Introduction

Electronegativity is an important concept in modern chemistry. Pauling defined electronegativity as the power of an atom in a molecule to attract electrons to itself [1]. Electronegativity is often described as an important principle for understanding the nature of the elements and the type of compounds they form with each other [2]. Electronegativities are used in physical, inorganic, and organic chemistry, for example in the determination of bond type [3]. In the open literature, methods of determining electronegativity are often provided [4] and that electronegativity scales can help us gain insight into variation of elemental electronegativity through the periodic table [5]. Texts usually concentrate on the qualitative concept [6], but there is no discussion on quantitative applications of electronegativity or electronegativity scales. This work shows that elemental electronegativities can be used in determining dipole moments, bond lengths, and bond dissociation energies.

Results and discussion

Data sources

Individual measurements of bond energies and bond lengths are published in many journals such as journals of the Royal Society of Chemistry or journals of the American Chemical Society. It is outside the scope of this work to systemically go through the many journals. However, the most recent compilations of bond energies and dipole moments can be found in the CRC Handbook of Chemistry and Physics [7], which contains extensive tables of bond energies, bond lengths, and dipole moments. Comprehensive experimental data on bond lengths (inter-nuclear separation) are also available in Crystal Data Determinative Tables third edn, vols 1 and 2 [8] and Crystal Data Determinative Tables third edn, vols 3 and 4 [9].

Most of the experimental values of bond energies used in this work taken from the CRC Handbook of Chemistry and Physics. Other sources are also consulted including Essentials of Inorganic Chemistry 1 [10]. Where bond energies of Main Groups 13, 14, 15, and 16 elements provided by the CRC Handbook are energies for multiple bonds and single-bond energies are quoted from the publication Periodicity and the P Block Elements [11] for part of the calculations after analysis show that they are reliable. Most of the values published in these publications agree well with each other and only occasionally display slight differences from each other.

Calculations of dipole moments

Molecules with a permanent dipole are polar molecules. Usually, permanent dipoles occur when molecules have two atoms bonded together which have different electronegativities. Molecules act as electric dipoles if the charge distribution in them corresponds to a separation of regions of partial positive and negative charges. The dipole moment µ is equal to qr, where q is the charge and r is the distance separating the charges + q and –q [10]. Simplistically, when two different atoms share a bond, there is a build-up of charge occurring in the overlap region in between the two atoms. When the two atoms have different sizes, the centre of the charge and overlap region lies closer to one atom than the other and contribute to the dipole moment of the molecule.

To study the influence of electronegativity on dipole moments, starting with a list of alkali halides and other halides with known dipole moments [7] were numerically analyzed. It was found that for a particular element X which forms part a halide XHa, the larger the size of the halogen Ha, the greater the dipole moment and the bigger contribution from the halogen to the dipole moment. Elements in period 2 on the periodic table behave differently from other elements. Some are most electro-negative and they are smaller than other elements, because there are only two electron shells surrounding the nuclei. Hence, they may not always form part of a series with other members of a particular group. After a few trials, it was found that two equations can calculate the value of the dipole moment of diatomic molecules fairly accurately.

The first equation is

$$ D_{\mu } = k_{\mu } \varepsilon . $$
(1)

Since the numerical analysis showed that for alkali halides, the dipole moments display the trend iodides > bromides > chlorides > fluorides and the dipole moments range from 6 to 11with the coefficients kµ being likely to be less than 9. Simple trial and error showed that they are (kµ) equal to 8.2, 6, 5, and 2.8, respectively for alkali iodides, bromides, chlorides, and fluorides. The coefficients kµ for other diatomic molecules are likely to be lower than 5. kµ was found to be 4 for alkaline earth oxides and 1.24 for hydrides.

When the difference in sizes of the two atoms contribute to the dipole moment of the molecule, the dipole moment can be calculated by Eq. (2)

$$ D_{\mu } = k_{\mu } \varepsilon + k_{{\text{r}}} r_{{\text{a}}} . $$
(2)

The coefficient kµ for Eq. (2) are 1, 2, and 0.1, respectively, for oxides, sulfides, and fluorides. For both equations, Dµ is the dipole moment and shown in Debye units. In SI unit, one debye unit is equal to 3.33564 × 10–30 Cm. kµ is a constant appropriate for each series similar compounds and ε is the difference in electronegativity value between the two atoms in the molecule and is dimensionless. In the second equation, when the difference in size of the two atoms contributes to the dipole moment of the molecule, kr is a second constant to be multiplied the difference ra in the covalent radii of the two atoms. The CRC Handbook provides dipole moments in Debye units and the calculated dipole moments displayed in Tables 1 and 2 are shown in the same unit. The electronegativity values [12] (Tables 3, 4, 5, 6), which are used in the calculations in this work, were derived from corrected ionization energies [13, 14] that were adjusted for pairing, exchange, and orbital energies. A complete list of electronegativity values are shown in Table 7. These electronegativity values had been used to calculate covalent bond lengths of compounds with single or double bonds. The calculated bond lengths when compared to observed values showed very good agreement in every case [12]. Second, these values are used rather the Pauling set, because the Pauling values are provided to only one decimal place.

Table 1 Examples of comparison of calculated and measured dipole moments
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Table 2 Further examples of calculated dipole moments
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Table 3 Examples of homo-nuclear bond dissociation energies
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Table 4 Examples of hetero-nuclear bond dissociation energies
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Table 5 Examples of observed and calculated bond lengths in the gas phase
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Table 6 Examples of inter-metallic bonding
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Table 7 Complete list of electronegativity values
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Examples of comparison of calculated and measured dipole moments are shown in Table 1. As shown in the table, there is fairly good agreement between measured calculated values and it is demonstrated that a fluoride (LiF) is not part of the series of fluorides in Table 1 and required a different constant kµ. In general, elements in the second period of the Periodic Table behave differently in many instances including bond energies as shown in the section of bond energies.

There is fairly good agreement in all cases and as discussed above oxygen and fluorine may not always form part of the usual series.

The dipole moment is determined only by the difference in electronegativity when the molecule involves only groups 1, 2, 16, or 17, because the elements in these groups are most electro-positive or electro-negative and usually with the dipole moments being high. Otherwise, the difference in size of the two atoms can influence the value of the dipole moment. When the difference in size of the two atoms contribute to the dipole moment of the molecule, the dipole moment can be calculated by Eq. (2). Table 2 shows some examples of these cases. The covalent radii employed in the calculations were previously derived from average bond lengths [12].

Calculations of bond dissociation energies

In general, the length of a multiple bond is shorter than a single bond and bond energy is inversely proportional to the inter-nuclear distance (i.e., the bond length), and that that bond length is also a function of electronegativity [15]. An equation was proposed by Ives [16] for calculating bond dissociation energies of hetero-nuclear molecules. This equation was improved and incorporated a term for bond order [17] (in units of kJ mol.−1) to calculate bond energies. The dissociation energies of hetero-nuclear bonds is calculated by Eq. (3)

$$ E_{{}} = \, [B_{{\text{N}}} \times \, \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \, \times \, \left( {E_{{\text{A}}} + E_{{\text{B}}} } \right) \, \times \, \left( {{1 } + \, \left( {r_{{\text{A}}} {-}r_{{\text{B}}} } \right)^{{2}} /D_{{{\text{AB}}}}^{{2}} } \right) \, \times \, ({1 } + \, (\varepsilon_{{\text{A}}} - \varepsilon_{{\text{B}}} )^{{2}} /{4})]. $$
(3)

The bond dissociation energy of homo-nuclear bonds is calculated by the simpler Eq. (4)

$$ E = \, [B_{{\text{N}}} \times \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 4$} \times R_{{\text{y}}} \times 0.{529}/(r_{{\text{A}}} \times {2}) \times ({1 } + \varepsilon_{{\text{A}}}^{{2}} /{4})]. $$
(4)

BN is the bond order, Ry is the Rydberg constant and is equal to 1312 kJ mol−1 and 0.529 Å is the classical Bohr radius (where 1 Å is 10–10 m) and rA is the covalent radius of the atom, and εA is the electronegativity of the element.

Here, EA and EB are dissociation energies of the diatomic molecules of elements A and B and rA and rB are the covalent radii of A and B, respectively. εA and εB are the electronegativities of atoms A and B and DAB is the bond length. Table 3 shows examples of bond dissociation energies of homo-nuclear molecules and Table 4 shows examples of bond dissociation energies of hetero-nuclear molecules. Please note that some of the bond energy data listed in Tables 3 and 4 have previously published in Tables 1, 2, 3, and 4 of the work Bond order and bond energies [17]. Please also note that some of the experimental bond energies reported in the CRC Handbook show values to varying number of decimal places. In the tables below, they are rounded to 2 decimal places.

As illustrated in Tables 3 and 4, there is good agreement between experimental and calculated values for both single and multiple bonds in most cases.

Calculation of bond lengths (inter-nuclear distances)

A simple equation for calculating inter-nuclear distances of covalent bonds was developed from an equation which calculated intern-nuclear distances of groups 1 and 2 ionic compounds [18] with a high degree of accuracy. This equation (Eq. (5)) [12] is expressed as

$$ L\left[ {{\text{AB}}} \right] \, = \, \left\{ {\left[ {\text{A}} \right]^{1} + \, \left[ {\text{B}} \right]^{1} } \right\}^{1} {-} \, 0.{1}\left[ {{\text{abs}}\left( {x_{{\text{A}}} {-}x_{{\text{B}}} } \right)^{0.5} } \right], $$
(5)

where L[AB] = the inter-nuclear separation between atoms A and B. [A] = the covalent radius of A, [B] = the covalent radius of B and xA = the electronegativity values of the atom A, and xB = the electronegativity values of the atom B.

Experimental bond lengths of gaseous molecules, which are not published in the CRC Handbook, are available by two reference texts, one by Herzberg [19] and a second by Herzberg and Huber and Herzberg [20]. Experimental bond lengths of polyatomic molecules are available in Journal of Physical and Chemical Reference Data [21].

Intermetallic compounds can be divided into two types, one group can be considered as simple interstitial compounds where the smaller metal atoms simply occupy the holes in between the larger metal atoms and the inter-nuclear distance between the smaller and larger atoms is the sum of the two metallic radii or bigger. In the second group the inter-nuclear distances between the metal atoms are less than the sum of the two particular metallic radii of the two atoms and the inter-nuclear distances can be calculated by the above equation. The electronegativity differences between the metals are usually small and we assume that the metals spheres overlap slightly as in covalent bonds.

Examples of bond lengths of compounds in the gaseous state in Angstroms (1 Å = 10–10 m) (inter-nuclear separations) calculated by the above equation are shown in Table 5. A comparison between calculated and observed values shows that there is good agreement in all cases.

Table 6 shows examples of bonding in inter-metallic compounds. The bond lengths were calculated and published in Table 10 of Ref. [3]. The metallic radii used in the calculations were previously derived [3].

Discussion

As shown in Table 2, dipole moments of some molecules are influenced by the difference in size between the atoms forming the bond. The atomic volume is a true measure of the size of an element [22]. The atomic volume of many metals is precisely known. In general, with the exception of manganese, metals have one or two simple allotropic forms. However, some non-metals such as sulfur or selenium have complex forms and elements such as fluorine and chlorine are gases and the solid forms show fairly complex allotropic structures and difficult to determine their atomic volume. Hence, this work substituted the true size with a set of derived covalent radii. When it is possible to determine their atomic volumes it is most probable that the calculated dipole moments will show better agreement with the observed values. Equations (1) and (2) discussed in the section Dipole Moments in this work show that the dipole moments can be determined by both the electronegativity and size of an atom. For example, both carbon and selenium have the same electronegativity value of 2.0, i.e., no difference in electronegativity. However, carbon monoselenide (CSe) has a dipole moment of 1.99. The size of the two atoms are different and hence generated a dipole moment and using Eq. (2) with kr equals 5 the calculated value of the dipole moment is 2.06.

The expressions described above for calculation of bond energies are applicable only to mainly covalent bonds. Hence, dissociation energies of many species cannot be accurately calculated by these two expressions. Elements in the second row the Periodic Table may behave differently from the other elements in the other row of the Table. The experimental bond energy for fluorine is 158.22 kJ mol−1 and the calculated value is 614.64 kJ mol−1, much less than the calculated energy. It appears a bond order of 0.25 is needed for fluorine provide a calculated value to agree with the experimental. The most probable reason is this small bond energy because of the small size of the fluorine atom. There is much decreased overlap of bonding orbitals with appreciable inter-nuclear repulsion and the relatively large electron–electron repulsion of the lone pair pairs which are much closer together in the F2 molecule. BrF and ClF are listed in Tables 4 and 5 and are examples showing the same set of electronegativities can be used to calculate accurately different properties and that the equations used are robust.

The calculated bond lengths in general agree very well with the measured bond lengths, because the derived radii are reliable. This work illustrates the importance and usefulness of utilizing a specific scale of electronegativity values. They can be used to calculate bond lengths, bond energies, and dipole moments accurately. Electronegativity is used to explain many reactions in organic chemistry [23]. This work may stimulate new work in other areas of chemistry.

Conclusions

Electronegativity is an important chemistry concept and values of electronegativity can be used both qualitatively and quantitatively. Distinct and novel contributions of the current work enable new equations to be formulated in this work to reliably calculate dipole moments and improve our understanding of dipole moments. Other equations used in this work shows that a consistent set of electronegativity values with appropriate equations can calculate bond lengths and bond dissociation energies. Comparison between calculated and experimental values shows very good agreement in the majority of cases. This work shows the potential of using electronegativity values in different areas of chemistry, in particular in organic chemistry, such as in investigating new addition reactions and the behavior of aromatic compounds. Unique aspects and advancements introduced by this manuscript can stimulate the study and uses of electronegativity values.