UDK 54 ISSN 1580-2949 Original scientific article/Izvirni znanstveni članek MTAEC9, 45(6)503(2011) NEW DISCOVERED PARADOXES IN THEORY OF BALANCING CHEMICAL REACTIONS NOVOODKRITI PARADOKSI V TEORIJI URAVNOTEŽENJA KEMIJSKIH REAKCIJ Ice B. Risteski 2 Milepost Place # 606, Toronto, Ontario, Canada M4H 1C7 ice@scientist.com »Failure to distinguish carefully between mathematical and metam^them^tical statements leads to paradoxes.« M. Kac, S. M. Ulam, Mathematics and Logic, Dover Pub. Inc., Mineola 1992, p. 126. Prejem rokopisa - received: 2011-01-24; sprejem za objavo - accepted for publication: 2011-09-21 In this work are given new paradoxes and fallacies discovered in the theory of balancing chemical reactions. All the counterexamples showed that so-called »chemical procedures« of balancing chemical reactions given in earlier chemical literature are inconsistent. Balancing chemical reactions is a mathematical procedure independent of chemistry. In order to avoid the appearance of paradoxes, chemical reactions must be considered as a formal system founded by virtue of well-defined mathematical model. The results obtained in this work affirmed that the usage of traditional ways of balancing chemical reactions is limited. They may be used only for balancing some elementary chemical equations. In other words, foundation of chemistry looks for a new approach of balancing chemical reactions, which must be completely different than current »chemical procedures«. This work is a collection and analysis of some paradoxes and fallacies which appeared in the theory of balancing chemical reactions. Keywords: chemical reactions, paradoxes, balancing, fallacies Predstavljeni so novi paradoksi in zmote, odkrite v teoriji uravnoteženje kemijskih reakcij. Vsi protiprimeri so pokazali, da so tako nekonsistentne tako imenovane kemijske procedure uravnoteženja kemijskih reakcij, navedene v zgodnjih virih. Uravnoteženje kemijskih reakcij je matematična procedura, neodvisna od kemije. Da bi se izognili nastajanju paradoksov, je treba kemijske reakcije formulirati kot formalen sistem na podlagi dobro definiranega matematičnega modela. Rezultati v tem delu potrjujejo, da je uporaba tradicionalnih načinov uravnoteženja kemijskih reakcij omejena. Uporabljamo jo le za uravnoteženje nekaterih elementarnih kemijskih reakcij. Z drugih besedami, temelj kemije išče nove približke za uravnoteženje kemijskih reakcij, ki se morajo razlikovati od sedanjih kemijskih procedur. To delo je izbor in analiza nekaterih paradoksov in zmote, ki so se pojavile v teoriji uravnoteženja kemijskih reakcij. Ključne besede: kemijske reakcije, paradoksi, uravnoteženje, zmote mathematical principles, but also there not exists »gen- "I TIV.TT'TJ/~\'r\T X X ■ O eral mathematics« and speaking more accurately it is not In this section we shall discuss the balancing of possible to exist. chemical reactions from scientific viewpoint. It is an es- For instance, in mathematical logic in the 20th cen- sential precondition for better understanding of our dis- tury lots of paradoxes were discovered,1 and mathemati- course about paradoxes connected with traditional ways cians thought that only there are possible antinomies and of balancing chemical reactions. that other parts of mathematics are in safety. Reality Before opening this discussion about balancing showed that it is not true. In mathematical analysis lots chemical reactions, we would like to give a few remarks of counterexamples were found.2,3 Also certain about the name of so-called course »general chemistry«. counterexamples were found in probability & statistics.3, Why? The reply is very simple, because this course 4 In topology as a contemporary mathematical discipline treats the balancing of chemical reactions as its subject. a great number of counterexamples were detected too.5 It At the beginning of our exposition we want to say that does not mean that other mathematical disciplines are the name »general chemistry« is not appropriately cho- without contradictions. No! Just the opposite, in almost sen. Generally speaking, »general chemistry« does not all the branches of mathematics different kinds of exist, and on top of all it is not possible to exist, because counterexamples are detected6. These facts show that the principles of this particular chemistry are weak and mathematical principles are not general, i. e., they hold do not hold for all parts of chemistry. They have only only in certain part of mathematics. These are the causes particular meaning and nothing more. In other words, it why there is not »general mathematics«. If we take into means that its principles are not general. This is just one account the fact that chemical reaction is a basic issue in thing. Another thing, chemistry is founded by virtue of chemistry and according to its definition (see: Definition 2.2, in the section 2), then it follows that chemistry is founded on mathematical principles. If there is not »general mathematics«, then how is possible to exist »general chemistry«? Simply speaking, it is impossible! In order to correct this irrational name of chemistry, it is necessary to choose an appropriate name which will be more suitable for that particular chemistry. For example, the names basic chemistry and elementary chemistry fit for that chemistry. In chemical as well as mathematical journals there are lots of published papers which treat the problem of balancing chemical reactions. Specially, in chemical journals are considered many different so-called »chemical ways« for balancing chemical reactions, but unfortunately all of them offer only particular procedures for balancing of some simple chemical reactions. These »chemical ways« very often have negative consequences for chemistry. For instance, they produce fallacies or absurd results, because most of them work on erroneous principles, but not on true principles. Balancing chemical reactions is not a simple procedure as some traditionally oriented chemists think or as they want to be. In7 the author emphasized very clearly, that balancing chemical reactions is not chemistry; it is just linear algebra. Balancing chemical reactions is a basic matter of chemistry, if not one of its most important issues, and it plays a main role in its foundation. Indeed, it is a subtle question, which deserves considerable attention. This topic has always excited nature as deeply as no other question in chemistry. The balancing of chemical reactions has acted upon human mind in such stimulating and fruitful way as hardly any other idea, but also this subject needs an explanation as no other concept. The concept of balancing chemical reactions in chemistry is tracked and transformed from traditionalism to intuitionism in the 19*^ century, from intuitionism to irrationalism in the first half of 20'h century, from irrationalism to particularism in the second half of 20"^ century, while the particularism is sharpened by Moore-Penrose matrix8, 9 concept into generalism and transformed into formalism, which takes on the status of a paradigm in the 21st century. To the question of balancing chemical reactions the mathematicians, chemists and computer engineers with different conceptions will not always give the same answer. A mathematician, a chemist and a computer engineer will answer each in a different way. Some of them will admit that perhaps the others are right in a certain sense and will try to reinterpret the others' procedures in their own language. But in general everyone, more or less, will remain convinced that, in fact, still only he is right. Since chemistry is not immune from contradictions, also we found several absurd results there when balancing chemical reactions. In this work only just these absurd contradictions are studied. Sometimes the mistakes in reasoning come because our experience with one situation causes us to assume that the same reasoning will hold true in a related but different situation. This kind of mistake can occur at a very simple level or at a more complex one. At a simple level, the most common conclusion is that we know that we have to reject the reasoning, although it may be difficult to say why. At the more complex level, we may conclude that the reasoning must be accepted even when the results seem to contradict our notion of how the real world works. Scientific research and experience have shown, when the results of reasoning and mathematics conflict with experience, then there is probably a fallacy of some sort involved10. In the literature11-13 there are a great number of definitions for fallacy, but we shall mention merely few of them. For us is important only deductive fallacy. A deductive fallacy is a deductive reason that is illogical. In philosophy, the term logical fallacy suitably refers to a formal fallacy. It is defined as a defect in the structure of deductive reason, which makes the reason invalid. Most textbooks echo the standard treatment of fallacy, as a reason, which seems to be valid but is not so14. According to Maxwell15, a fallacy leads by guile to a wrong but plausible conclusion. In mathematics the word fallacy could also refer to a truthful result obtained by wrong reasoning. In looking at wrong thinking about easy ideas, we found some cases in which reasoning about balancing chemical equations is wrong. As long as we cannot recognize what the fallacy is, the situation is a paradox. In some cases, as we shall see, the paradox is entirely inside chemistry. For most paradoxes that are inside chemistry, elimination of the fallacious reasoning produces a purified chemistry that is a better description of the real chemistry than the contaminated chemistry was. In philosophy there is a bunch of different definitions of paradoxes. According to Sainsbury,16 a paradox is an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises. Rescher17 defines a paradox as a set of propositions that are individually plausible but collectively inconsistent. Chihara18 provides a similar definition: a paradox is an argument that begins with premises that appear to be clearly true, that proceeds according to inference rules that appear to be valid, but ends in contradiction. Quine19, for example, offers the following definition (using the term antinomy, instead of a paradox): An antinomy produces a self-contradiction by accepted ways of reasoning. It establishes that some tacit and trusted pattern of reasoning must be made explicit and hence-forward be avoided or revised. In similar spirit, Koons20 defines paradox as an inconsistency among nearly nonrevisable principles that can be resolved only by recognizing some essential limitation of thought or In the next section we shall consider the paradoxes which appeared in different parts of chemistry. 2 PRELIMINARIES The word paradox in professional works has almost the same meaning as the word contradiction. Chemistry as other natural sciences is not immune of paradoxes. Unlike other natural sciences, in chemistry paradoxes appeared some time later, and it has some of them, while other sciences are overfull of such contradictions. Now, we shall mention the well-known paradoxes in chemistry. They are: □ The Prussian blue paradox:21 The reaction between ferric and ferrocyanide ions to form Prussian blue and ferrous and ferricyanide ions to form Turnbull's blue are profoundly influenced by the occurrence of the ionic re-dox equilibrium: Fe+3 + [Fe(CN)g]-4 l Fe+2 + [Fe(CN)g]-3, which is largely displaced toward the right. □ Feigl's paradoxes:22 • Hydrogen peroxide as a reducing agent (Oxidizing agents undergo mutual reduction) 2 KMnO4 + 3 H2O2 ^ 2 KOH + 2 MnO2 + 2 H2O + 3 O2, NaOCl + H2O2 ^ NaCl + H2O + O2, 2 [Fe(CN)g]-3 + H2O2 + 2 OH-^ 2 [Fe(CN)6]-4 + 2 H2O + O2, Au+3 + 3 H2O2 + 6 OH-^ 6 H2O + 3 O2 + Au. • Sulfurous acid brings about oxidations H2SO3 + 2 H2S ^ 3 H2O + 3 S, or or SO2 + 2 H2S ^ 2 H2O + 3 S, 2 Ni(OH)2 + SO2 O2 ^ NiSO4 + Ni(OH)4. Nitric acid is not an oxidant HNO2 + 3 HN3 ^ 2 H2O + 5 N2. Oxidation of aluminum at room temperature 3 HgCl2 + 2 Al ^ 2 AlCl3 + 3 Hg, 3 HgI2 + 2 Al ^ 2 AlI3 + 3 Hg, 3 HgS + 2 Al ^ Al2S3 + 3 Hg. Nonvolatile oxides of tin and antimony are made to disappear by heating them with a volatile compound SnO2 + 4 HI ^ SnI4 + 2 H2O. An acid sets a base free from a salt H3BO3 + 4 KF ^ KBF4 + 3 KOH. • Permanganate is not capable of oxidizing oxalic acid 5 H2C2O4 + 2 KMnO4 + 3 H2SO4 ^ 2 MnSO4 + 10 CO2 + K2SO4 + 8 H2O. • Ammonium polysulfide brings about an oxidation SnS + (NH4)2S2 ^ (NH4)2SnS3. (NH4)2SnS3 + 2HCl ^ 2 NH4Cl + H2S + SnS2. □ Levinthal's paradox:23 The length of time in which a protein chain finds its folded state is many orders of magnitude shorter than it would be if it freely searched all possible configurations. □ Quantum chemistry paradoxes:24 • Preponderant configurations. • Relevant symmetry. • Watson effect. • High ionization energies of the partly filled shells. • Continua of penultimate ionization. • Continua of translational energy. • Questions of time-scale. • Quantum mechanics pretends to be valid for other systems than electrons. • Assembly properties and repeated small entities. □ Campbell's paradoxes:25 • A catalyst is a substance which increases the rate of a reaction without entering into it. • System tends to a minimum in potential energy. • The entropy of a shuffled deck of cards is greater than that of a new deck. • Energy is the ability to do work. □ Paradoxes of spin-pairing energy in gadolinium (III)26: The spin-pairing parameter D = 9E1/8 for 4f q separates the averages of all states (So) and (S0-1) to the extent 2DS0 where Gd+3 has D = 0.80 eV. Hartree-Fock (flexible radial functions) have previously been performed for each of the four S, producing a value of D = 1.09 eV but the contributions to D from kinetic energy T, electron-nuclear attraction Q, and interelectronic repulsion C with ratios - 1:6:(- 4) distributed Tc (3.5), Tf (4.5), Qc (- 7), Qf (13), Ccc (3.5), Ccf (- 7.9), Cf (0.4) indexed c (closed nl shells) andf (4/). Pragmatic D = 0.80 eV corresponds to 1.84 times the calculated contribution from Cffand to - 0.184 times the sum of C integrals. Additional complications are expected from the correlation energy -100 eV. □ Structure-Activity Relationship (SAR) paradox27: Exceptions to the principle that a small change in a molecule causes a small change in its chemical behavior are frequently profound. □ Helium paradoxes:28 The relatively high 4He/21Ne, 3He/22Ne and 4He/CO2 ratios in midocean ridge basalts suggest that it is the midocean ridge basalt reservoir that is He-rich and that the high ratio 3He/4He in midocean ridge basalts is due to excess 4He, not a deficit in the 'primordial' isotope 3He. □ Temperature dependence of AG° and the equilibrium constant ^e«:29 The sign of AS° determines the temperature dependence of AG°, it is AH° that is responsible for the shift in Keq with temperature. □ The q/T paradox:30 Which 'contains more heat', a cup of coffee at 95 °C or a liter of icewater? □ pH paradox:31 pH = - lg[H+]. □ Parrondo's paradox32: A mathematical concept known as Parrondo's paradox is the unexpected situation in which two specific losing strategies can, by alternating between them, produce a winning outcome. □ Parrondo's paradox motivated the development of many new computational models of chemical systems, in which thermal cycling problem is studied.33 By these kinetics systems compare the rates of formation of products under temperature-cycling and steady-state conditions. Also, these computational models of thermal cycling announce new applications in chemistry, biochemistry and chemical engineering. More essentially, by these models one obtains knowledge that some simple chemical systems might behave paradoxically, and that forced oscillating conditions may induce an outcome. However, these paradoxes are not alone and there are more which appears in the theory of balancing of chemical equations. Just these paradoxes are main research object in this work. 3 A NEW CHEMICAL FORMAL SYSTEM Chemists must introduce a whole set of auxiliary definitions to make the chemistry work consistently. The more abstract the theory is, the stronger the cognitive power is. What does it mean a chemical equation? The reply of this question lies in the following descriptive definition given in a compact form. Definition 2.1. Chemical equation is a numerical quantification of a chemical reaction. Let X be a finite set of molecules. Definition 2.2. A chemical reaction on X is a pair of formal linear combinations of elements of X, such that r s p: XaijXj bjyj (1 < i < m) (2.1) j=1 with aij, bij > 0. j=1 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. Definition 2.3. Each chemical reaction p has a domain Domp = {x G XI a. >0} (2.2) Definition 2.4. Each chemical reaction p has an im- age Definition 2.5. Chemical reaction p is generated for some X G if both aij > 0 and bij > 0. Definition 2.6. For the case as the previous definition, we say x is a generator of p. Definition 2.7. The set of generators of p is thus Domp n Imp. Often chemical reactions are modeled like pairs of multisets, corresponding to integer stoichiometric constants. Definition 2.8. A stoichiometrical space is a pair (X, R ), where R is a set of chemical reactions on X. It may be symbolized by an arc-weighted bipartite directed graph r(X, R) with vertex set X U R , arcs x ^ p with weight aij if aij > 0, and arcs p ^ y with weight bij if bij > 0. Let us now consider an arbitrary subset A C X. Definition 2.9. A chemical reaction p may take place in a reaction combination composed of the molecules in A if and only if Domp C X. Definition 2.10. The collection of all feasible reactions in the stoichiometrical space (X, R), that can start from A is given by R A = {p G R I Domp C A}. In34 is proved the following proposition. (2.4) Proposition 2.11. Any chemical equation may be presented in this form r m s m X x, nYij=x yj n wij (2.5) j=1 where xj (1 < j < r) andyj (1 < j < s) are unknown rational coefficients, Yi and üi (1 < i < m) are chemical elements in reactants and products, respectively, aij (1 < i < m; 1 < j < r) and bij (1 < i < m; 1 < j < s) are numbers of atoms of elements Yi and 0, respectively, in j-th molecule. Definition 2.12. The nullity of the reaction matrix A is nullityA = n - r, (2.6) Imp = {y G XI b^ > 0} (2.3) where n is the total number of reaction molecules and by r = rankA the rank of the matrix A is denoted. Definition 2.13. For any chemical reaction these criteria hold: 1° if nullityA = 0, then the reaction is unfeasible, 2° if nullityA = 1, then the reaction is unique, and 3° if nullityA > 1, then the reaction is non-unique. We shall define a fallacy in this way. Definition 2.14. A wrong result attached with a seemingly logical explanation of why the result is correct is a fallacy. A new definition for a paradox should look like this. Definition 2.15. A paradox is a seemingly true assertion that leads to an inconsistency or a situation, which resists intuition. What does it mean the term non-stoichiometric? Briefly, it means that a substance may participate in two different reactions simultaneously, in which case the relative amounts of two products would bear no fixed ratio to one another. According to this, now we can define a non-stoichiometric reaction as a real vector space. Definition 2.16. A non-stoichiometric reaction is a vector space in which a given set of vector-molecules as reactants gives final vector-molecules as products whose molecular proportions are variable in a continuous sense. 4 NEW PARADOXES AND FALLACIES In this section we shall present chronologically the new paradoxes and fallacies, which we discovered in the theory of balancing chemical reactions. I. Steinbach gave this statement35: While chemical equations may balance algebraically, they are not necessarily stoichimetrically exact. In order to illustrate the above statement, »as the correct stoichiometric equations«, he »balanced« the following chemical reactions 2 KMnO4 + 5 H2O2 + 4 H2SO4 ^ 2 KHSO4 + 2 MnSO4 + 8 H2O + 5 O2, (3.1) K2Cr2O3 + 5 H2O2 + 5 H2SO4 ^ 2 KHSO4 + Cr2(SO4)3 + 9 H2O + 4 O2, (3.2) NaOBr + H2O2 ^ NaBr + H2O + O2, KClO + KClO2 ^ KClO3 + KCl, 3 HClO3 ^ HClO4 + Cl2 + 2 O2 + H2O, 4 KClO3 + 16 HCl ^ 4 KCl + 7 Cl2 + 8 H2O + 2 ClO2. (3.3) (3.4) (3.5) (3.6) He said: Obviously there are an infinite number ofso-lutions for the coefficients. Only a few of the total possible solutions are given. Next, he stated: when equations are balanced by either the valence-change or the ion-electron methods, the coefficients obtained are the correct stoichiometric ones. He finished his article like this: Equations (3.1) ^ (3.6) have no stoichiometric meaning, and it is doubtful whether they have any real significance other than that they contain the short-hand suggestion of the reactants used and the products obtained. They do, however, have a definite suggestion that further investigation would be most desirable. It is by examples such as these that the wide divergence between the stoichiometric equation and the actual mechanism of a chemical reaction is so poignantly revealed. All the statements mentioned above are paradoxical. Why these statements are inconsistent will be explained in the following text. Now, we shall make a very clean distinction what is what! The first statement is completely wrong, because the algebraic method has not any restriction of its usage. It holds for every chemical reaction, while other so-called »chemical methods« hold only for some particular cases. From the general solution of the reaction (3.1), x1 KMnO4 + x2 H2O2 + 2xj H2SO4 ^ xj KHSO4 + xj MnSO4 + (3xJ/2 + x2) H2O + (5xJ/4 + x2/2) O2, (VxJ, x2 G ^) we can see that it is a correct two-parametric stoichio-metric reaction, but not as stated Steinbach. It is just one thing. Another thing, he »offered« only a particular solution of (3.1) for XJ = 2 and x2 = 5. Immediately, after publication of his article35, Hall36 pointed out, the statement for the reaction (3.1) that is a »good« example of variable coefficients is absolutely wrong. It has been known to be stoichiometric for many, many years and was studied by C. F. Schönbein, who found that particular reaction as expressed by (3.1). Remark 3.1. Schönbein's particular solution is obtained under certain experimental conditions and it does not mean that for other different condition other solutions will not be possible! Chemical reaction (3.2) is balanced incorrectly! The coefficients of the above reaction correspond to this chemical reaction xJ K2Cr2O7 + x2 H2O2 + 5xJ H2SO4 ^ 2xJ KHSO4 + xJ Cr2(SO4)3 + (4xJ + x2) H2O + (3xJ + x2)/2 O2, (VxJ, x2 G ^) for x1 = 1 and x2 = 5. The reaction that is offered by Hall36 K2Cr2O7 + 3 H2O2 + 5 H2SO4 ^ 2 KHSO4 + 7 H2O + 3 O2, is balanced wrongly too! The general solution of the reaction (3.3) is xJ NaOBr + x2 H2O2 ^ xJ NaBr + x2 H2O + (xJ + x2)/2 O2, (VxJ, x2 G ^) and Steinbach found a particular solution for XJ = x2 = 1. Again, we had not any limitations with the usage of the algebraically method. Also, in this case the method worked perfectly. Chemical reaction (3.4) has this general solution xJ KClO + x2 KClO2 ^ (xJ + 2x2)/3 KClO3 + (2xJ + x2)/3 KCl, (VxJ, x2 G ^) and the above particular solution corresponds for x1 = x2 = 1. The general solution of the chemical reaction (3.5) is xi HaO3 ^ X2 HaO4 + (x1 - x2)/2 a2 + (5xi - 7x2)/4 O2 + (xi - x2)/2 H2O, (xi > 7x2/5) (3.7) For xi = 3 and x2 = 1, as a particular case from (3. 7) immediately follows (3.5). For xi = x2, from (3.7) one obtains this elementary reaction 2 HaO3 + O2 ^ 2 HaO4. For xi = 7x2/5, chemical reaction (3.7) transforms into 7 HClO3 ^ 5 HClO4 + Cl2 + H2O. If xi < x2, then (3.7) becomes xi HClO3 + (x2 - xi)/2 Cl2 + (7x2 - 5xi)/4 O2 + (x2 - xi)/2 H2O ^ x2 HClO4, (xi < x2). If x2 < xi < 7x2/5, then chemical reaction (3.7) attains this form x1 HClOj + (7x2 - 5x1)/4 O2 ^ x2 HClO4 + (x1 - x2)/2 Cl2 + (x1 - x2)/2 H2O, (x2 < xi < 7x2/5). For the reactions (3.3), (3.4) and (3.5) Lehrman37 with the mentioned particular cases gave a comprehensive construction of ad infinitum equations. The last reaction (3.6), between potassium chlorate and hydrochloric acid, has a general solution x1 KClOj + x2 HCl ^ x1 KCl + (- 3x1 + 5x2/2)/4 Cl2 + x2/2 H2O + (3x1 - x2/2)/2 ClO2, (3.8) (6x1/5 < x2 < 6x1). For x1 = 4 and x2 = 16 from the chemical reaction (3.8) as a particular solution follows (3.6). For xi = 5x2/6, from (3.8) one obtains this elementary reaction 5 KClOj + 6 HCl ^ 5 KCl + 3 H2O + 6 ClO2. For xi = x2/6, then (3. 8) becomes KClOj + 6 HCl ^ KCl + 3 Cl2 + 3 H2O. If xi > 5x2/6, then (3. 8) transforms into x1 KClOj + x2 HCl + (3x1 - 5x2/2)/4 Cl2 ^ x1 KCl + x2/2 H2O + (3xi - x2/2)/2 ClO2, (xi > 5x2/6). (3. 9) We would like to emphasize here that all the reactions are balanced by the well-known algebraically method, and all the two-parametric coefficients are correct. It is completely different from the third Steinbach's statement given above, which gives advantage to the valence-change or the ion-electron methods. If we take into account the very well-known rule that equations for consecutive reactions may be added and equations for concurrent reactions may not be added, then obviously Steinbach brings up the old, very old er- ror involved in adding equations for concurrent reactions. It is his biggest mistake. What happens when the reactions are not unique according to Steinbach? In that case, he »balanced« them as an infinite number of reactions, which do not express actual stoichiometric relations. They are »derived« by combining together the reactions for concurrent reactions, in which coefficients in each of the two or more reactions are previously multiplied by different numbers. Sure, these reactions are incorrect, as they do not correspond to the real stoichiometric relations among the substances, which are involved. In some cases, there is no constant relation among the quantities of substances expressed by his reactions. The ratio of the quantities will depend upon chemical conditions (concentration, temperature, etc.). The above Steinbach's reactions, in fact, are oxidation-reduction reactions for which we found the general solutions by using of algebraic method. Every one of them in fact represents two chemical reactions, not one as Steinbach stated. Therefore, these reactions are not non-stoichiometric reaction! Standen in his article38 named these Steinbach's reactions as bizarre non-stoi-chiometric equations. McGavock in39 gave a note on errata in the Stein-bach's article35. In order to explain what a non-stoichiometric reaction represents, McGavock considered an example from his book40. Example 3.2. It is the cracking reaction xl C8H18 ^ x2 C2H4 + x3C3H6 + x4CH4. (3.10) Now, we shall show that the above reaction (3.10) is not a non-stoichiometric reaction. From the scheme given below 18 H1 8 H42 2 4 II II II C II = = 2 = = 4 äT C 8 2 3 1 H 18 4 6 4 follows this vector equation xiVi = x2V2 + x3V3 + x4V4, I. e., or x1 (8, 18)T = x2 (2, 4)T + x3 (3, 6)T + x4 (1, 4)T, (8x1, 18xj)T = (2x2 + 3x3 + x4, 4x2 + 6x3 + 4x4)T. From the system of linear equations 8xj = 2x2 + 3x3 + x4, 18xj = 4x2 + 6x3 + 4x4, one obtains X4 = X1 and X3 = (7x1 - 2x2)/3. Now, balanced chemical equation has this general solution xi CgHi, ^ x2 C2H4 + (7xi - 2x2)/3 C3He + xi CH4, (xi > 2x2/7). (3.11) The vectors Vi, V2, V3 and V4 of the molecules of the above chemical reaction are linearly dependent and they generate an infinite number of vector spaces V^ over i. e., one obtains an infinite number of solutions [xi, x2, (7x1 - 2x2)/3, x1], (x1 > 2x2/7), that means that the chemical reaction (3. 10) is non-unique. Figure 1: Plane 7x1 - 2x2 - 3x3 = 0 in ^3 However, it is not possible, either on an algebraic or empirical basis, to exclude nonintegral values for the coefficients. Each point of the plane 7x1 - 2x2 - 3x3 = 0, given on the Figure 1, represents a triad of positive, nonintegral values. Infinity of such points corresponding to infinity of triads of coefficients in the above equation exists. By virtue of this finding, it is asserted that this cracking reaction does not represent a non-stoichio-metric reaction. It contradicts McGavock's statement. McGavock in39 obtained this reaction CSH1S 2 C2H4 + C3H6 + CH4. (3.12) Actually, the reaction (3.12) is a particular solution of (3.11), for x1 = 1 and x2 = 2. The above McGavock's »non-stoichiometric« reaction (3.12) according to Standen38 can also be written as two reactions: CSH1S CiH^A + CH, C7H14 ^ 2 C2H4 + C3H6. Before the reaction (3.12) be regarded as a genuine non-stoichiometric reaction, it must be shown that it is one reaction. It might be a combination of these two reactions: 2 C8H,8 3 C8H,8 7 C2H4 + 2 CH4, 7 C3H6 + 3 CH4. Also, this Standen's counterexample shows that chemical reaction (3.12) is not non-stoichiometric reaction. By this and the Definition 2.16, we proved that McGavock's presentation for non-stoichiometric reaction is an ordinary fallacy. II. Porges in his article41 wrote: After examinning a great many chemical equations, one concludes that most of them are of the type in which the number of compounds involved exceeds the number of elements by unity. This Porges' statement does not represent any criterion for balancing chemical equation and it is completely wrong. In fact, it is only an ordinary paradox! The counterexamples given below show it. Example 3.3. Let us consider this elementary chemical reaction x1 BiCl3 + x2 H2O ^ x3 HCl + x4 BiClO. Here, we have a case when the number of involved compounds does not exceed the number of elements by unity, i. e., in this case four elements are involved in four molecules. The above chemical reaction reduces to a system of four linear equations in four unknown variables and the chemical reaction has a unique solution. The balanced reaction has this form BiCl3 + H2O ^ 2HCl + BiClO. Example 3.4. For instance, in the chemical reaction x1 Cu2S + x2 HNO3 ^ x3 Cu2SO4 + x4 NO + x5 H2O, five elements are involved in five molecules. The above reaction has a unique solution x1 = 3, xi = 8, x3 = 3, x4 = 8, x5 = 4. Example 3.5. In this particular reaction x1 K4Fe(CN)6 + x2 K2S2O3 ^ x3 KCNS + x4 K2SO4 + x5 K2S + x6 FeS, are involved six elements and same number of molecules, i. e., we have six linear equations in six unknown variables. The chemical reaction has a unique solution x1 = 2, x2 = 12, x3 = 12, x4 = 9, x5 = 1, x6 = 2. Example 3.6. The chemical reaction x1 AgPF6 + x2 Re(CO)5Br + x3 CH3CN ^ x4 AgBr + x5 [Re(CO)5(CH3CN)]PF6, contains nine elements in five molecules and it reduces to a system of nine linear equations in five unknown variables. The chemical reaction has a unique solution x1 = x2 = — = x5 = 1. The mentioned four counterexamples contradict to the above Porges' statement. By this, we refuted his statement. In the same article41 there are fallacies too. For instance, he reduced the chemical reaction x1 FeCl2 + x2 K2Cr2O7 + x3 HCl ^ x4 FeCl3 + x5 KCl + x6 CrCl3 + x7 H2O, to the following system of linear equations (3.13) Xi = x4, 2xj + x3 = 3x4 + x5 + 3x6, 2x2 = X5, 2x2 = X6, (3.14) 7x2 = X7, x3 = 2x7. Immediately from (3.14), he obtained x1 = 6x2, x2 = x2, x3 = 14x2, x4 = 6x2, x5 = 2x2, x6 = 2x2, x7 = 7x2. (3.15) Next, he said: It is evident that the general integral solution of (3.14) is derived from x2 = k, where k is any positive integer, and consequently although (3.14) has an infinite number of integral solutions, all of them derive from x2 = k, and consist merely of multiplies of values for the respective variables established by (3.15). All solutions other than for k =1, are therefore trivial chemically as well as mathematically. Unfortunately, the above statement is fallacious! The last sentence All solutions other than for k =1, are therefore trivial chemically as well as mathematically, is incorrect. Previous Porges stated that x2 = k, where k is any positive integer, and after that he took k = 1. It is wrong! Why? To this question a very simple answer will follow like this. The reaction (3.13) has a unique solution. If we substitute (3.15) into (3.13), and after that if we divide the relation (3.13) by an arbitrary real number x2 ^ 0, immediately follows 6 FeCl2 + K2Cr2O7 + 14 HCl ^ 6 FeClj + 2 KCl + 2 CrClj + 7 H2O. Actually, Porges considered the general solution (3.15) of the system (3.14) separately of (3.13), what is wrong. The reaction (3.13) and the general solution (3.15) of the system (3.14) must be considered as one whole, because they are connected with each other. Obviously, we did not introduce any constant k, as it was done previously by Porges. Our approach is completely different than Porges' wrong way he used. The same remark also holds for the second Porges' reaction As2S3 + 3 (NH4)2S ^ 2 (NH4)3AsS3. (3.16) considered in41. The third considered reaction in41 is given by this expression x1 HAuCl3 + x2 K4Fe(CN)6 ^ x3 KAu(CN)4 + x4 KAu(CN)2 + x5 KAu(CN)2Cl2 + x6 KCl + x7 HCl + x8 [4Fe(CN)3-3Fe(CN)2]. (3.17) For its »solution« the author offered these expressions x1 = 12k - 2, x2 = 7k, x3 = 2, x4 = 8k - 1, x5 = 4k - 3, x6 = 16k + 2, x7= 12k -2, x8 = k, (3.18) where k takes on all positive integral values. Yet (3.18) is merely a particular solution, but it is not the general solution of (3.17). The general solution of the reaction (3.17) is given by this expression 14x1 HAuCl3 + 14x2 K4Fe(CN)6 ^ (24x2 - 14xi) KAu(CN)4 + (7x1 + 4x2) KAu(CN)2 + (21xi - 28x2) KAu(CN)2Cl2 + (56x2 - 14x1) KCl + 14x1 HCl + 2x2 [4Fe(CN)3-3Fe(CN)2], (4x2/3< xi<12x2/7). (3.19) The balanced reaction (3.19) is an expression of two parameters, but it is not one-parametric expression as it is given in41. In fact, it is another Porges' fallacy. Remark 3.7. Intentionally, we omitted from consideration other particular solutions of (3.19) because we had into account that our work has a limited size. Before ending his article41, Porges posed the following three questions: 1° Is there, for equations of this third type which admits infinitely many distinct, not multiple solutions, a least action principle similar to that in mechanics? 2° Of the unlimited number of ways of balancing such an equation as (3.17), is that corresponding to the solution in least integer of the linear equations the one invariably indicated by the laboratory work, which is, after all, the real criterion? 3° Further, what would a minimum solution be - that for which the square root of the arithmetic mean of the squares of the variables is less than for any other solution? To date we did not meet in chemical literature any reply on these questions. It is a challenge for us to try to give appropriate answers on the above questions. Sure, the answers will be given in a rough form, because a comprehensive replay looks for a special article dedicated only on that particular subject. The answer on the first question should be like this. No! In chemistry there is not a least action principle similar to that in mechanics, because the balancing chemical equation has not any tangent point with mechanics, just it is connected with linear algebra. In fact, balancing chemical equations is not chemistry; it is just linear algebra7. On the other hand, the term a least action associate on the dynamism of 18'h century as metaphysics was traced by Leibniz and Boscovich. Later Boyle gave an explicit formulation of chemistry in a coherent metaphysical scheme. A comprehensive study about metaphysics of chemical reaction is given in42. The second question is interesting for a discussion, and it can be answered negatively in this way. Unlimited number of ways of balancing such a reaction as (3.17) is not corresponding to the solution in least integer of the linear equations the one invariably indicated by the laboratory work, which is, after all, the real criterion. This case, in fact, boils down to the continuum problem. It is an extremely hard problem, which is a stumbling block for mathematicians as well as chemists. In other words, this case can sink into the third question. As the third question is posed, it contains a wrong formulation for minimum solution. It is one more Porges' fallacy. One can reply the last question like this. When the question is posed it was unbelievable for that time. It was impossible because the mathematical methods needed for its proof were unknown. For instance, Moore-Penrose pseudoinverse matrix8,9 was discovered ten years later in 1955 and its first application in chemistry43 appeared more than two decades later in 1978. Also, then was unknown the general problem of balancing chemical equations44. From today viewpoint, by using of Moore-Penrose pseudoinverse matrix the minimal solution is obtained in45,46. Now, a new question arises: why not look for a topology of solutions of chemical equations, instead of finding minimal solutions? It is a much better question, than non-unique equation to be reduced to a minimal case. The question is new, and it is extremely hard. Sure, it will be a challenge for the next research. Porges finished his article41 on this way: In the very few equations of the third type encountered by the writer, the balancing in least integers was always that for which k was also least, and was identical with the laboratory balancing; but given a general solution not minimized by smallest admissible value of k - theoretically not impossible - then what? It represents one last Porges' fallacy! We shall build the reply to this question on the concept of the continuum. In fact, the word continuum is recognizable as the name used by Cantor to refer to the real line. From the expression (3. 19), we can see that this kind of chemical equations reduces to the Cantor's continuum problem. This problem is simply condensed in the following question: How many points are there on the straight line in Euclidean space? In other words, the question is: How many different sets of integers do there exist?'4'' This problem is neither simple nor easy; it needs a wide explanation. It shows that balancing chemical equations is a main object in Foundation of Chemistry, which lies in an intertwined mixture of topology, abstract algebra, linear algebra, axiomatic set theory, mathematical logic, computability theory and proof theory. To explain a little more fully this idea we must first discuss the concept of a formal system. Why? Simply, chemical equation must be treated only as a formal system, if we like to avoid appearance of paradoxes. In an opposite case we shall have paradoxes as these mentioned in this section. A formal chemical system consists of a finite set of symbols and of a finite number of rules by which these symbols can be combined into formulas or statements. That kind of formal system is given in second section of this work. A number of such statements are nominated as axioms and by repeated applications of the rules of the system one obtains an ever growing body of provable statements. A proof of a given statement is a finite sequence of statements that starts with an axiom and ends with the preferred statement. The sequence is such that every transitional statement is either an axiom or is derivable by the rules of the system from statements that lead it. Thus, a statement that a sequence of formulas does or does not represent a proof of formula is »not« a statement in the formal system itself. It is a statement »about« the system and such statements are often referred to as »metamathematical«. For the first time a formal generalized inverse matrix approach for balancing chemical equation is introduced in44. Balancing chemical equations as a matrix well-defined formal system is given in the works45,46,48-50. III. Another paradox in the theory of balancing chemical equations is the following Standen's statement38: It would seem that examples could not be found where the number of mathematical equations actually exceeds the number of variables; for if the mathematical equations were inconsistent, the whole thing would be an impossibility, while if they were consistent it would indicate that the chemical equation had been appropriately broken down into its terms. To prove the above Standen's absurdity, we shall use the following counterexamples. Example 3.8. The chemical reaction x1 CoCl2 + x2 Na3PO4 ^ x3 NaCl + x4 Co3(PO4)2, contains five elements involved in four molecules, i. e., in this case the chemical reaction reduces to a system of five linear equations in four unknown variables and the chemical reaction has a unique solution. The balanced reaction has this form 3 CoCl2 + 2 Na3PO4 ^ 6 NaCl + Co3(PO4)2. Example 3.9. In this particular reaction x1 Cu(NH3)4Cl2 + x2 KCN + x3 H2O ^ x4 [K2Cu(CN)3 NH4Cl KCl] + x5 KCNO + x6 NH3, are involved seven elements in six molecules, i. e., we have seven linear equations in six unknown variables. The chemical reaction has a unique solution x1 = 2, x2 = 7, x3 = 1, x4 = 2, x5 =1, x6 = 6. Example 3.10. For instance, the chemical reaction x1 AgPF6 + x2 Re(CO)5Br + x3 CH3CN + x4 K2S2O3 ^ x5 KBr + x6 [Re(CO)5(CH3CN)]PF6 + x7 Ag2S + x8 K2O + x9 SO2, has eleven elements involved in nine molecules. The above chemical reaction reduces to a system of eleven linear equations in nine unknown variables, whose unique solution is x1 =4, x2 = 4, x3 = 4, x4 = 3, x5 = 4, x6 = 4, x7 = 2, x8 =1, x9 = 4. By the last three counterexamples we showed that the above Standen's38 statement is an absurd. IV. Next, we shall consider the Blakley's51 paradox. Among other chemical reactions, he considered hydroly- sis of two organic substances: bradykinin and grami-cidin-S. Example 3. 11. The following reaction Xi C2H5NO2 + X2 C3H7NO3 + x3 CgHj4N4O2 + X4 C5H9NO2 + x5 C9HjjNO2 ^ X6 H2O + X7 C50H73N15O11. (3.20) was studied in51. Blakley considered balancing of the above chemical reaction by a matrix approach using a module basis. He »proved« that »hydrolysis (3.20) of bradykinin is unique«. It represents only an empirical discovered relationship, which is wrong. We shall prove the absurdity of his statement by using the well-known algebraic method for balancing chemical equations. By the way, we shall show that this method is powerful and its usage is not limited as some traditional oriented chemists think. Let us consider the scheme of the chemical reaction (3.20). 1, where n is the total number of reaction molecules, then the chemical reaction is possible and it has an infinite number of solutions. Let us prove it. One can reduce the chemical reaction (3.20) to the following system of linear equations 2 xj + 3 x2 + 6 x3 + 5 x4 + 9 x5 = 50 x7, 5 xj + 7 x2 + 14 x3 + 9 x4 + 11 x5 = 2 x6 + 73 x7, (3.21) xj + x2 + 4 x3 + x4 + x5 = 15 x7, 2 xi + 3 x2 + 2 x3 + 2 x4 + 2 x5 = x6 + 11 x7. The general solution of the system (3.21) is given by the following expressions x4 = 66 x1/23 + 93 xJ23 - 45 x3/23, x5 = - 14 x1/23 - 26 x2/23 + 43 x3/23, (3.22) x6 = 95 xJ23 + 137 x2/23 - 24 x3/23, x7 = 5 xi/23 + 6 x2/23 + 6 x3/23, where xi > 0 (1 < i < 3) are arbitrary real numbers. After substitution of (3.22) in (3.20), one obtains a balanced chemical reaction xi C2H5NO2 + x2 C3H7NO3 + x3 C6Hj4N4O2 + (66 xJ23 + 93 x2/23 - 45 x3/23) C5H9NO2 + (- 14 x1/23 - 26 x2/23 + 43 x3/23) C9H11NO2 ^ (95 x1/23 + 137 x2/23 - 24 x3/23) H2O + (5 xJ23 + 6 x2/23 + 6 x3/23) C50H73N15O11. (3.23) From (3.23) follows this system of inequalities 22x1 + 31x2 - 15x3 > 0, - 14x1 - 26x2 + 43x3 > 0, (3.24) 95x1 + 137x2 - 24x3 > 0, 5x1 + 6x2 + 6x3 > 0. From the first and the second inequality of (3.24) immediately follows this expression 14 x1/43 + 26 x2/43 < x3 < 22 x1/15 + 31 x2/15. (3.25) The expression (3.25), the third and the fourth inequality of (3.24) are necessary and sufficient conditions for (3.23) to hold. For instance, if we substitute x1 = x2 = 1 and x3 = 2 in (3.23), then as a particular case appears this reaction C2H5NO2 + C3H7NO3 + 2 CgH14N4O2 + 3 C5H9NO2 + 2 C9H11NO2 ^ 8 H2O + C50H73N15O11, for which Blakley51 stated that it is unique, but it is not true. Now, we shall give two more particular cases for which (3.23) holds. Let x1 = x2 = x3 = 1, then from (3.23) one obtains the reaction 23 C2H5NO2 + 23 C3H7NO3 + 23 C6H14N4O2 + 114 C5H9NO2 + 3 C9H11NO2 ^ 208 H2O + 17 C5oH73N15O11. Let x1 = x2 = 1 and x3 = 3, then (3.23) becomes 23 C2H5NO2 + 23 C3H7NO3 + 69 C6H14N4O2 + 24 C5H9NO2 + 89 C9H11NO2 ^ 160 H2O + 29 C5oH73N15O11. The other particular cases of (3.23) are not considered, because we took into account the Remark 3.7. Example 3.12. In51 the hydrolysis of gramicidin-S was studied, given by the following reaction x1 C5H9NO2 + x2 C5H11NO2 + x3 C6H13NO2 + x4 C9H11NO2 + x5 C5H12N2O2 x6 C60H92N12O10 + x7 H2O. (3.26) For the chemical reaction (3.26) if we write its stoichiometric scheme, then one obtains CS O 2 O2 N 2 1 NO23 s: O2 N2 (N (N 60 O C5 C5 C6 C5 C6 H2 C 5 5 6 9 5 -60 0 H 9 11 13 11 12 -92 -2 N 1 1 1 1 2 -12 0 O 2 2 2 2 2 -10 -1 from where follows the stoichiometric matrix "5 5 6 9 5 -60 0 9 11 13 11 12 -92 -2 11112 -12 0 _2 2 2 2 2 -10 -1 A = with r = rankA = 4. Since nullity A = n - r = 7- 4 = 3>1, then chemical reaction is possible and it has an infinity number of solutions. Let us prove it. The system of linear equations obtained from (3. 26) is 5 Xi + 5 X2 + 6 X3 + 9 x4 + 5 X5 - 60 x6 = 0, 9 xi + 11 X2 + 13 x3 + 11 x4 + 12 x5 - 92 x6 - 2 x7 = 0, (3.27) x1+ x2 + x3 + x4 + 2 x5 - 12 x6 = 0, 2 x1 +2 xj + 2 x3 + 2 x4 + 2 x5 - 10 x6 - 1 x7 = 0. The general solution of (3.27) is given by the expressions xj = - 4 x4 + 20 x6 - x7, x2 = 7 x4 - 57 x6 + 9 x7/2, x3 = - 4 x4 + 35 x6 - 5 x7/2, x5 = 7 xg - x7/2, (3.28) where x4, x6 and x7 are arbitrary real numbers. After substitution of (3.28) in (3.26), the balanced chemical reaction has this form (-4 x4 + 20 x6 - x7) C5H9NO2 + (7 x4 - 57 x6 + 9 x7/2) C5H11NO2 + (- 4 x4 + 35 x6 - 5 x7/2) c6h13no2 + x4 C9H11NO2 + (7 x6 - x7/2) C5H12N2O2 ^ x6 C60H92N12O10 + x7 H2O, (3.29) From (3.29) follows this system of inequalities - 4 x4 + 20 x6 - x7 > 0, 7 x4 - 57 x6 + 9 x7/2 > 0, (3.30) - 4 x4 + 35 x6 - 5 x7/2 > 0, 7 x6 - x7/2 > 0. From the second and the third inequality of (3.30) immediately follows this expression - 14 x4/9 + 114 x6/9 < x7 <-8 x4/5 + 14 x6. (3.31) The expression (3.31), the first and the fourth inequality of (3.30) are necessary and sufficient conditions for (3.29) to hold. For instance, if we substitute x4 = 2, x6 = 1 and x7 = 10 in (3.29), then as a particular case appears this reaction 2 c5h9no2 + 2 c5hjjno2 + 2 C6Hj3NO2 + 2 c9hjjno2 + 2 c5hj2n2o2 ^ C60H92N12O10 + 10 H2O. for which Blakley51 stated that it is unique, but it is not true. Now, we shall give more two particular cases for which holds (3.29). Let x4 = x6 = 9 and x7 = 102, then from (3.29) one obtains the reaction 42 c5h9no2 + 9 c5hjjno2 + 24 c6hj3no2 + 9 c9hjjno2 + 12 C5Hj2N2O2 ^ 9 c6oh92ni2oio + 102 h2o. Let x4 = 27, x6 = 21 and x7 = 250, then (3.29) becomes 62 c5h9no2 + 117 c5hiino2 + 2 c6hi3no2 + 27 c9hiino2 + 22 C5Hi2N2O2 ^ 21 c6oh92ni2oio + 250 h2o. The other particular cases of (3.29) are not considered because we took into account the Remark 3.7. V. Das in his article52 applied the partial equation method for balancing chemical equations. There he wrote: if the number of reactants and products is equal to or less (or at most one more) than the total number of elements involved in the chemical equation, then there will be only one way of balancing a chemical equation. For the first two cases this statement is a paradox! The next two counterexamples given below contradict to the above statement. Example 3.13. The following chemical reaction xi NO2 + x2 HClO ^ x3 HNO3 + x4 HCl, has involved four elements in four molecules, but it has not a unique solution as stated above. It is an unfeasible reaction, because x1 = x2 = xs = x4 = 0. Now, we shall give another counterexample. Example 3.14. In the reaction xi KIO2 + x2 Pb(NO3)2 ^ x3 KNO3 + x4 PbI2, five elements are involved in four molecules. This reaction has only a trivial solution x1 = x2 = xb = x4 = 0. It shows that this reaction is unfeasible. VI. Next, we shall elaborate another very interesting paradox. Garcia53 gave a »half-reaction method« for balancing chemical equations. He described his »method« on this way. The chemical reaction is divided into two half-reactions and each one is balanced independently. These two balanced half-reactions are added together to get the correct stoichiometry of the reaction. One half-reaction is formed with compounds that contain the same elements other than oxygen and hydrogen. The remaining compounds, and others if it is necessary, constitute the other half-reaction. Example 3.15. This chemical reaction xi FeS2 + x2 HNO3 ^ x3 Fe2(SO4)3 + x4 NO + x5 H2SO4, (3.32) he »balanced« like this 2 FeS2 + 10 HNO3 ^ Fe2(SO4)3 + 10 NO + H2SO4 + 4 H2O. (3.33) Unfortunately reaction (3.33) is quantitatively and qualitatively different from the reaction (3.32). Actually, the reaction (3.33) is augmented reaction (3.32) by four water molecules. Reactions (3.32) and (3.33) belong to different types of reactions, and according to it, they are incompatible. The reaction (3.32) belongs to the type of unfeasible reactions, because its vectors of molecules do not generate a vector space V over For our next analysis we shall use the newest method7 for balancing chemical equations founded by virtue of theory of complex finite dimensional vector spaces. We chose it, because in this particular Garcia's case, it was the most suitable method for comparative analysis of chemical reactions which belong to different classes. Application of this method confirmed its scientific supremacy. From (3.32) one obtains the scheme given below 2 S FeS O3 N H O (S O N O4 S2 = V1 = 2 V = V3 = V4 = V5 Fe 1 0 2 0 0 S 2 0 3 0 1 H 0 1 0 0 2 N 0 1 0 1 0 O 0 3 12 1 4 The vector equation of reaction (3.32) is xiVi + x2V2 = x3V3 + x4V4 + x5V5, I. e., xi (1, 2, 0, 0, 0)T + x2 (0, 0, 1, 1, 3)T = x3 (2, 3, 0, 0, 12)T + x4 (0, 0, 0, 1, 1)T + x5 (0, 1, 2, 0, 4)T, or (xj, 2x1, x2, x2, 3x2)T = (2x3, 3x3 + x5, 2x5, x4, 12x3 + x4 + 4x5)T. The system of linear equations x1 = 2x3, 2xi = 3x3 + x5, x2 = 2x5, x2 = x4, 3x2 = 12x3 + x4 + 4x5, is inconsistent, because one obtains the contradiction x5 = x1/2 and x5 = - x1. It means that the vectors V1, V2, V3, V4 and V5 of molecules of the chemical reaction (3.32) are linearly independent and they do not generate a vector space V over By this we showed that the chemical reaction (3.32) is unfeasible. On the other hand, the rank of the reaction matrix A of the chemical reaction (3.32) is [10 2 0 I2 0 3 0 r = rankA = rank1 0 1 0 0 |0 1 0 1 L 0 3 12 1 01 11 2| =5 0| 4J According to the algebraic criterion (2.6) for balancing chemical reactions, the reaction (3.32), has nullityA = n - r = 5- 5 = 0, that means that the reaction (3.32) is unfeasible. Both proofs, vector and algebraic, confirmed the same, that the reaction (3.32) is unfeasible. It contradicts the Garcia's »procedure« named as »half-reaction method« for balancing chemical reactions. Now, we shall consider the reaction (3.33) in its unbalanced form x1 FeS2 + x2 HNO3 ^ x3 Fe2(SO4)3 + x4 NO + x5 H2SO4 + x6 H2O (3.34) Now, we need the stoichiometric scheme for the above chemical reaction. From this particular reaction (3.34), we shall derive very easy required stoichiometric scheme O4 4 2 O3 (S O4 S2 e NO O S 2 O2 Fe II H II II N II II II = = 2 = = 4 = = 6 V1 V V3 V4 V5 V Fe 1 0 2 0 0 0 S 2 0 3 0 1 0 H 0 1 0 0 2 2 N 0 1 0 1 0 0 O 0 3 12 1 4 1 From the above scheme one obtains this vector equa- tion I. e., x1V1 + x2V2 = x3V3 + x4V4 + x5V5 + x6V6 xi (1, 2, 0, 0, 0)T + x2 (0, 0, 1, 1, 3)T = x3 (2, 3, 0, 0, 12)T + x4 (0, 0, 0, 1, 1)T + x5 (0, 1, 2, 0, 4)T + x6 (0, 0, 2, 0, 1)T, or (x1, 2x1, x2, x2, 3x2)T = (2x3, 3x3 + x5, 2x5 + 2x6, x4, 12x3 + x4 + 4x5 + x6)T. The solution of the system of linear equations is x1 = 2x3, 2x1 = 3x3 + x5, X2 = 2x5 + 2x6 X2 = x4, 3x2 = 12x3 + x4 + 4x5 + x6, x2 = 5x1, x3 = x1/2, x4 = 5x1, x5 = x1/2 and x6 = 2x1. 34 FeS2 + 130 HNO3 -+ 106 NO + 17 H2SO4 ^ 17 Fe2(SO4)3 + 12 (NH4)2O. (3.35) If we substitute (3.35) in (3.34), and after that, if we divide the reaction by x\/2 one obtains (3.33). The vectors vi, v2, v3, V4, V5 and V6 of reaction molecules are linearly dependent and they generate a vector space V over By this we confirmed that the reaction (3.33) has a unique solution. Similarly as in the previous equation analysis, now we can go to the next step. Now we can calculate the rank of the reaction matrix A of the reaction (3.33). Therefore, we can express its rank in this way n I2 r = rankA = rank | 0 |0 0 3 0 0 12 0 0 0 1 1 01 oj 2| =5 0 j 1 According to the algebraic criterion (2.6) for balancing chemical equations, the reaction matrix A has nullityA = n - r = 6- 5 = 1. By this, again we showed that chemical reaction (3.33) has a unique solution. The analysis of the reaction (3.33), established that it belongs to the type of solvable equations, which have a unique solution. This comparative analysis confirmed that Garcia's half-reaction simple »method«53 is completely wrong. Generally speaking, his so-called »method« cannot recognize the type of reaction, and much less to decide if the chemical equation is solvable or not. To support it, we shall give a dozen of counterexamples, where water molecules in Garcia's procedure of reaction extension (3.33), may be substituted by other molecules of the elements involved in the reaction (3.32), as it is exposed by the following reactions 2 FeS2 + 8 HNO3 ^ Fe2(SO4)3 + 8 NO + H2SO4 + 3 H2, 2 FeS2 + 14 HNO3 ^ Fe2(SO4)3 + 14 NO + H2SO4 + 6 H2O2, 10 FeS2 + 34 HNO3 ^ 5 Fe2(SO4)3 + 22 NO + 5 H2SO4 + 12 NH2, 14 FeS2 + 50 HNO3 ^ 7 Fe2(SO4)3 + 38 NO + 7 H2SO4 + 12 NH3, 10 FeS2 + 34 HNO3 ^ 5 Fe2(SO4)3 + 22 NO + 5 H2SO4 + 6 N2H4, 10 FeS2 + 34 HNO3 ^ 5 Fe2(SO4)3 + 26 NO + H2SO4 + 4 (NH4)2SO3, 14 FeS2 + 50 HNO3 ^ 7 Fe2(SO4)3 + 38 NO + H2SO4 + 6 (NH4)2SO4, 4 FeS2 + 16 HNO3 ^ 2 Fe2(SO4)3 + 13 NO + 2 H2SO4 + 3 NH4O, Neither one of the above reactions nor chemical reaction (3.33) is equivalent to the chemical reaction (3.32). Therefore, reactions (3.32) and (3.33) are incompatible. In a similar way, Garcia considered the following two reactions53. Example 3. 16. The following chemical reaction x1 CrI3 + x2 KOH + x3 Cl2 ^ x4 K2CrO4 + x5 KIO3 + x6 KCl, (3.36) he »balanced« like this 2 CrI3 + 52 KOH + 21 Cl2 ^ 2 K2CrO4 + 6 KIO3 + 42 KCl + 26 H2O. (3.37) On top of all he said: This method is appropriate to balance any kind of reaction, even those that include complex ions or reactions of compounds with oxidation numbers difficult to determine. By the same analysis which we used in the previous counterexample, very easy we shall show the absurdity of his statement. Reaction (3.36) belongs to the type of unfeasible reaction. Its stoichiometric matrix A has rank r = rankA = 6 and its nullityA = n - r = 6- 6 = 0 verifies that it is an unfeasible reaction. Also, this reaction generates an inconsistent system of linear equations which has only a trivial solution xt = 0, (1 < i < 6). Thus, this algebraic criterion verifies that the reaction (3.36) is unfeasible too. Reaction (3. 37) generates a consistent system of linear equations which has a unique solution given in (3.37). Also, the nullityA = n - r = 7- 6 = 1, shows that this equation has a unique solution. Therefore, reactions (3.36) and (3.37) are incompatible, because they are two completely different types of reactions - the first one is an unfeasible reaction, while the second one is a unique reaction. Now, we shall mention just two chemical reactions, where water molecules in Garcia's »procedure« of reaction extension (3.37), are substituted by other molecules of the elements involved in the reaction (3.36): 2 CrI3 + 26 KOH + 8 Cl2 ^ 2 K2CrO4 + 6 KIO3 + 16 KCl + 13 H2, 2 CrI3 + 26 KOH + 21 Cl2 ^ 2 K2CrO4 + 6 KIO3 + 16 KCl + 26 HCl. Neither one of the above reactions nor chemical reaction (3.37) is equivalent to the chemical reaction (3.36). Therefore, reactions (3.36) and (3.37) are incompatible. In the same paper, Garcia considered two ionic reactions too. Unfortunately, also in these particular cases the same absurdity appears again. Example 3.17. For example, he »balanced« an unfeasible reaction x1 IO3- + x2 Br- ^ x3 IO2- + x4 Br2, (3.38) where xi = 0 (1 < i < 4) as a unique reaction IO3- + 2 Br- + 2 H+ ^ IO2- + Br2 + H2O. (3.39) Reactions (3.38) and (3.39) are two completely different reactions and they are incomparable! A second ionic example that Garcia considered is given in the next example. Example 3.18. The reaction Xi ClO- + X2 P4 ^ X3 H2PO4- + X4 Cl-, (3.40) is »balanced« like this 10 ClO- + P4 + 2 H2O + 4 OH-^ 4 H2PO4- + 10 Cl-. (3.41) Where is the hydrogen atom in the left side of (3.40)? Reaction (3.40) is an absurd, because it does not contain hydrogen atom in its left side. With this kind of reactions current chemistry does not work! Reaction (3.41) does not dependent on (3.40) and it represents a solvable equation. In other words, reactions (3.40) and (3.41) are completely different types of reactions and they are incompatible. VII. In54 the so-called »formal balance numbers« (FBN) are introduced like this: Formal balance numbers are an aid that may grossly facilitate the problem of balancing complex redox equations. They may be chosen as being equal to the traditional values of oxidation numbers, but not necessarily. An inspection of the redox equation may suggest the optimal values that are to be assigned to formal balance numbers. In most cases, these optimal values ensure that only two elements will šchange their state' (i. e. the values of the formal balance numbers), allowing the use of the oxidation number technique for balancing equations, in its simplest form. Just as for oxidation numbers, the algebraic sum of the formal balance numbers in a molecule/neutral unit is 0, while in an ion it is equal to its charge (the sum rule). It was quickly detected that the »procedure« given in54 boils down to using of well-known unconventional oxidation numbers, which previously were advocated by Toth55 and Ludwig56. Consider this sentence from previous definition: They may be chosen as being equal to the traditional values of oxidation numbers, but not necessarily. It is a paradox! If the »formal balance numbers« can be the same as oxidation numbers or not, then the whole definition is illogical. This definition represents only a contradictory premise, which does not have any correlation with balancing chemical equations. It is just one thing. Another thing, the above definition does not speak anything about balancing chemical reactions in a chemical sense of the word, or their solution in a mathematical sense. Recent research7 confirmed that a chemical equation can be balanced if and only if it generates a vector space. That is a necessary and sufficient condition for balancing a chemical equation! The so-called called »formal balance numbers«, which actually are the same as the well-known oxidation numbers, do not represent any criterion for balancing chemical equations. Also the author of54 asserted that his »procedure« is probably the fastest of all possible methods! Obviously the author omitted to prove it. Perhaps, his statement is valid (if he can prove it?) in some metachemistry, but from a viewpoint of current mathematics and chemistry it is not true. Why? The reason is very simple. In mathematics, as well as in chemistry there is neither a definition for speed of equation solution nor its unit, and according to it, it is impossible to compare which method is faster. It is just one thing; another thing is that the definition of so-called »formal balance numbers« (FBN)54 is paradoxical and it produces only inconsistent procedure for balancing chemical equations. According to it, the author's assertion of54 is an absurd. The »procedure«54 founded by virtue of so-called »formal balance numbers« (FBN), with several counterexamples was refuted in7. VIII. Ten Hoor in57 obtained this result: C + x O2 ^ 2(1 - x) CO + (2x - 1) CO2, (3.42) (1/2 < x < 1) if the coefficient of a product is allowed to be equal to zero. Taking x equal to its smallest or largest extreme value, equation (3.42) reduces to or C + 1/2 O2 ^ CO, C + a ^ CO (3.43) (3.44) respectively. The above statement is wrong. The reaction (3.42) holds if only if 1/2 < x < 1, like it is shown on Fig. 2, but not as it is given in57. Figure 2: The interval 1/2 < x < 1 In (3.42) x does not have any extreme value, because it is presented by the following linear functions: x, 2 - 2x and 2x - 1. None of these functions have extrema, since their second derivatives are equal to zero. Then on what basis ten Hoor57 states that for smallest or largest extreme value of x the reaction (3.42) reduces to (3.43) and (3.44) respectively? The chemical reaction (3.42) has two subgenerators 2 - 2x and 2x - 1 which generate the following particular cases: 1° For x =1, then (3.42) reduces to C + O ^ CO 2° For X = 1/2, then (3.42) becomes C + 1/2 O2 ^ CO. 3° For X < 1/2, then (3.42) transforms into C + X O2 + (1 - 2x) CO2 ^ 2(1 - x) CO, 4° For X >1, then from (3.42) one obtains C + X O2 + (2x - 2) CO ^ (2x - 1) CO2, 5° For 1/2 < X < 1, holds this reaction C + X O2 ^ 2(1 - X) CO + (2x - 1) CO2. Next, for the reaction (3.42) we shall determine its minimal coefficients by using of Moore-Penrose generalized inverse matrix46. From the chemical reaction (3.42) follows this scheme O O C O C C C 1 0 -1 -1 O 0 2 -1 -2 According to the above scheme, the reaction matrix A of (3.42) has this form [1 0 -1 -H _ 0 2 -1 -2_ A= 1/2 -1/6 -1/3 1/3 -1/3 0 / 0 -1/6 -1/6. Let us mention them. In their article59 they provided the following wrong definition: a chemical equation is a written representation of a chemical reaction, showing the reactants and products, they physical states, and the direction in which the reaction proceeds. According to the above definition a chemical equation will show like this a A(s) + b B (g) c C(s) + d D(g). (3.47) For instace, if r = s = 2 in (2.1), then as a particular case appears the reaction (3.47). Actually, it is not a definition for a chemical equation, just opposit it is a definition for a chemical reaction. Obviously the authors can not distinguish what is a chemical reaction and what is its chemical equation. These two things are different entities given by the Definitions 2.2 and 2.1 (in a descriptive form), i.e., 2.11 (in an analitical form), respectively. It is just one reason what the above definition is wrong. In order to be balanced certain chemical reaction is not necessary to be known its reactants and products as it is described in the above definition! It holds if only if reaction is given in a conventional form, but in an opposite case it does not hold. For instace, the chemical reaction (3.47) given in a convntional form can be presented in an algebrical free form too The Moore-Penrose generalized inverse matrix A+ is A+ = AT(A AT)-1 = The reaction (3.42) reduces to this matrix form Ax = 0, (3. 45) where x = (X1, X2, X3, X4)T is the vector of the coefficients of (3.42), 0 = (0, 0)T is the zero vector and T denoting transpose. The general solution of the matrix equation (3.45) is x = (I - A^A)«, (3.46) where a is an arbitrary vector and I is a unite matrix. For a = (1, 1, 1, 1)T, one obtains x„i, = (4/3, 1, 2/3, 2/3)T. Then the reaction (3.42) with its minimal coefficients attains this form 4/3 C + O2 ^ 2/3 CO + 2/3 COj, but not as ten Hoor asserted in57. By this proof we have shown that his statement is paradoxical. Also, the assumptions 1 and 2 which ten Hoor used in57 are completely wrong, because carbon burns according to the Boudouard's reaction58. Wrong suppositions can not lead to correct results. IX. Authors of the article59 studied several chemical reactions, but unfortunately there are given lots of erroneous results. a A(s) + b B(g) + c C(s) + d D(g) = 0. (3.48) After determination of its coefficients, one obtains that some of them have a negative sign and others have a positive sign. Positive coefficients stay in front of reac-tants and negative coefficients stay in front of products of reaction, that means that chemical reaction is self-adaptive. For example, in48 are balanced chemical reactions given in an algebrical free form. Next, we shall give an another reason why the above definition is wrong. Chemical equation does not have arrow mark as a reaction, just sign for equality. It is a main difference between chemical reaction and chemical equation. More accurately speaking, any chemical reaction has a chemical equation, but the opposite does not hold. Why? Next, we shall give an explanation about it by a new example. Let us balance the following chemical reaction X1 Pb^Os + X2 C ^ X3 Pbo,987O + X4 Pb3O4 + x5 CO + x6 CO2. (3.49) From the above reaction follows this system of linear equations 2 Xi = 0.987 X3 + 3 X4, 3 X1 = X3 + 4 X4 + X5 + 2 X6, X2 = X5 + X6. (3.50) In order to avoid fractional coefficients of the system (3.50), we shall multiply its first equation by 1000, such that one obtains this system of linear equations 2000 x1 = 987 x3 + 3000 x4, 3 x1 = x3 + 4 x4 + x5 + 2 x6, (3.51) x2 = x5 + x6. The systems (3.50) and (3.51) are equivalent and they have same solution. Now, according to the system (3.51) the chemical reaction (3.49) will transform into this particular form x1 Pb2000O3 + x2 C ^ x3 Pb987O + x4 Pb3000O4 + x5 CO + x6 CO2. (3.52) Do the expression (3.52) is a correct chemical reaction? No! It represents only an ordinary chemical absurd, because the molecules Pb2000O3, Pb987O and Pb3000O4 do not exist in chemistry. This is the cause why from chemical equation does not follow chemical reaction. Chemical reactions (3.49) and (3.52) in a mathematical sense both are equivalent reactions, since they reduce to the same system of linear equations, but in a chemical sense they are not equivalent reactions. That means, that math. equivalency ^ chem. equivalency. In other words, from a mathematical point of view, the systems (3.50) and (3.51) both are equivalent, but from a chemical view point they are not, since they generate different chemical reactions. The above explanation, we can articulate roughly on this way: performing of reaction is a chemical subject, and its balancing is a mathematical topic. This is the reason why balancing of chemical reactions is pure mathematical matter, but not a chemical issue. Next, we shall continue with the balancing of the reaction (3.49), because its general solution is necessary for a comparative analysis of other particular chemical reactions. The general solution of the system (3.50) is x4 = 2 x1/3 - 0.329 x3, x5 = - x1/3 + 2 x2-0.316 x3, (3.53) x6 = x1/3 - x2 + 0.316 x3, where x;, (1 < i < 3) are arbitrary real numbers. Balanced reaction has this form x1 Pb2O3 + x2 C ^ x3 Pb0,987O + (2 x1/3 - 0.329 x3) Pb3O4 + (- x1/3 + 2 x2-0.316 x3) CO + (x1/3 - x2 + 0.316 x3) CO2, (3.54) where x;, (1 < i < 3) are arbitrary real numbers. Since x4, x5 and x6 are > 0, then from (3.54) one obtains this system of inequalities 2 x1/3 - 0.329 x3 > 0, - x1/3 + 2 x2-0.316 x3>0, (3.55) x1/3- x2 +0.316 x3>0. From (3.55), we obtain this relation 3 x2 - 0.948 x3 < x1 < 6 x2 - 0.948 x3. (3.56) The inequality (3.56) is necessary and sufficient condition to hold the general reaction (3.54). Now, we can analyze the general reaction (3.54) for all possible values of x1, x2 and x3. As particular reactions of (3.54) we shall derive the following cases. 1° For x1 = 3, x2 = 2.4 and x3 = 4.5, from (3.53) follows x4 = 0.5195, x5 = 2.378 and x6 = 0.022, i.e., one obtains this particular reaction 3 Pb2O3 + 2.4 C ^ 4.5 Pb0 987O + 0.5195 Pb3O4 + 2.378 CO + 0.022 CO2. (3.57) 2° For x1 = 0, the reaction (3.54) transforms into this particular reaction 0.329 x3 Pb3O4 + x2 C ^ x3 Pbo 987O + (2 x2-0.316 x3) CO + (- x2 +0.316 x3)CO2, (3.58) (0.158 x3 < x2 < 0.316 x3). 3° For x2 = 0.158 x3, from (3.58) one obtains 0.329 Pb3O4 + 0.158 C ^ Pbo987O + 0.158 CO2. (3.59) 4° For x2 = 0.316 x3, the reaction (3.58) becomes 0.329 Pb3O4 + 0.316 C ^ Pbo987O + 0.316 CO. (3.60) 5° For x2 < 0.158 x3, from (3.58) follows 0.329 x3 Pb3O4 + x2 C + (0.316 x3 - 2 x2) CO ^ x3 Pbo,987O + (- x2 + 0.316 x3) CO2, (3.61) where x2 and xs are arbitrary real numbers. 6° For x2 > 0.316 x3, the reaction (3.58) becomes 0.329 x3 Pb3O4 + x2 C + (x2 - 0.316 x3) CO2 ^ x3 Pbo,987O + (2 x2 - 0.316 x3) CO, (3.62) where x2 and xb are arbitrary real numbers. 7° For x2 = 0, from (3.54) one obtains this particular reaction xi Pb2O3 + (xi/3 + 0.316 x3) CO ^ x3 Pb0987O + (2 xJ3 - 0.329 x3) Pb3O4 + (x1/3 +0.316 x3)CO2, (3.63) where x1 and xs are arbitrary real numbers. From (3.63) follows these inequalities xJ3 + 0.316 xs > 0 and 2 xJ3 - 0.329 xs > 0. From this system follows this inequality - xi/0.048 < x3 < 2 xi/0.987. (3.64) The reaction (3.63) holds if only if the inequality (3.64) is satisfied. 8° For x1 < 0.4935 xs, from (3.63) follows xi Pb2O3 + (0.329 x3 - 2 xJ3) Pb3O4 + (xJ3 + 0.316 xs) CO ^ x3Pb0 987O + (x1/3 +0.316 x3)CO2, (3.65) where x1 and xs are arbitrary real numbers. 9° For x1 = 0.4935 xs, from (3.63) follows 0.4935 Pb2O3 + 0.4805 CO ^ Pb0,987O + 0.4805 CO2. (3.66) 10° For x3 = 0, from (3.54) one obtains this particular reaction x1 Pb2O3 + x2 C ^ (2 x1/3) Pb3O4 + (- xJ3 + 2 x2) CO + (xJ3 - x2) CO2, (3.67) (3 x2 < x1 < 6 x2). 11° For x1 = 3 x2, from (3.67) follows 3 Pb2O3 + C ^ 2 Pb3O4 + CO. (3.68) 12° For x1 = 6 x2, from (3.67) one obtains 6 Pb2O3 + C ^ 4 Pb3O4 + CO2. (3.69) 13° For x1 < 3 x2, from (3.67) follows x1 Pb2O3 + x2 C + (x2 - xJ3) CO2 ^ (2 xJ3) Pb3O4 + (- xJ3 + 2 x2) CO, (3.70) where x1 and x2 are arbitrary real numbers. 14° For x1 > 6 x2, the reaction (3.67) transforms into x1 Pb2O3 + x2 C + (xJ3 - 2 x2) CO ^ (2 x1/3) Pb3O4 + (x1/3 - x2) CO2, (3.71) where x1 and x2 are arbitrary real numbers. The authors of the article59 gave this statement: Balancing the chemical equation (with three molecules) means finding the smallest whole numbers x1, x2 and x3 (as its coefficients). The above statement does not hold for every reaction. It has just particular meaning and holds if only if a chemical reaction has atoms with integers, in an opposite case, when the reaction contains atoms with fractional oxidation numbers, it does not hold. For example, see the previous reactions (3.57), (3.58) and so on. Also, in the article59 is »balanced« this reaction like this x1 Cu + x2 HNO3 ^ x3 Cu+2 + x4 NO2 + x5 NO3- + xg H2O, 3 Cu + 4 HNO3 ^ 3 Cu+2 + 2 NO + 2 NO3- + 2 H2O. (3.72) (3.73) The last reaction (3.73) is wrong. The correct form of balanced reaction (3.72) is x1 Cu + x2 HNO3 ^ x1 Cu+2 + (x2/2) NO2 + (x2/2) NO3- + (x2/2) H2O, (3.74) where x1, x2 > 0 are arbitrary real numbers. Next reaction was »balanced« in59 too x1 OH- + x2 SnO2-2 + x3 Bi(OH)3 ^ x4 Bi + x5 SnO3-2 + x, H2O, (3.75) in this form 0 OH- + 3 SnO2-2 + 2 Bi(OH) ^ 2 Bi + 3 SnO3-2 + 3 H,O. 3 +3 h2o. (3.76) The general form of the reaction (3.75) is x1 OH- + x2 SnO2-2 + (2x2/3 - xJ3) Bi(OH)3 ^ (2x2/3 - xJ3) Bi + x2 SnO3-2 + x2 H20, (x1 < 2x2). (3.77) For x1 = 0 and x2 = 3, from (3.77) immediately follows (3.76), like a particular reaction. Another reaction is »balanced« in59 x1 CH3CH20H + x2 Cr2O7-2 + x3 H+ ^ x4 CH3C02H + x5 Cr+3 + x, H20, (3.78) in this form 3 CH3CH20H + 2 Cr2O7-2 + 16 H+ ^ 3 CH3C02H + 4 Cr+3 + 11 H20. (3.79) The general form of the reaction (3.78) is x1 CH3CH20H + x2 Cr2O7-2 + (- 4x1 + 14 x2) H+ ^ x1 CH3C02H + 2x2Cr+3 + (- x1 + 7 x2)H2O, (3.80) (0 < x1 < 7x2/2). For x1 = 3 and x2 = 2, from (3.80) immediately follows (3.79), like a particular reaction. In59 the authors only determined the particular reactions (3.76) and (3.79), but considered reactions (3.75) and (3.78) have general forms with two arbitrary parameters, given by (3.77) and (3.80), respectively. X. As a last paradox we discovered in the theory of balancing chemical equations is the case considered below. The authors of the article60 studied this chemical reaction CHo,g8gOo,32 + w H20 + m 02 + 3.76 m N2 ^ x1 H2 + x2 CO + x3 C02 + x4 H20 + x5CH4 + 3.76 m N2. (3.81) Immediately from (3.81) we have seen its absurdity. In this reaction H2O appears as a reactant and as a product at the same time. Is it logical? No! It is just one thing. Another illogical thing is not reacted N2. Nitrogen appears in (3.81) as reactant 3.76m N2 and 3.76m N2 as a product at the same time. Also, the treatment of hydrogen as a product is illogical. Burning of CH0.686O0.32 presented by (3.81) does not give H2 as a product! We think that the reaction x1 CH0.686O0.32 + x2 O2 + x3 N2 ^ x4 CO + x5 C02 + x, H20 + x7 CH4 + x8 N203, (3.82) which represents a correct version of (3.81) is very interesting for chemistry. From (3.82) one obtains this scheme O0 86 H0 2 2 O O2 O2 H4 C O2 N2 C C H2 C N2 C 1 0 0 -1 -1 0 -1 0 H 0.686 0 0 0 0 -2 -4 0 O 0.32 2 0 -1 -2 -1 0 -3 N 0 0 2 0 0 0 0 -2 Reaction matrix A of (3.82) has this form "1.000 0 0 -1 -1 0 -1 0 0.686 0 0 0 0 -2 -4 0 0.320 2 0 -1 -2 -1 0 -3 10.000 0 2 0 0 0 0 -2 A = A+ = The reaction (3.82) reduces to the following system of linear equations x1 = X4 + x5+ x7, 0.686 X1 = 2 xg + 4 X7, (3.83) 0.32 xi + 2 x2 = x4 + 2 x5 + x6 + 3 x8, 2 x3 = 2 X8. The general solution of the system (3.83) is x5 = 1.977 x1/4 + X2/2 - 3 x3/4 - 3 x4/4, x6 = - 1.337 xi/2 + x2 - 3 X3/2 + x4/2, (3.84) x7 = 2.023 x1/4 - X2/2 + 3 x3/4 - x4/4, x8 = x3, where xi > 0 (1 < i < 4) are arbitrary real numbers. Since xi > 0 (5 < i < 8), then from (3.84) one obtains this system of inequalities 1.977 xi + 2 x2 - 3 x3 - 3 x4 > 0, - 1.337 x1 + 2 x2 - 3 x3 + x4 >0, (3.85) 2.023 x1 - 2 x2 + 3 x3 - x4 > 0. From (3.85) immediately follows this inequality 1.337 x1 - x4 < 2 x2 - 3 x3 < 2.023 x1 - x4. (3.86) Reaction (3.82) holds if and only if the condition (3.86) is satisfied. The reaction (3.82) contains three subgenerators which induce a topology of its solutions, but we omitted it, since it will be a subject of the author's next research. The other particular cases of (3.82) are not considered because we took into account the Remark 3.7. For instance, for x1 = 2, x2 = 4, x3 = 2 and x4 = 1, from (3.84) one obtains this particular solution x5 = 0.7385, x6 = 0.163, x7 = 0.2615 and x8 = 2. Then (3.82) becomes 2 CH0,686O0,32 + 4 O2 + 2 N2 ^ CO + 0.7385 CO2 + 0.163 H2O + 0.2615 CH4 + 2 N2O3. Now, we shall determine the minimal coefficients of the reaction (3.82). The Moore-Penrose generalized matrix A+ will have this form ■ 0.334436538275454 -0.037800281937753 -0164755899838609 0.019118215665491 0123566924878957 -0.014338661749118 -0338363333974391 0.077822527216911 -0.255985384055086 0.068263419384166 0.257141220018618 -0145193039626989 -0.071214743695069 -0183886228538830 . 0123566924878957 -0.014338661749118 -0.048377720464309 0.036283290348232' 0171516759260828 -0.128637569445621 -0.128637569445621 0.346478177084216 -0.003380429711109 0.002535322283332 -0.089138809341523 0.066854107006142 -0104876595295905 0.078657446471929 0.044141518588323 0.033106138941242 -0.128637569445621 -0153521822915784J The reaction (3.82) reduces to the matrix form (3.45), whose general solution is given by the expression (3.46). For a = (1, 1, 1, 1, 1, 1, 1, 1)T, one obtains xmin = (x1, x2, x3, x4, x5, x6, x7, x8)T, where x1 = 1.241594646560723466, x2 = 1.574780831709873263, x3 = 0.568914376217595052, x4 = 0.721001830633894027, x5 = 0.433611414778957395, xg = 0.251904161474584061, x7 = 0.086981401147872044, x8 = 0.568914376217595052. 5 DISCUSSION We cannot always trust chemical experiments! We cannot always trust mathematics either, for it can mislead us unless we define away the problem area. However, we surely can trust pure logic - no questionable experiments or unusual mathematical operations. Are they methods when somebody can find counterexamples on every step? Obviously, the answer is negative! It is merely a pale picture of the old chemically irrational traditionalism of the past chemistry. As it is showed by all counterexamples given in this work the traditional procedures for balancing chemical reactions are inconsistent. These particular procedures, we hope that they are the last traditional unsuccessful approach of balancing chemical reactions. Long time ago chemistry lost the battle with mathematics in sense of balancing chemical reactions. Before we finish this discussion, we would like to stress here, that our facts for arguing are founded by virtue of scientific results. From current view point of balancing chemical equations, we feel free to tell that traditional approach of balancing chemical reactions is only a history, and Jones' problem44, which was as one of the hardest problems of balancing chemical reactions is completely solved45. There exists a completely satisfactory ways of avoiding paradoxes7'34'45'46'48-50. The theory used is based on the idea of formal approach of balancing chemical reactions. In these works completely new general highly sophisticated methods are developed for balancing chemical reactions and their stability by virtue of the theory of generalized matrix inverse using Moore-Penrose, Drazin and von Neumann matrices. By these methods chemistry is cleaned from old traditional inconsistent procedures for balancing chemical reactions, such that is open a brand new direction for development of this topic in chemistry and its foundation on genuine principles. That is the newest trend in chemistry about this issue, which showed that traditionalism in chemistry is over. 6 CONCLUSION By this work the consideration of paradoxes in chemistry will begin very seriously as a special object and in any way it will increase researchers' carefulness to avoid the appearance of paradoxes. Sure, no perfect science! Appearance of paradoxes is always possible. It is more than certain, that this work opened doors for the next research in chemistry for its diagnostic of paradoxes and their resolution. It will accelerate the newest contemporary research in chemistry and it will destroy all barriers which hamper the development of chemistry and lay its foundation on genuine scientific principles. This work affirms: • that all formally provable mathematical methods are true if chemical reactions are considered as a consistent formal system, • that all mathematical truths can be formally provable, and • that the new branch Foundation of Chemistry proves the consistency and completeness of the formal approach of balancing chemical reactions and that it will be a special kind of chemistry, i.e., it will be a finite theory which contains only perfectly well known concepts with true axioms and positive conclusions. It affirms that the principles used in the mathematical approach of balancing chemical reactions, will not lead to contradictions. 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