REACTIONS: THERMODYNAMIC ASPECTS, KINETICS, AND CATALYSIS - Biomass as a Sustainable Energy Source for the Future - page 137

and for homogeneous reactions

1
V

dn i
dt =

dc i

dt = 1

dx i
dt

r = − − ð Eq : 5 : 7 Þ

V

Combining Equation (5.5), Equation (5.6), and Equation (5.7) and omitting subscript i :

r f = 1
V

dx

dt = k f V n 0

− x ð Eq : 5 : 8 Þ

ð x

= ð t
0

dx
n 0

k f dt ð Eq : 5 : 9 Þ

ð Þ − x

0

n 0
n 0

ln =k f t ð Eq : 5 : 10 Þ

ð Þ − x

x= n 0 n 0 x=n 0 e −k f t

−

e k f t ,

1 − ð Eq : 5 : 11 Þ

from which it is clear that the quantity of the reacting species decreases exponentially.

5.1.3 Bimolecular Second-Order Reactions

Second-order reactions can be of second order in one reactant or of first order in two
different reactants, i.e.,

2M ! products ð RX : 5 : 18 Þ

M 1 +M 2 ! products ð RX : 5 : 19 Þ

The reaction rate for the reaction in RX. 5.18 is given by

1
2

dc
dt =k f c 2

r f = − ð Eq : 5 : 12 Þ

or

2

n 0

1
2V

dx
dt =k f

ð Þ − x

V 2

V ð

2 =2 ð

x t

dx
n 0

k f dt ð Eq : 5 : 13 Þ

ð Þ − x

0 0

Vx
n 0 n 0

=2k f t

ð Þ − x