# 8.6: Untitled Page 195

## Chapter 8

Figure 8‐7. Concentration as a function of time, linear scale EXAMPLE 8.1. First Order, Reversible Reaction in a Batch Reactor A variation of the first order irreversible reaction is the first order reversible reaction described by the following chemical kinetic schema: k 1

Chemical kinetic schema:



A 

B

(1)

k 2

Here k is the forward reaction rate coefficient and k is the reverse 1

2

reaction rate coefficient. The net rate of production of species A can be modeled on the basis of the picture represented by Eq. 1 and this leads to a chemical reaction rate equation of the form

Chemical reaction rate equation:

R

  k c k c

(2)

A

1 A

2 B

Here we remind the reader that in this text we use arrows to represent pictures and equal signs to represent equations.

Given an initial condition of the form

I.C.

o

c

c ,

c

 0 ,

t  0

(3)

A

A

B

we want to derive an expression for the concentration of species A as a function of time for the batch reactor illustrated in Figure 8.1. Transient Material Balances

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Figure 8.1. Reversible Reaction in a batch reactor

We begin the analysis with the species mole balance for a fixed control volume

d

c dV

R dV

(4)

A

A

dt V

V

and express this result in terms of volume averaged quantities to obtain dc

A

  R

(5)

A

dt

The chemical kinetic rate equation given by Eq. 2 can now be used to write Eq. 5 in the form

dc

A

  k c   k c

(6)

1

A

2

B

dt

In order to eliminate  c  from this result, we note that the development B

leading to Eq. 5 can be repeated for species B, and the use of R   R

B

A

provides

dc

B

  R     R

(7)

B

A

dt

From Eqs. 5 and 7 it is clear that

dc

dc

B

A

 

(8)

dt

dt

indicating that the rate of increase of the concentration of species B is equal in magnitude to the rate of decrease of the concentration of species A. We can use Eq. 8 and the initial conditions to obtain

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