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Kinetic Rate Equations of Chemical Reaction Networks Chemical Reaction Networks

Kinetic Rate Equations of Chemical Reaction Networks Chemical Reaction Networks Reaction Kinetics Kinetic Rate Equations. Kinetics Modeling A simple example from inorganic chemistry: Hydrogen Combustion. What the highschool chemistry textbook said:

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Kinetic Rate Equations of Chemical Reaction Networks Chemical Reaction Networks

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  1. Kinetic Rate Equations of Chemical Reaction Networks Chemical Reaction Networks Reaction Kinetics Kinetic Rate Equations

  2. Kinetics Modeling A simple example from inorganic chemistry: Hydrogen Combustion What the highschool chemistry textbook said: 2 H2 + O22 H2O

  3. Kinetics Modeling A simple example from inorganic chemistry: Hydrogen Combustion What the highschool chemistry textbook said: 2 H2 + O22 H2O … and what really happens:

  4. Kinetics Modeling A simple example from inorganic chemistry: Hydrogen Combustion What the highschool chemistry textbook said: 2 H2 + O22 H2O … and what really happens: O 2 + energy  2 O H2 + energy  2 H

  5. Kinetics Modeling A simple example from inorganic chemistry: Hydrogen Combustion What the highschool chemistry textbook said: 2 H2 + O22 H2O … and what really happens: O 2 + energy  2 O H2 + energy  2 H O + H2  OH + H H+O 2 OH + O

  6. Kinetics Modeling A simple example from inorganic chemistry: Hydrogen Combustion What the highschool chemistry textbook said: 2 H2 + O22 H2O … and what really happens: O 2 + energy  2 O H2 + energy  2 H O + H2  OH + H H+O 2 OH + O OH + H2 H2O + H OH + OH H2O + O OH + H H2O

  7. Kinetics Modeling A simple example from inorganic chemistry: Hydrogen Combustion What the highschool chemistry textbook said: 2 H2 + O22 H2O … and what really happens: O 2 + energy  2 O H2 + energy  2 H O + H2  OH + H H+O 2 OH + O OH + H2 H2O + H OH + OH H2O + O OH + H H2O … and a lot more, involving O+, H+, OH+, OH _, photons …

  8. O2 H2 O H OH H2O

  9. O2 H2 O H 2 1 OH 3 H2O

  10. O2 H2 O H 2 1 OH 3 H2O

  11. 3 1 2 O + H2 OH + H H+O 2 OH + O OH + H2 H2O + H H2 Combustion Model A: A Chemical Circuit / Reaction Network Model O2 H2 O H 2 1 OH 3 H2O

  12. Kinetic Rate Equations for H2 Combustion Model A O2 H2 O H ___ ___ d [H2] d [H2] d [H2] d [H2] [H2] : H2-concentration k1 ,k3 : Forward reaction rate coefficients k1 , k3 : Backward reaction rate coefficients Forward: In Arrow-Directions Backward: Against Arrow-Directions d t d t d t d t 3 3 1 1 _ _ 1 2 OH ___ d [H2] + = _ Reactions 1 (Forward and Backward): d t 3 - k1 [O] [H2]+ k1 [H] [OH] = H2O _ Reactions 3 (Forward and Backward): - k3 [OH] [H2]+ k3 [H] [H2O] = ___ ___ Net Rate of H2-Production from all Reactions H2 participates in: Example: Rate of H2-Produdction

  13. _ _ _ _ _ ____ ___ ___ d [O] d [H2O] d [H2] = = = - k1[O] [H2] + k1[H] [OH] + k2[H][O2]- k2[O][OH] + k3[OH][H2]- k3[H][H2O] - k1[O] [H2] + k1[H] [OH] - k3[OH][H2]+ k3[H][H2O] d t d t d t _ ___ d [O2] = - k2[H] [O2] + k2[O] [OH] d t _ _ _ _ ___ ___ d [OH] d [H] = = + k1[O] [H2] - k1[H] [OH] - k3[OH][H2]+ k3[H][H2O] + k2[H] [O2] - k2[O] [OH] + k1[O] [H2] - k1[H] [OH] + k3[OH][H2]- k3[H][H2O] - k2[H] [O2] + k2[O] [OH] d t d t _ _ Kinetic Rate Equations for H2 Combustion Model A: All of them !

  14. = = = The Rate Functions R(t,y) for H2 Combustion Model A: R(t,y) (R1(t,y),R2(t,y), R3(t,y), R4(t,y), R5(t,y), R6(t,y)) where y (y1, y2, y3, y4, y5, y6) := ([H2], [O2], [H], [O], [OH], [H2O]) _ _ ___ d [H2] R1(t,y):= - k1[O] [H2] + k1[H] [OH] - k3[OH][H2]+ k3[H][H2O] d t _ ___ d [O2] R2(t,y):= - k2[H] [O2] + k2[O] [OH] _ _ d t ___ d [H] ___ R3(t,y):= + k1[O] [H2] - k1[H] [OH] + k3[OH][H2]- k3[H][H2O] - k2[H] [O2] + k2[O] [OH] _ d t _ _ ___ d [O] ___ R4(t,y):= - k1[O] [H2] + k1[H] [OH] + k2[H][O2]- k2[O][OH] = d t _ _ ___ d [OH] ___ R5(t,y):= + k1[O] [H2] - k1[H] [OH] - k3[OH][H2]+ k3[H][H2O] + k2[H] [O2] - k2[O] [OH] = _ d t _ ___ d [H2O] ____ R6(t,y):= + k3[OH][H2]- k3[H][H2O] = d t

  15. Kinetic Rate Equations as an ODE System There are N=6 coupled diff. eqs., for 6 dynamical variables y1(t), y2(t), y3(t), y4(t), y5(t), y6(t) where [X]t means “concentration of X at time t“: y1(t) := [H2]t , y1(TI) = y1o= initial conc. of H2 y2(t) := [O2]t , y2(TI) = y2o= initial conc. of O2 y3(t) := [H]t , y3(TI) = y3o= initial conc. of H y4(t) := [O]t , y4(TI) = y4o= initial conc. of O y5(t) := [OH]t , y5(TI) = y5o= initial conc. of OH y6(t) := [H2O]t , y6(TI) = y6o= initial conc. of H2O The 6 rate functions R1,R2,R3,R4,R5 and R6 are then given by the right-hand side expressions of the 6 combustion networkrate equations (see previous slide). In “vector” notation y (y1, …,y6) , R (R1, …,R6), kinetic rate equations and initial conditions can be written as: dy/dt = R(t,y) , y(TI) = y.o

  16. Ordinary Differential Equation (ODE) Solvers ODE system of N coupled diff. Eqs., starting at initial time TI : dy1/dt = R1(t, y1, …,yN), y1(TI) = y1o dy2/dt = R2(t, y1, …,yN), y2(TI) = y2o … … dyN/dt = RN(t, y1, …,yN) , yN(TI) = yN o Or, for short, in “vector” notation y:= (y1, …,yN) , R:= (R1, …,RN) : dy/dt = R(t, y) , y(TI) = y.o A simple example: The Euler Method Replace “dt” by “∆t” and “dy” by “∆y”: dy/dt ≈ ∆y/∆t y(t+∆t) ≈ y(t) + R(t, y(t)) ∆t For time interval [TI , Tf] choose “time increment” “∆t =: h” ∆t =: h = (Tf - TI ) / K, No. of time steps = K“very large” Then: y(tk+1) ≈ y(tk) + R(tk, y(tk)) h, tk := TI + k h , k=0,1,2, … K

  17. Let’s apply this to solve H2 Combustion Model A! We’ll use a “random” choice of rate coefficients. We’ll also use (see next slide) stoichiometric initial H2-O2-concentration ratios: [H2]o = 6 units , [O2]o = 3 units Thus, we should get a final H2O-concentration of [H2O]o = 6 units i.e., one H2O out for every H2in -- and no H2 or O2 or H or O or OH left over at the end. Well, see for yourself.... Please be patient: it takes a long time for these next slides to load!

  18. H2 Combustion Model A H2 concentration O2 time

  19. H2 Combustion Model A H2 H2O concentration O2 time

  20. Ooops!What happened here ? Only ~3.8 units of H2O are produced in the end ... but there should be 6 units! Where did the missing H- and O-atoms go ? See next slide ... Again, please be patient: it takes a long time for the next slide to load!

  21. H2 Combustion Model A H2 H2O concentration O2 H (x0.5) OH (x0.5) time

  22. OK, so H2 Combustion Model A is not very realistic: It does not convert all available reactants (H2, O2) into the final product H2O. So, let’s try a more realistic model: ... Again, please be patient: it takes a long time for the next slides to load!

  23. H2 Combustion Model B 3 6 5 O2 H2 O H 2 1 4 OH 7 H2O

  24. H2 Combustion Model B H2 H2O concentration O2 H (x0.5) OH (x0.5) time

  25. So H2 Combustion Model B works much better: It converts all the available reactants (H2, O2) into the final product H2O -- if you wait long enough. Exercise (Optional!): Write down, if you can... All 7 chemical reaction equations for Model B, in standard chemist’s notation (X+2Y3U+V etc.) 2) All 6 kinetic rate equations for Model B, with the contributions from all 14 reactions (7 Forward + 7 Backward) correctly included in the 6 rate functions.

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