# 8.5: Untitled Page 194

## 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

dc

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