# 9.6: Untitled Page 222

- Page ID
- 18355

## Chapter 9

2

*R*

*k c * *k * *k c c *

(9‐60)

*A*

I *A*

II III *A* *A*

and we use this result to show that the concentration of the reactive intermediate takes the form

2

*k c*

I *A*

*R*

*c*

*A*

(9‐61)

*A*

*k * *k*

*c*

*k * *k*

*c*

II

III *A *

II III *A*

Use of this result in Eq. 9‐55 leads to the net rate of production of C H given by 2

6

2

*k k c*

*R*

*k*

I

II *A*

*A* II

*R*

(9‐62)

C2H6

*k * *k*

*c*

II

III *A*

*k * *k c*

II

III *A *

Here we see that if the second term on the right hand side is negligible compared to the first term, we obtain the result given earlier by Eq. 9‐56. This indicates that the *assumption* given by *R*

0 is a reasonable substitute for the *restriction* given *A*

by

Restriction:

2

*R*

*k c *

(9‐63)

*A*

I *A*

When this inequality is imposed on Eq. 9‐62, we obtain the result given previously by Eq. 9‐56 *provided that* we are willing to assume that *small causes* *produce small effects* (Birkhoff, 1960). Even though Eqs. 9‐51 and 9‐63 lead to the same result, Eq. 9‐63 should serve as a reminder that neglecting something that is *small* always requires the crucial assumption that small causes produce small effects.

One important part of this analysis is the fact that the assumption concerning *c*

at entrances an exits cannot be extended into the reactor where finite values *A*

of the concentration of the *reactive intermediate* control the rate of reaction. This is clearly indicated by Eq. 9‐55. The situation we have encountered in this study occurs often in the transport and reaction of chemical species and can generalized as:

Sometimes a small quantity, such as *R*

*c*

*A* or *A* , can be

ignored and set equal to zero for the purposes of

analysis. Sometimes a small quantity cannot be ignored

and setting it equal to zero represents a serious mistake.

Knowing when small causes produce small effects requires experience, intuition, experiment and analysis. These are skills that are acquired steadily over time.

*Reaction Kinetics *

413

In this section we have examined the concepts of global, local, and elementary stoichiometry, along with the concept of mass action kinetics. We have made use of *pictures* to describe both elementary stoichiometry and elementary chemical kinetics, and we have illustrated how these pictures are related to *equations*. The concept of *local reaction equilibrium*, also known as the *steady‐state assumption* or the *steady state hypothesis* or the *pseudo steady‐state hypothesis*, has been applied in order to develop a simplified rate expression for the production of ethane and nitrogen from azomethane. The resulting rate expression compares favorably with experimental observations.

**9.2 Michaelis‐Menten Kinetics**

In Sec. 8.4 we presented a brief analysis of the cell growth phenomena that occurs in a chemostat. In addition, we presented the well‐known Monod equation that has been used extensively to model *macroscopic* cell growth. In this section, we briefly explore an enzyme‐catalyzed reaction that occurs in all cellular systems. Within a cell, such as the one illustrated in Figure 9‐9, hundreds of reactions occur. To appreciate the complexity of cells, we note that *Figure 9‐9*. Transport and reaction in a cell

a typical eukaryotic cell contains the following subcellular components: nucleolus, nucleus, ribosome, vesicle, rough endoplasmic reticulum, Golgi apparatus, Cytoskeleton, smooth endoplasmic reticulum, mitochondria, vacuole, cytoplasm, lysosome, and centrioles within centrosome (Segel, 1993). Obviously the cell is a busy place, and much of that business is associated with the enzyme-catalyzed reactions that produce intracellular material represented by species *D*

and extracellular material represented by species *C*. Species *D* provides the material that leads to cell growth as described in Sec. 8.4, while species *C*

provides desirable products to be harvested by chemical engineers and others.

414