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A New Topology of Solutions of Chemical Equations
A New Topology of Solutions of Chemical Equations
Journal of the Korean Chemical Society. 2013. Apr, 57(2): 176-203
Copyright © 2013, Korea Chemical Society
  • Received : April 04, 2011
  • Accepted : January 21, 2013
  • Published : April 20, 2013
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Ice B. Risteski

Abstract
In this work is induced a new topology of solutions of chemical equations by virtue of point-set topology in an abstract stoichiometrical space. Subgenerators of this topology are the coefficients of chemical reaction. Complex chemical reactions, as those of direct reduction of hematite with a carbon, often exhibit distinct properties which can be interpreted as higher level mathematical structures. Here we used a mathematical model that exploits the stoichiometric structure, which can be seen as a topology too, to derive an algebraic picture of chemical equations. This abstract expression suggests exploring the chemical meaning of topological concept. Topological models at different levels of realism can be used to generate a large number of reaction modifications, with a particular aim to determine their general properties. The more abstract the theory is, the stronger the cognitive power is.
Keywords
INTRODUCTION
In this work for the first time in scientific literature is induced a topology of solutions of chemical equations. This topology is developed by virtue of a new algebraic analysis of subgenerators of coefficients of chemical reaction and theory of point-set topology. 1 2
Why did we do it? Simply speaking, it was necessary because the theory of balancing chemical equations worked only on determination of coefficients of reactions, without taking into account interactions among them. Was it correct? No! It was an artificial approach, which was used by chemists, only in order to determine quantification relations among reaction molecules and nothing more.
That so-called traditional approach with a minor scientific meaning, did not provide complete information for reaction character, just it represented only a rough reaction quantitative picture. Chemists by that approach, or more accurately speaking so-called chemical techniques balanced only very simple chemical equations. Their procedures were inconsistent and produced illogical results. The author of this work refuted all of them in his previous comprehensive work. 3
We open this algebraic analysis by examining three senses which the word topology has in our discourse.
  • The first sense is that proposed when we say thattopology is the constructive theory of relations among sets. We notice that we draw constructive conclusions from our topological data. Progressively we become aware that constructive topological calculations conducted according to certain norms can be depended on if the data are correct. The study of these norms, or principles has always been considered as a branch of applied topology. In order to distinguish topology of this sense from other senses introduced later, we shall call itapplied topology.
  • In the study ofapplied topologyit has been found productive to use mathematical methods, i.e., to construct mathematical systems having some connection therewith. What such a system is, and the nature of the connections, are questions which we shall consider later. The systems so formed are obviously a proper subject for study in themselves, and it is usual to apply the termtopologyto such a study. Topology in this sense is a branch of mathematics. To distinguish it from other senses, it will be calledmathematical topology.
  • In both of its preceding senses,topologywas used as a proper name. The word is also frequently used as a common noun, and this usage is a third sense of the word distinct from the first two. In this sense a topology is a system, or theory, such as one considers in mathematical or applied topology. Thus we may have algebraic topology, geometric topology, differential topology, etc.
This is as far as it is desirable to go, at present, in defining mathematical topology . As a matter of reality, it is ineffective to try to define any branch of science by delimiting accurately its boundaries; rather, one states the essential idea or purpose of the subject and leaves the boundaries to fall where they can. It is a benefit that the definition of topology is broad enough to admit different shades of opinion. Also, it will be allowable to speak of topological systems, topological algebras , without giving an accurate criterion for deciding whether a given system is such.
There are, however, several remarks which it is suitable to make now to intensify and illuminate the above discussion.
In the first place, we can and do consider topologies as formal structures, whose interest from the standpoint of applied topology may lie in some formal analogy with other systems which are more directly applicable.
In the second place, although the distinction between the different senses of topology has been stressed here as a means of clarifying our thinking, it would be a mistake to suppose that applied and mathematical topology are completely separate subjects. In fact, there is a unity between them. Mathematical topology, as has been said, is productive as a means of studying applied topology. Any sharp line between the two aspects would be arbitrary.
Finally, mathematical topology has a regular relation to the rest of mathematics. For mathematics is a deductive science, at least in the sense that a concept of exact proof is fundamental to all part of it. The question of what constitutes an exact proof is a topological question in the sense of the preceding discussion. The question therefore falls within the area of topology; since it is relevant to mathematics, it is expedient to consider it in mathematical topology. Thus, the task of explaining the nature of mathematical strictness falls to mathematical topology, and indeed may be regarded as its most essential problem. We understand this task as including the explanation of mathematical truth and the nature of mathematics generally. We express this by saying that mathematical topology includes the study of the foundation of chemistry, as that of abstract balancing chemical equations .
The part of topology which is selected for treatment may be described as the constructive theory of point-set topological calculus. That this topological calculus is central in modern topology does not need to be argued. Also, the constructive aspects of this topological calculus are fundamental for its higher study. Moreover, it is becoming more and more obvious that mathematicians in general need to be conscious of the difference between the constructive and nonconstructive, and there is hardly any better manner of increasing this consciousness than by giving a separate treatment of the former.
The conventional approach to the topological calculus is that it is a formal system like any other; it is unusual only in that it must be formalized more strictly, since we cannot take topology for granted, and in that it can be explained in the statements of usual discussion. Here the point of view is taken that we can explain our systems in the more restricted set of statements which we form in dealing with some other (unspecified) formal system.
Since in the study of a formal system we can form assertions which can not be decided by the expedients of that system, this brings in possibilities which did not arise, or seemed merely pathological, in the conventional theories.
It is an explanation for the word topology used in our discourse.
PRELIMINARIES
How are things right now in the theory of balancing chemical equations? We shall try to give a comprehensive reply to this question from the view point of chemistry as well as mathematics.
There are two approaches, competing with each other, for balancing chemical equations: chemical and mathematical.
1° First, we shall explain the chemical approach for balancing chemical equations.
In chemistry, there are lots of particular procedures for balancing chemical equations, but unfortunately all of them are inconsistent. In order to avoid reference repetition, we intentionally neglected to mention chemical references here, because a broad list with them is given in. 4 5 These informal procedures were founded by virtue of so called traditional chemical principles and experience, but not on genuine principles. Since, these principles were not formalized, they very often generated wrong results. It was a main cause for the appearance of a great number of paradoxes in theory of balancing chemical equations. These paradoxes were discovered and analyzed in detail in. 3 Furthermore, it is true that from the beginning of chemistry to date chemists did not develop their own general consistent method for balancing chemical equations. Why? Probably they must ask themselves!
2° Next we shall explain the mathematical approach for balancing chemical equations.
Simply speaking, that which chemists did not do, mathematicians did.
The earliest reference with a mathematical method (frequently referred to as the algebraic method or the method of undetermined coefficients ) of balancing chemical equations is that of Bottomley 6 published in 1878. A textbook written by Barker 7 in 1891 has devoted some space to this topic too. Unfortunately, the method proposed by Bottomley, more than fifty years, was out of usage, because it and his author both were forgotten. Endslow 8 illustrated this method again in 1931. It is not surprising, therefore, that even today the method is not broadly familiar to chemistry teachers.
The next very important step which mathematicians made is the transfer of problem of balancing of chemical equations from the field of chemistry to the field of mathematics. Jones 9 by virtue of the Crocker’s article 10 in 1971 proposed the general problem of balancing chemical equations. He formalized the century old problem in a compact linear operator form as a Diophantine matrix equation. Actually it is the first formalized approach in theory of balancing chemical equations.
The Jones’ problem waited for its solution only thirty-six years. In 2007, the author of this article by using a reflexive g -inverse matrix gave an elegant solution 4 of this problem, which generalized all known results in chemistry and mathematics.
Krishnamurthy 11 in 1978 gave a mathematical method for balancing chemical equations founded by virtue of a generalized matrix inverse. He considered some elementary chemical equations, which were well-known in chemistry for a long time.
Das 12 in 1986 offered a method of partial equations for balancing chemical equations. He described his method by elementary examples.
In 1989 Yde 13 criticized the half reaction method and proved that half equations can not be defined mathematically, so that they correspond exactly to the chemists’ idea of half reaction. He wrote: The half reaction method of balancing chemical equations has severe disadvantages compared to alternative methods. It is difficult to define a half reaction exactly, and thus to define a corresponding mathematical concept. Furthermore, existence and uniqueness proofs of the solutions (balanced chemical equations) require advanced mathematics . It shows that balancing chemical equations is not a piece of cake as some chemists think or as they like it to be. Yde by this article announced the need for the formalization of chemistry. Baby is on the way, but is not born!
Also, Yde in his article 14 offered a mathematical interpretation of the gain-loss rule . He wrote: It does not look elegant! Neither does the proof of it! But there is hardly anything we can do about this, if we demand a full mathematical presentation. In fact, the point is that ‘it is complicated’ . Actually, he made a modern version of Johnson’s derivation of the oxidation number method. 15
Subramaniam, Goh and Chia in 16 showed that a chemical equation is equivalent to a class of linear Diophantine equations.
A new general nonsingular matrix method for balancing chemical equations is developed in. 17 It is a formalized method, which include stability criteria for the general chemical equation.
The most general results for balancing chemical equations by using the Moore-Penrose pseudoinverse matrix 18 19 are obtained in. 5 Also, this method is formalized method, which belongs to the class of consistent methods.
In 20 is developed a completely new generalized matrix inverse method for balancing chemical equations. The offered method is founded by virtue of the solution of a homogeneous matrix equation by using von Neumann pseudoinverse matrix. 21 23 The method has been tested on many typical chemical equations and found to be very successful for all equations. Chemical equations treated by this method possessed atoms with fractional oxidation numbers. Furthermore, in this work are analyzed some necessary and sufficient criteria for stability of chemical equations over stability of their reaction matrices. By this method is given a formal way for balancing general chemical equation with a matrix analysis.
Other new singular matrix method for balancing chemical equations which reduce them to an n × n matrix form is obtained in. 24 This method is founded by virtue of the solution of a homogeneous matrix equation by using Drazin pseudoinverse matrix. 25
The newest mathematical method for balancing chemical equations is proved in. 26 This method is founded by virtue of the theory of n -dimensional complex vector spaces. Such looks the picture for balancing chemical equations that mathematicians painted.
We would like to emphasize here that all of the previously mentioned contemporary matrix methods 4 5 17 20 24 26 are rigorously formalized and consistent. Only such formalized methods are not contradictory and work successfully without any limitations. All other techniques or procedures known in chemistry have a limited usage and hold only for balancing simple chemical equations and nothing more. Most of them are inconsistent and produce only paradoxes.
Into a mathematical model must be introduced a whole set of auxiliary definitions to make the chemistry work consistently. Just this kind of set will be constructed in the next section.
Only on this way chemistry will be consistent and resistant to paradoxes appearance.
A NEW CHEMICAL FORMAL SYSTEM
In this section we shall develop a new chemical formal system founded by virtue of principles of a point-set topology.
Let
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is a finite set of molecules.
Definition 2.1. A chemical reaction on
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is a pair of formal linear combinations of elements of
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, such that
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with aij , bij ≧ 0
The coefficients xj , yj satisfy three basic principles (corresponding to a closed input-output static model)
  • the law of conservation of atoms,
  • the law of conservation of mass, and
  • the reaction time-independence.
What does it mean a chemical equation ? The reply of this question lies in the following descriptive definition given in a compact form.
Definition 2.2. Chemical equation is a numerical quantification of a chemical reaction .
In 5 is proved the following proposition.
Proposition 2.3. Any chemical equation may be presented in this algebraic form
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where xj , (1≤ j n ) are unknown rational coefficients, Ψ i aij and Ψ i bij , (1≤ i m ) are chemical elements in reactants and products, respectively, aij and bij , (1≤ i m ; 1≤ j n ; m n ) are numbers of atoms of elements Ψ i aij and Ψ i bij , respectively, in j-th molecule.
Definition 2.4. Each chemical reaction ρ has a domain
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Definition 2.5. Each chemical reaction ρ has an image
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Definition 2.6. Chemical reaction ρ is generated for some x
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, if both aij > 0 and bij > 0.
Definition 2.7. For the case as the previous definition, we say x is a generator of ρ .
Definition 2.8. The set of generators of ρ is thus Domρ Imρ .
Often chemical reactions are modeled like pairs of multisets, corresponding to integer stoichiometric constants.
Definition 2.9. A stoichiometrical space is a pair (
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), where
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is a set of chemical reactions on
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. It may be symbolized by an arc-weighted bipartite directed graph
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with vertex set
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, arcs x ρ with weight aij if aij > 0, and arcs ρ y with weight bij if bij > 0.
Let us now consider an arbitrary subset
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.
Definition 2.10. A chemical reaction ρ may take place in a reaction combination composed of the molecules in
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if and only if Domρ
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.
Definition 2.11. The collection of all possible reactions in the stoichiometrical space (
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), that can start from
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is given by
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Definition 2.12. Subgenerators of the chemical reaction (2.1) are the coefficient of its general solution
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where x k1 , x k2 , …, x k,n–r , ( n r ) are free variables.
Definition 2.13. For any subgenerator holds
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Definition 2.14. A sequence of vectors { x 1 , x 2 ,…, x k } is a basis of the chemical reaction (2. 1) if the vectors of solutions xi , (1≤ i k ) of (2. 2) are linearly independent and xi , (1≤ i k ) generate the vector space W of the solutions xi , (1≤ i k ).
Definition 2.15. The vector space W of the vectors of solutions xi , (1≤ i k ) of (2. 2) is said to be of finite dimension k, written dim W = k, if W contains a basis with k elements .
Definition 2.16. If W is a subspace of V, then the orthogonal complement W of (2.2) is
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Definition 2.17. The set X ⊂ℝ is a set of all the coefficients xj , (1≤ j n ) of the chemical equation (2.2) of the reaction (2.1).
Definition 2.18. Cardinality of the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1) is
CardX = | X | = n .
Definition 2.19. If X ⊂ℝ is a set of the coefficients xj , (1≤ j n ) of the chemical equation (2.2) of the reaction (2.1), then the power set of X , denoted by
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( X ), is the set of all subsets of X .
Definition 2.20. Cardinality of the power set
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( X ) of the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1) is
Card
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( X ) = 2 |X| = 2 n .
Definition 2.21. The set X ⊂ℝ of the coefficients xj , (1≤ j n ) of the chemical equation (2.2) of the reaction (2.1), is open, if it is a member of the topology .
Definition 2.22. The set X ⊂ℝ of the coefficients xj , (1≤ j n ) of the chemical equation (2.2) of the reaction (2.1), is called closed, if the complement ℝ\ X of X is an open set .
Definition 2.23. The interior of X is the union of all open sets contained in X ,
Int { X } = ∪ { Y X | Y open} = X o .
Definition 2.24. The exterior of X is the interior of the complement of X ,
Ext { X } = Int { Xc }.
Definition 2.25. The closure of X is the intersection of all closed sets containing X ,
Cl { X } = ∩ { Y X | Y closed} = X .
Definition 2.26. The boundary of X is
X = Cl { X } – Int { X } = X X o = Bd { X }.
Definition 2.27. A point x X is called isolated point of X if there exists a neighborhood Y of x such that Y X = { x }.
Definition 2.28. A point x X is called accumulation point or limit point of a subset A of X if and only if every open set Y containing x contains a point of A different from x , i.e.
Y open, x Y ⇒ { Y \ { x }} ∩ A ≠ ∅.
Definition 2.29. The set of accumulation points of X, denoted by X’, is called the derived set of X.
Definition 2.30. A class
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of subsets of X, whose elements are referred as the open sets, is called topology of the chemical equation (2.2) of the reaction (2.1) if the following axioms are satisfied
  • , where ∅ is an empty set,
The pair ( X ,
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) is called a topological space of solutions of the chemical equation (2.2) of the reaction (2.1).
Definition 2.31. A subset Y of the topological space ( X ,
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), of solutions of the chemical equation (2.2) of the reaction (2.1), is said to be dense in Z ⊂ ( X ,
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) if Z is contained in the closure of Y , i.e., B Cl { X }.
Definition 2.32. Let x be point in the topological space ( X ,
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), of solutions of the chemical equation (2.2) of the reaction (2.1). A subset
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of ( X ,
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) is a neighborhood of x if and only if
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is a superset of an open set Y containing x , i.e.,
x Y
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, Y open .
Definition 2.33. The class of all neighborhoods of x ∈ ( X ,
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), denoted by
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p , is called the neighborhood system of x .
Definition 2.34. Let Y be a non-empty subset of a topological space ( X ,
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). The class
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Y of all intersections of Y with
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- open subsets of X is a topology on Y; it is called the relative topology on Y or the relativization of
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to Y, and the topological space ( Y ,
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Y ) is called a subspace of ( X ,
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).
Definition 2.35. The discrete topology of the chemical equation (2.2) of the reaction (2.1) is the topology
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( X ) on X, where
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( X ) denotes the power set of X. The pair ( X ,
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) is called a discrete topological space of the chemical equation (2.2) of the reaction (2.1).
Definition 2.36. The indiscrete topology of the chemical equation (2.2) of the reaction (2.1) is the topology
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= {∅, X }. The pair ( X ,
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) is called an indiscrete topological space of the chemical equation (2.2) of the reaction (2.1).
Definition 2.37. The n-th complete Bell polynomial 27 is defined by
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where fr fr = (–1) r−1 ( r – 1)! and the summation is over all non negative integers satisfying the following conditions
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where ri , (1≤ i n ) are the numbers of parts of size i .
Definition 2.38. Let Yn ( x 1 , x 2 , …, xn ) denote the Bell polynomial with all fi set at unity. This particular Bell polynomial can be interpreted as an ordered-cycle indicator .
Definition 2.39. Let
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be the number of quasi-orders on the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.40. Let
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be the number of connected quasi-orders on the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.41. Let
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( n ) be the number of partial orders on the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.42. Let
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be the number of connected partial orders on the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.43. Let
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be the set of all topologies that can be defined on the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.44. Let
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be the set of all connected topologies that can be defined on the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.45. Let
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be the set of all
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- topologies that can be defined on the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.46. Let
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be the set of all connected
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- topologies that can be defined on the set X = { x 1 , x 2 , …, xn } of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.47.
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MAIN RESULTS
In this section we shall present our newest research results.
Theorem 3.1. Echelon form of the chemical equation (2.2) of the reaction (2.1) has one solution for each specification of n – r free variables if r < n.
Proof . According to the Theorem 4.2 from 5 the chemical reaction (2.1) reduces to (2.2), i.e., this system of linear equations
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The echelon form of the system (3.1) is
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where 1 < j 2 < ⋯ < jr and a 11 ≠ 0, a 2j2 ≠ 0, …, arjr ≠ 0, r n .
If we use mathematical induction for r = 1, then we have a single, nondegenerate, linear equation to which (3.2) applies when n r = 1. Thus the theorem holds for r = 1.
Now, suppose that r > 1 and that the theorem is true for a system of r – 1 equations. We shall consider the r – 1 equations.
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as a system in the unknowns x j2 ,…, xn . Note that the system (3.3) is in echelon form. By the induction hypothesis, we may arbitrary assign values to the ( n j 2 + 1) – ( r – 1) free variables in the reduced system to obtain a solution x j2 , …, xn . As in case r = 1, these values and arbitrary values for the additional j 2 – 2 free variables x 2 , …, x j2–1 , yield a solution of the first equation with
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Note that there are ( n j 2 + 1) − ( r − 1) + ( j 2 − 2) = n r free variables.
Furthermore, these values for x 1 , …, xn also satisfy the other equations since, in these equations, the coefficients x 1 , …, xj 2−1 are zero.
Theorem 3.2. Let echelon form of the chemical equation (2.2) of the reaction (2.1) has v free variables. Let xi , (1≤ i v ) be the solutions obtained by setting one of the free variables equal to one (or any nonzero constant) and the remaining free variables equal to zero. Then the solutions xi , (1≤ i v ) form a basis for solution space W of the chemical equation (2.2) of the reaction (2.1).
Proof . This means that any solution of the system (3.2) can be expressed as a unique linear combination of xi , (1≤ i v ). Thus, the dimension of W is dimW = v .
Theorem 3.3. Let chemical equation (2.2) of the reaction (2.1) is in echelon form (3.2). The basis of solution space W of chemical equation (2.2) of the reaction (2.1) are the solutions xi , (1≤ i n r ), such that dim W = n r .
Proof . The system (3.2) has n r free variables xi 1 , xi 2 , …, xi,n−r . The solution xj is obtained by setting xij = 1 (or any nonzero constant) and the remaining free variables are equal to zero. Then the solutions xi , (1≤ i n r ) form a basis of W and so dimW = n r .
Theorem 3.4. Let xi 1 , xi 2 , …, xir , be the free variables of the homogeneous system (3.2) of the chemical equation (2.2) of the reaction (2.1). Let xj be the solution for which xij = 1 and all other free variables are equal to zero. Solutions xi , (1 ≤ i r ) are linearly independent .
Proof . Let A be the matrix whose rows are the xi , respectively. We interchange column 1 and column i 1 , then the column 2 and column i 2 , …, and then column r and column ir , and obtain r × n matrix
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The above matrix B is in echelon form and so its rows are independent, hence rankB = r . Since A and B are column equivalent, they have the same rank, i.e., rankA = r . But A has r rows, hence these rows, i.e., the xi are linearly independent as claimed.
Theorem 3.5. The dimension of the solution space W of the chemical equation (2.2) of the reaction (2.1) is n r , where n is the number of molecules and r is the rank of the reaction matrix A .
Proof . If we take into account that
r = rankA = dim ( ImA )
and
n = dim n = dim ( DomA ),
then immediately follows
dimW = dim ( KerA ) = dim ( DomA ) − dim ( ImA ) = n r .
Corollary 3.6. If n = r, then dimW = 0, that means reaction (2.1) is impossible .
Corollary 3.7. If n = r + 1, then dimW = r + 1 − r = 1, that means that chemical equation (2.2) of the reaction (2.1) has a unique set of coefficients .
Corollary 3.8. If n > r + 1, then dimW > r + 1 − r > 1, that means that chemical equation (2.2) of the reaction (2.1) has an infinite number of sets of coefficients .
Remark 3.9. Those chemical reactions with properties of Corollary 3.8, we shall call continuum reactions, because they can be reduced to the Cantor’s continuum problem . 28
It shows that the balancing of chemical equations is neither simple nor easy matter. To date, these reactions were not seriously considered in scientific literature, or more accurately speaking these reactions were simply neglected, because their research looks for a very sophisticated and multidisciplinary approach. Just it was a challenge and main motive of the author of this work, to dedicate his research on these reactions.
Theorem 3.10. If
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is the class of subsets of consisting of and all subsets of of the form
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with n ∈ ℕ, then
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is a topology on and n open sets containing the positive integer n .
Proof . Since ∅ and
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, belong to
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,
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satisfies 1° of Definition 2.30. Furthermore, since
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is totally ordered by set inclusion,
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also satisfies 3° of Definition 2.30.
Now, let
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be a subclass of
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, i.e.,
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where I is some set positive integers. Note that I contains a smallest positive integer n 0 and
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which belongs to
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. We want to show that
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also satisfies 2° of Definition 2.30, i.e., that
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.
Case 1. If X
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, then
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, and therefore belongs to
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by 1° of Definition 2.30.
Case 2. If
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, then
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But the empty set ∅does not contribute any elements to union of sets; hence
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Since
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is a subclass of g ,
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is a subclass of
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, so the union of any number of sets in
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belongs to
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. Hence
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satisfies
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, and so
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is a topology on ℕ.
Since the non-empty open sets are of the form
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with n ∈ ℕ, the open sets contain the positive integer n are the following
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Theorem 3.11. Let
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be the topology on which consists of ∅and all subsets of of the form
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with n ∈ ℕ, then the derived set of Y = { y 1 , y 2 , … yn }, ( y 1 < y 2 < … < yn }) of the coefficients of the chemical equation (2.2) of the reaction (2.1) is Y’ = {1, 2, … yn }.
Proof . Observe that the open sets containing any point x ∈ℕ are the sets
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. If n 0 yn −1, then every open set containing n 0 also contains yn Y which is different from n 0 hence n 0 yn −1 is a limit point of Y . On the other hand, if n 0 yn −1 then the open set
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contains no point of Y different from n 0 . So n 0 yn −1 is not a limit point of Y . Accordingly, the derived set of Y is Y ’= {1, 2, …, yn }.
Theorem 3.12. If Y is any subset of a discrete topological space ( X ,
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), then derived set Y’of Y is empty .
Proof . Let x be any point in X . Recall that every subset of a discrete space is open. Hence, in particular, the singleton set G = { x } is an open subset of X . But
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Hence, x Y for every x X , i.e., Y = ∅.
Theorem 3.13. If Y is a subset of X, then every limit point of Y is also a limit point of X .
Proof . Recall that y Y if and only if { G \ { y }} ∩ Y ≠ ∅for every open set G containing y. But X Y therefore
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Theorem 3.14. A subset Y of a topological space ( X ,
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) is closed if and only if Y contains each of its accumulation points .
Proof . Suppose Y is closed, and let y Y , i.e., y Yc . But Yc , the complement of a closed set, is open; therefore y Y for Yc is an open set such that
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Thus Y Y if Y is closed.
Now assume Y Y ; we show that Yc is open. Let y Yc ; then y Y , so ∃ an open set G such that
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But, y Y ; hence G Y = { G \ { y }} ∩ Y = ∅.
So, G Yc . Thus y is an interior point of Yc , and so Yc is open.
Theorem 3.15. If Z is a closed superset of any set Y, then Y Z .
Proof . By Theorem 3.13, Y Z implies Y Z . But, Z Z by Theorem 3. 14, since Z is closed. Thus Y Z Z , which implies Y Z .
The last case, given by the Corollary 3.8., will be an object of research in the next section.
APPLICATION OF THE MAIN RESULTS
Let’s consider the reaction
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This reaction was an object of research in theory of metallurgical processes. There it was considered only from thermodynamic point of view. 29 30 This reaction was studied broadly, but only in some particular cases. Its general case will be an object of study just in this section.
On one hand, at once we would like to emphasize that this reaction belongs to the class of continuum reactions. It is according to the Remark 3.9. On other hand, it shows that it is a juicy problem which deserves to be studied and solved in whole.
Since the reaction (4.1) is very important for metallurgical engineering, chemistry and mathematics, just here we shall consider it from this multidisciplinary aspect. That aspect looks for a strict topological approach toward on total solution of (4.1). This total solution gives an opportunity to be seen both general solution of (4.1) and its particular solutions generated by the reaction subgenerators.
First, we shall look for its minimal solution which is crucial in theory of fundamental stoichiometric calculations and foundation of chemistry. For that goal, let construct its scheme.
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From the above scheme immediately follows reaction matrix
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with a rankA = 3.
It is well-known 11 that the reaction (4.1) can reduce in this matrix form
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where x = ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 ) T is the unknown vector of the coefficients of (4.1), 0 = (0, 0, 0) T is the zero vector and T denoting transpose.
The general solution of the matrix equation (4.2) is given by the following expression
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where I is a unit matrix and a is an arbitrary vector.
The Moore-Penrose generalized inverse matrix, for the chemical reaction (4.1), has this format
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For instance, by using the vector
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