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with the idea that the atomic matrix can always be expressed
in row reduced echelon form, and a uniqueness proof is
given in Section 3.8 of Noble.5 This means that we can use
a series of ‘‘elementary row operations’’ and column/row
operations to express Eq. 12 in the form (10) Axiom II : 2
3
3 R1
N1N 6
7
6 R2 7
7
N 2 N 76
76
76 R3 7
7
76
76 : 7
7
76
7
76
76 : 7
7
56
7
4 RN À 1 5
NTN
RN
23
0
67
607
67
¼ 607
ð12Þ
67
67
4:5
0 (11) A¼1 2
3
R
36 17
N1N
R
76 27
7
N2N 7 6
6 R3 7
7
7
: 76
76 : 7
6
7
7
: 76
76
:7
7
76
NWN 7 6
7 6 RT 7
7
7
0 76
76
7 6 RT þ 1 7
7
56
:
7
4:5
0
RN
23
0
607
67
67
607
¼6 7
ð14Þ
6:7
67
67
4:5
0 We refer to NJA as the ‘‘atomic species indicator,’’ and we
identify the array of coefﬁcients associated with NJA as the
‘‘atomic matrix.’’2 Here, we note that Axiom II has been
identiﬁed by Eqs. 2, 5, 8 and 11, and this is acceptable
because each one of these representations can be derived
from the others.3 The earliest reference that the authors have
found for Eq. 11 is given by Eq. 159A.3 of Truesdell and
Toupin.4
AIChE Journal February 2012 Vol. 58, No. 2 Here, we see that we have W rows of nonzero values and
T À W rows of zeros. This indicates that the rank of the
atomic matrix is r ¼ rank ¼ W and that we have T À W linearly dependent equations in the set of T equations. As the
row rank and the column rank must be the same, we have
N À W linearly dependent columns. The rank of the atomic
matrix may be less than T when two or more rows of the atomic Published on behalf of the AIChE DOI 10.1002/aic 539 matrix are a linear combination of each other. This occurs if
two or more atomic elements appear in each molecular species
in the same ratio. For example, in a system containing benzene
(C6H6) and acetylene (C2H2) the atomic matrix has r ¼ rank
¼ 1 with T ¼ 2. In compact notation we express Eq. 14 as
AÃ R ¼ 0
(15)
in which A* is the row reduced echelon form of the atomic
matrix.
The form of the atomic matrix given in Eq. 14 is crucial
to the efﬁcient application of Axiom II, and this form is
‘‘not universally identiﬁed’’ as the ‘‘row reduced echelon
form.’’ Sometimes one encounters the following deﬁnition of
a row reduced echelon matrix:
1 The ﬁrst nonzero element in any nonzero row is 1
(called a ‘‘leading 1’’).
2 The leading 1 in each nonzero row appears in a column in which every other element is 0.
3 In any two successive rows with nonzero elements, the
leading 1 of the lower row occurs farther to the right than
the leading 1 of the higher row.
4 Rows containing only zero elements are grouped
together at the bottom.
An example of an atomic matrix having these particular
characteristics is the following
2
3
12030405
60 0 1 7 0 8 0 97
7
A¼6
(16)
40 0 0 0 0 0 1 15
00000000
and we refer to this as simply a ‘‘row reduced form.’’
In the atomic matrix, one can interchange columns without affecting the result as long as the attendant rows of R
are also interchanged. In terms of Eq. 11, this column/
row interchange is expressed as
NJB RB ! NJD RD ; B; D ¼ 1; 2; :::; N and J ¼ 1; 2; :::; T
ð17Þ Returning to Eq. 16, we follow this rule
column #2 with column #3 to obtain
2
102304
60 1 0 7 0 8
0
C2 ! C3 : A ¼ 6
40 0 0 0 0 0
000000
We can now interchange column
provide
2
1003
60 1 0 7
C3 ! C7 : AÃ ¼ 6
40 0 1 0
0000 and interchange
0
0
1
0 3
5
97
7
15
0 (18) #3 with column #7 to
0
0
0
0 4
8
0
0 2
0
0
0 3
5
97
7
15
0 DOI 10.1002/aic (20)
In compact notation, this can be expressed as
!
RNP
¼0
½ I W
RP (21) Here, I represents the (T À W) Â (T À W) identity matrix, W represents the W Â [N À (T þ 1)] matrix of coefﬁcients, RNP represents the ‘‘nonpivot species’’ column matrix of net rates of production, and RP represents the
‘‘pivot species’’ column matrix of net rates of production.
Carrying out the matrix multiplication in Eq. 21 leads to
IRNP þ WRP ¼ 0 (22) We now deﬁne the pivot matrix as
P ¼ ÀW
to obtain the pivot theorem given by
RNP ¼ P RP Pivot Theorem: (23) Although Eqs. 2, 5, 11, and 12 often provide useful information, it is the pivot theorem that represents the most important result one can obtain from Axiom II. In the development
of the pivot theorem, the choice of pivot and nonpivot species
is not arbitrary. To obtain the row reduced echelon form
indicated by Eq. 14, it is a necessary condition that all the
atomic species be present in at least one nonpivot species.
This condition can be achieved by the initial arrangement of
the atomic matrix or by the operations indicated in Eq. 17.
When dealing with single independent reactions, such as
the complete combustion of ethane by ‘‘homogeneous reaction’’ with oxygen illustrated in Figure 1, one can ‘‘balance’’
the chemical reaction in terms of a ‘‘picture’’ by counting
atoms to obtain
1
7
3
C2 H6 þ O2 ! H2 O þ CO2
2
4
2 (19) which is the author’s version of a ‘‘row reduced echelon form.’’
This form follows directly if we replace Statement #3 in the
above definition of t...
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