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Unformatted text preview: 5.5. CHEMICAL KINETICS OF A SINGLE ISOTHERMAL REACTION 179 • The species concentrations of species involved in the reverse reaction decrease.
Here, three intermediate variables which are in common usage have been deﬁned. First one
takes the reaction rate to be β r ≡ aT exp N −E
RT ≡k (T ) ν′
ρk k k =1
forward reaction or β r = aT exp −E
RT ≡k (T ), Arrhenius rate N 1 1 −
ρk k k =1
forward reaction N ρkνk k =1 reverse reaction 1
Kc N ν ′′
ρk k k =1 reverse reaction law of mass action , . (5.280) (5.281) The reaction rate r has units of kmole/m3 /s.
The temperature-dependency of the reaction rate is embodied in k (T ) is deﬁned by what
is known as an Arrhenius rate law:
RT k (T ) ≡ aT β exp . (5.282) This equation was advocated by van’t Hoﬀ in 1884; in 1889 Arrhenius gave a physical justiﬁcation. The units of k (T ) actually depend on the reaction. This is a weakness of the theory,
and precludes a clean non-dimensionalization. The units must be (kmole/m3 )(1− k=1 νk ) /s.
In terms of reaction progress, one can also take
r= 1 dζ
V dt (5.283) The factor of 1/V is necessary because r has units of molar concentration per time and ζ
has units of kmoles. The over-riding importance of the temperature sensitivity is illustrated
as part of the next example. The remainder of the expression involving the products of the
species concentrations is the deﬁning characteristic of systems which obey the law of mass
action. Though the history is complex, most attribute the law of mass action to Waage and
Guldberg in 1864.1
Last, the overall molar production rate of species i, is often written as ωi , deﬁned as
ωi ≡ νi r.
˙ (5.284) 1 P. Waage and C. M. Guldberg, 1864, “Studies Concerning Aﬃnity, Forhandlinger: Videnskabs-Selskabet
i Christiania, 35. English translation: Journal of Chemical Education, 63(12): 1044-1047.
CC BY-NC-ND. 18 November 2011, J. M. Powers. 180 CHAPTER 5. THERMOCHEMISTRY OF A SINGLE REACTION As νi is considered to be dimensionless, the units of ωi must be kmole/m3 /s.
Study the nitrogen dissociation problem considered in an earlier example, see p. 142, in which at
t = 0 s, 1 kmole of N2 exists at P = 100 kP a and T = 6000 K . Take as before the reaction to be
isothermal and isochoric. Consider again the elementary nitrogen dissociation reaction
N2 + N2 ⇌ 2 N + N2 , (5.285) which has kinetic rate parameters of
a = 7.0 × 1021 β = cm3 K 1.6
mole s −1.6, E = (5.286)
mole (5.288) In SI units, this becomes
3 a = E = m3 K 1.6
cm3 K 1.6
= 7.0 × 1018
7.0 × 1021 (5.289)
(5.290) At the initial state, the material is all N2 , so PN2 = P = 100 kP a. The ideal gas law then gives at
ρN 2 t=0 =
= PN2 |t=0 = ρN2 t=0 RT, P |t=0
RT (5.292) 100 kP a
8.314 kmole K (6000 K )
= 2.00465 × 10−3
Thus, the volume, constant for all time in the isochoric process, is
= V= nN2 |t=0
2.00465 × 10−3 (5.291) kmole
m3 = 4.9884 × 102 m3 . (5.293)
(5.294) (5.295) Now the stoichiometry of the reaction is such that
− dnN2 = −(nN2 − nN2 |t=0 ) = 1
(nN − nN |t=0 ),
2 ρN (5.297) =0 =1 kmole nN
V (5.296) =
= 2(1 kmole − nN2 ),
1 kmole nN2
− ρN 2 ,
4.9884 × 102 m3
− ρN 2 .
2 2.00465 × 10−3
m3 CC BY-NC-ND. 18 November 2011, J. M. Powers. (5.298)
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This note was uploaded on 11/26/2011 for the course EGN 3381 taught by Professor Park-sou during the Fall '11 term at FSU.
- Fall '11