# 9.5: Untitled Page 221

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## Chapter 9

problem is to assume that the net rate of production of the *reactive intermediate* or the *Bodenstein product* can be approximated by

Local Reaction Equilibrium:

*R*

0

(9‐51)

*A*

This simplification is often referred to as the *steady‐state assumption* or the *steady* *state hypothesis* or the *pseudo steady state hypothesis*. These are appropriate phrases when kinetic mechanisms are being studied by means of a batch reactor; however, the phrase *local reaction equilibrium* is preferred since it is not process-dependent. Use of Eq. 9‐51 with Eq. 9‐50b leads to

I

II

III

2

*R*

*R*

*R*

*k c*

*k c*

*k c c*

0

(9‐52)

*A*

*A*

*A*

I *A*

II *A*

III *A A*

and from this we determine the concentration of the reactive intermediate to be 2

*k c*

I *A*

*c*

(9‐53)

*A*

*k * *k*

*c*

II

III *A*

We now make use of Eq. 9‐50c to express the net rate of production of ethane as I

II

III

*R*

*R*

*R * *R*

*R*

(9‐54)

*B*

*B*

*B*

*B*

C2H6

and application of Eqs. 9‐48c and 9‐48d provides the chemical reaction rate equation given by

*R*

*k c *

(9‐55)

C

2H6

II

*A*

At this point we use Eq. 9‐53 in order to express the net rate of production of ethane as

2

*k k c*

I

II *A*

*R*

C2H6

*k *

(9‐56)

*k*

*c*

II

III *A*

in which *c * represents the concentration of azomethane, (CH ) N . Here we *A*

3 2

2

can see that the two limiting rate expressions for high and low concentrations are given by

2

*k k c*

*k k / k*

*c , * high concentration

I

II *A*

I II

III

*A*

*R*

(9‐57)

C

2H6

2

*k*

*k*

*c*

II

III

*A*

*k c*

*, * low concentration

I *A*

411

which is consistent with the experimental results of Ramsperger (1927) illustrated by Eq. 9‐29. We can be more precise about what is meant by high concentration and low concentration by expressing these ideas as *c*

*k / k , *

high concentration

*A*

II

III

(9‐58)

*c*

*k / k , *

low concentration

*A*

II

III

Here we see that the relatively simple process suggested by Eq. 9‐27 is governed by the relatively complex rate equation indicated by Eq. 9‐56. The analysis leading to this result is based on *three* *concepts*: (A) local and elementary stoichiometry, (B) mass action kinetics, and (C) the approximation of local reaction equilibrium. The simplifying assumptions associated with this development are discussed in the following paragraphs.

Assumptions and Consequences

A reasonable *assumption* concerning the continuous stirred tank reactor shown in Figure 9‐5 is that only azomethane, ethane and nitrogen participate in the reaction. This assumption, in turn, leads to the chemical kinetic schema illustrated both in Eq. 9‐27 and in Figure 9‐6. Experimental measurement of the concentrations in the inlet and outlet streams might confirm the assumption that only (CH ) N , C H and N are present in the reactor. However, the 3 2

2

2

6

2

experimental determination of the reaction rate is not in agreement with Eq. 9‐28.

In reality, our analysis is based on the *restriction* that no *significant * amount of reactive intermediate enters or exits the reactor, and we state this idea as

at the entrance and

Restriction:

*c*

*c , c , c , *

(9‐59)

*A*

*A*

*B*

*C*

exit of the reactor

While the concentration of the reactive intermediate might be small compared to the other species, it is certainly not zero. If it were zero, Eq. 9‐55 would indicate that the rate of production of ethane would be zero and that is not in agreement with experimental observation.

Given that *c*

is not zero, one can wonder about the assumption (see *A*

Eq. 9‐51) that *R*

is zero. In reality, *R*

*A*

*A* must be *small enough* so that it can be

approximated by zero, and we need to know how small is small enough. To find out, we make use of Eq. 9‐50b to determine that the net rate of production of *A *

is given by

412