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Part II. Representation of the free physical particles and of the laws for preservation of electric charge and preservation of mass in chemical reactions.
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
E-mail: vpenev@interbgc.com; vpenev@bas.bg;
v_penev@mail.bg
Web site: http://www.clmc.bas.bg
ABSTRACT
The method for an entirely mathematical representation of chemical processes, based on the physical conceptual scheme is analyzed. The notion "free operator" is introduced. It is shown that the type and number of free physical particles in the chemical reactions is represented entirely mathematically through the corresponding three types of operators in Mendeleev's space VM(3). The application of the proposed conceptual and mathematical formalism is demonstrated on the example of an entirely mathematical representing of a particular simple chemical process. The laws of preservation of the electric charge and mass in chemical reactions are formulated in terms of the developed conceptual and mathematical formalism.
INTRODUCTION
In our previous publications [1 - 6] we presented the main elements of conceptual, logical and mathematical formalism, necessary for an entirely mathematical formulation of the static's of arbitrary chemical objects (both simple and complex). We presented also almost all basic elements of the formalism necessary for mathematical reformulating of the dynamics of arbitrary chemical objects based on the physical conceptual scheme for representation of changes.
In this paper we shall:
- analyze the method for constructing entirely mathematical and unique representations of chemical processes;
- construct the last missing elements of the mathematical formalism;
- demonstrate the application of the developed conceptual and mathematical apparatus on the example of an entirely mathematical representation of a particular simple chemical process;
- formulate the laws of preservation of the electric charge and mass in chemical reactions in terms of the developed mathematical apparatus.
In [6] we showed how to represent entirely mathematically and uniquely the type and the number of physical particles really or formally absorbed or emitted at a given moment of time by chemical objects participating in a chemical process. Now we shall analyze the method for an entirely mathematical representation of chemical processes and shall construct the last missing elements of the mathematical apparatus.
So, each chemical process proceeds always in a some volume V in the ordinary physical space VÅ(3). Further, in each chemical process always (i.e. at any moment of time) take part a countable set1 of chemical objects. That is why, in order to represent entirely mathematically an arbitrary chemical process through the physical conceptual scheme one has:
1. To chose an appropriate coordinate system of chemical processes KP(7)=(OP;u,v,w;z,r,j;t) in the 7 dimensional space VP(7) of chemical processes2.
2. To chose with respect to KP(7) the these six dimensional momentary cross sections [VS(6),t0] and [VS(6),t1] of the seven dimensional space VP(7) which correspond to the moments of time t0 and t1, accepted as the beginning and the end of the considered chemical process.
3. To define (with respect to KP(7) in VP(7)) two six dimensional momentary snapshots of the mathematical images of chemical and physical objects in the beginning and the end of the considered chemical process. The first snapshot must contain the mathematical images of chemical and physical objects in the momentary cross section [VS(6),t0], corresponding to the beginning t0 of the considered chemical process. The second snapshot must contain the mathematical images of chemical and physical objects in the momentary cross section [VS(6),t1], corresponding to the end t1 of the considered chemical process.
4. To represent (with respect to KP(7) in VP(7)) entirely mathematically and uniquely the considered chemical process by giving the corresponding equation (or the corresponding system of equations), i.e., to define entirely mathematically and uniquely the corresponding system of chemical causal effectual relationships between the mathematical images of chemical and physical objects depicted on the two 6 dimensional momentary snapshots.
This four step scheme describes in a general form the method for constructing an entirely mathematical and unique representation of arbitrary chemical process through the physical conceptual scheme. In order to clarify better the meaning of this method we shall comment some points:
A) It follows directly from the definition of the space VP(7) of chemical processes2 that:
CORROLARY "MS": In every moment t of time (t ÎVT(1)) the corresponding to this moment 6 dimensional space of chemical structures [VS(6)]t is such a 6 dimensional cross section [VS(6),t] of the 7 dimensional space of chemical processes VP(7), which is "perpendicular" to the axis of time (i.e., to the one dimensional space VT(1)).
From this corollary one can draw the following conclusion:
The point of intersection of every separate 6 dimensional cross section [VS(6),t] of VP(7) with the 1 dimensional space of time VT(1) (i.e., the corresponding moment t of time) plays the role of consecutive number of the momentary space of chemical structures [VS(6)]t corresponding to this moment, i.e., to the corresponding 6 dimensional cross section [VS(6),t] of VP(7) which is "perpendicular" to the axis of time.
Let's assume that the space of time
VT(1) is linearly ordered3. Then,
it follows directly from the last conclusion that the set of all momentary 6
dimensional spaces of chemical structures [VS(6)]t (i.e.,
the set of all 6 dimensional cross section [VS(6),t] of VP(7))
is linearly ordered (since the set of consecutive numbers t of the cross sections
[VS(6),t] of VP(7) is linearly ordered). It easily seen
that exactly this conclusion makes it possible to use the physical conceptual
scheme for representation of changes in constructing an entirely mathematical
representation of chemical processes. Indeed, if the space of time
VT(1) is not linearly ordered, one can impossible to order consequently
the separated momentary snapshots containing the mathematical images of chemical
and physical objects, participating in the given chemical process and consequently
one can not use the physical conceptual scheme for representation of
changes.
B) By definition [5] the first three coordinates (u(t),v(t),w(t)) of the 6 dimensional mathematical images (u(t),v(t),w(t);z(t),r(t),j(t)) of simple chemical objects represent (with respect to KP(7) in VP(7)) the position of these objects in the ordinary physical space VE(3) at the moment t, while the second three coordinates (z(t),r(t),j(t)) represent the species of the corresponding simple chemical objects, which at the moment t are in the corresponding points (u(t),v(t),w(t)) in VE(3). In other words, the second triad of coordinates (z(t),r(t),j(t)) of the 6 dimensional mathematical images represents the chemical charge (see [5]) of the corresponding point (u(t),v(t),w(t)) in VE(3) at the moment t.
C) According to the Theorem"CS" (see [5]), an entirely mathematical representation of each set of chemical objects can be constructed through defining the corresponding momentary distribution of chemical charges in VE(3), i.e., through defining (with respect to KP(7) in VP(7)) the coordinates (u(t),v(t),w(t);z(t),r(t),j(t)) of all simple chemical objects (both free and bonded in complex chemical objects), which at the moment t are in the considered set.
D) The representation of any chemical process is, in fact, a defining of the corresponding particular system of causal effectual relationships between the sets of chemical and physical objects depicted on the two 6 dimensional momentary snapshots, corresponding to the beginning and to the end of the considered chemical process.
E) Namely the existence of a particular system of causal effectual relationships between the 6 dimensional momentary snapshots of the considered chemical and physical objects belonging to two different 6 dimensional momentary cross sections of the space VP(7) allow us to represent the set of chemical and physical objects at the end of the represented chemical process (depicted on the second 6 dimensional momentary snapshot) as a function of the set of chemical and physical objects at the beginning of this process (depicted on the first 6 dimensional momentary snapshot).
Resuming the last two comments in the language of mathematics we can formulate the following statement:
STATEMENT "CT": The constructing of an entirely mathematical unique representation of any arbitrary chosen chemical process is in fact constructing of the corresponding unique mathematical image (function)4 which transform a given set of mathematical images of chemical and physical objects, belonging to one 6 dimensional momentary cross section [VS(6),t0] of VP(7) into another particular set of mathematical images of chemical and physical objects, belonging to another 6 dimensional momentary cross section [VS(6),t1] of VP(7).
Continuing the comments we should note that:
F) The statement "CT" is very important, because it reveals clearly the relationship between the proposed model for entirely mathematical unique representation of chemical processes and the geometric conceptual scheme for representation of changes [6]. To prove this statement one should remember that the changes in the geometric scheme (in our case these are the chemical processes) are considered as geometric transformations which act on objects in a given space with the result that the initial objects (referred to as pro-images) are transformed into objects (refereed to as images) in the same space. In other words, the geometric transformations are unique mathematical mappings (functions) between a given set of geometric objects (pro-images) belonging to a given space and another set of geometric objects (images) belonging to the same space.
G) The
operators of translation and rotation 
, 
and 
in the Mendeleev's space VM(3) are used in constructing the
mathematical images mentioned in statement "CT", which represent in a unique
way the corresponding particular chemical processes.
H)
Besides the operators
,
and
in constructing the mentioned mathematical images, one makes also use
of different types of rotations and translations in the ordinary 3 dimensional
physical space VE(3). Further we shall use the following symbols:
(1) 
will stand for the operator of translation to a distance a along the
u-axis (i.e. along the dimension u) of VE(3);
(2) 
will stand for the operator of translation to a distance b along the
v-axis (i.e. along the dimension v) of VE(3);
(3) 
will stand for the operator of translation to a distance c along the
w-axis (i.e. along the dimension w) of VE(3);
(4) by the symbols 
, 
and 
we shall denote respectively
the operators of rotation in VE(3) by the corresponding angles a,
b and g.
I)
The translation operators ,
and , are used for a
mathematical description of the change in the position in the
space VE(3) of the simple chemical objects participating in an arbitrary
chosen chemical process. For that reason they act only on the first three coordinates
(u(t),v(t),w(t)) of the corresponding mathematical images. Their action (with
respect to KP(7) in VP(7)) is defined through:
| u(t1)
= []u(t0)
= u(t0) + a |
(1) |
| v(t1)
= []v(t0)
= v(t0) + b |
(2) |
| w(t1)
= []w(t0)
= w(t0) + c |
(3) |
In these expressions the coordinates (u(t0),v(t0),w(t0)) define (with respect to KP(7) in VP(7)) the position in the space VE(3) of a given simple chemical object at the moment t0, corresponding to the beginning of the considered chemical process, while (u(t1),v(t1),w(t1)) are its coordinates at the moment t1, corresponding to the end of this process.
J)
The rotation operators
, and
are used for mathematical representation of the change in the orientation
of complex chemical objects, considered as a whole, with respect to KE(3)
in VE(3).
We shall finish the comments with a simple analogy revealing clearly the meaning of the proposed method:
K) As a basis both the physical scheme for representation of changes and the proposed method for entirely mathematical and unique representations of chemical processes (as well as of the contemporary way used for describing chemical processes) stands the same principle as that used for making animated cartoons. To come round this idea it is sufficient to bear in mind that the role of separate "chemical animated frames" in the proposed method is played by the different 6 dimensional momentary snapshots of chemical and physical objects. On its turn, the chemical and physical objects on these snapshots play the role of particular "characters" on the corresponding "chemical animated frames". The role of "animation links" between the "characters" from the separate "chemical animation frames" is played by the corresponding particular mathematical images (functions), considered in Statement "CT". (Just this images show which "character" or "group of characters" from a "chemical animated frame" corresponds to a given "character" or "group of characters" from the preceding"chemical animated frame".) However, it should specially be noted that in the physical and chemical "animated cartoons" there is also a lot of strong restrictions in the freedom of "composing" both the individual "animated frames" and the "animation links" between them and also in the freedom of choosing at all the corresponding physical and chemical "animated plots". These restrictions are revealed e.g. in the fact that each "chemical animated cartoon" (i.e. the representation of each chemical process, no matter what is the kind of this representation) must obligatory obey such general "composition rules" as the laws of preservation of the electric charge and of the mass in chemical reactions, etc.
To here we formulated in a general form and analised the method for constructing entirely mathematical and unique representations of arbitrary chemical processes. Now we shall illustrate the use of this method on the example of representing a simple chemical process. At the same time, we shall introduce the remaining missing elements of the mathematical formalism and shall formulate in a general form the laws of preservation of the electric charge and of the mass in chemical reactions in terms of the proposed mathematical formalism.
Let us consider the process of k=fold ionization of some species of atoms A. This simple chemical process is represented by the following general chemical expression:
| A ® Ak+ + ke- | (4) |
Let us as an example for such a process consider that of a 4-fold ionization (k=4) of carbon C-14 (i.e. of the carbon isotope with 14 neutrons). This particular simple chemical process is represented by the following particular chemical expression (which does not account for the number of neutrons).
| C ® C4+ + 4e- |
(5)
|
According to the proposed method, this chemical process is represented entirely mathematically and uniquely through defining the corresponding set of two 6 dimensional momentary snapshots of the considered chemical and physical objects in the corresponding volume V in VE(3) (in which the represented chemical process proceeds). The first momentary snapshot corresponds to the moment t0, assumed as the beginning of the given process, and represents the left-hand side of (5), while the second momentary snapshot corresponds to the moment t1, assumed as the end of the process, and represents the right-hand side of (5).
In order to represent the species
of simple chemical objects C and C4+ let us make use the second generalized
mathematical model 
[1, 4]. Then the coordinates (z,r,j)
of their mathematical images with respect to Mendeleev's coordinate system KM(3)
in VM(3) would be (14,6,7p/4)
for the neutral atom C and (14,6,3p/4)
for the 4-fold ionized atom C4+. In this case on the first momentary
snapshot will be represented a 6 dimensional point, belonging to the 6 dimensional
momentary section [VS(6),t0] of VP(7) and with
coordinates (u(t0),v(t0),w(t0);14,6,7p/4)
with respect to KP(7), respectively. In other words, on this
momentary snapshot a neutral atom C-14, which at the moment t0 is
in the point with coordinates (u(t0),v(t0),w(t0))
in the volume V (VÎVE(3),
will be represented with respect
to KM(3) in VM(3) through the second triad of coordinates
(14,6,7p/4)).
Another 6 dimensional point will be represented on the second momentary snapshot,
belonging to the 6 dimensional momentary section [VS(6),t1]
of VP(7), whose coordinates with respect to KP(7) are
(u(t1),v(t1),w(t1);14,6,3p/4),
respectively. In other words, on this momentary snapshot a 4-fold ionized atom
C-14, which at the moment t1 is in the point with coordinates (u(t1),v(t1),w(t1))
in the volume V, will be represented with respect to KM(3) in VM(3)
through the second triad of coordinates (14,6,3p/4).
One can see from the particular chemical expression (5) that on the second momentary snapshot, besides the 6 dimensional point (u(t1),v(t1),w(t1);14,6,3p/4), four free electrons must also be represented. Let us assume that:
Definition "FO": By the term free operator we shall denote such an operator that is not applied to any object in the corresponding mathematical space.
Coming form this definition and the arguments used to the mathematical representation of the processes of emission and absorption of physical particles by chemical objects [6], one can accept three new rules in addition to the two ones formulated in [5, 6] for ascribing a chemical charge to the points in VE(3). The first of these three new rules is:
Rule "FE":
The
free electrons are represented mathematically
in a unique way in VP(7) through free quanta of rotation
around the axis Oz in VÌ(3), and
each free electron is represented through a free operator .
It becomes
clear from this rule that the four free electrons in the right-hand side
of the particular chemical expression (5) will be represented mathematically
in a unique way on the second momentary snapshot by four free quanta of
rotation
in VÌ(3).
So, if we write down in symbols all the aforesaid, we would find out the following mathematical representation of the particular chemical expression (5) with respect to KP(7) in VP(7):
| (u(t0),v(t0),w(t0);14,6,7p/4)
®
(u(t1),v(t1),w(t1);14,6,3p/4)
+ 4 |
(6) |
We should note at this point that in the mathematical expression (6) one can uniquely juxtapose to all quantities exactly specified numbers5 and exactly specified mathematical operations (in contrast with the chemical expression (5), for which this is not true). This fact is valid for the mathematical representations of all chemical processes (no matter what these processes are), constructed by using the proposed method. This is important, because it shows that the proposed method for entirely mathematical and unique representation of arbitrary chemical objects and processes would substantially simplify and accelerate the wide usage in chemistry of a lot of powerful cybernetic and computational methods and technologies, both theoretical and applied.
A second substantial difference between the mathematical and chemical representations of the considered particular simple chemical process is that in the mathematical expression (6) the number of neutrons is also represented in a unique way, whereas in the chemical expression (5) this number is not included at all. In other words, the mathematical representation allows to introduce in a unique way the particular isotopes (i.e. particular species of atoms) of the chemical elements taking part in a given chemical process (in contrast to the chemical representation existing so far, which allows to introduce in a unique way only the chemical elements participating in a given chemical process). Thus, the entirely mathematical representation of the chemical processes has a higher "resolution" as compared with the chemical representation existing so far6.
Let us continue the analysis of the
considered particular simple chemical process. Each one of the four free electrons,
represented on the second 6 dimensional momentary snapshot through the four
free operators , could
further either be absorbed by some simple chemical object in the volume V or
leave this volume. In terms of the developed mathematical formalism this means
that each one of the four free operators
could further be applied to the j-coordinates
of the mathematical image of some simple chemical object in the volume V, or
leave this volume at all (i.e., not to be further applied at all to the j-coordinates
of any simple chemical object in the considered volume V). On the other hand,
the set of chemical and physical objects in the volume V must obligatory obey
the law of preservation of electric charge. It becomes clear from this
statement and the preceding arguments that:
Statement "EK": The following condition for preservation of negative electric charge in the corresponding volume V must be obeyed for each chemical process:
 |
(7) |
where: the symbol 
stands for the operation summation7 in the volume
V; the symbols 
and 
stand for the sums of the j-coordinates
of the mathematical images of all the simple chemical objects of interest in
the volume V at the moments t0 and t1 (these two terms
in (7) represent the sums of oxidation states of all the simple chemical
objects of interest in the volume V at the beginning and at
the end of the represented chemical process)8;
the symbols 
and 
denote the sums of all free operators
which at the moments t0 and t1 are in the volume V (these
two terms in (7) represent the number of all free electrons, which at the moments
t0 and t1 are in V); the term r
represent the number of all free electrons, which during the time period of
interest have entered the volume V coming from outside and the term s
represent the number of all free electrons left the volume V during the same
time period.
In order to complete the representation the physical law for preservation of electric charge and the chemical law for preservation of the mass in chemical reactions in terms of the proposed mathematical formalism one should say how to represent mathematically in a unique way the free protons and neutrons. By analogy with Rule "FE" we introduce the following two rules:
Rule "FP":
Free
protons are represented mathematically in a unique way in VP(7)
through free quanta of translation along the axis Or
in VÌ(3), and each one free proton is represented via one free
operator .
Rule "FN":
Free
neutrons are represented mathematically in a unique way in VP(7)
through free quanta of translation along the axis Oz in VÌ(3),
and each one free neutron is represented via one free operator .
Using these rules one can formulate two more important statements:
Statement "PK": For every chemical process the following condition for preservation of the positive electric charge in the corresponding volume V in VE(3) must be obeyed:
 |
(8) |
where: the symbols 
and 
denote the sums
of the r-coordinates
of the mathematical images of all the simple chemical objects of interest in
the volume V at the moments t0 and t1 (these two terms
represent the sums of numbers of protons of all the simple chemical objects
of interest in the volume V at the beginning and at the end
of the represented chemical process)9; the
symbols 
and 
denote the sums of all free operators
which at the moments t0 and t1 are in the volume V (these
two terms represent the number of all free protons, which at the moments t0
and t1 are in V); the term m
represent the number of all free protons, which during the time period of interest
have entered the volume V coming from outside and the term n
represent the number of all free protons which left the volume V during the
same time period.
Statement "NK": For every chemical process the following condition for preservation of the number of neutrons in the corresponding volume V in VE(3) must be obeyed:
 |
(9) |
where: the symbols 
and 
denote the sums
of the z-coordinates of the mathematical images of all the simple chemical objects
of interest in the volume V at the moments t0 and t1 (these
two terms represent the sums of numbers of neutrons of all the
simple chemical objects of interest in the volume V at the beginning
and at the end of the represented chemical process)9;
the symbols 
and 
denote the sums of all free operators
which at the moments t0 and t1 are in the volume V (these
two terms represent the number of all free neutrons, which at the moments t0
and t1 are in V); the term t
represent the number of all free neutrons, which during the time period of interest
have entered the volume V coming from outside and the term h
represent the number of all free neutrons left the volume V during the same
time period.
On the basis of the last three statements one can draw the following important conclusions:
I. The expressions (7) and (8) from Statements "EK" and "PK" are discrete mathematical analogs (concerning the corresponding electric charge) of Ostrogradski-Gaus's theorem, representing mathematically the law of preservation of the electric charge in classical electrodynamics.
II. Considered together, as a system, the expressions (7) and (8) from the Statements "EK" and "PK" represent the law of preservation of the electric charge in chemical processes in terms of the developed mathematical formalism. In other words, Statements "EK" and "PK" represent the fact that in chemical processes neither positive nor negative electric charge can arise or disappear.
III. Considered together, as a system, the expressions (7), (8) and (9) from the Statements "EK", "PK" and "NK" represent the law of preservation of the mass in chemical reactions in terms of the developed mathematical formalism. In other words, the system of Statements "EK", "PK" and "NK" represents the fact that in chemical processes no physical mass can arise or disappear.
This publication completes in fact the representation of the basic elements of the conceptual, logical and mathematical formalism, necessary for a further entirely mathematical reformulation of the fundamentals of chemistry based on the physical conceptual scheme for representation of changes.
NOTES
1 The chemical objects are discrete. It follows directly from this fact that the set of chemical objects taking part in arbitrary chemical process is always countable. Just the fact that this set is countable makes possible to use further the operation summation instead of operation integration.
2 The space VS(6) and the coordinate system KS(6) of chemical structures are defined as follows: [5]: VS(6) = VM(3) x VE(3) and KS(6) = KM(3) x KE(3), where VM(3) and KM(3) stand respectively for the Mendeleev's space and the Mendeleev's coordinate system, defined in [1, 4], while VÅ(3) and KÅ(3) are respectively the ordinary 3 dimensional physical space and the properly chosen coordinate system in it. On its turn, the space VP(7) and the coordinate system KP(7) of chemical processes are defined as follows [6]: VP(7) = VS(6) x VT(1) = VT(1) x [VÌ(3) x VÅ(3)] and KP(7) = KS(6) x KT(1) = [KM(3) x KE(3)] x KT(1), where VT(1) and KT(1) stand respectively for 1 dimensional space of time and a properly chosen coordinate system therein.
3 For the definition of the notion linearly ordered set see e.g. ([7], p. 382).
4 In mathematics the notions function and map (mapping) are synonymous.
5 The quantities u(t0), v(t0), w(t0), u(t1), v(t1), w(t1) are of numbers.
6 The problem concerning the higher resolution of the spatial mathematical models of different species of simple chemical objects has been discussed in detail in ([1], ch. 3, §7).
7 We remind that namely the countability of the set of chemical objects allows us to use the operation summation (in the volume V) instead the operation integration (over volume V).
8
In both generalized mathematical models 
and
the j-coordinates
represent the oxidation state, i.e., the real or formal electric charge
of the corresponding simple chemical object. Therefore, the change
in the j-coordinates
in both generalized models uniquely represents the change of the real or formal
electric charge of the corresponding simple chemical objects.
Indeed, in accordance with Corollary 23, formulated in ([1], p. 117-118): in
both generalized mathematical models
and
the change of oxidation state of the given species of simple
chemical objects leads to a rotation of their mathematical images around the
axis Oz at the corresponding angle 
,
where h and t are the oxidation states of the given simple chemical
object at the beginning and at the end of the corresponding process of ionization
or recombination [1, 4]. It is clear from this Corollary that: (1) each
change of the oxidation state of given simple chemical object represents in
both generalized models 
and 
by the corresponding
exactly specified change in its j-coordinates; (2) all the processes
of ionization and recombination of the simple chemical objects can be represented
mathematically and uniquely in the Mendeleev's space VM(3) by
the rotation operators around the axis Oz at the corresponding angles.
9
We remind that in all constructed mathematical models P,
Q, G,
and of different
species of simple chemical objects the r-coordinates
of the mathematical images represent the number of protons and the z-coordinates
represent the number of neutrons in the corresponding species of simple chemical
objects.
REFERENCES
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