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.12
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 that topology 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 it applied topology. In the study of applied topology it 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 term topology to such a study. Topology in this sense is a branch of mathematics. To distinguish it from other senses, it will be called mathematical topology. In both of its preceding senses, topology was 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.45 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 Bottomley6 published in 1878. A textbook written by Barker7 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. Endslow8 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. Jones9 by virtue of the Crocker’s article10 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 solution4 of this problem, which generalized all known results in chemistry and mathematics.
Krishnamurthy11 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.
Das12 in 1986 offered a method of partial equations for balancing chemical equations. He described his method by elementary examples.
In 1989 Yde13 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 article14 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 in16 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 matrix1819 are obtained in.5 Also, this method is formalized method, which belongs to the class of consistent methods.
In20 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 methods4517202426 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 is a finite set of molecules.
Definition 2.1. A chemical reaction on is a pair of formal linear combinations of elements of , such that
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, andthe 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.
In5 is proved the following proposition.
Proposition 2.3. Any chemical equation may be presented in this algebraic form
where xj, (1≤j≤n) are unknown rational coefficients, Ψiaij and Ψibij, (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 Ψiaij and Ψibij, respectively, in j-th molecule.
Definition 2.4. Each chemical reaction ρ has a domain
Definition 2.5. Each chemical reaction ρ has an image
Definition 2.6. Chemical reaction ρ is generated for some x∈, 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 (), where is a set of chemical reactions on . It may be symbolized by an arc-weighted bipartite directed graph with vertex set , 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 .
Definition 2.10. A chemical reaction ρ may take place in a reaction combination composed of the molecules in if and only if Domρ ⊆.
Definition 2.11. The collection of all possible reactions in the stoichiometrical space (), that can start from is given by
Definition 2.12. Subgenerators of the chemical reaction (2.1) are the coefficient of its general solution
where xk1, xk2, …, xk,n–r, (n＞r) are free variables.
Definition 2.13. For any subgenerator holds
Definition 2.14. A sequence of vectors {x1, x2,…, xk} 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
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 = {x1, x2, …, 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 (X), is the set of all subsets of X.
Definition 2.20. Cardinality of the power set (X) of the set X = {x1, x2, …, xn} of the coefficients of the chemical equation (2.2) of the reaction (2.1) is
Card(X) = 2|X| = 2n.
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} = Xo.
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− – Xo = 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 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
The pair (X, ) 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, ), of solutions of the chemical equation (2.2) of the reaction (2.1), is said to be dense in Z ⊂ (X, ) 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, ), of solutions of the chemical equation (2.2) of the reaction (2.1). A subset of (X, ) is a neighborhood of x if and only if is a superset of an open set Y containing x, i.e.,
x ∈ Y ⊂ , Y open.
Definition 2.33. The class of all neighborhoods of x ∈ (X, ), denoted by p, is called the neighborhood system of x.
Definition 2.34. Let Y be a non-empty subset of a topological space (X, ). The class Y of all intersections of Y with -open subsets of X is a topology on Y; it is called the relative topology on Y or the relativization of to Y, and the topological space (Y, Y) is called a subspace of (X, ).
Definition 2.35. The discrete topology of the chemical equation (2.2) of the reaction (2.1) is the topology (X) on X, where (X) denotes the power set of X. The pair (X, ) 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 = {∅, X}. The pair (X, ) 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 polynomial27 is defined by
where fr ≡ fr = (–1)r−1(r – 1)! and the summation is over all non negative integers satisfying the following conditions
where ri, (1≤i≤n) are the numbers of parts of size i.
Definition 2.38. Let Yn (x1, x2, …, 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 be the number of quasi-orders on the set X = {x1, x2, …, xn} of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.40. Let be the number of connected quasi-orders on the set X = {x1, x2, …, xn} of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.41. Let (n) be the number of partial orders on the set X = {x1, x2, …, xn} of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.42. Let be the number of connected partial orders on the set X = {x1, x2, …, xn} of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.43. Let be the set of all topologies that can be defined on the set X = {x1, x2, …, xn} of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.44. Let be the set of all connected topologies that can be defined on the set X = {x1, x2, …, xn} of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.45. Let be the set of all -topologies that can be defined on the set X = {x1, x2, …, xn} of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.46. Let be the set of all connected -topologies that can be defined on the set X = {x1, x2, …, xn} of the coefficients of the chemical equation (2.2) of the reaction (2.1).
Definition 2.47.
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 from5 the chemical reaction (2.1) reduces to (2.2), i.e., this system of linear equations
The echelon form of the system (3.1) is
where 1 ＜ j2 ＜ ⋯ ＜ jr and a11 ≠ 0, a2j2 ≠ 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.
as a system in the unknowns xj2,…, xn. Note that the system (3.3) is in echelon form. By the induction hypothesis, we may arbitrary assign values to the (n – j2 + 1) – (r – 1) free variables in the reduced system to obtain a solution xj2, …, xn. As in case r = 1, these values and arbitrary values for the additional j2 – 2 free variables x2, …, xj2–1, yield a solution of the first equation with
Note that there are (n − j2 + 1) − (r − 1) + ( j2 − 2) = n − r free variables.
Furthermore, these values for x1, …, xn also satisfy the other equations since, in these equations, the coefficients x1, …, xj2−1 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 xi1, xi2, …, 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 xi1, xi2, …, 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 i1, then the column 2 and column i2, …, and then column r and column ir, and obtain r×n matrix
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 = dimℝn = 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 is the class of subsets of ℕ consisting of ∅ and all subsets of ℕ of the form with n ∈ ℕ, then is a topology on ℕ and n open sets containing the positive integer n.
Proof. Since ∅ and , belong to , satisfies 1° of Definition 2.30. Furthermore, since is totally ordered by set inclusion, also satisfies 3° of Definition 2.30.
Now, let be a subclass of , i.e., where I is some set positive integers. Note that I contains a smallest positive integer n0 and
which belongs to . We want to show that also satisfies 2° of Definition 2.30, i.e., that .
Case 1. If X ∈ , then
, and therefore belongs to by 1° of Definition 2.30.
Case 2. If , then
But the empty set ∅does not contribute any elements to union of sets; hence
Since is a subclass of g , is a subclass of , so the union of any number of sets in belongs to . Hence satisfies , and so is a topology on ℕ.
Since the non-empty open sets are of the form with n ∈ ℕ, the open sets contain the positive integer n are the following
Theorem 3.11. Let be the topology on which consists of ∅and all subsets of ℕ of the form with n ∈ ℕ, then the derived set of Y = {y1, y2, … yn}, (y1 < y2 < … < 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 . If n0 ≤yn −1, then every open set containing n0 also contains yn ∈ Y which is different from n0 hence n0 ≤yn −1 is a limit point of Y. On the other hand, if n0 ≥yn −1 then the open set contains no point of Y different from n0. So n0≥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, ), 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
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
Theorem 3.14. A subset Y of a topological space (X, ) 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
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
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
This reaction was an object of research in theory of metallurgical processes. There it was considered only from thermodynamic point of view.2930 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.
From the above scheme immediately follows reaction matrix
with a rankA = 3.
It is well-known11 that the reaction (4.1) can reduce in this matrix form
where x = (x1, x2, x3, x4, x5, x6, x7, x8)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
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
For instance, by using the vector
as an arbitrary chosen vector, A and A+ determined previously, by virtue of (4.3) one obtains the minimal solution of the matrix equation (4.2) given by
where x1 = 1954, x2 = 1854, x3 = 518, x4 = 1093, x5 = 1096, x6 = 55, x7 = 901 and x8 = 898.
Balanced chemical reaction (4.1) with minimal coefficients has this form
Now, we shall look for sets of solutions of the reaction (4.1). From (4.1) immediately follows this system of linear equations
which general solution is
where xi, (1 ≤i ≤5) are arbitrary real numbers. Actually, the expressions (4.5) represent the set {x6, x7, x8} of subgenerators of the reaction (4.1).
If we substitute (4.5) into (4.1), then one obtains balanced reaction
in its general form, where xi, (1 ≤i ≤5) are arbitrary real numbers.
According to the Definition 2.17, the set of the coefficients of (4.1) is
where xi, (1 ≤i ≤5) are arbitrary real numbers.
If we take into account the Definition 2.18, then the cardinality of the set X, given by (4.7), is CardX = |X| = 8, and according to the Definition 2.20, follows Card, that means that the power set (X) of the set X of the coefficients of chemical reaction (4.1) contains 256 members, which are subsets of X, i.e.,
where xi, (1 ≤i ≤5) are arbitrary real numbers.
According to the Definition 2.35, the discrete topology of the reaction (4.1) is the topology on the set X, where X and are given by (4.7) and (4.8), respectively. The pair (X, ) is the discrete topological space of the chemical reaction (4.1).
If we take into account the Definition 2.36, then the indiscrete topology of the chemical reaction (4.1) is
where xi, (1 ≤i ≤5) are arbitrary real numbers.
The pair
is an indiscrete topological space of the chemical reaction (4.1), where xi, (1 ≤i ≤5) are arbitrary real numbers.
Now, we shall consider the following class of subsets
of
where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
Let’s check axioms of the Definition 2.30.
1° Since X and ∅ belong to , it means that the first axiom of the Definition 2.30 is satisfied.
2° From the Definition 2.30, follows
We obtain that arbitrary union of any pair of sets of belongs to . It shows that axiom 2° of Definition 2.30 is satisfied.
3° Now, we shall determine arbitrary intersections of any number of sets of the class .
where xi, (1 ≤ i ≤5) are arbitrary real numbers.
We obtain that arbitrary intersection of any number of sets of the class belongs to . That means, that is satisfied the axiom 3° of Definition 2.30.
By this we showed that the class is topology on X since it satisfies the necessary three axioms of Definition 2.30.
Consider the topology on X, given by (4.11) and (4.12), respectively and the subset Y = {3x2, 3x3, 2x1 − 3x3 − x4 − x5} of X. Observe that 3x3 ∈ X is a limit point of Y since the open sets containing 3x3 are {3x3, 2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} and X, and each contains a point of Y different from 3x3, i.e., 2x1 − 3x3 − x4 − x5. On the other hand, the point 3x2 ∈ X is not a limit point of Y since the open set {3x2}, which contains 3x2, does not contain a point of Y different from 3x2. Similarly, the points – 13x1 + 6x2 + 18x3 + 5x4 + 2x5 and 11x1 − 3x2 − 15x3 − 4x4 − x5 are limit points of Y and the point 2x1 − 3x3 − x4 − x5 is not limit point of Y.
So
is the derived set of Y, where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
The closed subsets of X are
that is, the complements of the open subsets of X. Note that there are subsets of X, such as {3x3, 2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5, 11x1 − 3x2 − 15x3 − 4x4 − x5}, which are both open and closed, and there are subsets of X, such as {3x2, 3x3}, which are neither open nor closed.
Accordingly
where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
Therefore the set {3x2, 2x1 − 3x3 − x4 − x5} is a dense subset of X, but the set {3x3, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5} is not, where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
Consider the topology (4.11) on (4.12) and the subset Y = {3x3, 2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5} of X.
The points 2x1 − 3x3 − x4 − x5 and − 13x1 + 6x2 + 18x3 + 5x4 + 2x5 are each interior points of Y since 2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5 ∈ {2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5} ⊂ Y, where {2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5} is an open set. The point 3x3 ∈ Y is not an interior point of Y; so Int{Y} = {2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5}. Only the point 3x2 ∈ X is exterior to Y, i.e., interior to the complement Yc = {3x2, 11x1 − 3x2 − 15x3 − 4x4 − x5} of Y; hence Ext{Y} = Int{Yc} = {3x2}. Accordingly the boundary of Y consists of the points 3x3 and 11x1 − 3x2 − 15x3 − 4x4 − x5, i.e., Bd{Y} = {3x3, 11x1 − 3x2 − 15x3 − 4x4 − x5}, where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
Consider the topology on X, given by (4.11) and (4.12), respectively and the subset Y = {3x2, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} of X.
Observe that
Hence the relativization of to Y is
where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
From (4.5) and the Definition 2.13, follows this system of inequalities
From (4.13), one obtains the relation
The expression (4.14) is a necessary and sufficient condition to hold the general reaction (4.6).
Example 4.1. For x3 = 1/3, x4 = 1 and x5 = 1 from the first inequality of (4.13) follows x1 = 2, then from (4.14) immediately follows x2 = 3 and from (4.5) one obtains x6 = 1/3, x7 = 5/3 and x8 = 1, such that the particular reaction of (4.6) has a form
Now we shall find the dimension and the basis of the solution space W of the system (4.4) generated by the chemical reaction (4.1).
The system (4.4) reduces to this form
The system (4.15) has a three (nonzero) equations in eight unknowns; and hence the system has 8 − 3 = 5 free variables which are x4, x5, x6, x7 and x8. Thus dimW = 5.
To obtain a basis for W, one sets
1° x4 = 1, x5 = x6 = x7 = x8 = 0 in (4.15), such that the solution is x1 = (− 1, 0, − 1, 1, 0, 0, 0, 0),
2° x4 = 0, x5 = 1, x6 = x7 = x8 = 0 in (4.15), such that the solution is x2 = (− 4, 0, − 3, 0, 1, 0, 0, 0),
3° x4 = x5 = 0, x6 = 1, x7 = x8 = 0 in (4.15), such that the solution is x3 = (− 12, 1, − 9, 0, 0, 1, 0, 0),
4° x4 = x5 = x6 = 0, x7 = 1, x8 = 0 in (4.15), such that the solution is x4 = (3, 1, 2, 0, 0, 0, 1, 0),
5° x4 = x5 = x6 = x7 = 0, x8 = 1 in (4.15), such that the solution is x5 = (6, 1, 4, 0, 0, 0, 0, 1).
The set W = {x1, x2, x3, x4, x5} is a basis of the solution space W.
Since AWT and WAT are zero matrices of format 3×5 and 5×3, respectively, that means that the orthogonal complement W⊥ of (4.1) is
Now, we shall consider some particular cases of the reaction (4.6) generated by its generators.
I. Particular cases of (4.6) generated by the subgenerator 2x1 − 3x3 − x4 − x5
Here, we would like to emphasize that with considered particular cases the chemical reaction (4.6) do not lose its generality.
1° If x1 = (3x3 + x4 + x5)/2, then the reaction (4.6) transforms
where xi, (2 ≤ i ≤ 5) are arbitrary real numbers.
From (4.16) follow these inequalities
and
or
Reaction (4.16) is possible if and only if (4.17) holds.
Example 4.2. For x3 = x4 = 1 and x5 = 2/3 from (4.17) one obtains x2 = 3/2, such that particular reaction of (4.16) has a form
2° If x2 = (x3 + x4 + 3x5)/4, then the reaction (4.16) becomes
where xi, (3 ≤ i ≤ 5) are arbitrary real numbers.
3° If x2 > (x3 + x4 + 3x5)/4, then the reaction (4.16) transforms into
where xi, (2 ≤ i ≤ 5) are arbitrary real numbers.
From (4.20) one obtains
The system of inequalities (4.21) holds if and only if
Expression (4.22) is a necessary and sufficient condition to hold (4.20).
Example 4.3. For x2 = 3, x3 = 1 and x5 = 1, from (4.22) one obtains x4 = 1, such that particular reaction of (4.20) has a form
4° If x2 = (x3 + x4 + 3x5)/2, then (4.20) becomes
where xi, (3 ≤ i ≤ 5) are arbitrary real numbers.
5° If x2 > (x3 + x4 + 3x5)/2, then (4.20) holds.
6° If x2 < (x3 + x4 + 3x5)/2, then (4.20) becomes
where xi, (2 ≤ i ≤ 5) are arbitrary real numbers.
7° If x2 = (x3 + x4 + 3x5)/4, then from (4.20) follows (4.19).
8° If x2 > (x3 + x4 + 3x5)/4, then from (4.20) one obtains (4.16).
9° If x2 < (x3 + x4 + 3x5)/4, then (4.20) becomes
10° If x2 < (x3 + x4 + 3x5)/4, then the reaction (4.16) transforms into (4.26).
11° For x2 = (x3 + x4 + 3x5)/2, then (4.16) becomes (4.24).
12° If x2 > (x3 + x4 + 3x5)/2, then (4.16) transforms into (4.20).
13° If x2 < (x3 + x4 + 3x5)/2, then (4.16) holds.
14° If x1 > (3x3 + x4 + x5)/2, then (4.6) becomes
where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
Reaction (4.27) is possible if and only if holds this system of inequalities
The above inequality is a necessary and sufficient condition to hold reaction (4.27).
Example 4.4. For x3 = 2/3, x4 = x5 = 1 and x1 = 3, from (4.17) one obtains x2 = 2, such that particular reaction of (4.27) has a form
15° If x1 = (6x2 + 18x3 + 5x4 + 2x5)/13, then (4.27) becomes
where xi, (2 ≤ i ≤ 5) are arbitrary real numbers.
From the above reaction (4.30) one obtains this system of inequalities
From (4.31) immediately follows this expression
which is a necessary and sufficient condition to hold (4.30).
Example 4.5. For x3 = x4 = x5 = 1, from (4.32) one obtains x2 = 3/2, such that particular reaction of (4.30) has a form
16° If x2 = (x3 + x4 + 3x5)/4, then (4.30) transforms into (4.19).
17° If x2 > (x3 + x4 + 3x5)/4, then (4.30) holds.
18° If x2 < (x3 + x4 + 3x5)/4, then (4.30) becomes
where xi, (2 ≤ i ≤ 5) are arbitrary real numbers.
19° If x2 = (x3 + x4 + 3x5)/4, then (4.34) transforms into (4.19).
20° If x1 > (6x2 + 18x3 + 5x4 + 2x5)/13, then (4.27) holds.
21° If x1 < (6x2 + 18x3 + 5x4 + 2x5)/13, then (4.27) becomes
where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
22° If x1 = (3x3 + x4 + x5)/2, then from (4.27) follows
where xi, (2 ≤ i ≤ 5) are arbitrary real numbers.
23° If x1 > (3x3 + x4 + x5)/2, then holds (4.27).
24° If x1 < (3x3 + x4 + x5)/2, then (4.27) becomes (4.35).
25° If x1 = (3x2 + 15x3 + 4x4 + x5)/11, then from (4.27) follows
where xi, (2 ≤ i ≤ 5) are arbitrary real numbers.
The reaction (4.37) holds if and only if
26° If x2 = (x3 + x4 + 3x5)/2, then (4.37) transforms into (4.24).
27° If x2 > (x3 + x4 + 3x5)/2, then (4.37) holds.
28° If x2 < (x3 + x4 + 3x5)/2, then (4.37) becomes
where xi, (2 ≤ i ≤ 5) are arbitrary real numbers.
29° If x1 > (3x2 + 15x3 + 4x4 + x5)/11, then (4.27) holds.
30° If x1 < (3x2 + 15x3 + 4x4 + x5)/11, then (4.27) becomes (4.35).
The reaction (4.35) holds if and only if the following system of inequalities is satisfied
From (4.40) immediately follows this inequality
x1 < (6x2 + 18x3 + 5x4 + 2x5)/13.
The last inequality is a necessary and sufficient condition the reaction (4.35) to be possible.
31° If x1 = (3x3 + x4 + x5)/2, then (4.35) becomes (4.20).
32° If x1 = (3x2 + 15x3 + 4x4 + x5)/11, then from (4.35) one obtains (4.39).
33° If x1 = (6x2 + 18x3 + 5x4 + 2x5)/13, then (4.35) transforms into (4.34).
34° If x1 < (3x3 + x4 + x5)/2, then (4.6) becomes (4.35).
II. Particular cases of (4.6) generated by − 13x1 + 6x2 + 18x3 + 5x4 + 2x5
35° If x1 = (6x2 + 18x3 + 5x4 + 2x5)/13, then (4.6) transforms into (4.30).
36° If x1 > (6x2 + 18x3 + 5x4 + 2x5)/13, then from (4.6) one obtains (4.27).
37° If x1 < (6x2 + 18x3 + 5x4 + 2x5)/13, then (4.6) becomes (4.35).
III. Particular cases of (4.6) generated by the subgenerator 11x1 − 3x2 − 15x3 − 4x4 − x5
38° If x1 = (3x2 + 15x3 + 4x4 + x5)/11, then (4.6) transforms into (4.37).
39° If x1 > (3x2 + 15x3 + 4x4 + x5)/11, then from (4.6) one obtains
40° If x1 < (3x2 + 15x3 + 4x4 + x5)/11, then (4.6) becomes (4.35).
Next, we shall analyze some particular cases of the general reaction (4.6) for xi = 0, (1 ≤ i ≤ 5).
As particular reactions of (4.6) we shall derive the following cases.
If x1 = 0, then from (4.6) one obtains this particular reaction
If x2 = 0, then from (4.6) follows
where x1, x3, x4 and x5 are arbitrary real numbers.
In the reaction (4.43) the molecules arrangement is given in an implicit form. To be find their explicit arrangement, from (4.43) one obtains this system of inequalities
or
where x3, x4 and x5 are arbitrary real numbers.
If we substitute (4.44) in (4.43), then one obtains
41° If x1 > (15x3 + 4x4 + x5)/11, then from (4.43) follows this reaction
where x1, x3, x4 and x5 are arbitrary real numbers.
From (4.46) one obtains this system of inequalities
or
where x1, x3, x4 and x5 are arbitrary real numbers.
Last inequality is a necessary and sufficient condition to hold (4.46).
42° If x1 = (3x3 + x4 + x5)/2, then from (4.46) follows this reaction
43° If x1 > (3x3 + x4 + x5)/2, then from (4.46) transforms into
where x1, x3, x4 and x5 are arbitrary real numbers.
From (4.48) one obtains this system of inequalities
or
where x1, x3, x4 and x5 are arbitrary real numbers.
Last inequality is necessary and sufficient condition to hold (4.48).
44° If x1 < (3x3 + x4 + x5)/2, then from (4.46) follows this reaction
where x1, x3, x4 and x5 are arbitrary real numbers.
From (4.49) one obtains this system of inequalities
or
where x1, x3, x4 and x5 are arbitrary real numbers.
Last inequality is a necessary and sufficient condition to hold (4.49).
45° If x1 = (18x3 + 5x4 + 2x5)/13, then from (4.46) follows this reaction
46° If x1 > (18x3 + 5x4 + 2x5)/13, then from (4.46) follows (4.48).
47° If x1 < (18x3 + 5x4 + 2x5)/13, then from (4.46) one obtains (4.49).
48° If x1 = (15x3 + 4x4 + x5)/11, then (4.46) transforms into
49° If x1 > (15x3 + 4x4 + x5)/11, then from (4.46) follows (4.48).
50° If x1 < (15x3 + 4x4 + x5)/11, then from (4.46) one obtains (4.49).
51° If x1 = (18x3 + 5x4 + 2x5)/13, then (4.48) transforms into (4.50).
52° If x1 > (18x3 + 5x4 + 2x5)/13, then holds (4.48).
53° If x1 < (18x3 + 5x4 + 2x5)/13, then from (4.48) follows (4.49).
54° If x1 = (3x3 + x4 + x5)/2, then from (4.48) one obtains (4.47).
55° If x1 > (3x3 + x4 + x5)/2, then holds (4.48).
56° If x1 < (3x3 + x4 + x5)/2, then from (4.48) follows (4.49).
57° If x1 = (15x3 + 4x4 + x5)/11, then (4.48) transforms into (4.51).
58° If x1 > (15x3 + 4x4 + x5)/11, then holds (4.48).
59° If x1 < (15x3 + 4x4 + x5)/11, then from (4.48) follows (4.49).
60° If x1 = (3x3 + x4 + x5)/2, then (4.49) transforms into (4.47).
61° If x1 > (3x3 + x4 + x5)/2, then from (4.49) one obtains (4.48).
62° If x1 < (3x3 + x4 + x5)/2, then holds (4.49).
63° If x1 = (15x3 + 4x4 + x5)/11, then (4.49) becomes (4.51).
64° If x1 > (15x3 + 4x4 + x5)/11, then from (4.49) one obtains (4.48).
65° If x1 < (15x3 + 4x4 + x5)/11, then holds (4.49).
66° If x1 = (18x3 + 5x4 + 2x5)/13, then from (4.49) follows (4.50).
67° If x1 > (18x3 + 5x4 + 2x5)/13, then from (4.49) one obtains (4.48).
68° If x1 < (18x3 + 5x4 + 2x5)/13, then holds (4.49).
69° If x1 = (3x3 + x4 + x5)/2, then from (4.43) follows (4.47).
70° If x1 > (3x3 + x4 + x5)/2, then from (4.43) one obtains (4.48).
71° If x1 < (3x3 + x4 + x5)/2, then (4.43) becomes (4.49).
72° If x1 = (18x3 + 5x4 + 2x5)/13, then (4.43) transforms into (4.50).
73° If x1 > (18x3 + 5x4 + 2x5)/13, then from (4.43) follows (4.48).
74° If x1 < (18x3 + 5x4 + 2x5)/13, then from (4.43) one obtains (4.49).
75° If x1 = (15x3 + 4x4 + x5)/11, then from (4.43) follows (4.51).
76° If x1 > (15x3 + 4x4 + x5)/11, then (4.43) transforms into (4.46).
77° If x1 < (15x3 + 4x4 + x5)/11, then from (4.43) follows (4.49).
If x3 = 0, then from (4.6) one obtains this particular reaction
where x1, x2, x4 and x5 are arbitrary real numbers.
From (4.52) one obtains this system of inequalities
or
where x1, x2, x4 and x5 are arbitrary real numbers.
The inequality (4.54) is a necessary and sufficient condition to hold the reaction (4.52).
78° If x1 = (x4 + x5)/2, then from (4.52) one obtains this particular reaction
where x2, x4 and x5 are arbitrary real numbers.
79° If x1 > (x4 + x5)/2, then from (4.52) follows
where x1, x2, x4 and x5 are arbitrary real numbers.
80° If x1 < (x4 + x5)/2, then from (4.52) one obtains
where x1, x2, x4 and x5 are arbitrary real numbers.
81° If x1 = (6x2 + 5x4 + 2x5)/13, then from (4.52) follows this reaction
where x2, x4 and x5 are arbitrary real numbers.
The reaction (4.58) has the following three subgenerators
A necessary and sufficient condition to hold the above inequalities and the reaction (4.58) is
82° If x1 > (6x2 + 5x4 + 2x5)/13, then (4.52) becomes
where x1, x2, x4 and x5 are arbitrary real numbers.
The reaction (4.59) contains these subgenerators
The reaction (4.59) is possible if and only if are satisfied the above three inequalities and if and only if this inequality holds
83° If x1 < (6x2 + 5x4 + 2x5)/13, then from (4.52) one obtains
where x1, x2, x4 and x5 are arbitrary real numbers.
From the reaction (4.60) follows this system of inequalities
The above system of inequalities is satisfied if and only if
where x1, x2, x4 and x5 are arbitrary real numbers.
Last inequality is a necessary and sufficient condition to hold the reaction (4.60).
84° If x1 = (3x2 + 4x4 + x5)/11, then from (4.52) follows this particular reaction
where x2, x4 and x5 are arbitrary real numbers.
The reaction (4.61) holds if and only if this inequality is satisfied
85° If x1 > (3x2 + 4x4 + x5)/11, then the reaction (4.52) transforms into (4.59).
86° If x1 < (3x2 + 4x4 + x5)/11, then from (4.52) one obtains (4.60).
Let’s consider (4.55). This reaction contains the following two subgenerators 4x2 − x4 − 3x5 > 0 and − 2x2 + x4 + 3x5 > 0.
From them immediately follows
The inequality (4.62) is a necessary and sufficient condition to hold (4.55).
87° If x2 = (x4 + 3x5)/4, then (4.55) transforms into
88° If x2 > (x4 + 3x5)/4, then from (4.55) one obtains
where x2, x4 and x5 are arbitrary real numbers.
89° If x2 < (x4 + 3x5)/4, then from (4.55) follows
where x2, x4 and x5 are arbitrary real numbers.
90° If x2 = (x4 + 3x5)/2, then (4.55) becomes
91° If x2 > (x4 + 3x5)/2, then (4.55) transforms into (4.64).
92° If x2 < (x4 + 3x5)/2, then from (4.55) follows (4.65).
Now, we shall consider the reaction (4.56). This reaction contains the following three subgenerators 13x1 − 6x2 − 5x4 − 2x5, 2x1 − x4 − x5 and 11x1 − 3x2 − 4x4 − x5. According to the Definition (2.13), immediately follows this system of inequalities
or
where x1, x2, x4 and x5 are arbitrary real numbers.
The expression (4.68) is a necessary and sufficient condition to hold the reaction (4.56).
93° If x1 = (6x2 + 5x4 + 2x5)/13, then from (4.56) one obtains (4.58).
94° If x1 > (6x2 + 5x4 + 2x5)/13, then (4.56) transforms into (4.59).
95° If x1 < (6x2 + 5x4 + 2x5)/13, then (4.56) becomes (4.60).
96° If x1 = (x4 + x5)/2, then from (4.56) one obtains (4.55).
97° If x1 > (x4 + x5)/2, then (4.56) holds if (4.68) is satisfied.
98° If x1 < (x4 + x5)/2, then (4.56) becomes (4.57).
99° If x1 = (3x2 + 4x4 + x5)/11, then from (4.56) follows (4.61).
100° If x1 > (3x2 + 4x4 + x5)/11, then (4.56) becomes (4.59).
101° If x1 < (3x2 + 4x4 + x5)/11, then from (4.56) follows (4.60).
Let’s consider the reaction (4.57). From this reaction follows the system of inequalities
The system of inequalities (4.69) holds if and only if
where x1, x2, x4 and x5 are arbitrary real numbers.
The inequality (4.70) is a necessary and sufficient condition to hold the reaction (4.57).
Next, we shall consider particular cases of (4.57).
102° If x1 = (x4 + x5)/2, then from (4.57) one obtains (4.64).
103° If x1 > (x4 + x5)/2, then (4.57) transforms into
where x1, x2, x4 and x5 are arbitrary real numbers.
104° If x1 < (x4 + x5)/2, then the reaction (4.57) holds.
105° If x1 = (3x2 + 4x4 + x5)/11, then from (4.57) one obtains (4.61).
106° If x1 > (3x2 + 4x4 + x5)/11, then (4.57) transforms into (4.59).
107° If x1 < (3x2 + 4x4 + x5)/11, then from (4.57) follows (4.60).
108° If x1 = (6x2 + 5x4 + 2x5)/13, then from (4.57) one obtains (4.58).
109° If x1 > (6x2 + 5x4 + 2x5)/13, then from (4.57) follows (5.59).
110° If x1 < (6x2 + 5x4 + 2x5)/13, then (4.57) transforms into (4.60).
If x4 = 0, then from (4.6) one obtains this particular reaction
where x1, x2, x3 and x5 are arbitrary real numbers.
From (4.72) one obtains this system of inequalities
or
where x1, x2, x3 and x5 are arbitrary real numbers.
The inequality (4.74) is a necessary and sufficient condition to hold the reaction (4.72).
111° If x1 = (3x3 + x5)/2, then (4.72) transforms into
where x2, x3 and x5 are arbitrary real numbers.
Reaction (4.75) is possible if and only if holds the following inequality
The inequality (4.76) is a necessary and sufficient condition to hold (4.75).
112° If x2 = (x3 + 3x5)/4, then (4.75) obtains this form
113° If x2 > (x3 + 3x5)/4 then (4.75) transforms into
where x2, x3 and x5 are arbitrary real numbers.
The above reaction is possible if and only if holds this inequality
The inequality (4.79) is a necessary and sufficient condition to hold (4.78).
114° If x2 < (x3 + 3x5)/4, then from (4.75) one obtains
where x2, x3 and x5 are arbitrary real numbers.
The reaction (4.80) is possible if and only if holds the following inequality
The inequality (4.81) is a necessary and sufficient condition to hold (4.80).
115° If x2 = (x3 + 3x5)/2, then from (4.75) follows
116° If x2 > (x3 + 3x5)/2, then (4.75) becomes (4.78).
117° If x2 < (x3 + 3x5)/2, then (4.75) holds.
118° If x1 > (3x3 + x5)/2, then (4.72) holds.
119° If x1 < (3x3 + x5)/2, then from (4.72) one obtains
where x1, x2, x3 and x5 are arbitrary real numbers.
The reaction (4.83) is possible if and only if the inequality (4.74) is satisfied.
120° If x1 = (6x2 + 18x3 + 2x5)/13, then (4.72) becomes
where x2, x3 and x5 are arbitrary real numbers.
From the above reaction (4.84) one obtains this system of inequalities
From (4.85) immediately follows this inequality
which is a necessary and sufficient condition to hold (4.84).
121° If x2 = (x3 + 3x5)/4, then (4.84) obtains this form
122° If x2 > (x3 + 3x5)/4, then (4.84) holds.
123° If x2 < (x3 + 3x5)/4, then from (4.84) follows
where x2, x3 and x5 are arbitrary real numbers.
124° If x1 > (6x2 + 18x3 + 2x5)/13, then (4.72) transforms into
where x1, x2, x3 and x5 are arbitrary real numbers.
From (4.87) follows this system of inequalities
or
where x1, x2, x3 and x5 are arbitrary real numbers.
The inequality (4.89) is a necessary and sufficient condition to hold the reaction (4.87).
125° If x1 < (6x2 + 18x3 + 2x5)/13, then from (4.72) one obtains
where x1, x2, x3 and x5 are arbitrary real numbers.
From (4.90) one obtains this system of inequalities
or
where x1, x2, x3 and x5 are arbitrary real numbers.
The inequality (4.92) is a necessary and sufficient condition to hold the reaction (4.91).
126° If x1 = (3x2 + 15x3 + x5)/11, then from (4.72) follows this reaction
where x2, x3 and x5 are arbitrary real numbers.
Reaction (4.93) is possible if and only if holds this inequality
Actually, the inequality (4.94) is a necessary and sufficient condition to hold (4.93).
127° If x1 = (x3 + 3x5)/2, then (4.93) becomes (4.82).
128° If (4.94) holds, then holds (4.93) too.
129° If x2 < (x3 + 3x5)/2, then (4.93) transforms into
where x2, x3 and x5 are arbitrary real numbers.
130° If x1 > (3x2 + 15x3 + x5)/11, then (4.72) becomes
where x1, x2, x3 and x5 are arbitrary real numbers.
The subgenerators of the reaction (4.96) are given by the following expression
A necessary and sufficient condition to hold the inequalities (4.97) and the reaction (4.96) is
131° If x1 < (3x2 + 15x3 + x5)/11, then (4.72) transforms into
where x1, x2, x3 and x5 are arbitrary real numbers.
From the reaction (4.99) immediately follows this system of inequalities
The system of inequalities (4.100) holds if and only if this inequality is satisfied
where x1, x2, x3 and x5 are arbitrary real numbers.
If x5 = 0, then from (4.6) one obtains this particular reaction
where x1, x2, x3 and x4 are arbitrary real numbers.
From (4.102) follows this system of inequalities
or
where x1, x2, x3 and x4 are arbitrary real numbers.
The inequality (4.104) is a necessary and sufficient condition to hold the reaction (4.102).
Now we shall consider some particular cases of (4.102).
132° If x1 = (3x3 + x4)/2, then (4.102) transforms into
where x2, x3 and x4 are arbitrary real numbers.
The reaction (4.105) is possible if and only if this inequality holds
133° If x2 = (x3 + x4)/4, then from (4.105) one obtains
134° If x2 > (x3 + x4)/4, then the reaction (4.105) holds.
135° If x2 < (x3 + x4)/4, then the reaction (4.105) transforms into this form
where x2, x3 and x4 are arbitrary real numbers.
A necessary and sufficient condition to hold the reaction (4.108) is to be satisfied this inequality
136° If x2 = (x3 + x4)/2, then the reaction (4.105) reduces to this particular reaction
137° If x2 > (x3 + x4)/2, then (4.105) becomes
where x2, x3 and x4 are arbitrary real numbers.
The reaction (4.111) is possible if and only if is satisfied this inequality
138° If x2 < (x3 + x4)/2, then from (4.105) one obtains
where x2, x3 and x4 are arbitrary real numbers.
The reaction (4.113) is possible if and only if this inequality is satisfied
139° If x1 > (3x3 + x4)/2, then (4.102) holds.
140° If x1 < (3x3 + x4)/2, then from (4.102) one obtains
where x1, x2, x3 and x4 are arbitrary real numbers.
The reaction (4.115) holds if and only if this inequality is satisfied
where x1, x2, x3 and x4 are arbitrary real numbers.
141° If x1 = (6x2 + 18x3 + 5x4)/13, then from (4.102) one obtains
where x2, x3 and x4 are arbitrary real numbers.
The reaction (4.117) is possible if and only if the following inequality is satisfied
142° If x2 = (x3 + x4)/4, then (4.117) becomes (4.107).
143° If x1 > (6x2 + 18x3 + 5x4)/13, then from (4.102) follows
where x1, x2, x3 and x4 are arbitrary real numbers.
The reaction (4.118) holds if and only if this inequality is satisfied
where x1, x2, x3 and x5 are arbitrary real numbers.
144° If x1 < (6x2 + 18x3 + 5x4)/13, then (4.102) becomes
where x1, x2, x3 and x4 are arbitrary real numbers.
The reaction (4.120) is possible if and only if this inequality is satisfied
where x1, x2, x3 and x4 are arbitrary real numbers.
145° If x1 = (3x2 + 15x3 + 4x4)/11, then from (4.102) one obtains
where x2, x3 and x4 are arbitrary real numbers.
The above reaction (4.122) has only one subgenerator 2x2 − x3 − x4, that means that it is possible if and only if
146° If x2 = (x3 + x4)/2, then (4.122) becomes (4.110).
147° If x1 > (3x2 + 15x3 + 4x4)/11, then (4.102) holds.
148° If x1 < (3x2 + 15x3 + 4x4)/11, then (4.102) becomes
where x1, x2, x3 and x4 are arbitrary real numbers.
The reaction (4.124) holds if and only if the inequality (4.116) is satisfied.
By this we shall finish this section, where we considered some of the important particular cases.
AN EXTENTION OF THE RESULTS
In this section we shall extend topological and chemical results obtained in the previous section. Actually, here we shall develop an explicite topological calculus for the coefficients of the chemical reaction (4.6) for an another topology and will be considered balancing of the chemical reaction (5.3) which posses atoms with fractional oxidation numbers. Also for the reaction (5.6) we shall develop a comprehensive topological calculus.
First, for the reaction (4.6) we shall consider the topology
on
where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
Now, we shall determine
1° the derived sets of Y = {2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} and Z = {3x5},
2° the closed subsets of X,
3° the closure of the sets {3x1}, {3x5} and {2x1 − 3x3 − x4 − x5, 11x1 − 3x2 − 15x3 − 4x4 − x5},
4° which sets in 3° are dense in X?
5° the interior points of the subset T = {3x1, 3x5, 2x1 − 3x3 − x4 − x5} of X,
6° the exterior points of T,
7° the boundary points of T,
8° the neighborhoods of the point 11x1 − 3x2 − 15x3 − 4x4 − x5 and of the point 2x1 − 3x3 − x4 − x5,
9° the members of the relative topology on S = {3x1, 2x1 − 3x3 − x4 − x5, 11x1 − 3x2 − 15x3 − 4x4 − x5}.
Let’s go forward.
1° Note that {3x1, 3x5} and {3x1, 3x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} are open subsets of X and that
and
and
Hence 3x1, 3x5 and 11x1 − 3x2 − 15x3 − 4x4 − x5 are not limit point of Y. On the other hand, every other point in X is a limit point of Y since every open set containing it also contains a point of Y different from it. Accordingly,
Note that {3x1}, {3x1, 3x5} and {3x1, 2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5} are open subsets of X and that
3x1 ∈ {3x1} and {3x1} ∩ Z = ∅, 3x5 ∈ {3x1, 3x5} and {3x1, 3x5} ∩ Z = {3x5}, 2x1 – 3x3 – x4 – x5, – 13x1 + 6x2 + 18x3 + 5x4 + 2x5 ∈ {3x1, 2x1 – 3x3 – x4 – x5, – 13x1 + 6x2 + 18x3 + 5x4 + 2x5} and {3x1, 2x1 – 3x3 – x4 – x5, – 13x1 + 6x2 + 18x3 + 5x4 + 2x5} ∩ Z = ∅.
Hence 3x1, 3x5, 2x1 − 3x3 − x4 − x5 and − 13x1 + 6x2 + 18x3 + 5x4 + 2x5 are not limit point of Z = {3x5}. But 11x1 − 3x2 − 15x3 − 4x4 − x5 is a limit point of Z since the open sets containing 11x1 − 3x2 − 15x3 − 4x4 − x5 are {3x1, 3x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} and X and each contains the point 3x5 ∈ Z different from 11x1 − 3x2 − 15x3 − 4x4 − x5. Thus Z’ = {11x1 − 3x2 − 15x3 − 4x4 − x5}.
2° A set is closed if and only if its complement is open. Hence write the complement of each set in :
3° The closure Cl{X} of any set X is the intersection of all closed supersets of X.
The only closed superset of {3x1} is X the closed supersets of {3x5} are
and the closed supersets of {2x1 − 3x3 − x4 − x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} are {2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5, 11x1 − 3x2 − 15x3 − 4x4 − x5}, {3x5, 2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} and X.
Thus,
4° A set Y is dense in X if and only if Cl{Y} = X; so {3x1} is the only dense set.
5° The points 3x1 and 3x5 are interior points of T since
where {3x1, 3x5} is an open set, i.e., since each belongs to an open set contained in T. Note that 2x1 − 3x3 − x4 − x5 is not an interior point of T since 2x1 − 3x3 − x4 − x5 does not belong to any open set contained in T. Hence Int{T} = {3x1, 3x5} is the interior of T.
6° The complement of T is
Neither − 13x1 + 6x2 + 18x3 + 5x4 + 2x5 nor 11x1 − 3x2 − 15x3 − 4x4 −x5 are interior points of Tc since neither belongs to any open subset of Tc = {−13x1 + 6x2 + 18x3 + 5x4 + 2x5, 11x1 − 3x2 − 15x3 − 4x4 − x5}. Hence, Int{Tc} = x2, i.e., there are no exterior points of T.
7° The boundary Bd{T} of T consists of those points which are neither interior nor exterior to T. So
8° A neighborhood of 11x1 − 3x2 − 15x3 − 4x4 − x5 is any superset of an open set containing 11x1 − 3x2 − 15x3 − 4x4 − x5. The open sets containing 11x1 − 3x2 − 15x3 − 4x4 − x5 are {3x1, 3x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} and X. The supersets of {3x1, 3x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} are {3x1, 3x5, 11x1 − 3x2 − 15x3 − 4x4 − x5}, {3x1, 3x5, 2x1 − 3x3 − x4 − x5, 11x1 − 3x2 − 15x3 − 4x4 − x5}, {3x1, 3x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5, 11x1 − 3x2 − 15x3 − 4x4 − x5} and X; the only superset of X is X. Accordingly, the class of neighborhoods of 11x1 − 3x2 − 15x3 − 4x4 − x5, i.e., neighborhood system of 11x1 − 3x2 − 15x3 − 4x4 − x5 is
The open sets containing 2x1 − 3x3 − x4 − x5 are {3x1, 2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5}, {3x1, 3x5, 2x1 − 3x3 − x4 − x5, − 13x1 + 6x2 + 18x3 + 5x4 + 2x5} and X. Hence the neighborhood system of 2x1 − 3x3 − x4 − x5 is
9°, so the members of are
In other words
where xi, (1 ≤ i ≤ 5) are arbitrary real numbers.
Observe that {3x1, 2x1 − 3x3 − x4 − x5} is not open in X, but is relatively open in S, i.e., is T S — open.
Now, we shall determine a minimal solution of the equation
The scheme for the reaction (5.3) is
From the above scheme, follows reaction matrix
with a rankA = 3.
The Moore-Penrose matrix has this form
By using the vector
as an arbitrary chosen vector, A and A+ determined previously, by virtue of (4.3) one obtains the minimal solution of the chemical equation (5.3) given by
where
Balanced equation (5.3) with minimal coefficients is
A particular case of reaction (5.3) for x1 = x2 = 0, x3 = − c1, x4 = c3, x5, = x6 = 0, x7 = − c2 and x8 = c4 is considered in31.
Next, we shall look for sets of solutions of the reaction (5.3).
From (5.3) immediately follows this system of linear equations
which general solution is
where x4, x5, x6, x7 and x8 are arbitrary real numbers.
Balanced chemical reaction (5.3) has this form
where x4, x5, x6, x7 and x8 are arbitrary real numbers.
The reaction (5.6) holds if and only if
i.e.,
where x4, x5, x6 and x7 are arbitrary real numbers.
For instance, if x4 = x5 = x6 = 1, then from (5.8) one obtains x7 < 6.4805. Let it be x7 = 6. Then from (5.7) follows x8 > 0.24025, i.e., x8 = 0.3. Now, the reaction (5.6) will have this particular form
For (5.6) we shall consider the topology of closed sets
on
where xi, (4 ≤ i ≤ 8) are arbitrary real numbers.
The above topology (5.10) on the set (5.11) is a collection or class of subsets that obey the axioms of the Definition 2.30.
The complements of the closed sets are defined as open sets. The open sets of the topology are the collection of subsets given by
We would like to emphasize that the same set of all combinations of subsets can support several topologies. For instance, the subsets of the topology
are closed. Hence 21 is a different topology made on the same set of points, X. The open sets of this topology are
Actually, there are lots of different ways to define topologies. A subset can be open, or closed, or both, or neither relative to a particular topology. For instance, with respect to the topology given by the closed sets,
The topology of closed sets given by this collection
has its dual as the topology of open sets
Let X is given by (5.11) and let
Now, we shall find a topology on X generated by Y.
First, we shall compute the class Z of all finite intersections of sets in Y:
Taking unions of members of Z gives the class
which is a topology on X generated by Y.
Here made research showed that on the set X of the coefficients xj, (4 ≤ j ≤ 8) of the chemical reaction (5.6) can be generated many topologies. A general topological problem shall be given in the next section.
AN OPEN PROBLEM
According to the obtained results in this research, we shall propose the following problem.
Problem 6.1. How many topologies can be generated on the set X ⊂ ℝ of all the coefficients xj, (1 ≤ j ≤ n) of the chemical equation (2.2) of the reaction (2.1)?
The above problem is a completely new problem in topology and chemistry too. Sure that this problem is not a daily particular problem, just the opposite, it is a very hard scientific problem, which we shall try to solve it in this section.
Actually, the problem reduces to finding the number of partial orders on a finite set.
Solution. Now, we shall prove the following theorem, which will be necessary for solution of the problem.
Theorem 6.2. The following relations hold
Proof. We shall only prove 1° because the proof of 3° is similar, and 2° and 4° are the inverses of 1° and 3° respectively.
Easy one can note that each partially ordered set of n elements induces a partition of n, simply by considering the cardinalities of the connected components of the given partially ordered set. There are
distinct ways (up to isomorphism) of distributing n distinct elements into r(π) parts (where there are ri parts of size i). On each of these r(π) parts we can set up any partial ordering we wish and the resulting partial ordering on X will all be distinct, since different groups of distinct elements are involved. Thus, the theorem follows immediately.
Now, by an example we shall clarify the meaning of the above Theorem 6.2.
Example 6.3. Let’s consider the case for n = 3.
On the next four tables are presented the topologies which are calculated on same way as in Example 6.3 for 1 ≤ n ≤ 16.
Remark 6.4. The considered problem is solved only for some particular cases, but its complete solution really is extremely hard. Unfortunately, to date we did not find its explicit general solution which can be used for all values of n.
Table 1.Partial orders/-topologies
Table 2.Connected partial orders/ c-topologies
Table 3.Quasi-orders/ topologies
Table 4.Connected quasi-orders/ topologies
CONCLUSION
Since the traditional approach of balancing chemical equation produces only paradoxes, it was abandoned and substituted with modern consistent methods.
The modern methods for balancing chemical equations work only in well-defined chemical systems. For that purpose we introduced a new formal chemical system, which is a main prerequisite of chemistry to be consistent.
The well-known classical approach of direct reduction of hematite with a carbon here is generalized by the reactions (4.1) and (5.3), which possess atoms with integers and fractional oxidation numbers, respectively. For these reactions are determined their general and minimal solutions. The minimal solutions are determined by the author’s method.5 From the reaction (4.1), its particular cases are analyzed, such that it did not lose its generality.
Also, these reactions are determined and their subgenerators analyzed by use of elementary theory of inequalities. 32
By these chemical reactions it is showed that topological calculus is very easily applicable in chemistry and metallurgy, which gives a good opportunity for their extension toward a modern way founded by virtue of point-set topology.
Here developed topologies are generated for some subsets of solutions of reactions (4.1) and (5.3).
This article will accelerate research in the theory of chemical equations and will give topology more one application, such that the old stereotypical approach in chemistry and its foundation will be substituted with a new sophisticated topological calculus.
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