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6.16: Untitled Page 139

  • Page ID
    18272
  • Chapter 6

    R

     C2H5OH 

    R

    C H

    1

    1

    1

    0

    0

    2 

    2

    4

    0

      R CH CHO 

     

    0

    1

    1 1 

    3

    1

    3 

     

    0

    (6)

     

    R

    0

    0

    1

    0 1

    1 

    H O

     

    2

    0

     

    R H2

    R C4 H6 

    Here we note that our original choice of non‐pivot species, C H OH , 2

    5

    C H and H O , has been changed by the application of Eq. 5 that leads 2

    4

    2

    to the non‐pivot species represented by C H OH , C H and CH CHO .

    2

    5

    2

    4

    3

    At this point we make use of some routine elementary row operations to obtain the desired row reduced echelon form

    R

    C2 H5OH

    R

    C2 H4

     1

    0

    0

    1

    1

    1

      

    0

      R CH

    3 CHO

     

    0

    1

    0 1

    0

    2

     

      0

    (7)

    R

     

    H

    2O

    0

    0

    1

    0

    1

    1

    0

     

    R

    H2

    R

    C

    4 H6

    Given

    this

    representation

    of

    Axiom

    II

    we

    can

    apply

    a

    column / row partition illustrated by

    (8)

    which immediately leads to

    Stoichiometry

    257

    1

    0

    0 

    R

      R

    C

    1

    1

    1

    0

    2H5OH

    H

    2O 

     

    0

    1

    0

    R

     

    1

    0

    2

    R

    0 (9)

    C

     

    2H4

    H2

    0

    0

    1 

     0

    1

    1 

    0

     

    R

      R

    

     

    CH

    

    3CHO

    C

    4H6 

    non‐pivot

    pivot

    submatrix

    submatrix

    Here the non‐pivot submatrix is the unit matrix that maps a column matrix onto itself as indicated by

    1

    0

    0 

    R

    R

    C

    2H5OH

    C

    2H5OH

    0

    1

    0

    R

    R

    (10)

    C2H4

    C2H4

    0

    0

    1 

      R

    R

    

    CH

    3CHO

    CH

    3CHO 

    non‐pivot

    submatrix

    Substitution of this result into Eq. 9 provides the following simple form

    R

      R

    C

    1

    1

    1

    2H5OH

    H

    2O 

    R

    1

    0

    2

    R

    (11)

    C

    2H4

    H2

     0 1

    1 

    R

      R

    CH

    

    3CHO

    C

    4H6 

    pivot

    submatrix

    From this we extract a representation for the column matrix of non‐pivot species in terms of the pivot matrix of stoichiometric coefficients and the column matrix of pivot species. This representation is given by

    R

    

    R

    C

    1

    1

    1

    2H5OH

    H

    2O 

    R

    1

    0 2

    R

    (12)

    C

    2H4

    H2

     0

    1 1 

    R

    R

    CH

    

    3CHO

    C

    4H6 

    

     

    pivot matrix

    column matrix

    column matrix

    of non‐pivot species

    of pivot species

    This is a special case of the pivot theorem in which we see that the net rates of production of the pivot species are mapped onto the net rates of production of the non‐pivot species by the pivot matrix. The matrix multiplication indicated in Eq. 12 can be carried out to obtain R

      R

    R

    R

    (13a)

    C2H5OH

    H2O

    H2

    C4H6

    258