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8.6: Untitled Page 195

  • Page ID
    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.

    index-374_1.png

    Transient Material Balances

    365

    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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