ABSTRACT
A !k1 B !k2 C
d[A] dt ¼ k1[A]
d[B] dt ¼ k1[A] k2[B]
d[C] dt ¼ k2[B]
Integrated rate equations:
[A] ¼ [A]0 exp (k1t) [B] ¼ [B]0 exp (k2t)þ
k1[A]0 k2 k1 ( exp (k1t) exp (k2t))
[C] ¼ [C]0 þ [B]0(1 exp (k2t)þ [A]0 1þ k1 exp (k2t) k2 exp (k1t)
k2 k1
3. Three irreversible consecutive monomolecular reactions
A !k1 B !k2 C !k3 D
Differential rate equations:
d[A] dt ¼ k1[A]
d[B] dt ¼ k1[A] k2[B]
d[C] dt ¼ k2[B] k3[C]
d[D] dt ¼ k3[C]
Integrated rate equations:
[A] ¼ [A]0 exp (k1t) [B] ¼ [B]0 exp (k2t)þ
k1[A]0 k2 k1 { exp (k1t) exp (k2t)}
[C] ¼ [C]0 exp (k3t)þ [B]0 k2 exp (k2t)
k3 k2 k2 exp (k3t)
k3 k2
þ [A]0 k1k2 exp (k1t)
(k3 k1)(k2 k1) k1k2 exp (k2t)
(k3 k2)(k2 k1)þ k1k2 exp (k3t)
(k3 k2) (k3 k1)
4. One equilibrium monomolecular reaction
A ! k1
B
d[A] dt ¼ k1[A]þ k2[B]
d[B] dt ¼ k1[A] k2[B]
Integrated rate equations ([A]0 6¼ 0, [B]0¼ 0):
[A] ¼ [A]0 k1 þ k2 {k2 þ k1 exp (k1t) exp (k2t)} ¼
[A]0 k1 þ k2 {k2 þ k1 exp ((k1 þ k2)t}
[B] ¼ [A]0k1 k1 þ k2
[A]0k1 k1 þ k2 exp (k1t) exp (k2t) ¼
[A]0k1 k1 þ k2 {1 exp ((k1 þ k2)t)}
5. One equilibrium followed by one irreversible consecutive reaction
B !k3 C
Differential rate equations:
d[A] dt ¼ k1[A]þ k2[B]
d[B] dt ¼ k1[A] k2[B] k3[B]
d[C] dt ¼ k3[B]
Integrated rate equations ([A]0 6¼ 0, [B]0¼ [C]0¼ 0):
[A] ¼ [A]0 l2 l1 {(l2 k1) exp (l1t) (l1 k1) exp (l2t)}
[B] ¼ [A]0k1 l2 l1 { exp (l1t) exp (l2t)}
[C] ¼ [A]0 1 l2
l2 l1 exp (l1t)þ l1
l2 l1 exp (l2t)
l1 ¼ 0:5(p q) l2 ¼ 0:5(pþ q)
l1l2 ¼ k1k3 p ¼ k1 þ k2 þ k3 ¼ l1 þ l2 q ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 4k1k3ð Þ
p
6. Two reversible consecutive monomolecular reactions
A ! k1
B ! k1
C
d[A] dt ¼ k1[A]þ k2[B]
d[B] dt ¼ k1[A] k2[B] k3[B]þ k4[C]
d[C] dt ¼ k3[B] k4[C]
Integrated rate equations ([A]0 6¼ 0, [B]0¼ [C]0¼ 0):
[A] ¼ [A]0{T1 þ T2 exp (l2t)þ T3 exp (l1t)} [B] ¼ [A]0{T4 þ T5 exp (l2t)þ T6 exp (l1t)} [C] ¼ [A]0 [A] [B]
T1 ¼ k2k4 l2l1
T2 ¼ k1(l2 k3 k4) l2(l2 l1)
T3 ¼ k1(k3 þ k4 l1) l1(l2 l1)
T4 ¼ k1k4 l2l1
T5 ¼ k1(k4 l2) l2(l2 l1)
T6 ¼ k1(l1 k4) l1(l2 l1)
l1 ¼ P Q2
l2 ¼ P þ Q2 P ¼ k1 þ k2 þ k3 þ k4 Q ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 4(k1k3 þ k2k4 þ k1k4)
p
7. Reversible consecutive and irreversible parallel monomolecular reactions
d[A] dt ¼ k1[A]þ k2[B] k3[A]
d[B] dt ¼ k1[A] k2[B] k4[B]
d[C] dt ¼ k3[A]þ k4[B]
Integrated rate equations ([A]0 6¼ 0, [B]0¼ 0, [C]0¼ 0):
[A] ¼ [A]0 g1 g2
{(k2 þ k3 g1) exp (g1t) (k2 þ k3 g2) exp (g2t)}
[B] ¼ [A]0k1 g2 g1
{ exp (g1t) exp (g2t)}
[C] ¼ [A]0 1 g2 k4 g2 g1
exp (g1t) g1 þ k4 g1 g2
exp (g2t)
g1g2 ¼ k1k3 þ k2k4 þ k3k4 g1 þ g2 ¼ k1 þ k2 þ k3 þ k4
8. Reversible parallel and consecutive monomolecular reactions
Differential rate equations:
d[A] dt ¼ k1[A]þ k2[B] k5[A]þ k6[C]
d[B] dt ¼ k1[A] k2[B] k4[B]þ k3[C]
d[C] dt ¼ k5[A]þ k4[B] k6[C] k3[C]
Integrated rate equations ([A]0 6¼ 0, [B]0¼ 0, [C]0¼ 0):
[A] ¼ [A]0 b
g1g2 þ ag1 g
g2 g1 exp (g1t)þ
g22 ag2 þ b g2 g1
exp (g2t)
[B] ¼ [A]0 «
g1g2 þ k1g1 «
g2 g1 exp (g1t)þ
« k1g2 g2 g1
exp (g2t)
[B] ¼ [A]0 d
g1g2 þ k5g1 d
g2 g1 exp (g1t)þ
d k5g2 g2 g1
exp (g2t)
a ¼ k2 þ k3 þ k4 þ k6 b ¼ k2k4 þ k2k6 þ k3k6 « ¼ k1k4 þ k1k6 þ k4k5 d ¼ k1k3 þ k2k5 þ k3k5
g1g2 ¼ bþ «þ d g1 þ g2 ¼ aþ k1 þ k5
9. Irreversible bimolecular reaction
Aþ B!k C
The differential rate equations are
d[A] dt ¼ d[B]
dt ¼ k[A] [B]
d[C] dt ¼ k[A] [B]
and the integrated rate equations are for [A]0 6¼ 0, [B]0 6¼ 0 and [C]0¼ 0:
[A] ¼ ([B]0 [A]0) [B]0 [A]0
exp (([B]0 [A]0)kt) 1 1
[B] ¼ ([A]0 [B]0) [A]0 [B]0
exp (([A]0 [B]0)kt) 1 1
[C] ¼ [A]0 [A]0 [B]0
[B]0 exp (([B]0 [A]0)kt) [A]0 þ 1
A special case is for [A]0¼ [B]0, [C]0¼ 0:
[A] ¼ [B] ¼ [A]0 1þ [A]0kt
[C] ¼ kt[A] 2 0
1þ [A]0kt
and for 2A ! C with [C]0¼ 0:
[A] ¼ [A]0 1þ 2[A]0kt
[C] ¼ kt[A] 2 0
1þ 2[A]0kt
10. Reversible bimolecular reaction
Aþ B ! k1
C
d[A] dt ¼ k1[A] [B]þ k2[C]
d[B] dt ¼ k1[A] [B]þ k2[C]
d[C] dt ¼ k1[A] [B] k2[C]
For [A]0 6¼ 0, [B]0 6¼ 0 and [C]0¼ 0, the integrated equations are
[A] ¼ g1 g2R R 1
[B] ¼ ([A]0 [B]0 þ g1) R([A]0 [B]0 þ g2) R 1
[C] ¼ [A]0 þ g1 R([A]0 þ g2) 1 R
g1g2 ¼ [A]0k2 k1
g1 þ g2 ¼ [B]0 [A]0 þ k2 k1
R ¼ [A]0 þ g1 [A]0 þ g2
exp (k1(g2 g1)t)
A special case is when [A]0¼ [B]0, and [C]0¼ 0:
[A] ¼ g1 g2R R 1
[B] ¼ g1 Rg2) R 1
[C] ¼ [A]0 þ g1 R([A]0 þ g2) 1 R
g1g2 ¼ [A]0k2 k1
g1 þ g2 ¼ k2 k1
R ¼ [A]0 þ g1 [A]0 þ g2
exp (2k1(g2 g1)t)
Another special case is for
2A ! k1
C
and the integrated equations for [C]0¼ 0 are
[A] ¼ g1 g2R R 1
[C] ¼ [A]0 þ g1 R([A]0 þ g2) 2(1 R)
g1g2 ¼ [A]0k2 2k1
g1 þ g2 ¼ k2 2k1
R ¼ [A]0 þ g1 [A]0 þ g2
exp (2k1(g2 g1)t)
11. First-order kinetics with respect to one product and one reactant (autocatalysis). This is the situation where a product reacts with the reactant; it is thus autocatalytic.
