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Mathematical
representations of chemical processes:
Part I. Conceptual schemes for representing changes; space and coordinate
system; method.
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
Two main conceptual schemes for representing changes, physical and geometrical, are formulated and discussed. The space VP(7) and the coordinate system KP(7) of the chemical processes are introduced. Formulated and analyzed is the method for constructing of entirely mathematical and unique representations of arbitrary chemical processes. Part of the missing mathematical apparatus (mathematical formalism) is defined and constructed. It is shown that the processes of absorption and emission of different kinds of physical particles (i.e. electrons, protons and neutrons) of the simple chemical objects are mathematically represented in VP(7) via the action of the corresponding three kinds of operators on the mathematical images of the simple chemical objects in the Mendeleev's space VM(3).
In our previous publications [1, 2, 3, 4, 5] the following was done:
- Conceptual, logical and mathematical instruments, necessary for entirely mathematical reformulation of the static's of simple and complex chemical objects, were constructed and analyzed;
- A method for construction of entirely mathematically and unique representations of arbitrary chosen chemical objects (both simple and complex) was demonstrated.
In this article the following is being presented:
- The two basic conceptual schemes (physical and geometrical) for representation of changes are formulated and discussed;
- The necessary definitions and a part of the mathematical apparatus for construction of entirely mathematical representations of chemical processes using physical conceptual scheme are formulated;
- The method for construction of these mathematical representations are formulated;
- Entirely mathematical representation of absorption and emission processes of different kinds of physical particles (i.e. electrons, protons and neutrons) from the simple chemical objects is given.
Well known are two basic conceptual schemes for spatial representation of changes, namely the physical (or dynamical) and the geometrical. The difference between the two schemes originates from the different meaning of the category "change" in physics and geometry. In the physics "the changes" are considered mainly as processes passing with time in the ordinary 3-dimensional physical space VÅ(3). Thus, the categories "time" and "causality" play fundamental role in the physical conceptual scheme. (These categories play fundamental role in the languages of all sciences using the category "change" in the meaning considered above.) The physical scheme for representation of changes was used already by Newton in the mathematical formulation of the classical mechanics in order to represent the motion of mechanical objects. The same conceptual scheme has lately been used in other branches of physics for constructing the kinematics and the dynamics of the corresponding physical systems, i.e., the apparatus for description of the changes that pass in those systems with time. The essence of the physical scheme for representation of changes, in general, can be reduced to unique assignment of: 1) the corresponding set of subsequent "momentary shots" of physical quantities, presenting the system of objects of our interest (correspondingly the status of these objects); 2) of the causal effectual relations between the physical quantities depicted on the particular "momentary shots" belonging to this set.1
The geometrical conceptual scheme for representation of changes, in contrast to the physical one, does not involve the categories time and causality at all. The reason for this is simple. In geometry the category change is considered above all as transformation. On its turn, the various types of geometrical transformations are considered as mathematical operations acting on the corresponding objects in a given mathematical space with the result that the initial objects are transformed into objects of the same space. Here we should emphasize on the fact that geometry does not pay any attention to non geometrical (i.e. physical, chemical or whatever else) qualitative interpretation neither of the corresponding abstract mathematical spaces nor of the objects and operations in these spaces. In other words, in geometry no attention is paid to whether a given abstract 3-dimensional space is mathematical model of the ordinary 3-dimensional physical space VÅ(3) or it is mathematical model of another, qualitatively different space for example - the 3-dimensional space of colors. It is of no matter also whether a given 1-dimensional space is mathematical model of "physical time" or it is mathematical model of "the set of real numbers".
It is seen from the aforesaid, that the geometrical conceptual scheme for representation of changes is much more general and of much larger applicability than the physical (dynamical) scheme. On the other hand, on present stage of development of chemistry the usage of the physical scheme as a conceptual basis for mathematical representation of chemical processes seems to be much more advisable even due to the fact that the language of chemistry at present is much closer to the language of physics than to that one of geometry. We assume that only just after the reformulation of chemistry entirely in spatial mathematical terms one should think for its strict reformulation in geometrical terms, i.e. for its real geometrization. However, we think that the solution of this problem is far away into the future.
So, the method for entirely mathematical representation of chemical processes, proposed below, is based on the physical conceptual scheme for representation of changes.
After choosing the appropriate conceptual scheme we shall formulate some necessary definitions:
DEFINITION "P". By the term process we shall denote only such type of changes in a given structure2 in which only the composition or the construction or both are changed.
DEFINITION "M". By the term motion of a given structure we shall denote such a change in which neither the composition, nor the construction of this structure are changed.
DEFINITION "SCP". By the term simple chemical process we shall denote every process in which simple chemical objects2 are formed or destructed, or any species transformation takes place.
DEFINITION "CCP". By the term complex chemical process (or chemical reaction) we shall denote every process in which complex chemical objects2 are formed or destructed, or any species transformation takes place.
It becomes clear from these definitions and from the physical conceptual scheme that in order to represent entirely mathematically and uniquely an arbitrary chosen chemical process one must firstly construct a suitable space and an appropriate coordinate system in this space.
Let us define the 7-dimensional space VP(7) as a Cartesian product of the 6-dimensional space of chemical structures VS(6) and 1-dimensional space of time VT(1), i.e. VP(7) = VS(6) x VT(1) = [VE(3) x VM(3)] x VT(1). Further we shall denote VP(7) - space of chemical processes.3
Let us construct in VP(7) a 7-dimensional coordinate system KP(7)=(OP;z,r,j;u,v,w;t) which is a Cartesian product of the 6-dimensional coordinate system of chemical structures KS(6)=(OS;z,r,j;u,v,w) and an appropriately chosen coordinate system KT(1)=(OT;t) in the space of time VT(1). In other words, let KP(7) = KS(6) x KT(1) = [KE(3) x KM(3)] x KT(1), where: (i) the projection of the 7-dimensional point OP onto VM(3) coincides with its natural zero point OÌ, which is by definition [1,4,5] the origin of the coordinate system KM(3); (ii) the projection of OP onto the ordinary physical space VE(3) coincides with the origin OE of the coordinate system KE(3); (iii) the projection of OP onto the space of time VT(1) coincides with the point OT, chosen as origin of the coordinate system of time KT(1).
We shall further refer to KP(7) as coordinate system of the chemical processes. It can be easily checked that such a choice of the origin OP of KP(7) assures the conceptual correlation [5] between the origins of the coordinate systems KM(3), KE(3) and KT(1), and consequently between the spaces VM(3), VE(3) and VT(1). Indeed, the coordinates of the point OP with respect to KP(7) are (0,0,0;0,0,0;0). This means that for initial moment OT such a moment of time (t=0) has been chosen, in which in the point OE of the physical space VE(3) there is no simple chemical object at all. It becomes clear from this statement that for origin OP of the coordinate system KP(7) one must always chose the "chemical zero" (i.e. the "chemical vacuum" , the "chemical nothingness"). In other words, the 7-dimensional point OP represents always the zero (the vacuum, the nothingness) considered from a chemical point of view. It becomes clear from this conclusion that the chemical objects and processes are always described with respect to the "chemical zero" (i.e. the "chemical vacuum" , the "chemical nothingness").
Until to now we have chosen the conceptual scheme, we have formulated the necessary definitions and we have introduced the space and the coordinate system for constructing an entirely mathematical representations of the chemical processes. Now we shall formulate and analyze the method for constructing this representation and shall determine the still missing elements of the mathematical apparatus.
Starting from the physical conceptual scheme and the existing at present way for description of the chemical reactions (including radio-chemical ones) one can easily realize that:
STATEMENT 1: In order to represent entirely mathematically and uniquely an arbitrary chosen chemical process using the physical conceptual scheme we need such a mathematical apparatus allowing in each arbitrary chosen moment t of time to represent mathematically and uniquely both: (1) the species and the position of all chemical objects of interest in a preliminarily specified volume V of the ordinary physical space VE(3); (2) the type and the number of all physical particles (i.e. the number of all electrons, protons and neutrons) which in the given moment t of time participate explicitly (i.e. as individual physical particles not combined in simple chemical objects) in the considered chemical process.
We shall clarify the meaning of this statement by several comments:
1. In the representation of each chemical process (independently of type of representation) one is interested only of those chemical and physical objects that take part in that particular process.
2. By "preliminarily specified volume" V (VÎVE(3)) we mean the volume in which the represented chemical process is passing.
3. To represent uniquely an arbitrary chosen complex chemical object means to define in a unique way its stereochemical formula, i.e. to define in a unique way both the species of all simple chemical objects, constructing the considered complex chemical object, and the mutual disposition of these simple chemical objects in VE(3). (The first requirement in Statement 1 results from that.)
4. In the so far existing ways for description of chemical reactions (including radiochemical ones) one takes into account the type and the number (of each type) of the physical particles participating explicitly at a given moment of time in the corresponding chemical reaction. In other words, one takes into account both: a) the type and the number (of each type) of physical particles absorbed or emitted by the corresponding chemical objects at a given moment of time; b) the type and the number (of each type) of free physical particles, necessary for obeying the law of preservation of the electric charge and the law of preservation of the mass in the volume V. This results in the second requirement in Statement 1. (We must note that this requirement does not obey us at all to define the positions in VE(3) of all physical particles mentioned above.)
5. It becomes clear that the second requirement in Statement 1 results from the following necessity: the searched apparatus should allow entirely mathematical and unique representation of both - the particular system of chemical causal effectual relations4 and of the set of general physical and chemical laws which should be obeyed (and are obeyed) by each system of chemical chemical causal effectual relationships.
From the aforesaid one can conclude that:
I. To represent entirely mathematically and uniquely an arbitrary chosen chemical process passing in a volume V in the ordinary physical space VE(3), means in fact to define entirely mathematically and uniquely what follows:
- the corresponding series of "momentary shots" of the chemical and physical objects of interest in the specified volume V (VÎVE(3));
- the corresponding systems of particular causal effectual relations between the sets of chemical and physical objects depicted on the individual "momentary shots" of this series.
In the preceding publication (see [5], Theorem "CS") we have shown that:
Each set of arbitrary chosen chemical objects (both simple and complex) can be represented entirely mathematically and uniquely through defining of the corresponding invertible image Á of a given volume V in the ordinary physical space VE(3) onto the Mendeleev's space VM(3), i.e. Á : V ® VM(3), where VÎVE(3).
The image Á is built using the following two rules:
Rule
À: To each point in the volume V (VÎVE(3))
in which there is some simple chemical object one juxtaposes the mathematical
image in VM(3) of the species of this simple
chemical object obtained by any of both generalized mathematical models 
or 
, constructed
in [1, 4].
Rule B: To each point in the volume V in which there is none any simple chemical object one juxtaposes the natural zero point of VM(3).
As noted in [5], these two rules (i.e. the image Á) define in fact a unique procedure for ascribing an exactly specified "chemical charge" to each point of the ordinary 3 dimensional space VE(3). With the term "chemical charge"of a given point in VE(3) we denote the species of the corresponding simple chemical object, that occupies in a particular moment this point in VE(3).
When proving the Theorem "CS" [5] we shown how to represent entirely mathematically and in a unique way all chemical objects (both simple and complex), which at a given moment are in an arbitrary chosen volume V in VE(3). Now we shall develop this part of the mathematical formalism that is necessary for representing the processes of absorption and emission of the different types of physical particles from chemical objects. For the purpose we shall firstly remind the conceptual scheme for entirely mathematical and unique representation of simple chemical processes [1, 4]:
1. Each simple chemical process can be represented through the corresponding set of three basic types of simple chemical processes in every of which only one of the three types of physical particles building the simple chemical objects is changed.
2. The simple chemical processes in which only the number of the neutrons in the simple chemical objects is changed can be represented mathematically in VÌ(3) through operators of translation along the axis Oz acting only on the z-coordinates of the mathematical images of these objects.
3. The simple chemical processes in which only the number of the protons in the simple chemical objects is changed can be represented mathematically in VÌ(3) through operators of translation along the axis Or acting only on the r-coordinates of the mathematical images of these objects.
4. The simple chemical processes in which only the number of valent electrons in the simple chemical objects is changed really or formally (i.e. only the oxidation state5 of the simple chemical objects is changed) can be represented mathematically in VÌ(3) through operators of rotation around the axis Oz which act only on the j-coordinates of the mathematical images of these objects.
5. Each simple chemical process can be represented entirely mathematically and uniquely in VÌ(3) through the corresponding set of three different types of operators, acting on the corresponding coordinates of mathematical images in VÌ(3) of simple chemical objects participating in the considered simple chemical process.
One can see from this conceptual scheme that:
The entirely mathematical and unique representation of the type and the number of the physical particles participating at a given moment in a particular chemical process is related with the corresponding translations and rotations in the Mendeleev's space VÌ(3).
On the other hand, for all constructed mathematical models of different species of simple chemical objects [1, 4], the follows statements are valid:
1. All translations and rotations in the Mendeleev's space VÌ(3), which are of physical or chemical meaning, are always quantized.
This statement allows us to introduce the following symbols:
for the quantum of translation along the axis Oz in VÌ(3);
for the quantum of translation along the axis Or
in VÌ(3); 
for the quantum of rotation around the axis Oz in VÌ(3).
One can see from the manner of constructing
the mathematical models P, Q, G, 
and 
[1,
4] that :
|
= 1 and = 1,
for all the models P, Q, G,
and ; |
|
= 2p/8, for the approximate model  ; |
|
= 2p/32, for two more precise models 
and  , and
for two generalized models
and . |
|
2. The changes in the number of neutrons are represented entirely mathematically and uniquely through translations along the axis Oz in VÌ(3) and for all constructed models:
-
the increase in the number of
neutrons by n is represented through a n-fold implementation
of the translation quantum
along the direction of increasing z-coordinates;
-
the decrease in the number of
neutrons by n is represented through a n-fold implementation
of the translation quantum
along the direction of decreasing z-coordinates.
- the increase in the number of
protons by m is represented through a
m-fold implementation of the translation quantum
along the direction of increasing r-coordinates;
- the decrease in the number of
neutrons by m is represented through a m-fold implementation
of the translation quantum
along the direction of decreasing r-coordinates.
- the increase in the number of
valent electrons by h (i.e. the negative increase in the oxidation
state by h) is represented through h-fold implementation of
the rotation quantum ,
corresponding to the model under consideration, along the direction of increasing
j-coordinates;
- the decrease in the number of
valent electrons by h (i.e. the positive increase in the oxidation
state by h) is represented through h-fold implementation of
the rotation quantum ,
corresponding to the model under consideration, along the direction of decreasing
j-coordinates.
CORROLARY
I: The processes of absorption or emission
of n neutrons by a given simple chemical object are represented entirely
mathematically and uniquely through the action of the translation operator
()n on the z-coordinate
of the mathematical image of this simple chemical object in VÌ(3).
As a result from the action of this operator, the coordinate z at a moment t1
after the absorption or emission of n neutrons, is:
z(t1)
= [( |
(4) |
where: z(t0) is the z-coordinate of the simple chemical object at a moment t0 before the absorption or emission of n neutrons; the integer number n is positive (n>0) for processes of absorption and negative (n<0) for processes of emission of neutrons.
CORROLARY
II: The
processes of absorption or emission of m protons by a given simple chemical
object are represented entirely mathematically and uniquely through the action
of the translation operator ()m
on the r-coordinate
of the mathematical image of this simple chemical object in VÌ(3).
As a result from the action of this operator, the coordinate r
at a moment t1 after the absorption or emission of m protons,
is:
r(t1)
= [( |
(5) |
where: r(t0) is the r-coordinate of the simple chemical object at a moment t0 before the absorption or emission of m protons; the integer number m is positive (m>0) for processes of absorption and negative (m<0) for processes of emission of protons.
CORROLARY
III: The processes of real or formal absorption
or emission of h valent electrons (i.e. the processes leading
to a change in the oxidation state of a
given simple chemical object by h) are represented
entirely mathematically and uniquely through the action of the rotation operator
()h on the j-coordinate
of the mathematical image of this simple chemical object in VÌ(3).
As a result from the action of this operator, the coordinate j
at a moment t1 after the absorption or emission of h
valent electrons, is:
j(t1)
= [( |
(6) |
where: j(t0) is the j-coordinate of the simple chemical object at a moment t0 before the absorption or emission of h valent electrons; the integer number h is positive (h>0) for processes of absorption and negative (h<0) for processes of emission of valent electrons.
In its turn, it becomes clear from the corollaries given above, that:
CORROLARY
IV: The type of the physical particles
absorbed or emitted by a given simple chemical object in the moment t is represented
entirely mathematically and uniquely through the corresponding type of operators ,
or in the Mendeleev's
space VÌ(3).
CORROLARY
V: The number of the physical particles
(of a given type) absorbed or emitted by a given simple chemical object
in the moment t is represented entirely mathematically
and uniquely through the specifying the integer exponent of power n,
m or h of the corresponding operator ()n,
()m or ()h.
CORROLARY VI: In the absorption processes the operators act along the positive direction of the corresponding axis, while in the emission processes they act along the negative direction of this axis.
Till now we showed how one can represent entirely mathematically and uniquely the type and the number, which at a given moment of time are really or formally absorbed or emitted by the chemical objects taking part in an arbitrary chemical process. In that way we construct almost all elements of the mathematical formalism, necessary for an entirely mathematical and unique representation of arbitrary chemical processes through the physical conceptual scheme. In a next publication we shall construct the remaining missing elements of the mathematical formalism and shall formulate in a general form the method for constructing the mathematical representations of chemical processes.
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/ |
| [5] | Penev, V., L. Konstantinov. Geometrization of the language of chemistry: Mathematical representation of the complex chemical objects. – Academic Open Internet Journal, vol. 11 (2004), Part 2: Chemistry – http://www.acadjournal.com/2004/v11/part2/p3/ |
NOTES
1 The definition of the causal effectual relationships in physics is usually performed through defining the corresponding equations (or systems of equations) relating with each other the momentary values of the physical quantities, depicted on the particular "momentary shots".
2 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 [4, 5]. 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 [4, 5]. 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 [4, 5]. By the term species of simple chemical objects we denote 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 [1, 4]. By the term complex chemical objects we denote each 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 VE(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 [1, 4].
3 The space VS(6) and coordinate system KS(6) of chemical structures are defined in [5]. The Mendeleev's space VM(3) and Mendeleev's coordinate system KM are defined in [1,4].
4 The description of each chemical process (independently of the form of this description) is, in principle, a representation of the corresponding system of chemical causal effectual relationships. This system is represented in the same way as in physics - through defining the chemical equation (or the system of chemical equations) of the considered chemical reaction.
5 By the term 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).