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6.5: Untitled Page 128

  • Page ID
    18261
  • Chapter 6

    R

    1

    N

    N

    N

    . . . . N

    N

    11

    12

    13

    1 ,N1 ,

    1 N

      R

     

    2

    0

    N

    N

    .

    . . . . N

    N

    21

    22

    2 ,N1

    2 N  

     

    R

    0

    3 

     

    N

    N

    .

    . . . .

    .

    .

    Axiom II:

     31

    32

     

      0

    (6‐22)

    .

    .

    .

    .

    . . . .

    .

    .  

     

    .

    .

     

    .

    .

    .

    . . . .

    .

    .  

    0

    R

      N1

    N

    N

    .

    . . . . N

    N

    T 1

    T 2

    T ,N1

    TN   R

    N

    Everything we need to know about the conservation of atomic species is contained in this linear matrix equation; however, we need this information in different forms that will be developed in this chapter. In our development we will find patterns associated with the atomic matrix and these patterns will be connected to the physical problems under consideration.

    6.2.1 Axioms and theorems

    In dealing with axioms and proved theorems, it is important to accept the idea that the choice is not necessarily unique. From the authors’ perspective, Eq. 6‐20 and Eq. 6‐22 represent the preferred form of the axiom indicating that atoms are neither created nor destroyed by chemical reactions. We have identified both as Axiom II; however, we have also identified Eq. 6‐2 as Axiom II.

    Equation 6‐2 indicates that mass is conserved during chemical reactions while Eq. 6‐20 indicates that atoms are conserved during chemical reactions. Surely both of these are not independent axioms, thus we should be able to derive one from the other. In the following paragraphs we show how the axiom given by Eqs. 6‐20 can be used to prove Eq. 6‐2 as a theorem.

    To carry out this proof, we first multiply Eqs. 6‐20 by the atomic mass of the th

    J atomic species leading to

    A N

    AW

    N R

     0 , J  1 , 2 ,...,T

    (6‐23)

    J

    JA

    A

    A  1

    Here we have used AW to represent the atomic mass of species J in the same J

    manner that we have used MW to represent the molecular mass of species A.

    A

    We now sum Eq. 6‐23 over all atomic species to obtain

    Stoichiometry

    235

    J T

    A N

    AW

    N R

     0

    (6‐24)

    J

    JA

    A

    J  1

    A  1

    Since the sum over J is independent of the sum over A, we can place the sum over J inside the sum over A leading to the form A N J T

    AW N

    R

     0

    (6‐25)

      J JA A

    A  1 J  1

    We now note that R is independent of the process of summing over all J , thus A

    we can take R outside of the first sum and express Eq. 6‐25 as A

    A N

    J T

    R

    AW N

     0

    (6‐26)

     

    A

    J

    JA

    A  1

    J  1

    At this point we need only recognize that the molecular mass of species A is defined by

    J T

    MW

    AW

    N

    (6‐27)

    A

    J

    JA

    J  1

    in order to express Eq. 6‐26 as

    A N

    R MW

     0

    (6‐28)

    A A

    A  1

    Use of the definition given by Eq. 6‐5 leads to the following proved theorem: A N

    Proved Theorem:

    r

     0

    (6‐29)

    A

    A  1

    This result was identified as Axiom II by Eq. 6‐2 and earlier by Eq. 4‐11 in our initial exploration of the axioms for the mass of multicomponent systems. It should be clear from this development that one person’s axiom might be another person’s proved theorem. For example, if Eq. 6‐29 is taken as Axiom II, one can prove Eq. 6‐20 as a theorem. The proof is the object of Problem 6‐4, and it requires the constraint that the net rates of production are independent of the atomic masses, AW . We prefer Eq. 6‐20 as Axiom II since it can be used to J

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    236