A New Generalized Matrix Inverse Method for Balancing Chemical ...

A New Generalized Matrix Inverse Method for Balancing Chemical Equations and their Stability 109
Corollary 4.9. Let A∈. Then stab(A) < 0 ⇔ m(A) < 0 for
some Lozinskiĭ measure m on n .
More results for stability criteria are obtained in works
[142, 143].
5. An Application of the Main Results
In this section will be applied above method on many chemical
equations for their balancing. All chemical equations balanced
here appear first time in professional literature and they are
chosen with an intention to be avoided to date all well known
chemical equations which were repeated many times in the
chemical journals for explanation of certain particular techniques
for balancing of some chemical equations using only
atoms with integer oxidation numbers.
1º First we will consider an infeasible reaction, i. e., the
case when the nullity of the reaction matrix is zero.
matrix is one. Here we will balance many special chemical
equations with a goal to show the power of the offered mathematical
method.
Example 5. 2. Consider this equation
x 1 [4Yb(CN) 3·3Yb(CN) 2 ] + x 2 CsRu(CN) 2 F 2 (5. 2)
+ x 3 CsRu(CN) 4 + x 4 CsHF 2 + x 5 [PtF 3·7H 2 O]
= x 6 [Pt(NH 3 ) 2 (C 5 H 4 ON)] 2 (NO 3 ) 2·2H 2 O
+ x 7 Cs 3.99 Yb(CN) 6 + x 8 HRuF 2.97 + x 9 NO 2 .
From the scheme given below
Example 5. 1. Consider chemical equation
x 1 Fe 2 (SO 4 ) 3 + x 2 PrTlTe 3 + x 3 H 3 PO 4 (5. 1)
= x 4 Fe(H 2 PO 4 ) 2·H 2 O + x 5 Pr 2 (SO 4 ) 3 + x 6 Tl 1.99 (SO 3 ) 3
+ x 7 Te 2 O 3 + x 8 H 2 O.
The reaction matrix
2 0 0 -1 0.00 0.00 0 0
3 0 0 0 -3.00 -3.00 0 0
12 0 4 -9 -12.0 -9.00 -3 -1
0 1 0 0 -2.00 0.00 0 0
0 1 0 0 0.00 -1.99 0 0 .
0 3 0 0 0.00 0.00 -2 0
0 0 3 -6 0.00 0.00 0 -2
0 0 1 -2 0.00 0.00 0 0
is obtained from this scheme
is derived the reaction matrix
⎡ 7.00 0 0 0 0.00 0.00 -1.00 0.00 0 ⎤.
⎢ 18.0 2 4 0 0.00 -10.0 -6.00 0.00 0 ⎥.
⎢ 18.0 2 4 0 0.00 -8.00 -6.00 0.00 -1 ⎥.
⎢ 0.00 1 1 1 0.00 0.00 -3.99 0.00 0 ⎥.
A = ⎢ 0.00 1 1 0 0.00 0.00 0.00 -1.00 0 ⎥.
⎢ 0.00 2 0 2 3.00 0.00 0.00 -2.97 0 ⎥.
⎢ 0.00 0 0 1 14.0 -24.0 0.00 -1.00 0 ⎥.
⎢ 0.00 0 0 0 1.00 -2.00 0.00 0.00 0 ⎥.
⎣ 0.00 0 0 0 7.00 -10.0 0.00 0.00 -2 ⎦.
The rank of the above matrix is r = 8. Since the nullity of
the reaction matrix is k = n - r = 8 - 8 = 0, then we have only a
trivial solution x = 0, that means that the reaction is infeasible.
2º Next, we will consider the case when the chemical reaction
is feasible and is unique, i. e., the nullity of its reaction
The rank of the above matrix is r = 8. Since the nullity of
the reaction matrix is k = n - r = 9 - 8 = 1, then we have a nontrivial
solution x ≠ 0, that means that the reaction is feasible.
Singular value decomposition of the matrix A is given by
the expression A = USV T , where