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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-

$C O + O_{2} \leftrightarrow C O_{2} + O$ $CO + O_2 harrCO_2 + O$

$O + H_{2} O \leftrightarrow O H + O H$ $O + H_2O harrOH + OH$

$C O + O H \leftrightarrow C O_{2} + H$ $CO + OHharrCO_2+H$

$H + O_{2} \leftrightarrow O H + O$ $H+O _2harr OH+O$

lll. DERIVATION OF REACTION RATE MANUALLY

Let $k_{f i} \to$ $k_(fi)rarr$ The rate of forwarding Reaction & $k_{r i} \to$ $k_(ri)rarr$ The rate of Reverse Reaction

where $i \to$ $irarr$ Reaction Index

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

1. Forward & Reverse reaction

$C O + O_{2} \leftrightarrow_{k_{r i}}^{k_{f i}} C O_{2} + O$ $CO+O_2harr_(k_(ri))^(k_(fi))CO_2+O$

$O + H_{2} O \leftrightarrow_{k_{r i}}^{k_{f i}} O H + O H$ $O+H_2Oharr_(k_(ri))^(k_(fi))OH+OH$

$C O + O H \leftrightarrow_{k_{r i}}^{k_{f i}} C O_{2} + H$ $CO+OHharr_(k_(ri))^(k_(fi))CO_2+H$

$H + O_{2} \leftrightarrow_{k_{r i}}^{k_{f i}} O H + O$ $H+O_2harr_(k_(ri))^(k_(fi))OH+O$

2. System of Coupled Ordinary Differential Equations

1. $\frac{d [C O]}{d t} = - k_{f 1} [C O] [O_{2}] + k_{r 1} [C O_{2}] [O] - k_{f 3} [C O] [O H] + k_{r 3} [C O_{2}] [H]$ $(d[CO])/(dt)=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]$

2. $\frac{d [O_{2}]}{d t} = - k_{f 1} [C O] [O_{2}] + k_{r 1} [C O_{2}] [O] - k_{f 4} [H] [O_{2}] + k_{r 4} [O H] [O]$ $(d[O_2])/(dt)=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f4)[H][O_2]+k_(r4)[OH][O]$

3. $\frac{d [C O_{2}]}{d t} = k_{f 1} [C O] [O_{2}] - k_{r 1} [C O_{2}] [O] + k_{f 3} [C O] [O H] - k_{r 3} [C O_{2}] [H]$ $(d[CO_2])/(dt)=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]+k_(f3)[CO][OH]-k_(r3)[CO_2][H]$

4. $\frac{d [O]}{d t} = k_{f 1} [C O] [O_{2}] - k_{r 1} [C O_{2}] [O] - k_{f 2} [O] [H_{2} O] + k_{r 2} [O H] [O H] + k_{f 4} [H] [O_{2}] - k_{r 4} [O H] [O]$ $(d[O])/(dt)= k_(f1)[CO][O_2]-k_(r1)[CO_2][O]-k_(f2)[O][H_2O]+k_(r2)[OH][OH]+k_(f4)[H][O_2]-k_(r4)[OH][O]$

5. $\frac{d [H_{2} O]}{d t} = - k_{f 2} [O] [H_{2} O] + k_{r 2} [O H] [O H]$ $(d[H_2O])/(dt)=-k_(f2)[O][H_2O]+k_(r2)[OH][OH]$

6. $\frac{d [O H]}{d t} = 2 k_{f 2} [O] [H_{2} O] - 2 k_{r 2} [O H] [O H] - k_{f 3} [C O] [O H] + k_{r 3} [C O_{2}] [H] + k_{f 4} [H] [O_{2}] - k_{r 4} [O H] [O]$ $(d[OH])/(dt)=2k_(f2)[O][H_2O]-2k_(r2)[OH][OH]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]+k_(f4)[H][O_2]-k_(r4)[OH][O]$

7. $\frac{d [H]}{d t} = k_{f 3} [C O] [O H] - k_{r 3} [C O_{2}] [H] - k_{f 4} [H] [O_{2}] + k_{r 4} [O H] [O]$ $(d[H])/(dt)=k_(f3)[CO][OH]-k_(r3)[CO_2][H]-k_(f4)[H][O_2]+k_(r4)[OH][O]$

lV. DERIVATION OF REACTION RATES USING COMPACT NOTATIONS

1. Defining the Indices

Fig- Reaction Indices

i	Reaction	Reaction Mechanism
1	R1	$C O + O_{2} \to C O_{2} + O$ $CO+O_2rarrCO_2+O$
2	R2	$O + H_{2} O \to O H + O H$ $O+H_2OrarrOH+OH$
3	R3	$C O + O H \to C O_{2} + H$ $CO+OHrarrCO_2+H$
4	R4	$H + O_{2} \to O H + O$ $H+O_2rarrOH+O$

Fig 2- Species Indices

j	Species
1	$C O$ $CO$
2	$O_{2}$ $O_2$
3	$C O_{2}$ $CO_2$
4	$O$ $O$
5	$H_{2} O$ $H_2O$
6	$O H$ $OH$
7	$H$ $H$

2. Stoichoimetric Coefficient Matrix

Let $v_{j i}^{'}$ $v_(ji)^'$ & $v_(ji)^('')$ Stoichiometric Matrix Of Reactants & Products repectively

$v_(ji)^' = [[1,1,0,0,0,0,0],[0,0,0,1,1,0,0],[1,0,0,0,0,1,0],[0,1,0,0,0,0,1]]$ and $v_(ji)^('')=[[0,0,1,1,0,0,0],[0,0,0,0,0,2,0],[0,0,1,0,0,0,1],[0,0,0,1,0,1,0]]$

Let $v_(ji)$ =Stoichiometric Coefficient Matrix of Products - Stoichiometric Coefficient Matrix of Reactants

$rArr v_(ji)= v_(ji)^('') -v_(ji)^'$

$rArr v_(ji)=[[-1,-1,1,1,0,0,0],[0,0,0,-1,-1,2,0],[-1,0,1,0,0,-1,1],[0,-1,0,1,0,1,-1]]$

3. Net Reaction Rate

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

$q_(i)=k_(fi)prod_(j=1)^N[X_j]^(v_(ji)^('))-k_(ri)prod_(j=1)^N[X_(j)]^(v_(ji)^(''))$

For the $1^(st)$ Reaction( $i=1$ )

$q_(1)=k_(f1)prod_(j=1)^N[X_j]^(v_(j1)^('))-k_(r1)prod_(j=1)^N[X_(j)]^(v_(j1)^(''))$
$q_1=k_(f1)[X_1]^(v_11')[X_2]^(v_21')[X_3]^(v_31')[X_4]^(v_41')[X_5]^(v_51')[X_6]^(v_61')[X_7]^(v_71')-k_(r1)[X_1]^(v_11'')[X_2]^(v_21'')[X_3]^(v_31'')[X_4]^(v_41'')[X_5]^(v_51'')[X_6]^(v_61'')[X_7]^(v_71'')$
$q_1=k_(f1)[X_1]^(1)[X_2]^(1)[X_3]^(0)[X_4]^(0)[X_5]^(0)[X_6]^(0)[X_7]^(0)-k_(r1)[X_1]^(0)[X_2]^(0)[X_3]^(1)[X_4]^(1)[X_5]^(0)[X_6]^(0)[X_7]^(0)$
$q_1=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]$

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

$q_2=k_(f2)[O][H_2O]-k_(r2)[OH]^2$
$q_3=k_(f3)[CO][OH]-k_(r3)[CO_2][H]$
$q_4=k_(f4)[O_2][H]-k_(r4)[O][OH]$

4. Net Production Rate

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

$omega_j=sum_(i=1)^Lv_(ji)q_(i)$

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-

$[[omega1],[omega2],[omega3],[omega4],[omega5],[omega6],[omega7]]=[[(d[CO])/dt],[(d[O_2])/dt],[(d[CO_2])/dt],[(d[O])/dt],[(d[H_2O])/dt],[(d[OH])/dt],[(d[H])/dt]]$

4.1 Net Production Rate of Species 1:[CO]

$omega_1=sum_(i=1)^Lv_(1i)q_(i)$
$omega_1=v_(11)q_(1)+v_(12)q_(2)+v_(13)q_(3)+v_(14)q_4$
$omega_1 = -q_1-q_3$
$omega_1=(d[CO])/(dt)=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]$

4.2 Net Production Rate of Species 2:[ $O_2$ ]

$omega_2=sum_(i=1)^Lv_(2i)q_(i)$
$omega_2=v_(21)q_(1)+v_(22)q_(2)+v_(23)q_(3)+v_(24)q_4$
$omega_2 = -q_1-q_4$
$omega_2=(d[O_2])/(dt)=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f4)[O_2][H]+k_(r4)[O][OH]$

4.3 Net Production Rate of Species 3: [ $CO_2$ ]

$omega_3=sum_(i=1)^Lv_(3i)q_(i)$
$omega_3=v_(31)q_(1)+v_(32)q_(2)+v_(33)q_(3)+v_(34)q_4$
$omega_3 = q_1+q_3$
$omega_3=(d[CO_2])/(dt)=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]+k_(f3)[CO][OH]-k_(r3)[CO_2][H]$

4.4 Net Production Rate of Species 4: [ $O$ ]

$omega_4=sum_(i=1)^Lv_(4i)q_(i)$
$omega_4=v_(41)q_(1)+v_(42)q_(2)+v_(43)q_(3)+v_(44)q_4$
$omega_4 = q_1-q_2+q_4$
$omega_4=(d[O])/(dt)=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]-k_(f2)[O][H_2O]+k_(r2)[OH]^2+k_(f4)[O_2][H]-k_(r4)[O][OH]$

4.5 Net Production Rtae of Species 5: [ $H_2O$ ]

$omega_5=sum_(i=1)^Lv_(5i)q_(i)$
$omega_5=v_(51)q_(1)+v_(52)q_(2)+v_(53)q_(3)+v_(54)q_4$
$omega_5 = -q_2$
$omega_5=(d[H_2O])/(dt)=-k_(f2)[O][HO_2]+k_(r2)[OH]^2$

4.6 Net Production Rtae of Species 6: [OH]

$omega_6=sum_(i=1)^Lv_(6i)q_(i)$
$omega_6=v_(61)q_(1)+v_(62)q_(2)+v_(63)q_(3)+v_(64)q_4$
$omega_6 = 2q_2-q_3+q_4$
$omega_6=(d[OH])/(dt)=2k_(f2)[O][H_2O]-2k_(r2)[OH]^2-k_(f3)[CO][OH]+k_(r3)[CO_2][H]+k_(f4)[O_2][H]-k_(r4)[O][OH]$

4.7 Net production Rate of Species 7: [H]

$omega_7=sum_(i=1)^Lv_(7i)q_(i)$
$omega_7=v_(71)q_(1)+v_(72)q_(2)+v_(73)q_(3)+v_(74)q_4$
$omega_7 = q_3-q_4$
$omega_7=(d[H])/(dt)=k_(f3)[CO][OH]-k_(r3)[CO_2][H]-k_(f4)[O_2][H]+k_(r4)[O][OH]$

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.