# 8.6: Untitled Page 195

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- 18328

## 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 *d* *c *

*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

*d* *c *

*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

*d* *c *

*B*

*R * *R *

(7)

*B*

*A*

*dt*

From Eqs. 5 and 7 it is clear that

*d* *c *

*d* *c *

*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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