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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

    dc

    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

    Transient Material Balances

    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

    index-377_1.png

    index-377_2.png

    index-377_3.png

    368