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

3.4 Generalized first-order kinetics* 51
• Conclusions:
(1) The largest (negative) eigenvalue ( 2 in this case) determines the shortest
time scale and thus the short-time evolution of the reactant concentrations!
(2) The smallest (negative) eigenvalue ( 1 in this case) determines the longest
time scale of the reaction, i.e., the evolution of the reactant concentrations
at long times. The smallest (negative) eigenvalue thus determines the net
reaction rate!
(3) Note that in the example the concentrations at long times are constant
(equilibrium!) and therefore 1 =0.
• Evaluation of the eigenvectors p 1 und p 2 by inserting the eigenvalues { 1 2 }
one by one into
K · P = Λ · P (3.81)
yields, for each and the respective p , a linear equation of the type
K · p = · p (3.82)
y
y
(K − ) · p =0 (3.83)
(− 1 − ) 1 + 2 2 =0 (3.84)
1 1 +(− 2 − ) 2 =0 (3.85)
• Inserting 1 =0:
− 1 1 + 2 2 =0 (3.86)
1 1 + − 2 2 =0 (3.87)
y
y
1 1 = 2 2 (3.88)
1 = 2 · (3.89)
2 = 1 · (3.90)
y
• Inserting 2 = − ( 1 + 2 ):
p 1 =
µ
2 ·
1 ·
(3.91)
(− 1 + 1 + 2 ) 1 + 2 2 =0 (3.92)
1 1 +(− 2 + 1 + 2 ) 2 =0 (3.93)
y
2 1 + 2 2 =0 (3.94)
1 1 + 1 2 =0 (3.95)