + I2 + Starch

(6) I2 + I- I3-

(7) I3- I2 + I-

(h1) I2 + H2O I- + H+ + IOH

(h2) I- + H+ + IOH I2 + H2O

16

CDIMA Fitting: No Light-Trial 1k(1)=1e-3, k(2)=1.1e3

17

CDIMA Fitting: No Light-Trial 2k(1)=1.16e-3, k(2)=1.45e3

18

CDIMA: Light on/offLight on:

(8) I2 + h I· + I·

(9) I· + I· I2

(10) I· + I- I2-

(11) I2- I· + I-

(12) I· +ClO2 IClO2

Recovery-Light off:

(13) IClO2 + H2O IO3- + Cl- + 2H+

(14) 5I- + IO3- + 6H+ 3I2 + 3H2O

(16) ClO2 + I2- ClO2

- + I2

19

CDIMA Fitting: Complete Sequence

20

Detection of eaq- in Reductive

Processes

• Electrochemical reductions

• Sodium amalgams

• Solid-phase reductants

• Heterogeneous reactions where yields may be decreased by scavenging

precursors such as eaq- and H·

21

Elementary Steps in the Generation of H2 from the Reduction of Water by

Magnesium

• Mg(s) 2eaq- + Mg2+

• eaq- + H+ H·

• eaq- + eaq

- H2 + 2OH-

• eaq- + H· H2 + OH-

• H· + H· H2

22

Benzoate vs. Ethanol with Trichloroacetate

Mg + 2H2O Mg(OH)2 + 1.24H· + 0.25eaq- 2.18e-3 s-1

eaq- + Cl3CCOO- Cl2C·COO + Cl- 8.5e10 M-1s-1

H· + C6H5COO- ·C6H6COO- 9.2e8 M-1s-1

H· + C2H5OH CH3C·HOH + H2 1.7e7 M-1s-1

23

Competitive Scavenging

0

10

20

30

0 200 400 600

Time (s)

V(H

2) (m

L)