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6.4: Untitled Page 127

  • Page ID
    18260
  • Chapter 6

    We refer to N

    as the atomic species indicator and we identify the array of JA

    coefficients associated with N

    as the atomic matrix (Amundson, 1966, page 54) .

    JA

    To illustrate the structure of the atomic matrix, we consider the complete oxidation of ethane illustrated in Figure 6‐1. That process provides the basis for the following visual representation of the atomic matrix: Molecular Species  C H

    O

    H O CO

    2

    6

    2

    2

    2

    carbon

     2

    0

    0

    1 

    (6‐15)

    hydrogen

    6

    0

    2

    0

    oxygen

     0

    2

    1

    2 

    This representation connects atoms with molecules in a convenient manner, and it is exactly what one uses to count atoms and balance chemical equations. There are two symbols that are useful for representing the atomic matrix. The first of these is given by 

     which has the obvious connection to Eq. 6‐14, while the

    JA

    N

    second is given by A which has the obvious connection to the name of this matrix. In this text we will encounter both representations for the atomic matrix as indicated by

    2 0 0 1

    2 0 0 1

    N   6 0 2 0 ,

    or

    A

    6 0 2 0

    (6‐16)

    JA

    0 2 1 2

    0 2 1 2

    In order to use the atomic species indicator, N

    , to construct an equation

    JA

    representing the concept that atoms are neither created nor destroyed by chemical reaction, we first recall the definition of R

    A

    net molar rate of production 

    R

     per unit volume of species A

    (6‐17)

    A

    owing to chemical reactions 

    which is consistent with the pictorial representation of R

    given earlier in

    CO2

    Figure 4‐1. Next we form the product of the atomic species indicator with R to A

    obtain

     number of moles of   net molar rate of production 

     

    N R

      J‐type atoms per moleper unit volume of species A (6‐18) JA

    A

    of molecular species A owing to chemical reactions

     

    

    Stoichiometry

    233

    A little thought will indicate that the product of N

    and R can be described as

    JA

    A

    net molar rate of production per unit 

    N R

     volume of J‐type atoms owing to the

    (6‐19)

    JA

    A

    molar rate of production of species A

    and the axiomatic statement given by Eq. 6‐13 takes the form (Truesdell and Toupin, 1960, page 473)

    A N

    Axiom II:

    N

    R

     0 ,

    J  1 , 2 ,...,T

    (6‐20)

    JA A

    A  1

    This equation represents a precise mathematical statement that atomic species are neither created nor destroyed by chemical reactions, and it provides a set of T equations that constrain the N net rates of production, R , A  1 , 2 ,...,N .

    A

    While Axiom II provides T equations, the equations are not necessarily independent. The number of independent equations is given by the rank of the atomic matrix and we will be careful to indicate that rank when specific processes are examined. If ions are involved in the reactions, one must impose the condition of conservation of charge as described in Appendix E. Some comments concerning heterogeneous reactions are given in Appendix F.

    The net rate of production of species A indicated by R can also be expressed A

    in terms of the creation and consumption of species A according to

     molar rate of creation of

    molar rate of consumption of 

    R

      species A per unit volume    species A per unit volume (6‐21) A

    owing to chemical reactions  owing to chemical reactions

    

    Here we need to think carefully about the description of R given by Eq. 6‐17

    A

    where we have used the word net to represent the sum of the creation of species A and the consumption of species A. This means that Eqs. 6‐17 and 6‐21 are equivalent descriptions of R and the reader is free to chose which ever set of A

    words is most appealing.

    If we make use of the atomic matrix and the column matrix of the net rates of production we can express Axiom II as

    234