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

3.5 Numerical integration 63
I
Exact solution using symbolic algebra software:
(1) To set-up the initial value problem for the program, 30 we enter the following a
1 × 6 matrix:
⎡
0 ⎤
= − 1
0 =+ 1 − 2
0 =+ 2
(3.160)
⎢ (0) = 0 ⎥
⎣
(0) = 0
⎦
(0) = 0
(2) To obtain the solution, we select the menu item "SolveODE, Exact". The program
gives the output:
Exact solution is:
⎡
⎢ () = 1 µ
⎤
− 1
0 1 − 0 1 2 + 0 2 − 2
1
1 2 − 1 2 − ⎥ 1
⎢
⎣
− 1
− 2
() = 0 1 − 0 1
2 − 1 2 − 1
() = 1 µ
− 1
0 1 2 − 0 2 − 1
1
1 2 − 1 2 − 1
⎥
⎦
(3.161)
(3) After simplifying, factorizing, and rewriting the output, we obtain
⎡
() = 0
¡ ¢ ⎤
2 − 1 + 1 − 2
− 2 − 1
2 − 1 ⎢
1
¡ ¢
⎣ () = 0
− 2
− − 1 ⎥
1 − ⎦
2
() = 0 − 1
(3.162)
We immediately recognize these results from section 2.
I Numerical solutions: Numerical integration is performed by the different symbolic
algebra programs using one of the methods outlined in the preceding subsection. We
have to specify the DE’s, the initial values, and the values for the rate constants. 31
(1) We adopt the rate constant values 1 =100 and 2 =100.
(2) We define the problem (now using capital letters):
⎡
0 ⎤
= −
0 =+ − 10
0 =+10
⎢ (0) = 1 ⎥
⎣
(0) = 0
⎦
(0) = 0
(3.163)
30 The different software packages differinthewaysinwhichtheproblemshavetobesetup.
31 SWP can handle the problem only if the rate constant values are explicitly entered in the DE’s.
Other software does better.