Biochemical Reactions and Enzyme Kinetics - Introduction to Biomedical Engineering - page 453

8.1.3 Single-Stage Reversible Chemical Reaction
Next, consider the single-stage reversible chemical reaction as given in Eq. (8.13):

ð 8 : 13 Þ

Here, the chemicals A and B react to form the product P , and P has a reverse reaction to

form A and B. The law of mass action describing this system is given by

q P ¼ K 1 q A q B K 1 q P
q A ¼ K 1 q A q B þ K 1 q P
q B ¼ K 1 q A q B þ K 1 q P

ð 8 : 14 Þ

Equation (8.14) is nonlinear, which can be solved using SIMULINK or also mathematically,
as shown in the following example.

EXAMPLE PROBLEM 8.1
Consider the reaction given in Eq. (8.14), with q A ð 0 Þ¼ 10, q B ð 0 Þ¼ 15, and q P ð 0 Þ¼ 0, K 1 ¼ 2,

and K 1 ¼ 3 : Solve for q P :

Solution
From Eq. (8.14), it is clear that

q P ¼ q A ¼ q B

and after integrating, we have

q P q P ð 0 Þ¼ q A ð 0 Þ q A ¼ q B ð 0 Þ q B ð 8 : 15 Þ

To solve for

q P , we eliminate q a and q B using Eq. (8.15) by substituting q A ¼ q A ð

0 Þþ q P ð 0 Þ q P

and

q B ¼ q B ð

0 Þþ q P ð 0 Þ q P into Eq. (8.14),giving

q P ¼ K 1 q A q B K 1 q P ¼ K 1 ð q P ð 0 Þþ q A ð 0 Þ q P Þ q P ð ð 0 Þþ q B ð 0 Þ q P Þ K 1 q P ð 8 : 16 Þ

Since q P ð 0 Þ¼ 0, we have

q P ¼ K 1 ð q A ð 0 Þ q P Þ q B ð ð 0 Þ q P Þ K 1 q P

¼ 210 ð q P Þ ð 15 q P Þ 3 q P

0
@

1
A

ð 8 : 17 Þ

53
2 q P þ

2
P

¼ 2 q 150

and after rearranging terms

dq P dq P

150 ¼

Þ ¼

2 dt ð 8 : 18 Þ

53
2 q P þ

ð q P 18 : 3 Þ q P ð 8 : 2

q

2

P

Once again, partial fraction expansion is used to rewrite Eq. (8.18) as

dq P
q P

dq P
q P

0 : 0989

Þ

¼ 2 dt ð 8 : 19 Þ

ð 18 : 3 ð 8 : 2 Þ

Continued