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6.8: Untitled Page 131

  • Page ID
    18264
  • Chapter 6

    One form of the atomic matrix for this group of molecular species can be visualized as

    Molecular Species  C H

    O

    H O CO

    2

    6

    2

    2

    2

    carbon

     2

    0

    0

    1 

    (1)

    hydrogen

    6

    0

    2

    0

    oxygen

     0

    2

    1

    2 

    nd

    a

    for this particular arrangement the atomic matrix is given by

    2 0 0 1

    N   6 0 2 0

    (2)

    JA

    0 2 1 2

    A simple calculation (see Problem 6‐5) shows

    t

    tha the rank of the

    matrix is

    three

    r  rank N   3

    (3)

    JA

    thus we have three equations and four unknowns. The three homogeneous equations that are analogous to Eqs. 6‐35 are given by

    2 R

     0  0  R

     0

    (4a)

    C2H6

    CO2

    6 R

     0  2 R

     0

     0

    (4b)

    C2H6

    H2O

    0  2 R

    R

     2 R

     0

    (4c)

    O2

    H2O

    CO2

    while the analogous matrix equation corresponding to Eq. 6‐22 takes the form

    R

    C2H6

    2 0 0 1

    0

    R

    O2

     

    6 0 2 0 

     

    0

    (5)

    R H O 

    2

    0 2 1 2

    0

     

     

    R

     CO

    2 

    It is possible to use intuition and the picture given by Eq. 6‐8 to express the net rates of production in the form

    1

    R

      R

    (6a)

    C2H6

    CO

    2

    2

    7

    R

      R

    (6b)

    O2

    C

    4

    O2

    Stoichiometry

    241

    3

    R

      R

    (6c)

    H2O

    CO

    2

    2

    however, the use of Eqs. 4 to produce this result is more reliable. Finally, we note that these results for the net rates of production can be expressed in the form of the pivot theorem that is described in Sec. 6.4. In terms of the pivot theorem that can be extracted from Eq. 5 we have

    R

    C

    

    2H6

    1 2

    R

    R

    (7)

    O

    7 4

    2

    CO

    2 

    

    

    3 2 

    R

    H

    column matrix

    

    2O 

    

    of pivot species

    pivot matrix

    column matrix

    of non‐pivot species

    This indicates that all the net rates of production are specified if we can determined the net rate of production for carbon dioxide, R

    . Indeed,

    CO2

    all the rates of production can be determined if we know any one

    of the fou

    r

    tes

    ra

    ; however, we have chosen carbon dioxide as the pivot species in order to arrange Eqs. 4 and Eq. 5 in the forms given by Eqs. 6 and 7.

    In the previous example we illustrated how Axiom II can be used to analyze the stoichiometry for the complete combustion of ethane. The process of complete combustion was described earlier by the single stoichiometric schema given by Eq. 6‐8 and the

    n

    coefficie ts that appeared in that schema

    are evident in

    Eqs. 6 and 7 of Example 6.1.

    6.2.5 Elementary row operations and column / row interchange operations In working with sets of equations such as those represented by Eq. 6‐22, we will make use of elementary row operations and column/row operations in order to

    ge

    arran

    the equations in a convenient form. Elementary row operations were described earlier in Sec. 4.9.1 and we list them here as they apply to the atomic

    matrix:

    I. Any row in the atomic matrix can be modified by

    multiplying or dividing by a non‐zero scalar without

    affecting the system of equations.

    II. Any row in the atomic matrix can be added or

    subtracted from another row without affecting the

    system of equations.

    242