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  1. Home/
  2. Himanshu Chavan/
  3. REACTION RATE FOR A MULTISTEP MECHANISM

REACTION RATE FOR A MULTISTEP MECHANISM

                                    REACTION RATE FOR A MULTISTEP MECHANISM   l. OBJECTIVES Derive the reaction rates for the given multistep mechanism using compact notations and compare the results with the ones derived…

    • Himanshu Chavan

      updated on 02 Jul 2021

                                        REACTION RATE FOR A MULTISTEP MECHANISM

     

    l. OBJECTIVES

    Derive the reaction rates for the given multistep mechanism using compact notations and compare the results with the ones derived manually.

     

    ll. REACTION MECHANISM

    In this project, we will be considering the following four mechanisms-

     CO+O2↔CO2+OCO+O2↔CO2+O

    O+H2O↔OH+OH

    CO+OH↔CO2+H

    H+O2↔OH+O

     

    lll. DERIVATION OF REACTION RATE MANUALLY

    Let kfi→The rate of forwarding Reaction &kri→The rate of Reverse Reaction

    • wherei→Reaction Index

    Therefore, the forward and reverse reaction for the four mechanisms can be represented as -

    1. Forward & Reverse reaction

    CO+O2↔kfikriCO2+O

    O+H2O↔kfikriOH+OH

    CO+OH↔kfikriCO2+H

    H+O2↔kfikriOH+O

     

    2. System of Coupled Ordinary Differential Equations

    1.d[CO]dt=-kf1[CO][O2]+kr1[CO2][O]-kf3[CO][OH]+kr3[CO2][H]

    2.d[O2]dt=-kf1[CO][O2]+kr1[CO2][O]-kf4[H][O2]+kr4[OH][O]

    3.d[CO2]dt=kf1[CO][O2]-kr1[CO2][O]+kf3[CO][OH]-kr3[CO2][H]

    4.d[O]dt=kf1[CO][O2]-kr1[CO2][O]-kf2[O][H2O]+kr2[OH][OH]+kf4[H][O2]-kr4[OH][O]

    5.d[H2O]dt=-kf2[O][H2O]+kr2[OH][OH]

    6.d[OH]dt=2kf2[O][H2O]-2kr2[OH][OH]-kf3[CO][OH]+kr3[CO2][H]+kf4[H][O2]-kr4[OH][O]

    7.d[H]dt=kf3[CO][OH]-kr3[CO2][H]-kf4[H][O2]+kr4[OH][O]

     

    lV. DERIVATION OF REACTION RATES USING COMPACT NOTATIONS

    1. Defining the Indices

    Fig- Reaction Indices

       i          Reaction            Reaction Mechanism    
               1          R1 CO+O2→CO2+O
               2          R2 O+H2O→OH+OH
               3          R3 CO+OH→CO2+H
               4           R4 H+O2→OH+O

    Fig 2- Species Indices

          j        Species 
         1 CO
         2 O2
         3 CO2
         4 O
         5 H2O
         6 OH
         7 H

     

    2. Stoichoimetric Coefficient Matrix

    Let v′ji & v′′ji Stoichiometric Matrix Of Reactants & Products repectively

    v′ji=[1100000000110010000100100001] and v′′ji=[0011000000002000100010001010]

    Let vji=Stoichiometric Coefficient Matrix of Products - Stoichiometric Coefficient Matrix of Reactants

    ⇒vji=v′′ji-v′ji

    ⇒vji=[-1-111000000-1-120-10100-110-10101-1]

     

    3. Net Reaction Rate

    The net reaction rate of a given species is given by-

    • qi=kfiN∏j=1[Xj]v′ji-kriN∏j=1[Xj]v′′ji

    For the 1stReaction(i=1)

    • q1=kf1N∏j=1[Xj]v′j1-kr1N∏j=1[Xj]v′′j1
    • q1=kf1[X1]v11′[X2]v21′[X3]v31′[X4]v41′[X5]v51′[X6]v61′[X7]v71′-kr1[X1]v11′′[X2]v21′′[X3]v31′′[X4]v41′′[X5]v51′′[X6]v61′′[X7]v71′′
    • q1=kf1[X1]1[X2]1[X3]0[X4]0[X5]0[X6]0[X7]0-kr1[X1]0[X2]0[X3]1[X4]1[X5]0[X6]0[X7]0
    • q1=kf1[CO][O2]-kr1[CO2][O]

    Similarly, for the remaining three reaction(i=2,3&4)

    • q2=kf2[O][H2O]-kr2[OH]2
    • q3=kf3[CO][OH]-kr3[CO2][H]
    • q4=kf4[O2][H]-kr4[O][OH]

    4. Net Production Rate

    The net production rate of a given species is given by-

    • ωj=L∑i=1vjiqi

    Where L is the total number of reactions in the mechanism for each species

    The matrix Representation of the net production rate is given by-

    [ω1ω2ω3ω4ω5ω6ω7]=[d[CO]dtd[O2]dtd[CO2]dtd[O]dtd[H2O]dtd[OH]dtd[H]dt] 

    4.1 Net Production Rate of Species 1:[CO]

    • ω1=L∑i=1v1iqi
    • ω1=v11q1+v12q2+v13q3+v14q4
    • ω1=-q1-q3
    • ω1=d[CO]dt=-kf1[CO][O2]+kr1[CO2][O]-kf3[CO][OH]+kr3[CO2][H]

    4.2 Net Production Rate of Species 2:[O2]

    • ω2=L∑i=1v2iqi
    • ω2=v21q1+v22q2+v23q3+v24q4
    • ω2=-q1-q4
    • ω2=d[O2]dt=-kf1[CO][O2]+kr1[CO2][O]-kf4[O2][H]+kr4[O][OH]

    4.3 Net Production Rate of Species 3: [CO2]

    • ω3=L∑i=1v3iqi
    • ω3=v31q1+v32q2+v33q3+v34q4
    • ω3=q1+q3
    • ω3=d[CO2]dt=kf1[CO][O2]-kr1[CO2][O]+kf3[CO][OH]-kr3[CO2][H]

    4.4 Net Production Rate of Species 4: [O]

    • ω4=L∑i=1v4iqi
    • ω4=v41q1+v42q2+v43q3+v44q4
    • ω4=q1-q2+q4
    • ω4=d[O]dt=kf1[CO][O2]-kr1[CO2][O]-kf2[O][H2O]+kr2[OH]2+kf4[O2][H]-kr4[O][OH]

    4.5 Net Production Rtae of Species 5: [H2O]

    • ω5=L∑i=1v5iqi
    • ω5=v51q1+v52q2+v53q3+v54q4
    • ω5=-q2
    • ω5=d[H2O]dt=-kf2[O][HO2]+kr2[OH]2

    4.6 Net Production Rtae of Species 6: [OH]

    • ω6=L∑i=1v6iqi
    • ω6=v61q1+v62q2+v63q3+v64q4
    • ω6=2q2-q3+q4
    • ω6=d[OH]dt=2kf2[O][H2O]-2kr2[OH]2-kf3[CO][OH]+kr3[CO2][H]+kf4[O2][H]-kr4[O][OH]

    4.7 Net production Rate of Species 7: [H]

    • ω7=L∑i=1v7iqi
    • ω7=v71q1+v72q2+v73q3+v74q4
    • ω7=q3-q4
    • ω7=d[H]dt=kf3[CO][OH]-kr3[CO2][H]-kf4[O2][H]+kr4[O][OH]

    V. CONCLUSIONS

    The system of coupled ordinary differential equations derived both manually and using a system of compact notation is the same.

    Thus, the method of compact notation can be used to automatically acquire the system of ODEs using a program, which can be further numerically solved to obtain the required reaction rates.

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