Unformatted text preview: 229 7.1. ISOTHERMAL, ISOCHORIC KINETICS The standard model for chemical reaction, which will be generalized and discussed in
more detail later, induces the following two ordinary diﬀerential equations for the evolution
of O and O2 molar concentrations:
= −2 a13 T β13 exp
dt ρO ρO ρM +2 a14 T β14 exp =k13 (T ) = a13 T β13 exp ρO2 ρM , (7.6) =k14 (T ) =r13 dρO2
dt −E 14
RT −E 13
RT =r14 ρO ρO ρM − a14 T β14 exp =k13 (T ) −E 14
RT ρO2 ρM . (7.7) =k14 (T ) =r13 =r14 Here we use the notation ρi as the molar concentration of species i. Also a common usage for
molar concentration is given by square brackets, e.g. ρO2 = [O2 ]. The symbol M represents
an arbitrary third body and is an inert participant in the reaction. We also use the common
notation of a temperature-dependent portion of the reaction rate for reaction j , kj (T ), where
The reaction rates for reactions 13 and 14 are deﬁned as
kj (T ) = aj T βj exp r13 = k13 ρO ρO ρM ,
r14 = k14 ρO2 ρM . (7.8) (7.9)
(7.10) We will give details of how to generalize this form later. The system Eq. (7.6-7.7) can be
written simply as
= −2r13 + 2r14 ,
= r13 − r14 .
Even more simply, in vector form, Eqs. (7.11-7.12) can be written as
= ν · r.
(7.12) (7.13) Here we have taken
ρ O2 (7.14) ν= −2 2
1 −1 (7.15) r= r13
r14 (7.16) CC BY-NC-ND. 18 November 2011, J. M. Powers. 230 CHAPTER 7. KINETICS IN SOME MORE DETAIL In general, we will have ρ be a column vector of dimension N × 1, ν will be a rectangular
matrix of dimension N × J of rank R, and r will be a column vector of length J × 1. So
Eqs. (7.11-7.12) take the form
dt ρO2 = −2 2
1 −1 r13
r14 (7.17) Note here that the rank R of ν is R = L = 1. Let us also deﬁne a stoichiometric matrix φ of
dimension L × N . The component of φ, φli represents the number of element l in species i.
Generally φ will be full rank, which will vary since we can have L < N , L = N , or L > N .
Here we have L < N and φ is of dimension 1 × 2:
φ= 1 2 . (7.18) Element conservation is guaranteed by insisting that ν be constructed such that
φ · ν = 0. (7.19) So we can say that each of the column vectors of ν lies in the right null space of φ.
For our example, we see that Eq. (7.19) holds:
φ·ν = 1 2 · −2 2
1 −1 =0 0. (7.20) The symbol R is the universal gas constant, where
R = 8.31441 J
mole K 107 erg
J = 8.31441 × 107 erg
mole K (7.21) Let us take as initial conditions
ρO (t = 0) = ρO , ρO2 (t = 0) = ρO2 . (7.22) Now M represents an arbitrary third body, so here
ρM = ρO2 + ρO . (7.23) Thus, the ordinary diﬀerential equations of the reaction dynamics reduce to
ρO ρO ρO2 + ρO
+2a14 T β14 exp
ρO2 ρO2 + ρO ,
= a13 T β13 exp
ρO ρO ρO2 + ρO
ρO2 ρO2 + ρO .
−a14 T β14 exp
= −2a13 T β13 exp
dt CC BY-NC-ND. 18 November 2011, J. M. Powers. (7.24) (7.25) ...
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- Fall '11
- Dynamics, Chemical reaction, J. M. Powers, ρo, ρO ρO ρM