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Home  / GENERAL CHEMISTRY Textbook / Chapter 6. Molecule structure / ***Bond Lengths

***Bond Lengths

Bond lengths have been defined experimentally for a great number of molecules. In most cases - the greater is the bonding energy, the smaller is the length of the bond.

The bonding energies for molecules in table 6.4-1 NaF, NaCl, and NaBr comprise 476 kJ/mol, 412 kJ/mol, and 368 kJ/mol respectively. The bond lengths in these compounds will be respectively: 1.9 Å, 2.3 Å, and 2.5 Å. That is, the molecules in the table have an expected (non-paradoxical) regularity: the stronger the bond, the shorter it is; the closer the atoms are attracted to each other, the shorter is the distance between them.

Analogously, the bonding energies Li-Li, Na-Na, K-K comprise 110 kJ/mol, 72 kJ/mol, and 49 kJ/mol, while the bond lengths of Li-Li, Na-Na, K-K are equal to 2.7 Å, 3.08 Å, and 3.9 Å respectively. And the energy of a single C-C bond, as indicated above, comprises 347 kJ/mol while its length is equal to 1.54 Å, i.e., smaller than the bond length of Li -Li whose energy amounts to 110 kJ/mol.

However, for some of the molecules in table 6.4-1 this unquestionable (non-paradoxical) dependence changes to the opposite (paradoxical) dependence, which causes a WHY question to arise.

According to experimental data, bonding energies of C-C, Cl-Cl, and F-F, comprise 347 kJ/mol, 239 kJ/mol, and 158 kJ/mol, while the bond lengths in row C-C, Cl-Cl, F-F comprise 1.54 Å, 2 Å, and 1.4 Å, respectively, that is, first it increases, then it decreases.

To prove that this anomaly is not incidental, on figure 6.5-1 we see the experimentally defined dependencies of single-bond lengths on the FIEs for elements of periods 2 and 3.

The curves on figure 6.5-1 show that the lengths of single bonds in molecules of the Li2-F2 and Na2 - Cl2 type decreases with the increase of the FIEs of the atoms to be bonded. In accorda0nce with the previously cited dependence, the energy of a single bond between identical atoms in elements of the 2nd and 3rd periods depend on the FIEs of the atoms to be bonded and are expressed by a parabola. 

6_5_1
Figure 6.5-1

During the increase of the FIEs up to the middle of the periods, the bonding energy increases with the increase of the FIEs; after which, with the further increase of the FIEs of the atoms to be bonded, the energy decreases. See figures 6.4-5 and 6.4-6

Because of the decrease in the bond's length when the bonding energy increases, it was expected that the dependence of the bond lengths on the FIEs should also be expressed by a parabola (a curve with the maximum). However, according to experimental data, the curve of the bond length's dependence on the FIE was expressed by a hyperbola.  

Indeed, the bond lengths decreased with the increase of the FIEs of the atoms to be bonded. That is, the paradoxical dependence, noted above, was not at incidental. This regularity can be considered as paradoxical on the basis of simple, logic, qualitative, reasoning like: the stronger the bond, the closer are the atoms to each other.

We have seen that such simple qualitative reasoning was contradictory in respect to dependencies that were necessary in the experiment. We also thought that experimentally defined dependencies qualitatively contradict the main precepts of the theory of chemical bond formation. However, the semi quantitative evaluation of the expected dependencies, on the basis of solving the system of algebraic equations, has shown that the qualitative but seemingly paradoxical dependencies (simple, logic reasoning) in reality is well desired, not at all paradoxical, but logic.

That's why, just as in the previous cases, by solving the system of equations 6.4-5 to 6.4-7 we can calculate the expected dependencies of the bond lengths on the FIEs of the atoms to be bonded for the given case. The results of the calculation are given in figure 6.5-2.

 

6_5_2
 Figure 6.5-2

These calculations show that the dependence of the bond length on the FIE, just as in the experiment, is expressed by a hyperbole.

In accordance with the calculations made via the same equations, a parabola expressed the dependence of the bonding energy on the FIEs of the atoms to be bonded. Therefore, the observed dependencies of the bonding energy and bond lengths on the FIEs, (which at first sight seemed paradoxical in regard to each other and to the theory) proved to be not at all beyond one's expectations as a result of their semi-quantitative evaluation on the basis of the solution of the system of equations (a more fundamental approach at second glance).

That is, if at first sight they seemed to contradict the theory, at second glance they become convincing proofs of the correctness of the theory of chemical bonding.

Having studied the materials given in sections 5 and 6, we have learned about the structure of atoms and that of simple molecules. We have understood why atoms are bonded into molecules, and how to define the energy necessary to impart to a dual-atomic molecule in order to break it into atoms.

The data given in figure 6.5-2 coincides, relative to the curves, since three curves are hyperbolas, though they differ greatly relative to absolute values. Thus, for example, in the case of a chlorine molecule  (Cl2) (IECl = 13.0), the experimental value is 3.4 times greater than what was calculated.

This discrepancy has a simple explanation. As already indicated, in the course of the calculation it was supposed that the atoms X in molecule (X-X) are hydrogen-like; that is, they have only one electron rotating in the one and only layer. The ionization energy of this electron, according to the supposition suggested for the calculation, was equal to the ionization energy of atom X in molecule X-X.

In reality, the atoms of all the periods (besides the 1st ¾ hydrogen and helium) contain inner electronic layers. According to the experiment, the distance between the nucleus and the electron in a hydrogen-like chlorine atom is 3.4 times greater than in a hydrogen atom.

According to the model, all the distances defined in the course of the calculation, are proportional to the distance between the nucleus and the electron, i.e., the distances between the nuclei and the bonding electrons (c), between the nuclei (2b), and between the electrons in molecules Cl2 - by 3.4 times greater than in molecule H2.

When calculating dual-atomic molecules with identical atoms or with atoms having an identical number of electronic layers (atoms of an identical period) we should expect close results relative to the calculated and experimental dependencies of bonding energies on IE (ionization energy) exactly what we see in figure 6.5-2

On the other hand, it may not be clear as to how molecules of the HF, HCl type, (where hydrogen atoms bond to atoms of other periods) are formed. In these atoms, as indicated above, the distance between the electron and the nucleus (at identical ionization energies) was more than three times greater than the distance between the nucleus and the electron in a hydrogen atom. This means that, according to the model for bonding molecules of the HF, HBr, HCl type cannot be formed. It is known that these molecules are stable; therefore the question about their formation, in the framework of the given model, is regarded as paradoxical.

Indeed, the radius of circle (a) where the bonding electrons rotate in molecule HCl, should, on the one hand, be equal to the radius of the circle where the bonding electrons rotate in molecule H2, while on the other hand, the radius of molecule Cl2 should be three times greater.

In the first approach, in answer to this paradoxical question, it was expected that the circle where the bonding electrons rotated, and relatively, all the geometric values of the molecule (a, 2b, and c) in this case should be averaged, i.e., greater than in H and smaller than in Cl.    

Thus, in molecule HCl, the bonding electrons are farther away from the hydrogen atom than is the H2 molecule, and closer to the nucleus in the chlorine atom than in a Cl2 molecule. We suppose that the change of distances causes the anomalous dependencies of the bonding energy on the ionization energy.

A comparison of the calculated and experimental dependencies for compounds of hydrogen with elements of the 2nd and 3rd periods (fig. 6.4-3) shows that, unlike all the former cited cases, here the calculated and experimental dependencies do not coincide. That is, as compared with the previously cited cases (where they coincided) these dependencies are actually anomalous.

In the first approach, the anomalous increase of distance between the hydrogen atom's nucleus and the electron (when bonding the hydrogen atom to elements of the 2nd and 3rd periods) one can imagine the bonding of the hydrogen atom to a charge of 1 proton unit, but well distanced from the nucleus of the electron, i.e., to a hydrogen atom that has a smaller ionization energy (compared with that of a normal atom).

On the other hand, the bonding electrons, situated closer to the chlorine atom than to the Cl2 molecule, cause the effective ionization energy of the chlorine atom to increase. The decrease of the hydrogen atom's dissociation energy (increase of the distance between the hydrogen and the bonding electrons) is connected with energy expenditure, while the approach of the bonding electrons to the chlorine atom is connected with energy gain that compensates the energy loss.

The difference between the effective FIEs of atoms H and Cl and the dipole moment of molecule HCl increases when the distance between the bonding electrons and the hydrogen atom increases, and the distance between the chlorine atom and the bonding electrons decreases.

6_5_3 
Figure 6.5-3

To double-check the correctness of this supposition, we calculated the energy dependence in bonds of H-X molecules (where X is an atom of the 2nd - 4th periods) on the potential energy of the first electron of atom X for two variants of ionization energy relative to the hydrogen atom: 8, and 6 eV.

The calculated dependencies are seen in figure 6.5-4 .

6_5_4 
Figure 6.5-4

The coincidence of the calculated and experimental curves and the closeness of the calculated and experimental data (when the hydrogen atom's ionization energy, taken for the calculation, was equal to from 8 eV to 10 eV) that coincided qualitatively and semi-quantitatively with the experimental dependencies, allows to confirm the correctness of the explanation relative to the anomalous dependence of bonding energy in dual-atomic molecules like H-X on the ionization energy of atom X.

The anomalous properties of molecule H-X have been known in chemistry for quite some time when studying the dependence of ionic covalent bonds on the first ionization energy of atoms A and B in molecules A-B.

The main part of dual-atomic molecules was well described quantitatively by a dependence according to which — the greater the difference in the ionization energy of atoms A and B in molecules A-B, the greater is the degree of ionization in molecules A-B.  

The degree of ionization was defined as the correlation of the experimental value of the molecule's dipole momentum to the calculated value of the dipole momentum. When defining the calculated dipole momentum, a supposition was made about the transition of one electron from atom A to atom B.

There is almost an identical ionization degree in H-X molecules that was accomplished at great differences in ionization energies of the bonding hydrogen atoms (see table 6.5-1)

Thus, for example, the ionization degrees for FCl and HBr have close values - 11 % and 12 % respectively - (table 6.5-1). At the same time, the difference between the ionization energies of the atoms in molecule FCl comprises 4.4 eV while the difference between the ionization energies of the atoms in molecule HBr comprises 1.8 eV.

The ionization degree in molecule HBr (12 %) corresponded to the ionization energy of the hydrogen atom at about 6 eV smaller, i.e., a value of about 7.5 eV.

Likewise, in molecule HI, the degree of ionization amounted to 5 when the difference in the ionization energies was about 3 eV. An almost identical degree of ionization, according to table 6.5-1, is accomplished in molecules BrCl (5) and ICl (6) where the difference in ionization energies comprises 1.2 eV and 2.5 eV relatively.

That is, the ionization energy of a hydrogen atom in HI corresponded to the ionization energy of a hydrogen atom (8 eV - 9.4 eV), i.e., values close to those that were received during the interpretation of the anomalous dependence of bonding energy on the ionization energy of the atoms to be bonded in molecule H-X.

 

TABLE  6.5 -1   

Dependence  of  Ionization  Degrees  on  the  Difference  of Ionization  Energies  (DIE)  of  atoms  A  and  B  in  molecules  A-B 

Molecules

DIE of

Atoms (eV)

Ionization

Degree, %

HI 3.1 5
BrCl 1.1 5
ICl 2.5 6
FCl 4.4 11
HBr 1.8 12
FBr 6.4 15
HCl 0 17
HF 4 41
KI 6 63
LiH 8 77
KF 13 83

The dependence of the ionization degree on the difference of ionization energies of atoms A and B in molecules A-B was defined for two cases: 1) when A = H and 2) when A ≠ H. B is never equal to H.

The same ionization degree, observed in molecules H-B, as compared to molecules A-B, is received with a greater difference in the ionization energies. Studies of properties of compounds containing hydrogen, have shown, that the ionic character of bonds with hydrogen, is about the same as in the case if hydrogen had ionization energy equal to 10 eV, that is, a value close to what we got when explaining the causes of the anomalous dependence of the bonding energy in molecules H-B with dependence on the ionization energy of the atoms to be bonded.

One and the same supposition based on substantiated models (via calculation and experiment) and the quantitatively close results, allow to confirm the correctness of the explanations relative to the anomalous values of the degree of ionic hydrogen compounds H-B by the great discrepancy of the radiuses of the initial atoms H and B.

6_5_5
  Figure 6.5-5 

It is of interest to recall the fact that the anomalous degree of the value of hydrogen has been touched upon in many textbooks from the 1960s to the 1980s. However, the material was given in these textbooks without an explanation about the cause of such anomalies. 

Chapter 6. Molecule structure >>   
Conclusions >>   
**Molecules formed of multi-electron atomS >>  
**Ionization energy of multi-electron atoms >>       
**Chemical Energy. FIEs of element and bonding energy >> 
**Chemical Bonding Energy >>
***Bond Lengths
Valence >>
Conclusions >>     
**Donor - acceptor Bond (DAB) >>   
Van der Waals Bond (VWB) >>    
Dynamic Bonds >>
Conclusions >> 
Chapter 6 Textbook Questions