Page 322 - Advanced thermodynamics for engineers

14.3 RATE CONSTANT FOR REACTION, k 311

and, for the backward direction

n n
d½A i Š Y 00 Y 00
¼ n n ij k bj ½A i Š ¼ Dn ij k bj ½A i Š : (14.11)
ij
ij
j 00 0 n n
ij
dt
b i¼1 i¼1
Hence the net rate of formation of A i is
" #
n n
Y Y
d½A i Š v 00 v 00
¼ Dv ij k fj ½A i Š k bj
ij ½A i Š ij : (14.12)
dt net i¼1 i¼1
Consider the following rate controlled reaction equation
k f
n a A þ n b B 5 n c C þ n d D; (14.13)
k b
the net rate of generation of species C is given by
d½CŠ
¼½k f ½AŠ ½BŠ k b ½CŠ ½DŠ Š: (14.14)
n c
n a
n b
n d
dt
Using the notation [A]/[A] e ¼ a, [B]/[B] e ¼ b,[C]/[C] e ¼ g, and [D]/[D] e ¼ d, where the suffix e
represents equilibrium concentrations, gives
d½CŠ=dt ¼ k f a b ½AŠ ½BŠ k b g d ½CŠ ½DŠ : (14.15)
n d
n a
n b
n c
n a n b
n c n d
e
e
e
e
At equilibrium
k f ½AŠ ½BŠ ¼ k b ½CŠ ½DŠ ¼ R j : (14.16)
n a
n d
n c
n b
e
e
e
e
Therefore the net rate is
d½CŠ=dt ¼ R j ½a b g d Š: (14.17)
n c n d
n a n b
14.3 RATE CONSTANT FOR REACTION, k
The rate constant for the reaction, k, is related to the ability of atoms or ions to combine. In combustion
engineering this will usually occur when two or more particles collide. Obviously the collision rate is a
function of the number of particles per unit volume, and their velocity of movement: i.e. their con-
centration and temperature. Most chemical reactions take place between two or three constituents
because the probability of more than three particles colliding simultaneously is too small. It has been
found experimentally that most reactions obey a law like that shown in Fig. 14.3.
This means that the rate constant for the reaction can be defined by an equation of the form
k ¼ Ae E=RT : (14.18)
This equation is called the Arrhenius equation. The factor A is called the pre-exponential factor, or
the frequency factor, and is dependent on the rate at which collisions with the required molecular
orientation occur. A sometimes contains a temperature term, indicating that the number of collisions is