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

3.4 Generalized first-order kinetics* 49
I Ansatz: The ansatz for solving Eq. 3.58 assumes that we can express A in terms of
another vector B, 25 A = P · B (3.59)
where P · B means the dot product, using a rotation matrix P which is defined so that
it diagonalizes K via the eigenvalue equation K·P = P · Λ , which by multiplication
from the left with P −1 yields
P −1 · K · P = Λ (3.60)
The eigenvalue matrix Λ is a diagonal matrix of the form
⎛ ⎞
1 0 0
Λ = ⎝ 0 2 0
0 0 . ⎠ (3.61)
..
As defined by Eq. 3.60, its elements on the diagonal { 1 2 } are the eigenvalues of
the rate constant matrix. ThematrixP is called the eigenvector matrix associated
with K. The columns of P are the respective eigenvectors
associated with the
respective eigenvalues .
I Solution: Inserting the ansatz
A = P · B (3.62)
into our matrix equation 26 ·
A = K · A (3.63)
we have
or
(P · B)
Muliplication of this equation from the left by P −1 gives
= K · P · B (3.64)
P · ·
B = K · P · B (3.65)
P −1 · P · ·
B = P −1 · K · P · B (3.66)
or
·
B = Λ · B (3.67)
This DE can be immediately integrated since Λ is diagonal. The solution is
B = Λ B 0 (3.68)
where Λ is a diagonal matrix with elements © 1 2 ª , and Λ B 0 means
element-wise multiplication. The result gives the time dependence of B, i.e., B()
starting from the initial values B 0 .
25 The components of B will be seen to be linear combinations of the components of the concentration
vector A. B is immediately obtained once we have the matrix of eigenvectors P (see below).
26 The dot above a concentration vector indicates differentiation by .