Vlerii Soolev,Ntlii Biln,Romn Dyhkovskyi,Egr Cseres Cn,Am Smolinski

a Department of Constraction,Geotechnics and Geomechanics,Dnipro University of Technology,Dnipro 49005,Ukraine

b Department of General and Structural Geology,Dnipro University of Technology,Dnipro 49005,Ukraine

c Department of Underground Mining,Dnipro University of Technology,Dnipro 49005,Ukraine

d Scientific Research Institute of the Center of Renewable Energy and Energy Efficiency,San Agustin National University Arequipa,Arequipa PE-04000,Peru

e Central Mining Institute,Katowice 40-166,Poland

Keywords:Underground coal mine Gas sorption Quantum-mechanical model Rock mass

ABSTRACT The purpose of this study was to develop a physico-mathematical model and technique for estimation of chemical bond stability depending on electric field intensity of an external point charge.A hypothesis for a possible physico-chemical mechanism of the formation of additional harmful gases in the rock destruction by blasting was proposed.The theoretical basis of the hypothesis is the method of theretical evaluation of bond energy depending on the distance to a point charge,the third Coulomb centre.The quantum-mechanical model for calculating the electronic terms of molecules makes it possible to solve problems associated with the determination of parameters of molecules under the action of various physical fields on the system under consideration.The model was approved for some diatomic molecules.The discrepancy between the experimental data and calculated data did not exceed 14%,which proves accuracy of the obtained results.The model can be used in the field of research into the causes of gas-dynamic phenomena in underground coal mines,in studies of the degree of stability of nanostructured components of coal under physical influences,and in the theoretical design of new compounds and structures in the field of nanomaterial science and nanotechnology.

1.Introduction

The provision of safety aspects is essential in coal production.However,the relevant effective solutions sometimes interfere with the strategy of new technologies implementation in the equipment-intensive operation [1-4].The contradiction is not obvious until,e.g.in the case of an abnormal and unpredictable behavior of rocks in response to physico-mechanical effects [5-7].Indeed,industrial gas-dynamic phenomena are formed as a result of physical and chemical processes and are manifested under predetermined combination of technological factors of extraction,geologic aspects and parameters of physical impacts [8-11].

The research related to the physico-chemical effects in mining processes is conditioned by the need of new basic knowledge about the nature and physical interactions of external force fields in nanoscale mineral systems [12-14].In this case,research can be divided into the following two main aspects:the first one using quantum mechanics and physical chemistry,and the second using statistical thermodynamics,classical physics,geomechanics etc.[15,16].The results of these investigations complement each other,forming a relatively complete scenario for the background and development of the various physical processes (e.g.gas-dynamic phenomenon)[9,10,17,18].

Possible mechanisms and special aspects of physicochemical effects caused by the change in the state of stability of chemical bonds in substances of different aggregative states have been discussed in the cited papers above.The most frequently detectable ones are the coal and gas outburst,rock and gas outburst,supernumerary methane emissions,spontaneous ignition and coal dust explosions,the formation of harmful gases in significant quantities from blasting operations,etc.Taking the physicochemical characteristics into account,it is necessary to form the unified physical concepts of the nature of each experimentally observed effect that does not have a convincing scientific interpretation.

Among the unresolved problems in the field of operations safety,the physical effect,which can initiate a decrease in stability and breaking of the chemical bond,i.e.initiate the development of chemical processes,is of special importance.Based on the analysis of the foreseeable physical scenario of reducing the substance stability as a result of the influence of some external impacts,the most likely reason that can underlie many effects may be identified.For instance,it is well known that rock destruction by blasting leads to an additional formation of harmful gases;their amount depends on the chemical and mineral composition of the rocks,their physico-mechanical properties,the temperature and the depth of bedding [19-25].According to the experimental data on explosions of one type of explosives in different rocks and different types of explosives in one rock,the total amount and composition of toxic gases increases to 220%and about 1000%accordingly[22].It may be concluded that that rocks have a greater influence on the yield of toxic gases than the chemical composition of explosives[26].

The second example is coal and gas outburst.The key issue here is the origin of a large amount of gas,exceeding the maximum possible one,which can ‘‘retain”the microstructure of coal [27,28].Since we are talking about an additional amount of gas,in fact,we are talking about the likely chemical reactions of the transition of the coal organic mass to gas [9,10].

The third example is the spontaneous ignition of new formed coal dust.At this point,the role of surface oxidation reactions,possibly catalytic ones,is obvious [15].From the examples given above,it follows that the breaking of old bonds and the formation of new ones probably relate to chemical reactions the most significantly caused by mass of the substance in the transformations.If electric charges are involved in the reactions as catalysts,then the reactions are low-energy,e.g.in the physical processes typical for the preparation and development of the explosion of an explosive charge [9,10,15,29-31].

The purpose of this study is the development of a physicomathematical model and the estimation techniques for the stability state of a chemical bond as a function of the strength of the electric field of an external point charge.

2.Theoretical results

It is supposed that the development of physical and chemical processes in rocks is essentially conditioned by the influence of kinetic parameters determining the catalytic properties of rocks,including the surface and the features of its functional state [15].It is also possible that the activity of local zones on the surface is produced by the presence of electric charges with a distribution density in accordance with the features of the surface relief.In this case,the course of the reactions does not need any additional stimulation by any external actions.

The need to estimate the stability of the chemical bond in the field of an external electric charge became apparent on grounds of the study of an elementary chemical act on solid new formed surfaces located in a gaseous medium [32].

Presenting the system under study as a static model allows us to suppose that by approaching of CO molecules (or NOX and others)to the surface carrying electrical charges (non-completed and deformed bonds,dislocations,adsorbed ions etc.),a part of molecules fall in the charge field.At the room temperature the electrical field strength at a range of 10-9m from a one-valence ion approximately equals 1.5 × 107V/cm.If the energy of a molecule is enough for it to approach on some critical distance,e.g.to an ion,in this case a break of its bonds occurs,and this process can go on in different ways.Let us consider the most probable dissociation reactions for a CO molecule:(1)CO →C+O;(2)CO →C++O-;(3)CO →C-+O+;(4)CO →C++O+e;(5)CO →C+O++e.

The minimum energy to pass to the molecule CO for breaking its bonds corresponds to the first reaction.So,from here on we shall take only the first reaction into account.Let us suppose also that the surface density of the ions rests constant.

To solve the problem,we shall use quantum mechanical regularities and results of works [33,34].In general,the proposed method of calculation differs from the known method for the construction of one-electron molecular orbits in the form of linear combinations of atom functions (method MO ЛКAO)by that as the basis are taken some polycentric functions in ellipsoidal coordinates and all the physical functions are calculated analytically.The constructed closed functions for the Green function allow solving the quant-mechanical problem with perturbations.The MO ЛКAO method does not give a possibility to make such calculations as a necessity appear in determining the numerical coefficients in every point of space.The difficulties appear likewise in the calculation of the energy of electron-electron interaction.

The main advantage of quant-mechanical calculations consists in the fact that the energy terms are easily calculated both in the ground state and in the excited one for chemical bonds.Here a possibility appears to follow the way of the dynamic process from the beginning of the state of a stable chemical bond to its gradual‘‘antibonding”till the moment of breaking.

For the evaluation of the proposed scenario,we will use the problem when the chosen chemical bond is under the action of some disturbance W(τ).The Schrödinger equation and interaction Hamiltonian of three particles(based on entering of Jacobi coordinates)will have such a form:

where Mt=M1+M2+M3is the mass of three interacting particles with the masses M1,M2and M3;1/M=1/M3+1/(M1+M2);Z1,Z2and Z3(Z2≥(Z1;Z3))the charges of interacting particles;R→the radius-vector of the centre of mass of the three-particles system connecting ions a and b;r→the radius-vector of the particle M3in the system of mass centre M1and M2;andandthe distance from the ions a and b,respectively,to the point of space under consideration.

We will take that M3≤(M1;M2).After the separation of the movement of the centre of three particles,the Schrödinger equationwill take the form:

where H0is the interaction Hamiltonian;Ψ the state vector of the quantum system;and Enthe full energy of three bodies in the centre of inertia of the system.H0in the atomic system of units(¯h=Z3=M3=1)is expressed in the following way:

The Eq.(5)can be rewritten in the ellipsoidal coordinate that allows us to divide the variables and transform it to the system of three ordinary differential equations of the second order:

where A is the division constant;Λ=0,1,2,...;μ=(ra-rb)/R,-1 ≤μ ≤1;ε=ER2/2;Z(±)=(ra± rb)/R;λ=(ra+rb)/R,1 ≤λ ≤∞;and Φ(φ)=exp(iΛφ).

The analysis for the Eqs.(7)and (8)permits to separate the model problem corresponding to the following equation:

where c=(λ2-1)/4+A/2 +ε/2+Z+/r.

The Green’s function for the operator in the Eq.(9)is constructed in [34].The solution of the model problem (Eq.(9))in the determination of k,n,λ is expressed via the Whittaker function fk,λ(t)by involving the Legendre function Yλ,n(μ):

The formalism developed here was earlier applied for the analytical calculation of molecules of hydrogen H2andLiH,of chemical bonds in solid state [33,34].

For the calculation of molecule CO it is possible to use the construction of the wave function in form of the determinant of the fourth order.In transitions of C and O from atom state in molecule state (CO and CO2),the internal two 1s-and 2s-electrons in the atom of carbon and two 1s-,2s-and 2p-electrons in the oxygen atom don’t change their energetic states.While forming carbon oxides from C and O atoms,the internal two 1s-electrons in the atom of carbon,taken two at a time from the energetic state 1s,2s and 2p from the atom of oxygen also don’t change their energetic states.Therefore,the internal atom electrons shield the nuclear charge,and the external ones take part in the formation of chemical bond.

To determine the effective potential φ′e（r ）and charge Z it is necessary to solve Poisson’s equation with the boundary conditions

where φiis the atom wave function.By solving the Eq.(10),the following potentials are obtained:

For 1s-electron:

For 2s-electron:

And for 2p-electron:

The electron energies,corresponding to the quant numbers k=0.5;λ=n=0;k=1.5;λ=n=0 will correspond to the energy determined from the expression

By the procedure offered here the shielding of nuclei C and O is made,and the electron terms are then calculated and constructed.

While determining the vibration spectrum for the chemical bond of the molecule CO the Schrödinger equation was used:

where A is the parameter determined by the dissociation energy of the chemical bond;=V(x)the Morse potential;R0the equilibrium state;x=(R-R0)/R0;α determined under the conditions of crossing the model and real potential curves with the axis R;and M=M1M2/(M1+M2)the reduced mass.

When constructing the dependence E(R),that spectrum of energy interaction of the molecule CO is of interest,which corresponds to its discrete part,i.e.when E ＜0.If we transform the Eq.(12):

and entering the notations n+s+1/2=s2=(-2R0ME)/α2φ2and substituting φ(t)=t-1/2φ(t),we will get the Whittaker equation:

As for connected conditions the wave function at infinity should tend to zero,so n=0,1,2,...,and the calculation of the energy of molecular wave spectrum is to be made by the following expression:

3.Results and disscusion

The use of the potential Morse as an approximation to the electron term of the molecule leads to a simple solution of the search problem of the vibrational spectrum of a molecule.The results of calculations of the vibrational levels of the CO molecule in the（0.5,0,0）and（1.5,0,0）states are given in [34].The correctness of the quantum mechanical model is confirmed by the data of the experimental values of the interatomic distances and the corresponding binding energies [35](Table 1).As an example,the results of calculation of the main electronic terms of the molecules N2,O2,CO,LiH,HCl,and Li2are shown in Fig.1,respectively.

As a result of calculations,there were obtained numerical values of CO molecule electron conditions in unperturbed state and in the presence of external charges of different signs and values(Fig.2).The calculations were made for the conditions with the temperature 0 K.The results of numerical simulation suggested that the possibility of a bond breakage increases either with the increase of distance between a molecule and an ion or with the increase of the ion charge,all other factors being equal.The molecule stability is determined by the function E(Z(±);R)under the condition of the charge field influence.A strong activating factor increasing the possibility of bond breakage is the temperature.The break of the CO bond in the field of bivalent ion takes place under the room temperature;in this case the curves 2 and 4 will be similar to the curve 3 which does not have a minimum(Fig.2).So,the probability of a bond breakage substantially increases with the temperature growth,all other factors being equal.The same influence is produced by the increase of pressure.

Table 1 Comparative characteristics of molecules.

Fig.1.Electronic terms of diatomic molecules.

Fig.2.The regularity of the CO molecule potential energy E(Z(±);R)change depending on the value and the sign of the surface charge Z(±):curve 1-molecule without perturbation;curve 2-molecule in the field of charge (-2)at the distance 5 × 10-10 m from the charge;curve 3-bond breakage;the molecule in the field of charge(-4)at the distance 2.5×10-10 m from the charge;curve 4-molecule in the field of charge (+3)at the distance 5 × 10-10 m from the charge.

Among different molecules,the energy(D0)of the CO molecule bond breakage has the maximum about 1074 kJ/mol;for CO+and CO2,the D0value is 810 and 540 kJ/mol,respectively;and for such nitric oxides as NO,NO+and NO-,the energy of bonds’ breakage will be 628,1045 and 500 kJ/mol,respectively[26].The given values of energy for the bond breakage of toxic molecules produced in explosion products show that the probability of molecular destruction increases with the decreasing of D0.The scenario of bond dynamics(the change of stable state)qualitatively does not change and completely corresponds to the dynamics of the CO bond in the field of the point electric charge.

In conformity with the calculations,if the surface charge increases,the antibonding effect in the chemical bond of molecules grows up,i.e.the molecule chemical bonds break with greater probability as compared with that of a smaller charge,all factors being equal.

The state of the surface as a kinetic factor is an exclusively individual quality of crystals(and of solid bodies in general),including not only an energetic component but also the quality of the longrang transmission of structural information.According to this fact one can suppose that the formation(or destruction)of organic and non-organic phases,all other factors being equal,will be conditioned by the physical and mechanical state of the surface.The origin of new phases,which are different in composition,aggregate state and chemical activity,is possible on different surfaces.

In technological cycles of explosive production related to chemical processes of dissolution,crystallization,creation of composites from mutually insoluble phases of different aggregate state,in one way or another,realizes the supposed mechanism of the flow of chemical reactions conditioned by the change of bonds energy in the field of electrical charges.In this case it is important not only to understand but to imagine clearly the character and regularities of the flow of a chemical process and to direct it correspondently.

In the process of rock destruction by blasting,a network of radial cracks is formed in which the explosion products penetrate under pressure.Under the mechanical loads,the new-formed cracks’ surfaces first enter in reaction with the explosion gases.These processes take the time of ≤10-4s.The mechanical paramagnetic centres (e.g.radicals)with high reactivities react with gas molecules forming new combinations.

The prospects for the development of explosive production with new complex of properties obviously will be connected with the development of precision technologies which are able to control their structures,physical-chemical and explosion properties.The new fundamental results in the field of physical chemistry,given in this work,are a part of knowledge necessary for the development of new technology.

4.Conclusions

A quantum-mechanical model of the dynamics of elementary acts of chemical reactions has been developed.The main theoretical difficulty was the solution of the problem of the behavior of a diatomic molecule in the field of the Coulomb centre.Regularities of changes in the bond energy of a molecule,depending on the distance between the molecule and the electric charge,the intensity of the external electric field,and other parameters,were established.To break a chemical bond,a sufficient condition is the effect of the electric charge field,in case of approaching the molecule to the distance at which the bond of the molecule is broken.A model has been developed for calculating the electron molecular terms,taking the screening of nuclear potentials by the electron nuclei of bound electrons into account.A promising area of use of the developed quantum-mechanical approach in the study of catalytic reactions occurring in systems ‘‘moving components-charges”is the physics and chemistry of solid fuels (coal gasification,utilization of methane and carbon oxides,production of new materials,including energy-saturated,etc.),and the physics and chemistry of rock processes (the origin and development of unstable states in local areas of the rock mass and coal seams,processes occurring in the rock mass with actively developing cracking,self-ignition of coal dust and gas).The proposed method for calculating the chemical bond energy (e.g.small molecules)in the field of electric charge (ion)intensity was tested when solving various physicochemical problems.Despite the fact that many problems solved using the proposed quantum-mechanical model,at first glance,have no relation to the physics of mining processes,yet one fundamental chemical act in the scenario of a decreasing degree of stability of a substance is one.This can be indirectly indicated by several examples below.In particular,the physico-mathematical model can be used in clarifying certain issues related to the mechanism of carbon dioxide destruction that plants absorb from the atmosphere and oxygen generation in photosynthesis scenarios.On the example of reactions occurring in plant matter,catalytic systems can be created for the disposal of harmful and toxic gases in various technological processes of chemical production,etc.

In the processes of mineral and ore formation occurring with the participation of carbon,the use of the developed quantummechanical models became a sufficient base for the development of fundamentally new ideas about the mechanisms of nucleation and the genesis of carbon-bearing phases.A fundamentally new point of view is expressed regarding the physicochemical concepts of the origin of coal,graphite and diamond,the causes and mechanisms of formation of outburst zones in rocks and coal,and the formation of gas from coal,etc.

Declaration of Competing Interest

The authors declare no conflict of interest.

Acknowledgements

The studies were accomplished within the framework of the project ‘‘Investigation of coal nanostructure as a source of coal mine methane”with a financial support of the Ministry of Education and Science of Ukraine according to the Order No.199 of February 10,2017.