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

3.4 Generalized first-order kinetics* 48
3.4 Generalized first-order kinetics*
3.4.1 Matrix method*
For coupled complex reaction systems with only first-order reactions, the solution of the
rate equations can be reduced to an eigenvalue problem (Jost 1974). The solution of
the rate equations, i.e., the calculation of the time-dependent concentration profiles,
requires finding the eigenvalues and eigenvectors of the rate constant matrix of the
system. 24
Eigenvalue problems are readily solved in practice using numerical methods (Press 1992).
We consider an example which we solved using conventional methods in Section 2.3.
I
Example:
The rate equations
A 1
1
À
2
A 2 (3.51)
[A 1 ]
= − 1 [A 1 ]+ 2 [A 2 ] (3.52)
[A 2 ]
=+ 1 [A 1 ] − 2 [A 2 ] (3.53)
canbewritteninamorecompactformasamatrix equation:
⎛ ⎞
[A 1 ] µ µ
⎜
⎝ ⎟ −1 +
[A 2 ]
⎠ =
2 [A1 ]
(3.54)
+ 1 − 2 [A 2 ]
With the concentration vector
and the rate constant matrix
K =
A =
µ
[A1 ]
[A 2 ]
µ
−1 + 2
+ 1 − 2
(3.55)
(3.56)
this becomes
·
A = K · A (3.57)
I
Generalized matrix rate equation for coupled first-order reaction systems:
·
A = K · A (3.58)
24 A brief overview on matrices and determinants is given in Appendix D.