# 8.5: Untitled Page 194

- Page ID
- 18327

## Chapter 8

*R*

CH4

0

1 0 1 2

*R*

H

0

Axiom II:

2

4 2 0 6

(8‐27)

*R*

0

CO

0 0 1 1

0

*R*

C

2H6O

Making use of the *row reduced echelon form* of the atomic matrix and applying the pivot theorem given in Sec. 6.4 leads to

*R*

CH

1

4

*R*

1 *R*

(8‐28)

H

2

C

2 H6 O

1

*R*

CO

in which C H O has been chosen as the *pivot species*. Hinshelwood and Asky 2

6

(1927) found that the reaction could be expressed as a first order decomposition providing a rate equation of the form

Chemical reaction rate equation:

*R*

*k c*

(8‐29)

C2H6O

C2H6O

If we let dimethyl ether be species *A*, we can express the reaction rate equation as *R*

*A*

*k cA *

(8‐30)

Use of this result in Eq. 8‐25 leads to

*d* *c *

*A*

*k * *c *

*A *

(8‐31)

*dt*

and we require only an initial condition to complete our description of this process. Given the following initial condition

I.C.

o

*c * *c , *

*t * 0

*A*

*A*

(8‐32)

we find the solution for *c * to be

*A*

o

*k t*

*c * *c e*

(8‐33)

*A*

*A*

This simple exponential decay is a classic feature of first order, irreversible processes. One can use this result along with experimental data from a batch reactor to determine the first order rate coefficient, *k*. This is often done by

*Transient Material Balances *

363

plotting the logarithm of

o

*c * */ c * versus *t*, as illustrated in Figure 8‐6, and *A*

*A*

noting that the slope is equal to *k *.

*Figure 8‐6*. Batch reactor data, logarithmic scale

If the rate coefficient in Eq. 8‐33 is known, one can think of that result as a *design* *equation. * The idea here is that one of the key features of the design of a batch reactor is the specification of the *process time. * Under these circumstances, one is inclined to plot

o

*c * */ c * as a function of time and this is done in *A*

*A*

Figure 8‐7. The situation here is very similar to the mixing process described in the previous section. In that case the *characteristic time* was the average residence time, *V / Q *, while in this case the *characteristic time* is the inverse of the rate

coefficient,

1

*k *. When the rate coefficient is known one can quickly deduce that

the *process time* is on the order of

1

3 *k*

to

1

4 *k *. Very few reactions are as simple

as the first order irreversible reaction; however, it is a useful model for certain decomposition reactions.

364