Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

3.4 Generalized first-order kinetics* 53
• B 0 = P −1 A 0 :
• Solution for B ():
• Solution for A ():
B 0 = P −1 A 0 = P −1 ·
µ
10
0
⎛
⎜
= ⎝
− 10
1
1 + 2
10
1
1 + 2
⎞
⎟
⎠ (3.107)
B = Λ B 0 (3.108)
µ ⎛ ⎞
1
−( 1 + 2
=
) −
0 ⎜ 10
0 1 ⎝
1 + 2
⎟
1
⎠ (3.109)
10
1 + 2
=
⎛
⎜
⎝
− 10
1
1 + 2
× −( 1+ 2 )
10
1
1 + 2
× 1
A = P · Λ P −1 · A 0 = P · Λ P −1 ·
µ µ
−( 1 + 2
= P ·
) 0
P
0 1
−1 10
·
0
= P ·
= P ·
=
=
⎜
⎝
⎞
⎟
⎠ (3.110)
µ
10
µ ⎛ 1
−( 1 + 2 ) −
0 ⎜ 10
0 1 ⎝
1 + 2
1
10
1 +
⎛ µ
⎞ 2
−( 1+ 2 )
1
− 10
1 + 2
1
µ
10
1
1 + 2
⎛
2
−( 1+ 2 )
10 + 10 1
⎜ 1 + 2 1 + 2
⎝ 1
−( 1+ 2 )
10 − 10 1
1 + 2
⎛
2 + 1 −( 1+ 2 )
10
⎜ 1 + 2
⎝ 1 − −( 1+ 2 )
10 1
1 + 2
1 +
⎞ 2
0
⎞
(3.111)
(3.112)
⎟
⎠ (3.113)
⎟
⎠ (3.114)
⎞
⎟
⎠ (3.115)
⎟
⎠ (3.116)
This result is identical to the result for [A ()] obtained in Section 2.3.
3.4.2 Laplace transforms*
The concept of Laplace and inverse Laplace transforms is extremely useful in chemical
kinetics because