# 8.7: Untitled Page 196

- Page ID
- 18329

## Chapter 8

*c *

o

*B*

*cA*

*cA *

(9)

This result allows us to eliminate *c * from Eq. 6 leading to *B*

*d* *c *

*A*

o

(

1

*k*

*k* 2) *cA*

*k* 2 *cA *

(10)

*dt*

One can separate variables and form the indefinite integral to obtain 1

o

ln (

1

*k*

*k* 2) *c*

*k* 2 *c*

*t*

*C*

(

1

(11)

1

*k*

*k* 2)

*A*

*A*

where *C * is the constant of integration. This constant can be determined 1

by application of the initial condition which leads to

*k *

1

*k* 2

*k*

*c*

2

ln

*A*

( *k * *k *) *t *

(12)

o

1

2

*k*

1

*c*

*k*

1

*A*

An explicit expression for *c * can be extracted from Eq. 12 and the result *A*

is given by

*k*

*k*

( *k * *k *) *t*

o

2

1

1

2

*c *

(13)

*A*

*cA*

*e*

*k *

1 *k* 2

*k* 1 *k* 2

It is always useful to examine any special case that can be extracted from a general result, and from Eq. 13 we can obtain the result for a first order, irreversible reaction by setting *k * equal to zero. This leads to 2

*k t*

o

1

*c *

*A*

*cA e*

*, *

*k* 2

0

(14)

which was given earlier by Eq. 8‐33.

Under *equilibrium conditions*, Eq. 2 reduces to

*k c*

*k c , * for *R * 0

(15)

1 *A*

2 *B*

*A*

and this can be expressed as

*c*

*K c , * at equilibrium

(16)

*A*

*eq B*

Here *K * is the *equilibrium coefficient* defined by *eq*

367

*K*

*eq*

*k* 2 *k* 1

(17)

The general result expressed by Eq. 13 can also be written in terms of *k* 1

and *K * to obtain

*eq*

*K*

*eq*

1

*k*

*K*

*t*

*c *

o

*c *

1 (1

)

*eq*

*e*

(18)

*A*

*A *1 *K*

1 *K*

*eq*

*eq*

When *K * 1 we see that this result reduces to Eq. 14 as expected. In the *eq*

design of a batch reactor for a reversible reaction, knowledge of the equilibrium coefficient (or equilibrium relation) is crucial since it immediately indicates the limiting concentration of the reactants and products.

**8.3 Definition of Reaction Rate **

If one assumes that the batch reactor shown in Figure 8‐4 is a *perfectly mixed,* *constant volume* reactor, Eq. 8‐25 takes the form

*perfectly mixed, *

*dc*

*A*

*R , *

*constant volume *

(8‐34)

*A*

*dt*

* batch reactor *

Often there is a tendency to think of this result as defining the “reaction rate”

(Dixon, 1970) and this is a perspective that one must avoid. Equation 8‐34

represents a special form of the *macroscopic mole balance for species* *A* and it does not represent a *definition* of *R *. In reality, Eq. 8‐34 represents a very attractive *A*

special case that can be used with laboratory measurements to determine the species *A* *net molar rate of production*, *R *. Once *R * has been determined *A*

*A*

experimentally, one can search for chemical kinetic rate expressions such as that given by Eq. 8‐30, and details of that search procedure are described in Chapter 9. If successful, the search provides both a satisfactory form of the rate expression and reliable values of the parameters that appear in the rate expression. To be convinced that Eq. 8‐34 is not a definition of the reaction rate, one need only consider the perfectly mixed version of Eq. 8‐24 which is given by *dc*

*c*

*dV*( *t*)

*perfectly mixed, *

*A*

*A*

*R , *

(8‐35)

*dt*

*V*( *t*)

*A*

*dt*

* batch reactor *

368