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9.16: Untitled Page 232

  • Page ID
    18365
  • Chapter 9

    present, and they usually are, it is appropriate to partition the mechanistic matrix into a stoichiometric matrix and a Bodenstein matrix as indicated by Eqs. 9‐110 through 9‐112. The general partitioning of Eq. 9‐127 can be expressed as

     R 

    S

    R

     

      M r 

    r

    (9‐129)

    M

     

     R

    B

    B 

     

    and this leads to forms analogous to Eqs. 9‐111 and 9‐112. We list the first of these results as

    Stoichiometric matrix:

    R

     S r

    (9‐130)

    in which S is the stoichiometric matrix composed of stoichiometric coefficients.

    The second of Eqs. 9‐129 is given by

    Bodenstein matrix:

    R

     B r

    (9‐131)

    B

    in which B represents the Bodenstein matrix. In general, the Bodenstein products are subject to the approximation of local reaction equilibrium that is expressed as

    Local reaction equilibrium:

    R

     0

    (9‐132)

    B

    and this allows one to extract additional constraints on the elementary chemical reaction rates. The result given by Eq. 9‐130 represents a key aspect of reactor design that can be expressed in more detailed form by

    B  K

    R

    S

    r ,

    A  1 , 2 , ..... N

    (9‐133)

    A

    AB B

    B  I

    Here S

    represents the stoichiometric coefficients, r represents the elementary AB

    B

    chemical reaction rates, and K represents the number of elementary reactions as indicated by Eq. 9‐31.

    9.5 Problems

    Section 9.1

    9‐1. Apply a column/row partition to show how Eq. 9‐6 is obtained from Eq. 9‐5.

    9‐2. Illustrate how a row/row partition leads from Eq. 9‐22 to Eq. 9‐23.

    Reaction Kinetics

    433

    9‐3. Use Eqs. 9‐34 through 9‐42 to obtain a local chemical reaction rate equation for species D.

    9‐4. Develop the local chemical reaction rate equation for species C on the basis of Eqs. 9‐34 through 9‐42.

    9‐5. Re‐write Eqs. 9‐40 and 9‐42 using the stoichiometric coefficients, 

    , 

    ,

    A I

    D II

    etc., in place of  ,  ,  , etc. Show that this change in the nomenclature leads to the form encountered in Eq. 6‐38.

    9‐6. Develop the representation for R given in Eqs. 9‐43.

    B

    9‐7. It is difficult to find a real system containing three species for which the reactions can be described by

    k

    1

    k 2

    A

    

    B  C

    (1)

    however, this represents a useful model for the exploration of the condition of local reaction equilibrium. The stoichiometric constraint for this series of first order reactions is given by

    R

    R R

     0

    (2)

    A

    B

    C

    since the three molecular species must all contain the same atomic species. For a constant volume batch reactor, and the initial conditions given by IC.

    o

    c

    c ,

    c

     0 ,

    c

     0 ,

    t  0

    (3)

    A

    A

    B

    C

    one can determine the concentrations and reaction rates of all three species as a function of time. If one thinks of species B as a reactive intermediate, the condition of local equilibrium takes the form

    Local reaction equilibrium:

    R

     0

    (4)

    B

    In reality, the reaction rate for species B cannot be exactly zero; however, we can have a situation in which

    R

     R

    (5)

    B

    A

    Often this condition is associated with a very large value of k , and in this 2

    problem you are asked to develop and use the exact solution for this process to determine how large is very large. You can also use the exact solution to see why

    434