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
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