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9.10: Untitled Page 226

  • Page ID
    18359
  • Chapter 9

    R

     

    A

    1

    0

    1

     

    R

    r

    B

    1

    1

    1

    I

     

    R

      

    C

    1

    1

    1

    r

    (9‐88)

    II

     

    R

    1

    0

     1

    D

      r III 

    R

     0

    1

    0

    E

    Often it is convenient to express this result in the following compact form R

     M r

    (9‐89)

    M

    in which R is the column matrix of all the net rates of production, M is the M

    mechanistic matrix (Björnbom, 1977), and r is the column matrix of elementary chemical reaction rates. These quantities are defined explicitly by

    R

     

    A

    1

    0

    1

    R

    r

    B

    1

    1

    1

    I

    R

      R

    M

     

    r

     

    C

    ,

    1

    1

    1

    ,

    r

    (9‐90)

    M

    II

    R

    1

    0

     1

    D

    r III 

    R

     0

    1

    0 

    E

    

    mechanistic matrix

    When reactive intermediates, or Bodenstein products, are present, the mechanistic matrix is decomposed into a stoichiometric matrix and a Bodenstein matrix and we give an example of this situation in the following paragraphs.

    Here it is crucial to understand that the column matrix on the left hand side of Eq. 9‐88 consists of the net molar rates of production of all species including the reactive intermediates or Bodenstein products. It is equally important to understand that the column matrix on the right hand side of Eq. 9‐88 consists of chemical reaction rates that are not net molar rates of production. Instead they are chemical reaction rates defined by Eqs. 9‐82d, 9‐84d and 9‐85d. The definitions of these chemical reaction rates can be expressed explicitly as

    r

    k c c

    I

    I A B

    r

    r

    k c c

    (9‐91)

    II

    II B C

    r

    k c c

     III 

     III C D

    The matrix representations given by Eq. 9‐88 and Eq. 9‐91 can be used to extract the individual expressions for R , R , R , R and R that are given by A

    B

    C

    D

    E

    Reaction Kinetics

    421

    Species A:

    R

      k c c k c c

    (9‐92a)

    A

    I A B

    III C D

    Species B:

    R

      k c c k c c k c c

    (9‐92b)

    B

    I A B

    II B C

    III C D

    Species C:

    R

    k c c k c c k c c

    (9‐92c)

    C

    I A B

    II B C

    III C D

    Species D:

    R

    k c c k c c

    (9‐92d)

    D

    I A B

    III C D

    Species E:

    R

    k c c

    (9‐92d)

    E

    II B C

    In addition to extracting these results directly from Eq. 9‐88 and Eq. 9‐91, we can also obtain them from the schemata illustrated by Eqs. 9‐81 in the same manner that was used in Sec. 9.1.1. In Chapter 6 we made use of the pivot matrix that maps the net rates of product of the pivot species onto the net rates of production of the non‐pivot species. In this development we see that the mechanistic matrix maps the elementary chemical reaction rates onto all the net rates of production.

    At this point we note that the row reduced echelon form of the mechanistic matrix illustrated in Eq. 9‐90 is given by

     1

    0

    1

    0

    1

    0

    *

    M

      0

    0

    1

    (9‐93)

    0

    0

    0

     0

    0

    0

    This indicates that two of the net rates of production are linearly dependent on the other three. From Eqs. 9‐92 we obtain

    R

      R

    (9‐94a)

    D

    A

    R

      R R

    (9‐94b)

    C

    A

    E

    while the net rates of production for species A, B and E are repeated here as R

      k c c k c c

    (9‐94c)

    A

    I A B

    III C D

    R

      k c c k c c k c c

    (9‐94d)

    B

    I A B

    II B C

    III C D

    R

    k c c

    (9‐94e)

    E

    II B C

    422