Bond Energy Equation Research Articles

Estimation of the Specific Energy Requirement for Size Reduction of Solids in Ball Mills

Design engineers generally use Bond Work Index and Bond energy equation to estimate specific energy requirements for ball milling operations. Morrell has proposed different equations for the Work Index and specific energy, which are claimed to have a wider range of application. In this paper an attempt has been made to provide a comparison of these two approaches and bring out their shortcomings. It has been shown that it is possible to develop a modified approach based on the well proven Kapur energy-size relationship and the size-mass balance mathematical modelling approach. Several important points have been illustrated using our own experimental data and a large amount of relevant data available in the literature.

The influence of electronegativity on triangular three-centre two-electron bonds: the relative stability of carbonium ions, π-complex chemistry and the −2 hβ effect

Using the HMO approximation, bond energy equations for triangular three-centre, two-electron [3 c-2 e] bonded species AB 2 + (π-complexes) and the isomeric systems (AB−B +, BB−A +, BA−B + and A ++BB) are described. The electronegativity difference (Δ χ) between the atoms or groups A and B is assumed to be related to the difference ( h) in their Coulomb integrals and the variation of relative energies with electronegativity difference is explored. The bond energy curve for π-complexes is displaced relative to those of the two-centre, two-electron [2 c-2 e] bonded species and this displacement accounts for the significant influence of electronegativity difference on reactions proceeding via [3 c-2 e] bonded intermediates or transition states. The origin of the displacement of the bond energy versus electronegativity difference curve for the π-complexes is identified as a −2 hβ BB term in the bond energy equation. In contrast to [2 c-2 e] bonds, this term makes the influence of electronegativity difference on triangular [3 c-2 e] bonds directional, that is, the bond energies of AB 2 + and BA 2 + are different in contrast to those of AB and BA. A more electronegative atom A destabilises the [3 c-2 e] bond by removing electron density from the bonding interaction BB ( β BB) whereas a less electronegative atom A will strengthen the bond by increasing the electron density between the atoms B. Reactions involving [3 c-2 e] AB 2 + bonds are classified as homo- or heteroprocesses and the influence of electronegativity difference on these discrete transformations is discussed in terms of the contribution of h, h 2 and −2 hβ functions to differences in bond energy. The analysis is extended to π-complexes with back-donation and the equivalence of the description of onium ions using either two [2 c-2 e] bonds or two [3 c-2 e] bonds is demonstrated. Extension of the analysis to 2-norbornyl cations suggests that, due to the shape of the bond energy versus electronegativity difference curve, this cation exists within a window of stability between the alternative isomers. 1,2-Disubstituted norbornyl cations are used as a model of π-complexes and the influence of substituent effects on relative stability is explored using the AM1 method. After allowance for resonance and hyperconjugation effects, the results are found to be consistent with the general conclusions of the simple HMO model.

Chemical hardness of metal ions in the gas phase: a thermochemical approach

The heterolytic dissociative version of Pauling's bond-energy equation, proposed earlier for an AB molecule, has now been generalised for molecules of type ABn in order to rank a number of multivalent cations in terms of their gas-phase chemical hardnesses. The chemical hardness of a cation is found to be related to its charge and size, and Klopman's frontier-orbital energy. The results are used to systematise the gas-phase affinity displayed between a monovalent anionic ligand (specifically, a halide) and a metal ion.

Pearson's hard–soft acid–base principle and the heterolytic dissociative version of Pauling's bond-energy equation

A correspondence between the γ parameter recently developed, which characterises the binding site of an ion, and Klopman's frontier-orbital energy has been established to show that γ does represent the hardness of an ion. It is shown that γ can be used to describe the polarity of an A–B bond where A is an acid and B a base. Also that Pearson's hard–soft acid–base principle means that acids and bases interact selectively in an exchange reaction of the type AB + CD ⇄ AC + BD so as to form bonds of similar, to the extent possible, polarity between themselves.

Pearson′s chemical hardness, heterolytic dissociative version of Pauling′s bond-energy equation and a novel approach towards understanding Pearson′s hard–soft acid–base principle

The empirical chemical hardness parameters developed recently by Pearson have been examined critically. It was found that these parameters normally characterise a group but not the corresponding ion as postulated. Equation (i) has been found quite successful when the heterolytic dissociation of D(A+B–)–½[D(A+A–)+D(B+B–)]= 2|Δγ|2(i) AB into A++ B– follows the general notion of electronegativity (i.e. the more electronegative part of AB dissociates as the anion), where D(A+B–) is the energy required to dissociate the AB bond into A++ B– and D(A+A–) and D(B+B–) are the heterolytic bond-dissociation energies of the molecules AA and BB respectively. Equation (i) has been used to define a new parameter γ, where |Δγ|=|γA+–γB–|. The γ parameters characterise the ions, and have been used to rank various monovalent anions and cations in a unified way to explain the hard–soft acid–base principle of Pearson. It was found that a reaction of type (II) proceeds from left to right in the gas phase if the AB + CD → AC + BD (ii)|ΔΔγ| value of the right-hand side is greater than that of the left. Out of some 282 gas-phase reactions considered, there are few exceptions. Emphasis has been placed on explaining the course of inorganic reactions. It is felt that the γ parameter can be identified with the chemical hardness of an ion.

Evaluation of group electronegativity by Pauling's thermochemical method

Pauling's bond energy equation, D(H-G)=[D(H-H) D(G-G)] 1/2+ K(Δ χ ) 2 (1), is applied to some 28 HG molecules, where H is hydrogen and G any group of atoms, to calculate the electronegativity (χ G ) of a group

On the existence of absolute Pauling electronegativities

From a reappreciation of the recently proposed Parr—Pearson scheme on absolute electronegativity and hardness, it follows that electron affinities can be considered as absolute Pauling electronegativities. The fundamental difference between Pauling and Mulliken electronegativities is demonstrated. A quantumchemical framework of Hückel-type is developed wherein these electron affinities act as one-electron energies (diagonal matrix elements). This novel definition of absolute Pauling electronegativities is shown to be consistent with Pauling's original bond energy equation. The validity of the deductions made implies a possible new look at chemical exchange forces in terms of intra-atomic charge-invariance effects (atom—antiatom interactions).

A new application of the Kratzer pseudopotential in theoretical chemistry

The Kratzer pseudopotential is applied to two center chemical systems. The resulting energy expressions are formally consistent with those obtained from standard semi-empirical quantum mechanical procedures. The advantages of the Kratzer expressions however are: (a) the validity of the Pauling—Sherman approximation for the resonance integral can be demonstrated; (b) one-electron energies are redefined unambiguously; (c) the equilibrium values of the coefficients a and b in the wave equation ψ A B = a ψ A + b ψ B are directly obtained as a function of these one-electron energies (diagonal matrix elements); (d) the empirical Pauling bond energy equation is reproduced in the first pproximation; (e) the electronegativity-equalization principle of Sanderson is shown to be completely equivalent with the final result of a variational treatment in semi-empirical quantum-mechanical methods. A stringent spectroscopic confirmation is provided by the theoretical reproduction of a lower order spectroscopic constant for a variety of 38 diatomics. These results are in favour of an anti-PPP method, of which the basis is algebraically related to the usual PPP method. A new expression is derived for the critical distance, determining the ionic-covalent crossing point.

On Drago's E-C Equation, Pearson's HSAB Rule and the Ionic Approximation to Chemical Bonding

Abstract An attempt is made to explain the E-C formalism for ionic interactions in terms of the ionic approximation to chemical bonding. Dravo's E-C equation is seen to be a first approximation to the bond energy equation as it is given by the ionic bonding approach. The meaning of the ratio C/E is discussed and its relation with the hardness and softness of interacting species, as these occur in Pearson's HSAB rule, shows that the electron affinity or electronegativity of elements completely determines the chemical behaviour of ionic species. This analysis illustrates the consistency of the ionic approximation to chemical bonding.

Calculation of bond energies in diatomic molecules

An extrapolation method is proposed for an approximative evaluation of covalent bonding powers of some elements from their electronegativity values. Using these values in the bond energy equation, obtained from the principle of electronegativity equalization [3], bond energies can be calculated with an accuracy, comparable with the one, obtained by Evans and Huheey [1], who included an electrostatic attraction energy term in the calculation of bond energies. Alterations in the covalent bonding power (electronegativity) of some elements in function of the nature of the bonding partner are demonstrated. The disagreement between the Pauling and Pearson rules for the description of heteropolar bond stability is discussed.

How Inaccurate is Pauling's Bond Energy Equation?

PEARSON1 recently pointed out that Pauling's bond energy equation does not give reasonable values of the enthalpy changes for a series of reactions, and he has also stressed a number of pitfalls of Pauling's electronegativities. The principle of soft and hard acids and bases (SHAB), on the other hand, predicts the correct sign of ΔH for those reactions and the general chemical patterns mishandled by the simple electronegativity concept.

Failure of pauling's bond energy equation

Failure of pauling's bond energy equation