Logical structure of the fundamentals of chemistry, conceptual schemes of its geometrization and spatial mathematical models of the sets of different species of simple chemical objects

Academic Open Internet Journal

www.acadjournal.com

Volume 2, 2000

Logical structure of the fundamentals of chemistry, conceptual schemes of its geometrization and spatial mathematical models of the sets of different species of simple chemical objects

Valentin Tzvetanov Penev
Ludmil Lubomirov Konstantinov
Marin Stoianov Marinov^†

Central Laboratory of Mineralogy and Crystallography
Bulgarian Academy of Sciences
Acad. Georgi Bonchev Str., bl. 107, 1113 Sofia, BULGARIA
E-mail: vpenev@interbgc.com; vpenev@bas.bg; v_penev@mail.bg
Web site: http://www.clmc.bas.bg

ABSTRACT

It is shown that the fundamentals of the language of chemistry can be logically divided into four relatively separated parts everyone of which introduce the corresponding group of basic chemical notions. The central notions of the four parts and other basic notions are defined. The system of five basic assumptions playing the role of axiomatics in chemistry is formulated. The conceptual schemes, used for geometrization of each part of the fundamentals of chemistry, are formulated. Spatial mathematical models of the sets of different species of atoms and different species of monatomic ions are briefly presented. As a result: (i) the problem of the geometrization of the language of chemistry is solved on a conceptual level; (ii) it is shown that it is possible to translate the structure of chemical notions and relations in entirely mathematical language.

INTRODUCTION

The problem of geometrization of the language of chemistry has been formulated, grounded and partially solved for the first time in [1]. In [2] we have presented briefly the formulation and grounding of this problem in a general form and outlined the way of its solving. The main aims of this paper are:

1) To define several basic notions and to reveal the general logical structure of the fundamentals of the language of chemistry;
2) To formulate a system of basic chemical assumptions as axiomatics in chemistry;
3) To formulate the conceptual schemes for geometrization of different parts of the foundamentals of chemistry and to present briefly the spatial mathematical models of different species¹ of simple chemical objects.

In other words the main aims of this paper are to present briefly the main parts of the solutions of the first three problems of the system of four basic problems, formulated in [2]. As a result, we shall solve in fact the problem of the geometrization of the language of chemistry on a conceptual level and, at the same time, shall briefly present some results discussed in detail in [1].

We shall note that the solution of the aforesaid three problems enables the transition of the language of chemistry from a “lower, inductive stage” to a ”higher, deductive stage” and to ”the most higher (supreme), axiomatic stage” of its evolution [3].

GENERAL LOGICAL STRUCTURE OF THE FOUNDAMENTALS OF CHEMISTRY

We shall define here some basic notions necessary to present the logical structure of the language of chemistry:

DEFINITION I. By the term simple chemical object we denote each chemical object composed of a unique atomic kernel and a corresponding electron shell.

DEFINITION II. By the term complex chemical object we denote each chemical object composed of more than one simple chemical object.

DEFINITION III. By the term atom we denote each electrically neutral simple chemical object, and by the term monatomic ion (or ionized atom) we denote each simple chemical object, which is electrically charged either really or formally.

It is evident from these definitions that the set of simple chemical objects includes only all species of atoms and monatomic ions. The set of complex chemical objects includes the species of polyatomic ions, simple and complex molecules, etc. The notions simple chemical object and complex chemical object are closely related to the notions structure, composition, and construction which are defined as follows:

DEFINITION A. By the term structure of a given complex object in a specified energy state we denote each unity of a particular composition and a particular construction which characterizes uniquely the object in this energy state.

DEFINITION B. By the term composition of a given complex object in a specified energy state we denote each set of relatively independent particular parts (e.g. simple chemical objects), which characterizes uniquely the object in this energy state.

DEFINITION C. By the term construction of a given complex object in a specified energy state we denote each particular mutual disposition of relatively independent parts (e.g. simple chemical objects), composing the object in the three-dimensional physical space, which characterizes uniquely the object in this energy state.

According to thermodynamics in the definitions A, B and C, energy state of a particular object means the the quantity of the internal energy of this object considered as a thermodynamic system (TDS).

From these definitions one can see that the simple chemical objects have a physical but not a chemical structure. Consequently, these objects are complex from a physical point of view and simple from a chemical point of view. That is why they are referred to as simple chemical objects. In contrast, the complex chemical objects have both physical and chemical structure and are complex from both physical and chemical point of view. That is why they are referred to as complex chemical objects.

The definitions given above allow to outline the general logical structure of the fundamentals of chemistry by the following scheme:

General logical structure of the fundamentals of chemistry

I. CHEMISTRY OF SIMPLE OBJECTS (chemistry of different species of atoms and different species of monatomic ions
1. STATICS of simple chemical objects This parts includes the notions necessary for describing different species of simple chemical objects.	2. DYNAMICS of simple chemical objects This parts includes the notions necessary for describing different processes of transformations of the species of simple chemical objects.
II. CHEMISTRY OF COMPLEX OBJECTS (chemistry of different species of polyatomic ions, simple and complex molecules etc.)
3. STATICS of complex chemical objects This parts includes the notions necessary for describing different species of complex chemical objects.	4. DYNAMICS of complex chemical objects This parts includes the notions necessary for describing different processes of formation, destruction and transformations of the species of complex chemical objects.

From a logical point of view this scheme shows that:

A) The fundamentals of chemistry consist of four relatively separated parts;

B) Every one of these four parts introduces a corresponding group of basic chemical notions;

C) Each group of notions is centered on a specific notion. In what follows this notion will be referred to as central notion of the corresponding group of notions;

D) There are complex relationships of horizontal and vertical subordination between the four separated parts of the fundamentals of chemistry. To support this statement we note that both the dynamics of simple chemical objects and the statics of the complex ones can only be developed if the statics of the simple chemical objects has previously been developed. On its turn, the dynamics of the complex chemical objects cannot be developed unless the other three parts of the chemistry fundamentals have previously been developed.

Let us see now which are the central notions in the four parts of the chemistry fundamentals and how they can be defined:

1. The central notion of the first part is species of simple chemical objects. We denote by this notion the set of all simple chemical objects of the same physical composition (and of the same physical construction if they are in equivalent energy states). Here, physical composition means the corresponding uniquely defined (quantitatively and qualitatively) set of neutrons, protons and electrons composing a given simple chemical object. Correspondingly, by physical construction we denote the overall wave function of the given simple chemical object.

2. The central notion in the second part is simple chemical process. We denote by this notion each process which transforms one species of simple chemical objects into another one. Examples for such processes are all processes of ionization and recombination of simple chemical objects.

3. The central notion in the third part is species of complex chemical objects. In contrast to the simple chemical object, each complex chemical object is a unity of chemical composition and chemical construction. The notion chemical composition means the set of different species of simple chemical objects, composing this particular complex chemical object. The notion chemical construction means the mutual disposition of the simple chemical objects, composing this particular complex chemical object in the three-dimensional physical space V_E(3). The notion species of complex chemical objects means the set of all complex chemical objects of equal chemical composition, chemical construction and of equivalent energy states.

4. The central notion in the fourth part of the fundamentals of chemistry is the notion complex chemical process. This notion means each process of: (i) formation of complex chemical objects; (ii) destruction of complex chemical objects; (iii) transformation of one species of complex chemical objects into another one.

Further we shall use the following abbreviations and specific chemical-mathematical symbols:

PS - Periodic System; SFPS - Short Form of the Periodic System; LFPS - Long Form of the Periodic System;
denotes the set of all species of simple chemical objects, while stands for a particular element of this set;
S denotes the set of all species of atoms, whereas s denotes a particular element of this set (i.e. a particular species of atoms);
denotes the set of all species of ionized atoms, while is used for a particular element of this set (i.e. a particular species of ionized atom);
N_n, N_p and N_e denote respectively the number of neutrons, protons and electrons in a particular species of simple chemical objects.

Using these specific chemical-mathematical symbols we can represent mathematically the relations between the different notions. For example, according to Definitions I and II, the set includes only the subsets S and . This means that the set is a union of the subsets S and :

(1)	, where and (and consequently )

According to the definitions given above, each simple chemical object is either electrically neutral or charged: if , then and, contrary: if , then . Therefore:

(2)

The condition for electric neutrality (N_e = N_p) is a criterion whether a particular species of simple chemical object () belongs to the corresponding subsets S or . In fact, according to Definition III, a given simple chemical object is an atom only if it is electrically neutral, and it is an ionized atom only if it is electrically charged:

(3)

and

Relations (1-3) thus describe the basic chemical notions through the symbols introduced above.

FIVE BASIC CHEMICAL ASSUMPTIONS

Now we shall formulate a system of basic chemical assumptions, playing the role of axiomatics in chemistry, and answer the following two questions: (1) What should be introduced firstly in chemistry? (2) What is the exact way to its introduction?

The logical analysis show that the language of chemistry is based on the system of five fundamental assumptions. The first four assumptions of this system are:

ASSUMPTION 1. There exists a set S of different species of atoms.

ASSUMPTION 2. There exist sets of different processes, transforming the species of simple chemical objects into other species of simple chemical objects. (Examples for such sets of processes are ionization, recombination, etc.)

ASSUMPTION 3. There exist sets of different species of complex chemical objects. (Examples for such sets are those of different species of polyatomic ions, of different species of simple molecules, of different species of complex molecules, etc.)

ASSUMPTION 4. There exist sets of different processes of formation of complex chemical objects, of destruction of complex chemical objects or of transformation of the species of complex chemical objects into different species of complex chemical objects. (Examples for such sets of processes are the formation of polyatomic ions, the destruction of simple molecules, chemical reactions of molecules, etc.)

From a logical point of view, each of these fundamental assumptions for existence is a central axiom of the corresponding part of the fundamentals of chemistry, because it introduces the central notion of this part and other basic notions, related to this central notion. Besides, the four assumptions are related with each other, and form a system of central axioms. To support these statements we shall note that the existence of the set of different species of ionized atoms follows directly from the assumption for existence of the set S of different species of atoms (i.e., from Assumption 1) and from the assumption for the existence of the set of different processes of ionization (the latter is a part of Assumption 2).

It is evident from the aforesaid that the set S of different species of atoms must be introduced first in chemistry. This is the answer of the first question formulated above.

According to Assumption 1 and 2, there exist different species of simple chemical objects. Furthermore, according to Definitions I-III and A-C, the simple chemical objects are structureless from a chemical point of view. That is why, they cannot be differentiated in species by their chemical structure. The only way to determine the species of such objects is by defining a system of relations between their sets of properties. Therefore, the only chemical way to define the set S is by defining the system of relations between sets of properties of different elements of S. This is exactly the way that is used in chemistry. Indeed, all the forms of PS are different particular representations of the system of relations between the sets of properties of different species of atoms. Therefore, from the logical point of view, the fifth fundamental assumption in the language of chemistry is:

ASSUMPTION 5. The set S is introduced in contemporary chemistry by different implicit three-dimensional forms of the Periodic System².

Therefore, the geometrization of the language of chemistry should be started by constructing mathematical representations of the three-dimensional forms of Periodic System (i.e. of the Mendeleev's Periodic law) which introduce the set S in chemistry. This is the answer of the second question formulated above.

SPATIAL MATHEMATICAL MODELS OF DIFFERENT SPECIES OF ATOMS

We have constructed and analysed [1] one approximate and two more accurate spatial mathematical models of the set S and Mendeleev's periodic law, based on particular tabular forms of PS, denoted as starting tables of the corresponding models. Common to these models is that they are constructed in the same space and coordinate system by the same method. The major difference between the three mathematical models is that they originate from three different tabular forms of PS: the approximate model originates from a simplified SFPS, denoted as T₀(see Fig. 1) while the two more accurate models and originate from two different 32-column tabular LFPS, denoted respectively as T₁ and T₂ (see Fig. 2 and Fig. 3).

On the example of the approximate model P the basic mathematical formalism for geometrization of the statics of simple chemical objects is introduced and the method for constructing spatial mathematical models of some basic chemical notions and relations is demonstrated. It should be pointed out that the approximate model and the simplified starting table T₀ must not be regarded beyond the purposes mentioned above. So, the approximate model is an intermediate working model towards more accurate mathematical models originating from “correct” versions of the Periodic System, presented by the starting tables T₁ and T₂.

The conceptual scheme used in constructing the mathematical models P, Q and G consists of the following four steps:

1. Construction of a space in which the species of different simple chemical objects (atoms and monatomic ions) can be represented uniquely;

2. Choice of coordinate systems in this space that would allow to represent uniquely the species of simple chemical objects;

3. Construction of the desired unique spatial mathematical model of the set S, which to present as well the corresponding starting form of PS. This step consists in the choice of a set of mathematical objects to represent uniquely the species of atoms in a previously chosen coordinate system in a previously constructed space. The set of these mathematical objects-images should be specified so that the system of relations between them to be a veracious mathematical model of the corresponding starting tabular form of PS;

4. Verification of the uniqueness and the veraciousness of the constructed spatial mathematical model. In this step one proves that:(a) the constructed mathematical objects really present uniquely the species of atoms with respect to the previously chosen coordinate system in the constructed space; (b) the relationships between the mathematical images of different species of atoms are really a veracious mathematical model of the regularities presented in the corresponding tabular form of PS.

The first two steps in this conceptual scheme are very important because they introduce the space and the coordinate system used later to construct the remaining mathematical models of the fundamentals of chemistry.

Let V_M(3) be such a three-dimensional metric Euclidean space onto the field of the real numbers, in which each dimension has a different qualitative meaning.

Furthermore, let K_M=(O;z,r,j) be a specially chosen cylindrical coordinate system in V_M(3), such that:

A) The three coordinates have the following different qualitative meaning: z represents the number of neutrons in the simple chemical objects; r represents the number of protons in the simple chemical objects; j represents the periodicity in the change of the chemical properties of different species of atoms. (This complex periodicity is represented with different degrees of accuracy and completeness in the different starting forms of PS.)

B) The origin O of K_M coincides with the natural zero point of V_M(3). (From a qualitative point of view the existence of natural zero point in the space V_M(3) means simply a lack of protons, neutrons, and chemically active electrons.)

The space V_M(3) and the coordinate system K_M are denoted as Mendeleev's space and Mendeleev's coordinate system. They are substantial in the geometrization of the language of chemistry by making it possible to construct unique spatial mathematical images of both the species of simple chemical objects and of simple chemical processes (i.e. processes transforming the species of simple chemical objects).

By defining V_M(3) and K_M we made the first two steps in the conceptual scheme. The third step consists in formulation of the corresponding Theorems and Lemmas, which are similar for the three models. For the case of the model originating from the starting table T₂ they are as follows³:

THEOREM 2.2. There exists such a unique spatial mathematical model of different species of atoms which, at the same time, is a veracious mathematical model of the starting table T₂.

This theorem plays the role of a basic statement for the model G. We prove this theorem by constructing this model with respect to K_M in V_M(3). For this purpose, we have defined an invertible map from S onto the corresponding set of points in the Mendeleev's space by:

LEMMA 2. Each species of atoms can be uniquely presented in the space V_M(3) by a corresponding point with cylindrical coordinates (z,r,j) in K_M, where:

(B1)	z is the number of neutrons in the nuclei of the presented species of atoms;
(B2)	r is the number of protons in the nuclei of the presented species of atoms;
(B3)	j = n(2p/32), where n = 1,...32, is the number of this column in the starting table T₂, which contains the presented species of atoms.

In the final, fourth, step one must prove that the set of points G is really a unique mathematical model of S and a veracious mathematical model of T₂. For this purpose we have proved that:

1. The rules (B1, B2, B3) define an invertible map;
2. The mathematical image of each chemical element consists of the mathematical images of all the species of atoms (i.e. isotopes) of this chemical element only;
3. The mathematical image of each column of T₂ consists of the mathematical images of all the chemical elements of this column only;
4. The order of chemical elements in each column in the starting table T₂ is analogous to that of their mathematical images in the mathematical image of this column;
5. The mathematical image of each period of T₂ consists of the mathematical images of all the chemical elements of this period only;
6. The order of chemical elements in each period of T₂ is identical to that of their mathematical images in the mathematical image of this period;
7. The order of columns and periods in T₂ is identical to that of their mathematical images.

In constructing and analyzing the three models we have achieved the following goals [1]:

a number of specific chemical-mathematical symbols were introduced;
the definitions of some basic chemical notions were revised;
the three- and two-dimensional images were determined for a number of basic chemical notions (chemical element, group and period of PS, maximal stoichiometric valence, etc.);
for each of the three models a corresponding set of statements of various logical ranks (definitions, axioms, lemmas, corollaries, etc.) was formulated;
a comparative analysis was carried out of the three spatial mathematical models, P, Q and G.

SPATIAL MATHEMATICAL MODELS OF DIFFERENT SPECIES OF MONATOMIC IONS

In the previous section we presented the conceptual scheme for the geometrization of that part of the fundamentals of chemistry which introduces the set S of different species of atoms and the corresponding chemical notions. According to Eq. (1), the set of different species of simple chemical objects includes as subset the set of different species of ionized atoms. That is why, in order to complete the geometrization of the statics of simple chemical objects, one should construct the spatial mathematical models of the various species of monatomic ions. The notion species of monatomic ions is closely related to the notions ionization, recombination and oxidation number (or oxidation state) which belong to the second part of the fundamentals of chemistry, namely the dynamics of simple chemical objects. As a result, the successful modeling of this notion requires correct definitions of the three related notions mentioned above. On the other hand, these are not correctly defined in the contemporary chemistry. That is why, we shall revise their definitions as follows:

DEFINITION IV. By the term ionization of a given simple chemical object (atom or ionized atom) we denote any process in which the number N_eof electrons of this object is changed (without changing the numbers N_n and N_p of its neutrons and protons) in such a manner that the object becomes (or remains) electrically charged. Otherwise, ionization is the process of transformation of an atom of a particular species to some species of ionized atom, or of a particular species of ionized atom to another species of ionized atom.

DEFINITION V. By the term recombination of a given ionized atom (i.e., of an electrically charged simple chemical object) we denote any process of change in the number N_e of electrons of this object (without changing the numbers N_nand N_p of its neutrons and protons) such that the atom becomes electrically neutral. In other words, the recombination is a process of transformation of some species of ionized atom to the corresponding species of atom.

DEFINITION VI. By the term oxidation number (or oxidation state) of a given monatomic ion we denote the number of electrons that should, either really or formally, be added or removed in order to transform this ion into an atom (i.e. into an electrically neutral simple chemical object).

It is seen from the last definition that the notion monatomic ion is a generalization of the notion atom, because each atom may be considered as an monatommic ion of oxidation number equal to zero.

Based on the above definitions, two different spatial mathematical models and of the set of different species of simple chemical objects were constructed and analyzed. Each of these models is a generalization of the corresponding more accurate mathematical model (Q or G) in the case of arbitrary species of simple chemical objects - atoms and monatomic ions. For the case of the generalized model the basic statement is:

THEOREM 3.2. There exists such a unique spatial mathematical model of the set of different species of simple chemical objects, which is a generalization of the model .

The first step in proving this theorem is to construct the generalized model with respect to K_M in the space V_M(3). For that purpose an invertible map from the set onto the corresponding sets of points of V_M(3) is defined by:

LEMMA 3. Each species of simple chemical objects can uniquely be presented in V_M(3) by a point of cylindrical coordinates (z, r,j) with respect to K_M, where z and r are the same as in Lemma 2, and , where: n = 1,...,32, is the number of that column in the starting table T₂, which contains the species of simple chemical objects when they are in electrically neutral state; the integer number t is equal to the oxidation number of the simple chemical objects.

By formulating this lemma we constructed the generalized mathematical model . The second step in proving the theorem shows that the corresponding set of points is a unique mathematical model of the set of different species of simple chemical objects.

The construction of the generalized model is similar to that of .

In constructing the two generalized models we have achieved the following goals [1]:

the list of specific chemical-mathematical symbols was expanded;
the relations between some basic chemical notions were formulated mathematically by these symbols;
a comparative analysis of the two generalized models and was carried out;
three- and two-dimensional mathematical images were determined for monatomic ions of equal oxidation number of different isotopes of one and a same chemical element;
three- and two-dimensional mathematical images of different chemical elements were determined;
for each of the two generalized models, and , a set of statements of various logical ranks (axioms, corollaries, definitions, etc.) was formulated. This set generalizes the corresponding set of statements for the models Q and G.

Particularly important is Corollary 23 ([1], р. 117-118) of the two generalized models, and . It states that the change in the oxidation number of a simple chemical object leads to rotation of its mathematical image around the axis Oz at an angle , where h and t are the oxidation numbers of the object at the beginning and at the end of the corresponding process of ionization or recombination. It is clear from this corollary, that all processes of ionization or recombination of simple chemical objects can be presented in a unique way in the space V_M(3) by the corresponding rotation operators of their mathematical images around the axis Oz.

The investigation of the constructed spatial mathematical models has been completed with an analysis of the qualitative meaning of the points and the distances between them in each of the models. The basic results are:

1) each point of the sets P, Q, G, and is a unique mathematical image of the corresponding species of simple chemical objects (i.e., of the corresponding relatively defined set of properties);

2) the distances between the points in each of these sets reveal the differences between the corresponding relatively defined sets of properties and the physical and chemical reasons for these differences.

CONCEPTUAL SCHEMES FOR THE GEOMETRIZATION OF THE OTHER PARTS OF THE FOUNDAMENTALS OF CHEMISTRY

In the previous sections we presented mathematical models of different species of simple chemical objects and Mendeleev's law (i.e. the geometrization of the statics of simple chemical objects). In what follows we shall formulate the conceptual schemes for geometrization of the other three parts of the fundamentals of chemistry.

Part II. DYNAMICS OF SIMPLE CHEMICAL OBJECTS

We already showed that the notion simple chemical process is a central notion for the second part of the fundamentals of chemistry, and that the species of simple chemical objects is uniquely determined by their physical composition. It follows from these statements that each simple chemical process can uniquely be presented as a linear composition of three independent basic kinds of simple chemical processes. Each of these processes changes only the number of one of the three species of physical particles constructing a given simple chemical object (neutrons, protons and electrons). On the other hand, in each of the constructed mathematical models the z, r and j-coordinates of the mathematical images represent the respective number of neutrons, protons and (implicitly) electrons in the corresponding species of simple chemical objects. Therefore, the three basic kinds of simple chemical processes can uniquely be presented in V_M(3)by operators acting only on the respective coordinates z, r and j of the corresponding mathematical images. Each simple chemical process which transforms in an arbitrary way the species of simple chemical objects can uniquely be represented mathematically as a composition of the three different kinds of operators listed above. This statement completes the conceptual scheme for geometrization of the dynamics of simple chemical objects.

Part III. STATICS OF COMPLEX CHEMICAL OBJECTS

The central notion of the third basic part of the language of chemistry is the species of complex chemical objects. In order to construct unique spatial mathematical images of the various species of complex chemical objects one has to construct:

1) The Cartesian product of the Mendeleev's space V_M(3) (in which the composition of complex chemical objects is uniquely represented) and the usual three-dimensional physical space V_E(3) (in which the construction of complex chemical objects is uniquely represented). Thereby we obtain a six-dimensional Euclidean space V_S(6) = V_M(3) x V_E(3) which is referred to as space of the chemical structures.

2) A special six-dimensional coordinate system K_S=(O_S;z,r,j;u,v,w) (denoted as coordinate system of chemical structures) in the space V_S(6). This coordinate system is constructed as a Cartesian product K_S=K_Mx K_E of the Mendeleev's coordinate system K_M=(O;z,r,j) and any coordinate system K_E=(O_E;u,v,w) in the usual physical space V_E(3), in whose origin O_E there is no simple chemical objects. It should be noted that the origin of the system K_S is a six-dimensional point O_S in the space of the chemical structures V_S(6). The projection of this point onto Mendeleev's space V_M(3) coincides with the natural zero of this space, i.e. with the origin O of Mendeleev's coordinate system K_M.

It is easily seen that all the species of complex chemical objects can uniquely be presented in V_S(6). For the purpose, it is necessary to present all the simple chemical objects composing a given complex chemical object by six-dimensional radii vectors with respect to the six-dimensional coordinate system of chemical structures K_S. The first three coordinates of each six-dimensional radius vector represent in a unique manner the species of the simple chemical objects constructing a given complex chemical object, i.e. they represent the chemical composition of the corresponding species of complex chemical objects. The last three coordinates of the radii vectors represent in a unique manner the mutual disposition in the usual physical space V_E(3) of the corresponding simple chemical objects constructing a given species of complex chemical objects, i.e. they represent the chemical construction of this species of complex objects. This latter item completes the conceptual scheme for the geometrization of the statics of complex chemical objects.

Part IV. DYNAMICS OF COMPLEX CHEMICAL OBJECTS

The central notion for the fourth part of the fundamentals of chemistry is the complex chemical process. Such is each process of: (i) formation of complex chemical objects; (ii) destruction of complex chemical objects; (iii) transformation of the species of complex chemical objects.

Therefore, in contrast to simple chemical processes (which transform the species of simple chemical objects only), the complex chemical processes can transform not only the species of the involved simple chemical objects, yet their mutual disposition in the usual physical space V_E(3). This statement outlines the way to construct the unique mathematical representations of all the species of complex chemical processes, namely:

1) Each transformation of the species of simple chemical objects can uniquely be represented by a corresponding linear combination of three basic species of operators acting onto the coordinates of the mathematical images of simple chemical objects in Mendeleev's space V_M(3).

2) Each transformation in the mutual disposition of simple chemical objects can uniquely be represented by a corresponding linear combination of operators of translation and rotation acting onto the points representing the position of these objects in the usual physical space V_E(3).

Therefore, each complex chemical process (i.e. each chemical reaction) can uniquely be represented mathematically in the space of chemical structures, V_S(6), by the corresponding linear combination of operators acting onto the coordinates of the six-dimensional radii vectors which represent the simple chemical objects taking part in a particular complex chemical process. This latter statement concludes the conceptual scheme for geometrization of the last part of the fundamentals of chemistry, the dynamics of complex chemical objects.

CONCLUSIONS

In this paper we have:

define several basic chemical notions and reveal the general logical structure of the fundamentals of chemistry;
formulate a system of basic chemical assumptions as axiomatics in chemistry;
present briefly the conceptual schemes for geometrization of different parts of the foundamentals of chemistry and some of the spatial mathematical models of different species of simple chemical objects, constructed in [1].

In other words we have present briefly the main parts of the solutions of the first three basic problems, formulated in [1, 2]. As a result, we have:

solve in fact the problem of the geometrization of the language of chemistry on a conceptual level;
showed that it is possible to translate the structure of chemical notions and relations in mathematical language which allows to use in chemistry all advantages of the mathematical formulations.

ACKNOWLEDGMENTS. The financial support of this work by the Bulgarian Ministry of Sciences and Education (under Contract X - 595/95 with National Research Fond), is gratefully appreciated. We would also like to thank Prof. D. Bonchev, Acad. S. Hristov, Acad. I. Kostov, Prof. O. Mekenian, Prof. H. Canety and Acad. I. Yuhnovsky for their support.

REFERENCES

[1]	Penev V., Geometrization of the Fundamentals of Chemistry. RIVA Publ. House, Sofia, 1997, 224 pp. (monograph in Bulgarian with summary in English)
[2]	Penev, V., L. Konstantinov, M. Marinov, The problem of geometrization of the language of chemistry, Academic Open Internet Journal, vol. 2 (2000), Part 2: Chemistry.
[3]	Blanche Robert, L' axiomatique, Paris, Press universitaires de France, 1965

NOTES

¹ Species in this study means the major subdivision of a genus or subgenus, regarded as the basic category of chemical classification (see for example Random House Webster's meanings of this term.)

² We state that the three-dimensional forms of the Periodic System are defined implicitely because the numbers of protons and electrons of different species of atoms (i.e. of different isotops) of a same chemical element are presented by two-dimensional forms of Periodic System, while the number of neutrons is presented in a separated table.

³ The presented in this paper basic statements, their logical classificational names (definition, theorem, lemma, corrolary, etc.) and numbering are quatations from the monograph [1].

FIGURE 1. Simplified short form of the Periodic System, denoted as starting table T₀.
This table represents only the most general characteristics of the Periodic law (i.e. the ordering of chemical elements in groups and periods, as well as the ordering of the groups and the periods), but does not represent the atoms of chemical elements belonging to the corresponding A or B subgroups. The approximate model originates from this table.

FIGURE 2. Long form of the Periodic System, denoted as starting table T₁.
In this table is presented the successive order of filling up of the electron shells of atoms as a function of the atomic number.
The asterisks denote: * - electron orbitals ns¹ and ns²; ** - electron orbitals np¹ and np²;
*** - electron orbitals (n-1)d¹ ® (n-1)d¹⁰; **** - electron orbitals (n-2)f⁰(n-1)d¹ ® (n-2)f¹⁴.
The more accurate model originates from this table.

FIGURE 3. Long form of the Periodic System, denoted as starting table T₂.
This table presents the type of chemical elements and their affiliation to subgroups a and b. The element He is put in the subgroup IIa as it belongs to the s-elements. The more accurate model originates from this table.