Geometrization of the language of chemistry: Mathematical representation of the complex chemical objects

Academic Open Internet Journal

www.acadjournal.com

Volume 11, 2004

Geometrization of the language of chemistry:
Mathematical representation of the complex chemical objects

Valentin Tzvetanov Penev
Ludmil Lubomirov Konstantinov

Central Laboratory of Mineralogy and Crystallography
Bulgarian Academy of Sciences
Acad. Georgi Bonchev Str., bl. 107, 1113 Sofia, BULGARIA
Fax: (+359 2) 9797 056; Phone: (+359 2) 97 97 055
E-mail: vpenev@interbgc.com, vpenev@bas.bg, v_penev@mail.bg
Web site: http://www.clmc.bas.bg

ABSTRACT

The space and the coordinate system, necessary to construct entirely mathematical representations of the complex chemical objects, are introduced. It is proved that using the method proposed in [1] one can construct entirely mathematical and unique representations of arbitrary complex chemical objects. It is shown that using this method an exactly defined "chemical charge" is in fact ascribed to each point of the ordinary three-dimensional physical space.

KEYWORDS: geometrization, theoretical chemistry, foundations of chemistry, mathematical models

INTRODUCTION

In the previous papers [2, 3, 4]:

A system of three initial hypotheses, concerning the existence of qualitatively differing evolutionary stages in the development of scientific disciplines and sciences was formulated. Based on these hypotheses the particular problem for geometrization of the language of chemistry was formulated and grounded in a general form. The way to solve this problem was outlined through formulation of a system of four basic goals that should be solved during the next evolutionary stage of the development of the language of chemistry. Several aspects of this system of goals were briefly discussed in order to clarify their meaning as well as to introduce the meaning context needed. A short review of the main results, presented in detail in the monograph [1], was done.

It was shown that the fundamentals of the language of chemistry could be logically distinguished into four relatively independent parts each one of introducing the corresponding group of basic chemical notions. For each of these parts the corresponding central notions of and related with it basic chemical notions were defined. The system of basic assumptions playing the role of axiomatical kernel of the language of chemistry was formulated. The conceptual schemes, used for geometrization of each part of the fundamentals of chemistry, were formulated. Spatial mathematical models of the sets of different species of atoms and the Periodic table as well as different species of monatomic ions were briefly presented. As a result: (i) the problem of the geometrization of the language of chemistry was solved on a conceptual level; (ii) it was shown that it would be possible to translate the structure of chemical notions and relations in entirely mathematical language.

The main goals of this paper are:

to introduce the spaces and coordinate systems, necessary for construction of entirely mathematical representations of the complex chemical objects;

to demonstrate the method for construction of such representations.

DEFINING THE SPACES AND COORDINATE SYSTEMS

Prior to construction entirely mathematical representations of complex chemical objects we shall recall some definitions formulated in [1, 4]:

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

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

DEFINITION C. By the term construction of a given complex object in a specified energy state we denote such a particular mutual disposition in the three-dimensional physical space V_E(3) of relatively independent parts (e.g. simple chemical objects), composing this complex object, 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 quantity of the internal energy of this object considered as a thermodynamic system (TDS).

DEFINITION I. By the term simple chemical object we denote each chemical object composed of a unique atomic kernel and the 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 "A&IA". By the term atom we denote each electrically neutral simple chemical object, and by the term monatomic ion (or ionised atom) we denote each simple chemical object, which is electrically charged either really or formally. In other words, an atom is each simple chemical object, which oxidation state is equal to zero, and a monatomic ion is each simple chemical object which oxidation state is different from zero.

From these definitions it becomes clear that each complex chemical object is a specific chemical structure built of a countable set of simple chemical objects. That is why in order to construct an entirely mathematical spatial representation of an arbitrarily chosen complex chemical object one should:

Construct such a mathematical space that allows each chemical structure to be represented entirely mathematically and uniquely as a unity of two qualities - chemical composition (i.e. chemical species of each of the building this structure simple chemical objects) and construction (i.e. mutual disposition of the building this structure simple chemical objects in the ordinary physical space V_E(3));

Choose in this space such a coordinate system with respect to which each chemical structure can be represented entirely mathematically and uniquely as a unity of the two above-mentioned qualities.

So, the necessary space V_S(6) is built as a Cartesian product of the ordinary 3 dimensional physical space V_E(3) and the Mendeleev's space V_M(3), introduced for the first time in [1, 4], i.e.: V_S(6) = V_M(3) x V_E(3). We shall further refer to V_S(6) as space of chemical structures.

Let us build in V_S(6) such a 6 dimensional coordinate system K_S(6)=(O_S,z,r,j;u,v,w), which is a Cartesian product of the Mendeleev's coordinate system K_M(3)=(O_M;z,r,j), introduced firstly in [1, 4], and such a 3 dimensional coordinate system K_E(3)=(O_E;u,v,w) in the ordinary physical space V_E(3), in the origin O_E of which there is no simple chemical objects. In other words, let K_S(6) = K_M(3) x K_E(3), where: the projection of the 6 dimensional point O_S onto the 3 dimensional Mendeleev's space V_M(3) coincides with its natural zero-point¹, i.e. with the 3 dimensional point O_Mwhich, by definition [1, 4] is the origin of the coordinate system K_M(3); the projection of the point O_S onto the ordinary 3 dimensional physical space V_E(3) coincides with the origin O_E of the coordinate system K_E(3). We shall refer further to K_S(6) as coordinate system of chemical structures.

From the definition of V_S(6) and K_S(6) as Cartesian products and from the qualitative meaning of V_M(3), K_M(3), V_E(3) and K_E(3) one can conclude that:

Each point in the space of chemical structures V_S(6) is a unity of six coordinates. This unity is represented through two ordered triads of coordinates each one with own qualitative meaning;

The second triad of coordinates of each point in V_S (6) is defined with respect to K_E(3) and represents the corresponding uniquely determined position in the ordinary 3 dimensional physical space V_E(3);

The first triad of coordinates of each point in V_S(6) is defined with respect to K_M(3) in V_M(3) and represents the species of this simple chemical object occupying the corresponding, uniquely specified through the second triad of coordinates position in the ordinary physical space V_E(3).

It becomes clear from the above conclusions that the 6 dimensional space of chemical structures, constructed in this way - V_S(6), is a unity of two 3 dimensional spaces which are qualitatively different and, at the same time, conceptually correlated. The necessity of conceptual correlation between the spaces building up V_S(6) originates from the fact that each chemical structure is a unity of two different and at the same time conceptually correlated qualities - chemical composition and construction in V_E(3).

Here, we shall pay a special attention to the requirement that no simple chemical objects should be present at the origin O_Eof the coordinate system K_E(3). This limitation on the choice of points in V_E(3) for origin of the coordinate system K_E(3) is in fact a requirement for a conceptual correlation between the origins of the coordinate systems K_M(3) and K_E(3). Namely the observation of this requirement guarantees the necessary conceptual correlation between the spaces V_M(3) and V_E(3). It is easy to see that if the requirement for a conceptual correlation between the origins of K_M(3) and K_E(3) is not fulfilled, the corresponding 6 dimensional point O_S (O_S = O_M x O_E) will not play the role of origin of the 6 dimensional coordinate system K_S(6), i.e. of a point which coordinates are (0,0,0;0,0,0) with respect to K_S(6). Indeed, let us choose as origin, O_E, of the coordinate system K_E(3) such a point in V_E(3) in which there is some, simple, chemical object. Then, the second triad of coordinates of the corresponding 6 dimensional point O_S, with respect to K_E(3) will by definition be equal to zero (because the projection of O_S onto V_E(3) coincides with the point O_Echosen by definition as origin of the coordinate system K_E(3)). At the same time according to the third conclusion, the first triad of coordinates of the corresponding 6 dimensional point O_Swill not be equal to zero, because in the corresponding position of V_E(3), represented by O_E, there will be some species of simple chemical object. But according to [1, 4] the presence of a simple chemical object of any species is always represented in V_M(3) by a point with non-zero coordinates with respect to K_M(3)).

It becomes clear that not any point of the 3 dimensional physical space can be chosen as origin of the coordinate system K_E(3) due to the necessity of a conceptual correlation between the spaces V_M(3) and V_E(3) manifested as a requirement for conceptual correlation between the origins of the coordinate systems K_M(3) and K_E(3). However, the question arises: is there at least one point in V_E(3) which fulfils the imposed requirement and thus to be chosen as the origin of K_E(3)? It is easy to see that at each moment of time the set of points in V_E(3), in which there are simple chemical objects is countable (discrete), whereas the set of points, in which there are no simple chemical objects is always continuous. In other words, at each moment of time the set of points in V_E(3) which fulfils the imposed requirement (and, consequently, able to be chosen as the origin of K_E(3)) is a continuum, whereas the set of points which does not fulfil it (and, consequently, not able to be chosen as the origin of K_E(3)) is always countable (discrete). Or, from the viewpoint of the requirement for a conceptual correlation between the spaces V_M(3) and V_E(3), at each moment of time the 3 dimensional physical space V_E(3) can, in fact, be considered as an empty continuum with point-like "holes", i.e. as a continuum in which only an countable set of points is occupied by simple chemical objects. By this we illustrated in fact that at each moment of time in the ordinary physical space V_E(3) there is a whole continuum of coordinate systems fulfilling the requirement for conceptual correlation.

MATHEMATICAL REPRESENTATION OF THE CHEMICAL STRUCTURES

With respect to the so defined coordinate system K_S(6) in V_S(6) one can easily construct an entirely mathematical and unique representation of an arbitrary complex chemical object. For this purpose it is sufficient to prove the following statement:

THEOREM "CS". Each chemical structure can be represented entirely mathematically and uniquely with respect to K_S(6) in V_S(6) through the corresponding unique, discrete and noninvertible function, which domain is the ordinary 3 dimensional physical space V_E(3) and it range (codomain) is the Mendeleev's space V_M(3).

This theorem can be easily proved using the method of demonstration via constructing. Indeed, let us consider an arbitrarily chosen chemical structure with known stoichiometric formula. Then, the sought unique discrete function representing the considered chemical structure is obtained merely by replacing in the stoichiometric formula of this structure the symbols of the simple chemical objects by the coordinates of the corresponding species of these objects in the Mendeleev's space V_M(3) by using any of the two generalized mathematical models or , developed in [1, 4].

It should be accentuated on the fact that all unique and discrete functions used to represent entirely mathematically the various chemical structures are noninvertible. To come round to the validity of this statement it is sufficient to remind that there are chemical structures which composition includes several (more than one) simple chemical objects of one and the same chemical species. In such structures, to the various points of the ordinary physical space V_E(3) one can juxtapose one and the same point of the Mendeleev's space V_M(3) that represent the species of the same chemical simple chemical objects.

The following can be drawn from this theorem:

1) At each moment of time any set of arbitrarily chosen chemical objects (both simple and complex) is represented entirely mathematically, uniquely and spatially (with respect to K_S(6) in V_S(6)) through establishing the corresponding noninvertible image² of the ordinary physical space V_E(3) onto Mendeleev's space V_M(3). The image itself is constructed using the following two rules:

Rule I: To each point of V_E(3), in which some simple chemical object exists, one juxtaposes the mathematical image in V_M(3) of this species of simple chemical objects obtained by any of the generalized spatial mathematical models or .

Rule II: To each point of V_E(3) in which there is no simple chemical object at all one juxtaposes the natural zero-point of V_M(3).

The qualitative meaning of these two rules is as follows: they define a unique procedure for attributing an exactly specified chemical charge to each point of the ordinary 3 dimensional physical space V_E(3). By the term chemical charge of a point of V_E(3) we mean the species of the corresponding simple chemical object, occupying this point at a given moment of time.

2) From a mathematical point of view chemistry deals with the description and investigation of one exactly defined subset of the set of all functions - the subset of these unique discrete and noninvertible functions, which describes the possible chemical objects.

3) The definition and classification of the structures of various complex chemical objects are in fact determination and classification of these types of unique discrete and noninvertible functions which have chemical meaning in V_S(6) (i.e. which are mathematical images of complex chemical objects).

4) From the Theorem "CS" follows also that:

COROLLARY "PC". All complex chemical objects of periodic construction can be represented entirely mathematically and uniquely with respect to K_S(6) in V_S(6) through the corresponding periodic unique discrete and noninvertible functions.

Therefore, it becomes clear that crystal chemistry deals with description and investigation of one exactly specified subset from the set of all unique discrete and noninvertible functions - the subset of the periodic unique discrete and noninvertible functions, describing the chemical objects with periodic construction.

REFERENCES

[1] Penev V. Tz. Geometrization of the fundamentals of chemistry. 1997, Riva Publ. House, Sofia, 224 pp. (in Bulgarian with an extended summary in English) (Full text and summary are available through the official site of the project "Geometrization of the language of chemistry" – http://www.clmc.bas.bg/staff/V_Penev/gcl/)

[2] Penev, V., L. Konstantinov, M. Marinov. 1999. Geometrization of the language of chemistry. Formulation of the problem. - Compt. rend. Acad. bulg. Sci., 52, 1-2, 49-52.

[3] Penev, V., L. Konstantinov, M. Marinov. 2000. The problem of geometrization of the language of chemistry. – Academic Open Internet Journal, Volume 2 (2000), Part 2: Chemistry – http://www.acadjournal.com/2000/v2/part2/p2/

[4] Penev, V., L. Konstantinov, M. Marinov. 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. - Academic Open Internet Journal, Volume 2 (2000), Part 2: Chemistry – http://www.acadjournal.com/2000/v2/part2/p3/

NOTES

¹ The Mendeleev's space V_M(3) has by definition a natural zero-point O_M, which qualitative meaning is the absence of any protons, neutrons and electrons. Namely this natural zero-point O_M is chosen as origin of the Mendeleev's coordinate system K_M(3).

² In mathematics the notions function, mapping and image are synonymous.