
Home / GENERAL CHEMISTRY Textbook / Chapter 6. Molecule structure Chapter 6. Molecule structureTHE HYDROGEN MOLECULEAccording to the model (see Fig 4.1) during the formation of a covalent chemical bond, there is an energy gain received at the expense of the transition of the electrons of one atom to the outermost shell of another. This energy gain can be defined by the affinity of hydrogen atoms to the electron. That is, by the value of about 0.72 eV · 2, where 0.72 eV is the affinity energy of one hydrogen atom relative to one electron. Simultaneously, during bond formation, the electrons and the nuclei approach each other, which leads to an energy loss. According to experimental data, the distance between the nuclei in a hydrogen molecule comprises 0.74Å. Relatively, the energy loss, at the expense of the mutual repulsion of the nuclei, is equal to about 9 eV. That is, a hydrogen molecule, according to this calculation, cannot be stable. The hydrogen molecule is formed of two atoms of hydrogen. During molecule formation, two electrons, hitherto having belonged to two different hydrogen atoms, begin rotating on a plane perpendicular to the axis connecting the nuclei (see figure 6.1.1) Attraction and Repulsion Forces of an AB Molecule In order to well realize the correctness of this model, let's calculate the energy of a hydrogen molecule as we did in the case of the atom. Here, also, the molecule's energy is equal to the sum of its electronic energies. According to the model, the electrons rotate around point E on a plane perpendicular to the axis connecting the nuclei A and B. The electrons' attraction forces to the nuclei are directed perpendicular to the plane of the circle where the electrons rotate and where they are mutually counterbalanced. Their mutual action is therefore equal to zero. That is, the problem of calculating the energy of a hydrogen molecule is reduced to the problem of defining the energy of a heliumlike atom (atom with 2 electrons). Previously (see 5.2) it has been shown that for this calculation it is necessary to know the nuclear charge since the energy of a heliumlike atom is defined by the equation: E_{HA} = 13.595(Z  0.25)^{2 }· 2 (6.1.1) It must be noted that there is no real positive charge at point E. The electrons are attracted to this point at the expense of forces F_{1}^{1}_{ }that are projections of forces F_{1} onto axis DC. These forces are identical to the forces that hold the electrons on the orbit of a heliumlike atom with charge Z. These forces prevent the breakaway of the electrons because of the mutual repulsion of F_{3} and the centrifugal forces, that is, 2F_{1}^{1} = F_{3}+F_{4 }where F_{3} are the interelectronic repulsion forces and F_{4} are the centrifugal forces. Here, as always, we use the system of calculations introduced by Bohr for the calculation of atomic systems. The system's energy and the linear parameters in these calculations are defined via the comparison of charges and distances (radii) with the electron's energy and the orbit's radius in the hydrogen atom. The electron's charge is accepted as a unit charge, while the radius of the hydrogen atom (0.529Å) serves as a unit of length. The unit force in this case is the interaction force of the proton with the electron in a hydrogen atom, equal to the distance of 0.529 Å. Thus, the nuclear charge can be resembled as Ze since the proton's charge is equal to that of the electron. Considering all the accepted designations, let's define the effective charge Z at point E of a hydrogen molecule (see figure 6.1.1) The electrons here are situated at points C and D while the nuclei (protons)  at points A and B. Point E is situated in the center around which the electrons rotate on a plane perpendicular to that of the drawing. The attraction forces of the electrons to the nuclei are designated in the figure by F_{1}. The repulsive forces between the nuclei are designated as F_{2}. The repulsive forces between the electrons are shown as F_{3} . The projections of F_{1} onto axis CD are indicated as F_{1}^{1}. The projections of F_{1} onto axis AB are indicated as F_{1}^{11}. The properties of hydrogen molecules do not change with time, so the distances between the electrons and the nuclei are constant. That is, forces F_{1}^{11}attracting the nuclei equal forces of internuclear repulsion. Forces F_{1}^{1}, acting upon the electrons, are also equal for the same reason. These forces act upon electrons C and D in contrary directions and are equal in value, i.e., their resultant force amounts to zero. The same is true of forces 2F_{1}^{1} and F_{3}+F_{4}. Now let's designate the radius of the electron's orbit as a (EC = a), the distance between the nucleus as 2b (AB = 2b), the distance between the electron and the nucleus as c (AC = c), and half of the force attracting the electron to point E as F_{5}. Thus we get: F_{5} = F_{1}^{1} + F_{1}^{1} = 2F_{1}^{1}. (6.1.2) On the other hand, since the forces acting upon each electron are equal to the forces in hydrogenlike atoms with a radius of a and a charge of Ze, we get: F_{5} = Ze^{2}/a^{2} (6.1.3) If we substitute the value of F_{5} in equation 6.1.2, we get: Ze^{2}/a^{2} = 2F_{1}^{1} (6.1.4) i.e., in order to define Z and the energy of a hydrogen molecule, we must solve this equation. F_{1}^{1} as already indicated, is the projection of F_{1} onto axis CD. According to trigonometry, (see figure 6.1.1)
F_{1}^{1} = F_{1} x cos∠ ECB (6.1.5) Then cos ÐECB = EC/CB or, according to the accepted designations, EC = a, and BC = c; therefore cos ÐECB = a/c. According to the Pythagorean Theorem and the accepted designations, EB = b: Cos ∠ ECB = a/(a^{2}+b^{2})^{0..5} = a(a^{2}+b^{2})^{0.5} Substituting the value of cos Ð ECB in equation 6.1.5 we get: F_{1}^{1} = F_{1} x a(a^{2}+b^{2})^{0.5} According to the accepted designations, BC = c F_{1} = e^{2}/c^{2} (6.1.6 ) since the positive charge at point B is equal to 1. Substituting the value of F_{1} from this equation, we get: F_{1}^{1} = e^{2}/c^{2} x a(a^{2}+b^{2})^{0.5} ; that is, c^{2} = a^{2}+b^{2} Thus we get the Pythagorean Theorem: F_{1}^{1} = e^{2}a(a^{2}+b^{2})^{0.5}/a^{2}+b^{2} = e^{2}a(a^{2}+b^{2})^{1.5} Substituting the value of F_{1}^{1} from this equation to equation 6.1.4, we get: Ze^{2}/a^{2} = 2e^{2}a(a^{2}+b^{2})^{1.5} By multiplying both parts of this equation by a^{2} and dividing them by e^{2}, we get: Z = 2a^{3}(a^{2}+b^{2})^{1.5} Now let's divide and multiply the value in brackets by a: Z = 2a^{3} · a^{3}(1+b^{2}/a^{2})^{1.5} = 2[1+(b/a)^{2}]^{1.5} (6.1.7) That is, in order to define the value of b, we must define the value of b/a for which we will make use of the following equation. According to figure 6.1, 2F_{1}^{11} = F_{2} or, according to trigonometry and figure 6.1, F_{1}^{11} = F_{1} cos ∠ CBE since cos ∠ CBE = b/(a^{2}+b^{2})^{1.5} according to trigonometry and the Pythagorean Theorem. Then, making use of the Coulomb Law we get: F_{1}^{11} = [e^{2}/(a^{2}+b^{2})] · b/(a^{2}+b^{2})^{0.5} = e^{2}b/(a^{2}+b^{2})^{1.5} (6.1.8) On the other hand, according to fig. 6.1 and the Coulomb Law: 2F_{1}^{11}= e^{2}/4b^{2}, that is, 2e^{2}b/(a^{2}+b^{2})^{1.5} = e^{2}/4b^{2} Now multiply both parts of the equation by the following: 4(a^{2}+b^{2}) · (a^{2}+b^{2})^{0.5 }/ b · e^{2} and we get: e^{2} · 4(a^{2}+b^{2})(a^{2}+b^{2})^{0.5} / 4 · b^{2} · b · e^{2} = = 2e^{2} · b · 4(a^{2}+b^{2})(a^{2}+b^{2}) / (a^{2}+b^{2})^{0.5}(a^{2}+b^{2})^{0.5} · b · e^{2}. We thus have: (a^{2}+b^{2})(a^{2}+b^{2})^{0.5} / b^{3} = 8. The square of both parts of the equation gives us: (a^{2}+b^{2})^{2} · (a^{2}+b^{2}) / b^{6} = 64 or: (a^{2}+b^{2})^{3} / b^{6} = 64. By deriving the cubic root on both parts of the equation, we get: (a^{2} + b^{2})/b^{2} = 4 or a^{2}/b^{2} + 1 = 4; a^{2}/b^{2} = 3; a/b = 3^{0,5} (6.18a) Substituting the values of b/a = 1/3^{0.5} in equation 6.1.7 we get: Z = 2(1+1/3^{0.5})^{1.5} = 2(1,333)^{1.5} = 1,299 Substituting the value of Z in equation 6.1.1, we finally get the value of the energy of a hydrogen molecule (E_{H2}), which is equal to: 1,317 kJ/mol.·2·(1,2990.25)^{2} =1,317 · 2 · (1,049)^{2} = 2,898kJ/mol. Calculating the energy of a hydrogen molecule, we get the data that helps us calculate the geometrical parameters of a molecule. The resulting charge acting upon the electrons is equal to 1.049 proton units. Therefore the radius of the electron's transition is by 1.049 smaller than that of a hydrogen atom, which is equal to 0.529Å. Respectively, the radius of the orbit (circle) in where the bonding electrons are rotating in a hydrogen molecule is equal to 0.504 Å (0.529/1.049 = 0.504). Since the distance between the nuclei is equal to 2b and b is equal to a/3^{0.5}, the distance between the nuclei is equal to 0.582(0.504 · 2/3^{0,5}) = 0.582 Å. The distance between the nuclei and the electrons is defined by the Pythagorean Theorem and is equal to (0.504^{2} + 0.291^{2})^{0.5} = 0.582Å. The scaled up values of the distances a, b, and c allow us to imagine the actual size of a hydrogen molecule shown on figure 6.1.1. These values allow us to define the energy of a hydrogen molecule and independently of the calculation via the virial theorem. The potential energy of a hydrogen molecule is calculated via comparison with that of a hydrogen atom whose potential energy is equal to 2,634 kJ/mol. Hydrogen atoms (whose electrons and protons are charged with identical absolute values and have contrary signs) are attracted to each other and are located at a distance of 0.529Å. The potential energy is calculated via the equation: E = q_{1}q_{2} / R where q_{1} and q_{2} are the charges in the particles and R is the distance between them. In atoms and molecules the positive and negative charges are equal in value just as in a hydrogen atom. That is why the energy of their Coulomb interaction is counter proportional only in reference to the distance between the charges in hydrogen molecules and atoms. The summed up potential energy of a hydrogen molecule is equal to the difference between the electrons' attraction energy to the nuclei and the interelectronic, internuclear repulsion energy. The attraction energy is: E_{atr} = 2,634 · 0.529 · 4 / 0.582 = 9,577 kJ/mol. The repulsion energy is equal to: E_{rep} = 2,634 · 0.529 / 0.582 +2,634 · 0.529 / 1.008 = 3,776 kJ/mol. The difference between the attraction energy (which defines the stability of the molecule) and the repulsion energy is 9,577  3,776 = 5,801 kJ/mol. The electrons' attraction energy to the nuclei in two hydrogen atoms comprises 2,634 · 2 = 5,268 kJ/mol. That is, the gain in attraction energy (i.e., potential energy) during molecule formation comprises 5,801  5,268 = 533 kJ/mol. That is, the nuclei in a molecule are more firmly bonded with the electrons, and in order to break a molecule into atoms, more energy is required. This has been proven experimentally. In order to break a hydrogen molecule into atoms, the molecule should be heated to a temperature of more than 3,000º. The total energy of a hydrogen molecule is equal to the difference between the kinetic and potential energies of the electrons. According to the virial theorem, the kinetic energy is equal to half of the potential energy of a system where only Coulomb forces are at work. Relatively, the total energy is equal to half of the potential energy. That is, the total energy of a hydrogen molecule is equal to 5,801/2 = 2900 kJ/mol. The total energy of two hydrogen atoms, as indicated above, is equal to 2,634 kJ/mol. That is, the energy gain, when forming a hydrogen molecule out of a hydrogen atom, comprises 2,900  2,634 = 266 kJ/mol. In the course of our calculations on the basis of equalizing forces, we found that the energy of a hydrogen molecule comprised 2,898 kJ/mol and that the potential energy was equal to 5,800 kJ/mol. In accordance with the virial theorem, the total energy of a hydrogen molecule was equal to 5,800/2 = 2,900 kJ/mol. That is, the total energy, calculated by both methods, coincided, thus proving that there are only common Coulomb interactions in the molecule. To check the correctness of the solution, let's check the correctness of the initial equations. The equality of the F_{1}^{1} forces is obvious from the equality of the F_{1} forces. Let's check the correctness of equation: 2F_{1}^{11} = F_{2} . According to the Coulomb Law, F_{2} = 1·1/(2b)^{2} since the charges at points A and B are equal to 1 proton unit. On the other hand, 2F_{1}^{11} = (1 · 1 · 2) / c^{2}. Let's substitute the values of b and c (b = 0.291Å; c = 0.582Å): F_{2} = 1/0.582^{2} = 2.952. 2F_{1}^{11} = 2 cosine ∠ ABC/0.582^{2} = 2 · b/c 0.582^{2} = 2 · 0.5/0.582^{2} = 2.952. Thus the equation 2F_{1} = F_{1}^{11} is correct. As we see, the forces in the described model, acting upon the nuclei and the electrons, are balanced via counteracting forces, i.e., the system is balanced. According to the virial theorem the energy of a molecule made up of electrons and nuclei (E_{mol}) is defined by the sum: E_{mol} = E_{kin}  E_{pot} where E_{kin} and E_{pot} are the electrons' kinetic energies and the system's potential energy (i.e., the electrons' attraction energy to the nuclei and the interelectronic repulsion energy). According to the virial theorem, 2E_{kin} = E_{pot} in absolute value. Thus, the value of 2,905 kJ/mol is equal to the electrons' kinetic energy and half of the molecule's potential energy. Then 2,640 kJ/mol corresponds to the double value of the electron's kinetic energy and the electron's attraction potential energy to the nuclei in a hydrogen atom. During the formation of the molecule, the kinetic energy of the electrons increases by 265 kJ/mol, while the absolute value of the potential energy increases by 2,905 · 2  2,640 · 2 = 530 kJ/mol. That is, the electrons in the molecules move more rapidly than they do in the atoms, though in the first case, they are more readily attracted to the nuclei. The energy gain at the expense of the greater attraction of the electrons to the nuclei is twice higher than the loss of energy caused by the electrons' kinetic energy increase. The molecules are formed with energy gain, which explains their stability at room temperature (» 20°C). Since the process of molecule formation proceeds with a discharge of energy, in order to break a molecule into atoms, i.e., accomplish the reverse process; the molecules should get some energy, which is calculated via the scheme in section 6.1. In this scheme we presume that the molecule is a system in which two nuclei are bonded by two electrons rotating in a circle whose plane is perpendicular to the axis connecting the nuclei. We also presume that the defining forces in the given system are the Coulomb and centrifugal forces. The correctness of these presumptions can be proven only by comparing the calculation data with those of the experiment. In accordance with the experimental data, the FIE of a hydrogen molecule is equal to 1,494 kJ/mol. As a result of breaking an electron off a hydrogen molecule, a positive hydrogen ion is formed (H_{2}^{+}). Chemical literature does not contain any experimental data on the second ionization energy of a hydrogen molecule. This is why, in order to compare the calculated data with the experimental data, it is necessary to calculate the energy of the positive hydrogen ion along the same scheme that we used to calculate the energy of a hydrogen molecule. We will find that the energy of a positive hydrogen ion, according to the following scheme, is now equal not to the heliumlike atom, but to the hydrogenlike atom with a charge of Z which is equal to the reduced charge at point E while Z can be calculated via the equation: Z = (N^{2}/2n) [(4n/N)^{2/3}  1]^{3/2}  Sn where N is the nuclear charge in proton units; n is the number of bonding electrons; Sn is the term that considers the interelectronic repulsion. In the case of one electron (H_{2}^{+}) Sn is equal to zero. The detailed deduction of this equation will be described later. See section 6.4, equation 6.413. When calculating via this equation, we find that: Z = (1^{2}/2) [(4/1)^{2/3}  1]^{3/2} = 0.5 (4^{0.666 } 1)^{1.5} = 0.93 Respectively, energy H_{2}^{+} is defined by the equation: E_{H2}^{+} = 1,317 · 0.93^{2} = 1,150 kJ/mol Molecule H_{2}^{+} can be represented as a molecule formed of a hydrogen atom and a proton. The total electronic energy of the initial components is equal to the FIE of the hydrogen atom, i.e., 1,317 kJ/mol. That is, according to calculations, during the formation of the H_{2}^{+} ion, there is no energy gain, but there is energy loss of 167 kJ/mol. That is, molecule H_{2}^{+}, according to the calculation, is extremely unstable. [The Encyclopedia of Inorganic Chemistry (1994) mentions this fact on page 1,463.] Relatively, when breaking away one electron from a hydrogen molecule, it breaks into a hydrogen atom and a proton. The total energy is equal to 1,317 kJ/mol. Thus, the experimentally defined electronic energy of a hydrogen molecule (E_{H2}) is defined by the equation: E_{H2} = 1,317 kJ/mol + 1,494 kJ/mol = 2,811 kJ/mol where 1,317 kJ/mol is the energy of a hydrogen atom; 1,494 kJ/mol is the FIE_{H2}. The calculated energy of the hydrogen molecule was equal to 2,900 kJ/mol. The discrepancy between the experimental and calculated data comprised 3.06 %. That is, [(2,900 kJ/mol  2,811 kJ/mol) / 2,900 kJ/mol] = 0.0306. Thus the calculated energy value of the hydrogen molecule proved to be by 3.06 % greater than the experimental data offered. As previously said in this section, the energy of a hydrogen molecule is defined, according to figure 6.1.1, as the energy of a heliumlike atom (a nucleus surrounded by two electrons). According to the calculation of heliumlike atoms via equation (6.11) we get: E_{hel} = 1,317 (Z  0.25)^{2} · 2. The energies of heliumlike atoms with nuclear charges equal to 1, 2, and 3 proton units, comprised 1,485; 8,025; and 19,825 kJ/mol respectively. In comparison, the experimentally defined energy of these atoms (the sum of the ionization energies of H¯; He; and Li^{+}) comprised 1,395; 7,607; and 19,090 kJ/mol respectively. In other words, the experimentally defined energy value for atoms of H¯; He; and Li^{+} was smaller than the calculated data by 6.1 %; 5.2 %; and 3.7 % respectively. As already noted above, the experimentally defined energy value of a hydrogen molecule was by 3.06 % smaller than the calculated one on the basis of the model, which proves quite convincingly that the model is a perfectly correct one. Besides the above cited experimental defining of the hydrogen molecule's energy via the ionization energy, there are other experimental ways of defining the same as well. The most common method is the thermal method where we define the energy necessary for the molecule to break the bonds between the atoms. It has been defined experimentally that in order to break up a hydrogen molecule into atoms, the hydrogen should get energy of 437 kJ/mol. Indeed, it might seem sufficient to add 437 kJ/mol to 2,640kJ/mol (energy of two hydrogen atoms) in order to get the experimental energy value of a hydrogen molecule. However, let's not hurry to the conclusion, but, for the time being, let's dwell in detail upon the experimental methods, which define the energy necessary to break the bond of a hydrogen molecule. To begin with, let's compare two problems. Problem №1. Define the energy necessary to break the bond between a magnet and a piece of iron. This problem can be readily solved with the help of an electric device that will tear the piece of iron off the magnet. Here the energy consumption can be calculated via the amount of electric energy used by this device during the process. Problem №2. In the case of a hydrogen molecule, we cannot separate the atoms in a molecule with the help of a device, and therefore we cannot directly measure the energy necessary to break the bond in this molecule. Here we must heat 100 ml of hydrogen, measure the amount of energy used in this process and the amount of broken hydrogen molecules received via reaction H_{2 }® 2H. The hydrogen molecules must be given enough energy (heat) necessary to break the bond. In the process of heating the hydrogen, the molecules' kinetic energies increase; the molecules start moving quicker straightforward and rotationally and the hydrogen atoms' nuclei begin to vibrate more readily causing collisions. In the course of these collisions between the molecules, caused by the exchange of energy between them, such molecules appear in which the average distances between the nuclei are greater than those in the initial (nonexcited) hydrogen molecules. The initial charges in these molecules become smaller, Z_{H2}, and the potential energies of these molecules decrease while their electronic energies become equal to those of two hydrogen atoms, causing the molecules to burst. Thus, when we measure the energy necessary for breaking the bond in a hydrogen molecule, we cannot measure such energy in the molecule as we can in the case of a magnet and the piece of iron. In the second case we have to impart energy to the system with more than 10^{5} hydrogen molecules. In this system, as a result of the exchange of energies between the molecules, there are molecules with sufficient energy accumulated so as to cause the burst of the molecules into atoms. That is, in order to convey sufficient energy to molecules to break them into atoms, we must excite (heat) the energy of other molecules, which do not break into atoms during the experiment. Unlike the case with the magnet and iron, we have to spend more energy than is normally necessary for the breaking of a molecule into atoms. This is why the experimentally defined value of 437 kJ/mol (expenditure of energy necessary to break a hydrogen molecule into atoms) exceeds the value of the difference in energy between that of the molecule and of the divided hydrogen atoms. How can we define the expenditure of energy on the heating of the nonbreaking molecules (i.e., on the nonproductive, useless expenditure of energy)? This can be done if we know how much energy (heat) is discharged in the course of the chemical reaction. How can we measure the heat of the reaction? For this we impart to the system energy whose value is equal to, say, the expenditure of electricity, and then we define the energies of the final products and the nonreacting initial products. The energy of the nonreacting products is equal to the products' initial amount multiplied by the heat capacity and by the temperature at which the experiment was conducted. The energy of the final products is equal to their heat capacity multiplied by the temperature at which the experiment was conducted. The energy imparted to the system, which we are to measure, is spent on heating the initial and final products up to the reaction temperature, and also on the energy increase of the hydrogen molecule's electrons up to the energy of the atoms' electrons. The value of this energy was determined in the course of a theoretical calculation. To compare the calculation of the molecule's energy with that of the experiment, we must add not 437 kJ/mol to the electronic energy, but the difference between these values plus the difference in the energies spent on the heating of the initial and final products up to the reaction temperature. To define this difference, we should find the difference between the heat capacities of the initial and final products. The heat capacity of a substance is the relation of the amount of heat (energy) received by the substance, or discharged at cooling, to the corresponding temperature change in the substance. If we take heat capacity in relation to one gram of substance, this is regarded as a unit of heat capacity. Relative to z atom, or g mol of substance, it is called either atomic or molar heat capacity. See the supplement for more details on heat capacity. Thus, the imparted and measured amount of energy (437 kJ/mol) has been spent on the increase of electronic energy (DE_{el}) and on the difference of the energies in the initial and final products: DE = E_{ini}  E_{fin} ; E_{ini} = C_{H2} · T_{p} ; E_{fin} = C_{H} · T_{p} ; where C_{H2} and C_{H} are the heat capacities for hydrogen (H_{2}) and hydrogen atoms. The nuclear heat capacities of hydrogen and two hydrogen atoms are practically equal. The expenditure of energy during the heating of molecular hydrogen up to a temperature that causes the breaking of the atoms (2,500°5,000°C) is well stipulated by the increase in the electrons' energies. The mechanism for the increase of the electrons' energies in a molecule have been described (See Gankins' How Chemical Bonds Form and Chemical Reactions Proceed  1998, p.441) This mechanism does not work with divided atoms, for in such atoms the energy of the electrons does not depend on the distance between the atoms. Experiments proved this judgment. According to experiments, the electronic heat capacity of atoms is equal to zero. To calculate the experimental value, which should be added to the electronic energy of two hydrogen atoms, we must subtract the value of C_{eH2} (T_{2 } T_{1}) from 437 kJ/mol. [C_{eH2} is the electronic heat capacity, T_{2} is the reaction temperature, T_{1} is the temperature at which the electronic freedom stages defrost or becomes noticeable.] According calculation (See Gankins' How Chemical Bonds Form and Chemical Reactions Proceed  1998, p.441) from the 437 kJ/mol, received from measuring the expenditure of energy for thermal bond breaking, we should subtract a value equal to about 200 kJ/mol That is, to get the experimental energy value of a hydrogen molecule (H_{2}), we should add not 437 kJ/mol, but a value smaller by 200 kJ/mol, i.e., 237 kJ/mol to the experimental value of the energies of two hydrogen atoms (2H). The experimentally defined energy value of a hydrogen molecule comprised 2,877 kJ/mol, while the calculated energy value of the same comprised 2,905 kJ/mol. Thus, the experimental value differs from the calculation by less than one percent. Such a coincidence allows us to say that the real hydrogen molecule is identical to the hydrogen molecule model from which the calculated data was received. In the calculation based on the model, only electrostatic interactions were considered. As the calculation results differed from the experimental data by less than 3%, we can say that the forces defining the formation of molecules out of atoms are electrostatic. Let's take the distance between the nucleus and the electron if we bond an electron to a hydrogen atom. The attraction of each electron to the nucleus decreases by 25%; the effective charge of the atom's nucleus decrease by 25%; the distance between the nucleus and the electron in a hydrogen atom, bonding the electron, will be 0.65Å. We have calculated the distance (C) between each of the bonding electrons and the nuclei in a hydrogen molecule, which is equal to 0.582Å. As previously indicated, a hydrogen atom with a radius of 0.529 Å, can bond (connect) an additional electron entering the unfilled outermost shell of the hydrogen atom which contains one electron. The distance between the electrons and the nuclei in a molecule is 0.582Å greater than the distance between the electron and the nuclei in an atom (0.529Å), but smaller than that between the electron and the nucleus when an electron is bonded to an H atom (0.65 Å). This proves that both electrons enter the outermost shell of the atoms being bonded. Thus, via simple arithmetic, on the basis of the virial theorem and the data on the electron's and the proton's charges, we have managed to calculate the electronic energy of a hydrogen molecule and the energy which is gained during chemical bond formation between two hydrogen atoms. We have also calculated the distances between the nuclei and the electrons in a hydrogen molecule and the distances between the nuclei. According to the calculated model, the angle between the attraction forces bonding the electrons to the nuclei of the hydrogen is equal to 60° C. Respectively, the projection of the attraction force bonding an electron of one of the nuclei to the attraction force of the same electron to another nucleus, force F_{CH }(Fig 6.1) is equal to half of the electronic force attraction to the nuclei. Relatively, the attraction force of the electrons to the atoms increases by 1.5 times, as compared to the atom to the outermost shell of another, that is, by the attraction of the nuclei of one atom to the electrons of another. Thus the electronic energy of a heliumlike atom with a nuclear charge of 1 and 1.5 proton units comprises, according to the calculation: 15.3 eV and 42.5 eV respectively. That is, when bonding only an electron to a hydrogen atom, the energy gain, according to the calculation, comprises 1.7 eV i.e., (15.3  13.6 = 1.7 eV). When an atom is bonded (electron + nucleus) during the formation of a hydrogen molecule, the energy gain is equal to 28.9 eV (42.5  13.6 = 28.9 eV) for each of the atoms being bonded, which fully compensates the internuclear repulsion energy. ;That is, the total energy gain during chemical bond formation, is conditioned, paradoxical as it may seem, to a greater extent by the mutual approach of the atoms' nuclei, than by the transition of the electrons of one atom to the outermost shell of another, that is, by the attraction of the nuclei of one atom to the electrons of another. Experimentally, the energy of an atom with two electrons and a nuclear charge of 1.5 proton units can be evaluated as an average value between the energy of a hydride ion (H^{}) and a helium atom (He), that is, by the value of (54.4eV) i.e., close to the previous calculation, and therefore — greater than the repulsion energy.
Chapter 6. Molecule structure Conclusions >> 