The SN3–SN2 spectrum. Rate constants and product selectivities for solvolyses of benzenesulfonyl chlorides in aqueous alcohols

Rate constants for a wide range of binary aqueous mixtures and product selectivities (S) in ethanol–water (EW) and methanol–water (MW) mixtures, are reported at 25 °C for solvolyses of benzenesulfonyl chloride and the 4‐chloro‐derivative. S is defined as follows using molar concentrations: S = ([ester product]/[acid product]) × ([water solvent]/[alcohol solvent]). Additional selectivity data are reported for solvolyses of 4‐Z‐substituted sulfonyl chlorides (Z = OMe, Me, H, Cl and NO₂) in 2,2,2‐trifluoroethanol–water. To explain these results and previously published data on kinetic solvent isotope effects (KSIEs) and on other solvolyses of 4‐nitro and 4‐methoxybenzenesulfonyl chloride, a mechanistic spectrum involving a change from third order to second order is proposed. The molecularity of these reactions is discussed, along with new term ‘S_N3–S_N2 spectrum’ and its connection with the better established term ‘S_N2–S_N1 spectrum’. Copyright © 2009 John Wiley & Sons, Ltd.     

A DFT study of transition structures and reactivity in solvolyses of tert‐butyl chloride,cumyl chlorides,and benzyl chlorides

DFT computations were performed on the S_N1 and S_N2 solvolyses of substituted cumyl chlorides and benzyl chlorides in ethanol and water, by increasing stepwise the C? Cl distance and by optimization. The total energy increases with the increase in the Cl? C distance in S_N1 reactions, while free energy of activation pass through maximum. To validate the results, the calculated free energies of activation were compared with data obtained by kinetic measurements. The structural parameters of the transition states were correlated with the Hammett substituent constants and compared with the data of hydrolyses of tert‐butyl chloride and methyl chloride, which proceed with known mechanisms. Conclusions on the mechanisms of the reactions were driven from the effect of substituents on free energies of activation. Cumyl chlorides substituted with electron‐donating (e‐d) groups solvolyze with S_N1 mechanism, while the reactions of substrates that bear electron‐withdrawing groups proceed with weak nucleophilic assistance of the solvent. Benzyl chlorides hydrolyze through an S_N2 pathway except those derivatives that have strongly e‐d groups, where the reaction has S_N1 character, but a weak nucleophilic assistance of the water should also be taken into consideration. Copyright © 2007 John Wiley & Sons, Ltd.     

Solvent network at the transition state in the solvolysis of hindered sulfonyl compounds

Alcoholysis rates of unhindered benzenesulfonyl chlorides (X‐ArSO₂Cl, X = H‐; 4‐Br‐; 4‐Me‐) are similar in methanol; the same behavior is also observed in ethanol, whereas the reactivity order in iso‐propanol is 4 Me‐ < H‐ < 4‐Br‐. On the other hand, alcoholysis of sterically hindered arenesulfonyl chlorides (X‐ArSO₂Cl) (X = 2,4,6‐Me₃‐3‐NO₂‐; 2,6‐Me₂‐4‐tBu‐; 2,4,6‐Me₃‐; 2,3,5,6‐Me₄‐; 2,4,6‐iPr₃‐; 2,4‐Me₂‐; 2,4,6‐(OMe)₃‐) in all studied alcohols show a significant increase in reactivity, the so‐called positive steric effect. Most of the substrates showed a reaction order b ~ 2 with respect to the nucleophile in methanol and ethanol, and b ~ 3 in iso‐propanol. The correlation between reactivity and the Kirkwood function (1/ξ) gives negative sensitivity (U) for all systems. All substrates showed high sensitivity to media nucleophilicity that depends on Σσ_X. Obtained results suggest the alcoholysis of benzenesulfonyl chlorides proceeds through S_N2 mechanism where the transition state (TS) involves the participation of 2–3 alcohol molecules; such a TS can be cyclic, in the case of unbranched alcohols, or linear, for alcohols with bulkier hydrocarbon groups like iso‐propanol. To include the number of alcohol molecules playing such a role in the TS, the following terminology is proposed: cS_N2s_n for S_N2 reactions involving n solvent molecules in a cyclic (c) TS, where “s” stands for the solvent and “n” is either the closest integer or half‐integer to the reaction order relative to the solvent or, in computational studies, the proposed number of solvent molecules taking part in the TS, whereas S_N2s_n is proposed when the TS is not cyclic. Copyright © 2016 John Wiley & Sons, Ltd.     

Positioning solvolyses within the SN2‐SN1 and SN3‐SN2 spectrum of reaction mechanisms

A simple linear regression (Q equation) is devised to position solvolyses within the established S_N2‐S_N1 spectrum of solvolysis mechanisms. Using 2‐adamantyl tosylate as the S_N1 model and methyl tosylate as the S_N2 model, the equation is applied to solvolyses of ethyl, allyl, secondary alkyl and a range of substituted benzyl and benzoyl tosylates. Using 1‐adamantyl chloride as the S_N1 model and methyl tosylate as the S_N2 model, the equation is applied to solvolyses of substituted benzoyl chlorides in weakly nucleophilic media. In some instances, direct correlations with methyl tosylate were employed. Grunwald–Winstein l values and kinetic solvent isotope effects are also used to locate solvolyses within the spectrum of mechanisms. Product selectivities (S) for solvolyses at 50 °C of p‐nitrobenzyl tosylate in binary mixtures of alcohol–water and of alcohol–ethanol for five alcohols (methanol, ethanol, 1‐propanol and 2‐propanol and t‐butanol) are reported and show the expected order of solvent nucleophilicity (RCH₂OH > R₂CHOH > R₃COH). The data support the original assignments establishing the N_OTs scale of solvent nucleophilicity. Copyright © 2016 John Wiley & Sons, Ltd.     

The importance of the ortho effect in the solvolyses of 2,6‐difluorobenzoyl chloride

The ortho effect of the chloro substituents in 2,6‐dichlorobenzoyl chloride sufficiently hindered attack on the acyl carbon such that an ionization mechanism was observed over the full range of solvents studied. We now compare this behavior with that of 2,6‐difluorobenzoyl chloride. The smaller fluoro substituents allow the dominant pathway to be addition–elimination (association–dissociation) in all solvents except those rich in fluoroalcohol, where ionization is dominant. Ranges of operation for both mechanisms had previously been observed for the parent benzoyl chloride but with a wider ionization range than for the 2,6‐difluoro derivative. This indicates that, relative to the parent, the electronic destabilizing influence of the fluorines on acyl cation formation outweighs the steric retardation to attack because of the presence of the two ortho‐fluorine atoms. An extended (two‐term) Grunwald–Winstein equation treatment of the solvolyses of 2,6‐difluorobenzoyl chloride is reported. Copyright © 2011 John Wiley & Sons, Ltd.     

Concerted SN2 mechanism for the hydrolysis of acid chlorides: comparisons of reactivities calculated by the density functional theory with experimental data

DFT computations have been performed in acetone and water solvents in order to investigate the mechanism of hydrolysis of acid chlorides. Acetyl chloride and chloroacetyl chloride hydrolyze via concerted, one‐step S_N2 mechanism, with the attack of water at the sp² hybridized carbon atom of the C?O group, and the transition state (TS) has distorted tetrahedral geometry. Solvent molecules act as general base and general acid catalysts. The TS of chloroacetyl chloride is tighter and less polar than the TS of acetyl chloride. The structure of the S_N2 TS for the hydrolysis of benzoyl chlorides changes with the substituents and the solvent. Tight and loose TSs are formed for substrates bearing electron withdrawing (e‐w) and electron donating (e‐d) groups, respectively. In acetone, only the e‐w effect of the substituents increase the reactivity of the substrates, and the change of the structure of the TSs with the substituents is small. In water, polar and very loose TSs are formed in the reactions of benzoyl chlorides bearing e‐d substituents, and the rate enhancing effect of both e‐d and e‐w groups can be computed at higher level of theory. Calculated reactivities and the changes of the structure of the TSs with substituents and solvent are in accordance with the results of kinetic studies. In S_N2 nucleophilic substitutions late/early TSs are formed if the attacking reagent is poorer/better nucleophile than the leaving group, and loose/tight TSs are formed for substrates bearing e‐d/e‐w substituents and in protic/aprotic solvents. Copyright © 2010 John Wiley & Sons, Ltd.     

Gas‐phase acidity of bis[(perfluoroalkyl)sulfonyl]imides. Effects of the perfluoroalkyl group on the acidity

The gas‐phase acidity (GA) values were determined for a number of perfluoroalkyl‐substituted sulfonylimides by measuring proton‐transfer equilibria using a Fourier transform ion cyclotron resonance (FT‐ICR) mass spectrometer. The GA scale below 286.5 kcal mol^?1 for (CF₃SO₂)₂NH was extended and partially revised. The GA value of (C₄F₉SO₂)₂NH which is currently the strongest acid was revised from 284.1 to 278.6 kcal mol^?1. The effect of fluorine atoms on the acidity of perfluoroalkyl‐substituted sulfonylimides was described with the following model where N_(α), N_(β), N_(γ), and N_(δ) are the numbers of fluorine atoms at α, β, γ, and δ position in R_fSO₂ (R_f = perfluoroalkyl group), respectively. This correlation indicates that the electron‐withdrawing ability of the R_fSO₂ group can be described in terms of the number of fluorine atoms in the perfluoroalkyl group corrected by taking into account their positions. Copyright © 2014 John Wiley & Sons, Ltd.     

Structure and reactivity in the hydrolyses of aliphatic carboxylic acid esters and chlorides

For hexanoic acid and its seven isomers, relative rates have been determined for acid catalysed esterification with methanol, and compared with those for saponification of the methyl esters. A good correlation between logarithms of relative rates for the two reactions is obtained, and it is suggested that the eight isomers provide a test set of compounds in which steric effects alone act on reactivity at the acyl carbon. A full set of steric parameters ( values) are presented. Rates of solvolyses of the acid chlorides of the isomers have been determined conductometrically in 3:1 wt:wt acetonitrile water. Logarithms of relative rates show a poor correlation with , and, taking into account the solvent dependence of the rates, the pattern excludes both rate‐limiting formation of a tetrahedral intermediate and rate‐limiting dissociation of chloride to form acylium ions. The remaining possibilities, a concerted process (A_ND) and rapid reversible formation of a hydrate followed by rate‐limiting dissociation of chloride (A_N + D) are considered. Copyright © 2007 John Wiley & Sons, Ltd.     

Propanolysis of arenesulfonyl chlorides: Nucleophilic substitution at sulfonyl sulfur

We have studied the mechanism of solvolysis of arenesulfonyl chlorides by propan‐1‐ol and propan‐2‐ol at 303‐323 K. Kinetic profiles were appropriately fit by first‐order kinetics. Reactivity increases with electron‐donating substituents. Ortho‐alkyl substituted derivatives of arenesulfonyl chlorides show increased reactivity, but the origin of this “positive” ortho‐effect remains unclear. Likely, ortho‐methyl groups restrict rotation around the C‐S bond, facilitating the attack of the nucleophile. No relevant reactivity changes have been found with propan‐1‐ol and propan‐2‐ol in terms of nucleophile steric effect. The existence of isokinetic relationships for all substrates suggests a single mechanism for the series. Solvolysis reactions of all substrates in both alcohols show isokinetic temperatures (T_iso) close to the working temperature range, which is an evidence of the process being influenced by secondary reactivity factors, likely of steric nature in the TS. Solvation plays a relevant role in this reaction, modulating the reactivity. In some cases, the presence of t‐Bu instead of Me in para‐ position leads to changes in the first solvation shell, increasing the energy of the reaction (ca. 1 kJ·mol^?1). The obtained results suggest the same kinetic mechanism of solvolysis of arenesulfonyl chlorides for propan‐1‐ol and propan‐2‐ol, as in MeOH and EtOH, where bimolecular nucleophilic substitution (S_N2) takes place with nucleophilic solvent assistance of one alcohol molecule and the participation of the solvent network involving solvent molecules of the first solvation shell.     

Enthalpies of formation and isomerization of aromatic hydrocarbons and ethers by G3(MP2)//B3LYP calculations

A computational method for estimation of the gas‐phase enthalpies of formation of aromatic hydrocarbons and ethers has been developed. The method is based on high‐level G3(MP2)//B3LYP calculations, atomization reactions, and structure‐dependent correction terms. By this method, enthalpies of formation Δ_fH_m°(g, 298.15 K) of 86 aromatic compounds were evaluated. The calculated enthalpies of formation raise questions of the reliability of several experimental enthalpies of formation reported in the literature. As an application of the computational enthalpies of formation, reaction enthalpies for several types of isomerization reactions of aromatic compounds were calculated. In cases in which experimental reaction enthalpies were available for comparison, the agreement between the computational and experimental data proved to be excellent. Copyright © 2008 John Wiley & Sons, Ltd.     

Influence of the ortho effect in the solvolyses of dichlorobenzoyl chlorides

The ortho‐effect of substituents upon the kinetics of reactions taking place at a reaction center attached to an aromatic ring has long been a topic of interest. For benzoyl chloride solvolyses, it was shown by Bentley and coworkers that the 2,6‐dimethyl‐derivative followed an ionization pathway with characteristics very similar to those for the solvolyses of p‐methoxybenzoyl chloride. We have carried out a Grunwald–Winstein equation treatment of the solvolyses of 2,6‐dichlorobenzoyl chloride, with similar sized chlorines replacing the methyl groups but now with an overall electron‐withdrawing influence of the ortho‐substituents. In this way the reactivity is moderated and the study can be extended to the important fluoroalcohol‐containing solvents. For the 30 solvents studied, an ionization pathway with a moderate nucleophilic solvation component is indicated. For comparison purposes, the treatment has also been applied to the 2,4‐, 3,4‐, and 3,5‐dichloro‐ derivatives. For the 2,4‐dichloro‐derivative, the two reaction channels are clearly visible and the solvents included for each channel are consistent with their solvent properties. Copyright © 2011 John Wiley & Sons, Ltd.     

Sulfonyl nitrenes from different sources: computational study of formation and transformations

The formation of sulfonyl nitrenes RSO₂N (R = Me, p‐Tol, CF₃) from different sources, such as sulfonyl azides RSO₂N₃, N‐halosulfonamides RSO₂NHHal, salts RSO₂NNaCl, N‐hydroxysulfonamides RSO₂NHOH and sulfonylimino‐λ³‐iodanes ArI = NSO₂R or ‐bromanes, by elimination of neutral molecules (N₂, HCl, NaCl, H₂O and ArHal, respectively) was studied theoretically at the density functional theory and second‐order Møller‐Plesset perturbation theory using the 6‐311++G(d,p) and cc‐pVTZ basis sets. The originally formed singlet nitrenes suffer nonradiative intersystem crossing to the triplet state (S₁ ? T₁) or, sometimes, undergo spontaneous barrierless pseudo‐Curtius rearrangement into the corresponding sulfonylamines RN = SO₂. The activation barriers decrease in the order (kcal/mol): RSO₂NHHal ~ RSO₂NHOH (60–70) ≥ RSO₂N(Na)Hal (40–50) > RSO₂N₃ (35) > RSO₂N = IPh (9–13) ≥ RSO₂N = BrC₆H₄CF₃ (4–8). Copyright © 2013 John Wiley & Sons, Ltd.     

The enthalpies of dissociation of the N?O bonds in two quinoxaline derivatives

The present work reports the first experimental thermochemical study of mono‐N‐oxides derived from quinoxaline, namely, 3‐methoxycarbonyl‐2‐methyl‐quinoxaline N‐oxide and 3‐ethoxycarbonyl‐2‐methyl‐quinoxaline N‐oxide. The values of the enthalpies of formation, in the condensed state, and of the enthalpies of sublimation, derived from static bomb calorimetry and Calvet microcalorimetry measurements, respectively, are combined to derive the standard molar enthalpies of formation in the gaseous phase for these two compounds. From the latter values, the first and second N? O bond dissociation enthalpies for the corresponding di‐N‐oxides have been obtained. The gas‐phase experimental results are also compared with calculated data obtained with a density functional theory approach. Copyright © 2008 John Wiley & Sons, Ltd.     

Correlation of the rates of solvolysis of diphenylphosphinyl chloride using an extended form of the Grunwald–Winstein equation

The specific rates of solvolysis of diphenylphosphinyl chloride ( 1 ) have been measured at 25.0 °C in 30 solvents. For six representative solvents, studies were made at several temperatures and activation parameters determined. These were used to calculate a value at 25.0 °C in 100% 2,2,2‐trifluoroethanol (TFE) from values at higher temperatures. The 31 solvents gave a reasonable extended Grunwald–Winstein plot, correlation coefficient (R) of 0.920, which improved to 0.956 when the four TFE–ethanol points were excluded. The sensitivities (l and m) were similar to those obtained for dimethyl phosphorochloridate and phosphorochloridothionate and for N,N,N′,N′‐ tetramethyldiamidophosphorochloridate. As with the three previously studied solvolyses, an S_N2 pathway is proposed for the solvolyses of 1 . Copyright © 2007 John Wiley & Sons, Ltd.     

Theoretical studies on atmospheric chemistry of HFE‐347mcc3: reactions with OH radicals and Cl atoms

Detailed theoretical investigation has been performed on the mechanism, kinetics and thermochemistry of the gas phase reactions of CF₃CF₂CF₂OCH₃ (HFE‐347mcc3) with OH radicals and Cl atoms using M06‐2X/6‐31 + G(d,p) level of theory. Reaction profiles are modeled including the formation of pre‐reactive and post‐reactive complexes at entrance and exit channels, respectively. Using group‐balanced isodesmic reactions, the standard enthalpies of formation for species are also reported. The calculated bond dissociation energy for C―H bond is in good agreement with previous data. The rate constants of the two reactions are determined for the first time in a wide temperature range of 250–1000 K. At 298 K, the calculated rate coefficients are in good agreement with the experimental results. The atmospheric life time of HFE‐347mcc3 is estimated to be 4.4 years. Copyright © 2014 John Wiley & Sons, Ltd.     

Determination of sulfoxide bond dissociation enthalpies of dibenzothiophene S‐oxide derivatives with computational methods

Photodeoxygenation of dibenzothiophene S‐oxide (DBTO) has been suggested as a clean way to generate atomic oxygen in solution. Sulfoxide bond dissociation enthalpies (BDEs) are important to the quantum yield and mechanism of this photodeoxygenation. In this study, BDE of substituted DBTO molecules with various functional groups were determined using M062X/aug‐cc‐pV(T + d)Z and MP2/aug‐cc‐pV(T + d)Z levels of theory. The sulfoxide BDE was determined using an isodesmic reaction. The observed effect of substitution was that functional groups with negative σ_para values (electron donating) strengthen the sulfoxide bond and that substituents with positive σ_para values (electron withdrawing) weaken the sulfoxide bond. The Mulliken charge on sulfur for DBTO and the corresponding dibenzothiophene were identified as a suitable indicator of the predicted S–O BDE. Moreover, steric and hydrogen bonding interactions were found to affect sulfoxide BDE for certain functional groups when located at the 1 or 4 positions of the substituted DBTO. Copyright © 2014 John Wiley & Sons, Ltd.     

Substituent effect on electron affinity,gas‐phase basicity,and structure of monosubstituted propargyl radicals and their anions: a theoretical study

The substituent effect of electron‐withdrawing groups on electron affinity and gas‐phase basicity has been investigated for substituted propargyl radicals and their corresponding anions. It is shown that when a hydrogen of the α‐CH₂ group or acetylenic CH in the propargyl system is substituted by an electron‐withdrawing substituent, electron affinity increases, whereas gas‐phase basicity decreases. The calculated electron affinities are 0.95 eV (CH?C? CH₂?), 1.15 eV (CH?C? CHF?), 1.38 eV (CH?C? CHCl?), 1.48 eV (CH?C? CHBr?) for the isomers with terminal CH and 1.66 eV (CF?C? CH₂?), 1.70 eV (CCl?C? CH₂?), 1.86 eV (CBr?C? CH₂?) for the isomers with terminal CX at B3LYP level. The calculated gas‐phase basicities for their anions are 378.4 kcal/mol (CH?C? CH₂:^?), 371.6 kcal/mol (CH?C? CHF:^?), 365.1 kcal/mol (CH?C? CHCl:^?), 363.5 kcal/mol (CH?C? CHBr:^?) for the isomers with terminal CH and 362.6 kcal/mol (CF?C? CH₂:^?), 360.4 kcal/mol (CCl?C? CH₂:^?), 356.3 kcal/mol (CBr?C? CH₂:^?) for the isomers with terminal CX at B3LYP level. It is concluded that the larger the magnitude of electron‐withdrawing, the greater is the electron affinity of radical and the smaller is the gas‐phase basicity of its anion. This tendency of the electron affinities and gas‐phase bacisities is greater in isomers with the terminal CX than isomers with the terminal CH. Copyright © 2009 John Wiley & Sons, Ltd.     

A new strategy for solid phase synthesis of a secondary amide library using sulfonamide linker via radical traceless cleavage

A new strategy for solid phase synthesis of a secondary amide library using sulfonamide linker via radical traceless cleavage is reported. Polystyrylsulfonyl chloride (1) reacted with primary amines to afford polystyryl-supported N-alkyl sulfonamides (2), which were acylated with acid chlorides and followed by radical cleavage with TiCl₄/Zn to afford secondary amides. It was interestingly found that the products released from acyl alkanesulfonamide resins are closely dependent on the substituents of benzene rings of alkyl or acyl groups on the resins. When the substituent on benzene ring of N-benzyl group of sulfonamides is an electron rich MeO-group, the products released from sulfonamide resins are dependent on the substitution position on benzene ring: para-MeO- to yield 1,2-bis (p-methoxylphenyl)ethane and N-p-methoxylbenzyl benzamide (30:1); ortho-MeO- to give 1,2-bis (o-methoxylphenyl)ethane and N-o-methoxylbenzyl benzamide (1:15); and meta-MeO- only to release N-m-methoxylbenzyl benzamide. Neither N-benzoyl sulfonamide resins on benzene ring with electron-drawing para-O₂N-, nor the one with electron-donating para-H₂N- could release any amide product, while the N-benzoyl sulfonamide resins on benzene ring with para-acetamido group released para-acetamidobenzamides. The conjugation effect to stabilize the radical groups in the radical cleaving process was observed.     

Theoretical calculations for neighboring group participation in gas‐phase elimination kinetics of 2‐hydroxyphenethyl chloride and 2‐methoxyphenethyl chloride

Theoretical calculation of the kinetics and mechanisms of gas‐phase elimination of 2‐hydroxyphenethyl chloride and 2‐methoxyphenethyl chloride has been carried out at the MP2/6‐31G(d,p), B3LYP/6‐31G(d,p), B3LYP/6‐31 + G(d,p), B3PW91/6‐31G(d,p) and CCSD(T) levels of the theory. The two substrates undergo parallel elimination reactions. The first process of elimination appears to proceed through a three‐membered cyclic transition state by the anchimeric assistance of the aromatic ring to produce the corresponding styrene product and HCl. The second process of elimination occurs through a five‐membered cyclic transition state by participation of the oxygen of o‐OH or the o‐OCH₃ to yield in both cases benzohydrofuran. The B3PW91/6‐31G(d,p) method was found to be in good agreement with the experimental kinetic and thermodynamic parameters for both substrates in the two reaction channels. However, some differences in the performance of the different methods are observed. NBO analysis of the pyrolysis of both phenethyl chlorides implies a C? Cl bond polarization, in the sense of C^δ+…Cl^δ?, which is a rate‐determining step for both parallel reactions. Synchronicity parameters imply polar transition states of these elimination reactions. Copyright © 2009 John Wiley & Sons, Ltd.     

Energetic and structural characterization of 2‐R‐3‐methylquinoxaline‐1,4‐dioxides (R = benzoyl or tert‐butoxycarbonyl): experimental and computational studies

The gaseous standard molar enthalpies of formation of two 2‐R‐3‐methylquinoxaline‐1,4‐dioxides (R = benzoyl or tert‐butoxycarbonyl), at T = 298.15 K, were derived using the values for the enthalpies of formation of the compounds in the condensed phase, measured by static bomb combustion calorimetry, and for the enthalpies of sublimation, measured by Knudsen effusion, using a quartz crystal oscillator. The three dimensional structure of 2‐tert‐butoxycarbonyl‐3‐methylquinoxaline‐1,4‐dioxide has been obtained by X‐ray crystallography showing that the two N? O bond lengths in this compound are identical. The experimentally determined geometry in the crystal is similar to that obtained in the gas‐phase after computations performed at the B3LYP/6‐311 + G(2d,2p) level of theory. The experimental and computational results reported allow to extend the discussion about the influence of the molecular structure on the dissociation enthalpy of the N? O bonds for quinoxaline 1,4‐dioxide derivatives. As found previously, similar N? O bond lengths in quinoxaline‐1,4‐dioxide compounds are not linked with N? O bonds having the same strength. Copyright © 2007 John Wiley & Sons, Ltd.