Valence, Oxidation Number, and Formal Charge

Valence, Oxidation Number, and Formal Charge • Terms such as valence, oxidation number and formal charge appear frequently in both elementary and advanced chemistry texts. • However,, it is evident from the literature that these terms are often viewed to be synonymous and that conclusions pertaining to such an interpretation may be misleading.  Valence N b off electrons Number l t th t an atom that t uses in i bonding. b di  Oxidation number The charge remaining on an atom when all ligands are removed heterolytically in their closed form, with the electrons being transferred to the more electronegative partner (homonuclear bonds do not contribute to the oxidation number).  Formal charge The charge remaining on an atom when all ligands are removed homolytically.  Coordination number The number of atoms bonded to the atom of interest. interest Valence • Two of the most important factors that provide a first order evaluation of the nature of a covalent molecule are the electron count (cf. the octet & eighteen‐electron rules) and the valence of each atom. • Whereas the term electron count is self‐evident (i.e., the total number electrons in the valence shell of an atom in a molecule) and used consistently, the word valence (aka valency) has many uses:  valence electrons  valence bond theory  valence shell electron pair repulsion theory (VSEPR)  bond‐valence model … the valence of a particular atom is expressed in terms of a sum of the valences of individual bonds, which in turn are calculated using empirically parameterized equations that require knowledge of the bond lengths from crystallography.  covalency … refers to the formation of a covalent bond.  electrovalency or ionic valency … refers to complete electron transfer to form ions, with positive electrovalency being used to express the number of electrons released by an element, while negative electrovalency describes the number of electrons added.) added ) • But when used on its own, to what does valence refer? • In view of the relationship between the valence of an atom and the molecular formulae of its compounds, the valence of an atom is often described as its combining power, a phrase that was first introduced by Frankland (1852) with respect to his atomicity concept. • Frankland s atomicity was later named quantivalenz by Hofmann (1865) and subsequently Frankland’s abbreviated to valenz by Kekulé (1867) and Wichelhaus (1868). • Sidgwick’s classic 1927 book on The Electronic Theory of Valency states that “Valency is a ggeneral term used to describe the power p which atoms p possess of combiningg with one another to form molecules”. • The valence of an atom in a covalent molecule is simply the number of electrons that an atom has used in bonding. • Mathematically, the valence of an atom in a molecule is equal to the difference between (i) the number of valence electrons in the free atom (i.e., the group valence) and (ii) the number of nonbonding electrons on the atom in the molecule. Valence = ‘number of electrons in valence shell of free atom’ ‐ ‘number of nonbonding electrons on atom in molecule’ • If the valence is less than the group valence, it indicates that there are nonbonding electrons on the atom. • Knowledge of the valence of an atom in a molecule is important because it tells us 1. whether the atom has enough electrons to support the number of atoms attached via 2‐center, 2‐electron bonds. 2. whether any electrons remain on the atom and are available for subsequent reactivity. e.g., the h valence l off nitrogen in NH3 is 3, which h h indicates d that h there h is a pair off electrons l on nitrogen that may be used to coordinate additional groups (such as BH3 or H+). • In transition‐metal chemistry, knowledge of the number of non‐bonding electrons, i.e., the dn configuration, fi ti i paramountt because is b it is i an important i t t componentt in i determining d t i i th the magnetic properties and electronic spectroscopy of a molecule. • If the valence of the atom appears to be greater than the group valence, valence it indicates that either the bonding cannot be described in terms of 2‐center, 2‐electron bonds but rather involves 3‐center, 2‐electron bonds. • An example of such a species with 3‐center, 3 center 2‐electron 2 electron bonds is CH5+. • As originally employed, the valence of an atom (A) refers to its combining power relative to that of hydrogen; as such, the valence of an atom in a molecule is explicitly and readily indicated by the molecular formula of the hydride AHn. e.g., the nitrogen atom in NH3 has a valence of 3. • However, valence has been taken to be synonymous with oxidation number, number of bonds and coordination number. • The origin of this confusion can be readily seen by consideration of the neutral AHn hydrides for which the valence, magnitude of the oxidation number, number of bonds, and coordination number each have the same value, n. e.g., the nitrogen atom in NH3 has a valence of 3, an oxidation number of ‐3, 3 bonds, and a coordination number of 3. • However, this relationship is purely coincidental ! e.g., consider [NH4]+ for which the nitrogen atom has used all five of its valence electrons (there are no lone pair electrons) and is pentavalent, but its oxidation number is ‐3, the number of bonds is 4, and the coordination number is 4. Valence & Oxidation Number • The oxidation o idation number n mber of an atom in a molecule molec le is based on a formalism that forces a covalent compound to possess complete ionic character and may be defined as Oxidation number = ‘charge on compound’ ‐ ‘charge on ligands’ note: an exception are homonuclear bonds, in which case the bond is broken homolytically and a single electron is transferred to each atom. • g assigned g to simple p monoatomic ligands g do not varyy from In manyy cases, the charges compound to compound. • A notable exception is provided by hydrogen for which both H+ and H have permissible closed‐shell configurations (1s0 and 1s2 , respectively). In this case, the charge assigned to h d hydrogen i determined is d t i d by b the th relative l ti electronegativity l t ti it off the th atom t t which to hi h it is i attached. tt h d • It is important to emphasize that the derived charge does not correspond to the charge on the atom in the molecule, but is hypothetical. • In this regard, it is unfortunate that the oxidation number is often described as a “formal charge” because this term has a very distinct and different meaning to that of oxidation number. • In many situations, the valence of an atom in a molecule may be coincidentally equal to its oxidation number. • p to break down. Specifically, p y, valence and There are several factors that cause this relationship oxidation number are not the same when either (i) homonuclear element–element bonds are present ee.g., g the valence of the central carbon in CMe4 is four, four but the oxidation number is zero because homonuclear element–element bonds are neglected in the determination of oxidation number. (ii) the ligands attached to the atom of interest are dissociated with opposite charges e.g., the valence of carbon in CH2Cl2 is four, but the oxidation number is zero because, in terms of electronegativity differences with respect to carbon, H is viewed as H+ and Cl is viewed as Cl−. (iii) the molecule is charged and the ligand is dissociated as a cation, e.g., H+ , the valence of nitrogen in NH4+ is five, but its oxidation number is ‐3. • Comparison of the valence, oxidation number, number of bonds, and coordination number for some simple molecules. • Examples of situations where the equivalence of valence to the magnitude of oxidation number, number of bonds, and coordination number break down. Valence & Number of Bonds • The equivalence eq i alence between bet een valence alence and number n mber of 2‐center, 2 center 2‐electron 2 electron bonds for molecules molec les of the type AHn breaks down when the atom in question bears a formal charge. e.g., consider the series of isoelectronic molecules BH4− , CH4 , and NH4+ ; while each of the molecules exhibits four bonds to the central atom, the valences of B , C , and N are 3, 4, and 5, respectively. Valence = ‘number of bonds’ + ‘formal charge’ • As such, h it is evident d that, h f neutrall molecules for l l off the h type AHn , the h coincidental d l equivalence between valence and number of bonds is a consequence of the central atom bearing no formal charge. Valence & Coordination Number (CN) • The coordination number n mber (CN) is simply simpl defined as the number n mber of atoms attached to the atom of interest in a molecule.For neutral molecules of the type AHn the CN of A is n and is equivalent to its valence. • The equivalence between valence and coordination number breaks down when a multiple bond is present. e.g., consider the series of molecules H3CCH3 , H2CCH2 , and HCCH . The valence of carbon in each molecule is 4, but the CNs are 4, 3, and 2, respectively. • In essence, when the central atom uses more than one of its electrons to bond to another atom, the coordination number of the central atom is reduced from the value of its valence. • Likewise, the relationship between valence and CN breaks down when the bond is a dative covalent bond where one of the atoms provides both electrons. e.g., the valence of B in three‐coordinate BH3 and four‐coordinate H3N‐BH3 is each 3. • From the opposite perspective, perspecti e it is evident e ident that the valence alence of the coordinating atom of a dative ligand increases by two units when it coordinates to a Lewis acidic site. • Thus, while the nitrogen in NH3 is trivalent, it becomes pentavalent once it uses its lone pair upon coordinating to a Lewis acid (e.g., BH3 or H+). Using Valence & Oxidation Number To Provide Insight into the Nature of a Molecule • It is pertinent to consider whether it is the valence or oxidation number that provides the greater insight into understanding the nature of a molecule. 1. The oxidation numbers of C span the range of ‐4 to +4. 2. Does the difference in the nature of CH4 and CCl4 warrant such a large change in oxidation number? ee.g., g CH4 (ON = 4) is not a widely used reducing agent, and CCl4 (ON = + 4) is not a widely used oxidizing agent. 3. CMe4 (ON = 0) has little in common with CH2Cl2 (ON = 0). 4. As such, the large variation in oxidation states cannot be regarded as providing much h insight i i ht into i t the th nature t off these th molecules. • A further illustration of how oxidation number may offer little insight into the chemistry of a molecule is provided by the fact that both Me2CCl2 and :CCl2 have an oxidation number of +2, but the chemistry of these molecules are very distinct. distinct • Thus, whereas Me2CCl2 is a stable species, the carbene :CCl2 is only known to exist as a reactive intermediate. • Likewise, Lik i the th carbon b atoms t i both in b th CH3CH3 and d CH3 have h an oxidation id ti number b off ‐3, 3 but b t the th former compound is a stable molecule while the latter is a highly reactive radical. • In contrast, the valence of an atom in a molecule does provide useful information. Thus, with two exceptions, all of the compounds listed in previous table exhibit a valence of 4 for carbon despite the fact that the oxidation numbers range from +4 to ‐4. carbon, ‐4 • The two exceptions are :CCl2 , with a valence of 2, and CH3 with a valence of 3. These exceptions are notable because they correspond to the two compounds that are not isolable under normal conditions. • A further example is the oxidation number of the α‐carbon atom in a series of alcohols. ON = ‐2 for CH3OH ON = ‐1 for MeCH2OH ON = 0 for Me2C(H)OH ON = +1 for Me3COH Each of the alcohols possess C in the same valence state and thereby reiterates the ability of valence to provide a greater appreciation of chemical reasonableness than does oxidation number. • The valence of an atom in a molecule provides a much more meaningful criterion for evaluating the chemical reasonableness of a molecule than does the oxidation number. • Thus, while the concept of oxidation numbers is of use in certain situations, such as the balancing of redox equations, it is less fundamental than the underlying concept of valence and its use in evaluating the chemical reasonableness of a molecule is only successful when the oxidation number and valence of an atom in a molecule are coincidentally the same. Knowledge of valence is essential because it allows one to ascertain whether ‐ the atom has enough electrons to support the number of atoms attached via 2‐center, 2‐ l b d bonds. electron • Compare CH4 and CH5+.  both contain C with an identical electronic configuration, ON = ‐4 and both obey the octet rule. • However, the valence of C in CH5+ is 6 even though C only has 4 valence electrons indicating that there is something unusual about this molecule! • Indeed, CH5+ is very unstable, but how could such a species exist if the valence of C were to be 6? Since the carbon atom in CH4 does not have any electrons available for protonation (unlike the nitrogen atom in NH3), ) the answer is quite simply provided by the fact that it is not the carbon atom itself that is protonated. • Rather, it is the C‐H bond that is protonated, thereby resulting in a 3‐center, 2‐electron interaction. • With a 3‐center, 2‐electron interaction, CH5+ may be regarded as a dihydrogen complex of CH3+ and so the valence of the carbon in this species remains 4. • Thus, because it is the C‐H bond that is protonated, Thus protonated the carbon atom in CH5+ contributes only 4 valence electrons to bonding the 5 hydrogen atoms. Knowledge of valence is essential because it allows one to ascertain whether reactivity. ‐ any electrons remain on the atom and are available for subsequent reactivity • The use of valence to provide an indication of whether there are any electrons available on the atom for further bonding, is provided by consideration of NH3. • Thus, the valence of nitrogen in NH3 is three, thereby indicating that there is a pair of electrons available for subsequent reactivity, such as protonation or coordination to a Lewis acid. id Calculation of formal charges for (XH4)q (X = B, C, N; q = ‐1, 0, +1) Comparison of Oxidation Number, Formal Charge, and “Actual” Charge • The formal charge of an atom in a given Lewis structure is the charge it would possess if the electrons in each covalent bond were to be distributed equally between the two bonded atoms such that each partner receives one electron. i.e., homolytic cleavage • The formal charge may, therefore, be defined as the difference between (i) the number of group p valence) and (ii) the number of electrons remainingg electrons in the free atom (i.e., the g on the atom in a molecule when all bonds are broken homolytically. Formal charge = ‘number of electrons in valence shell of free atom  ‘number ‘ b off electrons l t remaining i i after ft homolytic h l ti cleavage l off allll bonds’ b d’ • Since each 2‐center, 2‐electron bond contributes one electron to the electron count of an atom in a molecule, the electron count is equal to the number of electrons in the valence shell of the free atom plus the number of covalent bonds, bonds adjusted by the formal charge. charge • Thus, the formal charge may also be expressed as: g = ‘number of electrons in valence shell of free atom’ Formal charge + ‘number of bonds’  ‘electron count’ • While the concepts of oxidation number and formal charge are closely related in the sense that they both refer to a charge on an atom in a molecule, they have very different values because the charges are determined by two fundamentally different formalisms that respectively exaggerate the ionic and covalent character of the bonding. • Does the “actual” charge on an atom in a covalent molecule correspond more closely to the oxidation number or the formal charge? Electroneutrality Principle “Stable complexes are those with structures such that each atom has only a small electric charge. Stable M‐L bond formation generally reduces the positive charge on the metal as well as the negative charge and/or e‐ density on the ligand. The result is that the actual charge on the metal is not accurately reflected in its formal oxidation state” • Neither Oxidation Number or Formal Charge are accurate descriptions of electron density at the metal metal. • Charge on an atom is typically considered to be less than ±1 e.g., high oxidation state, early TM complexes are stabilized via π‐donation shifting electron density from π‐donor π donor ligands to the metal. metal This in part accounts for the extreme oxophilicity of early TM. Valence = ‘number of electrons in valence shell of free atom’ ‐ ‘number of nonbonding electrons on atom in molecule’ Valence = ‘number of bonds’ + ‘formal charge’ Oxidation number = ‘charge on compound’ ‐ ‘charge on ligands’ Formal charge = ‘number of electrons in valence shell of free atom  ‘number number of electrons remaining after homolytic cleavage of all bonds bonds’ Formal charge = ‘number of electrons in valence shell of free atom’ + ‘number number of bonds’ bonds  ‘electron electron count count’ MO Theory • MO theory assumes that electrons are delocalized across the molecular framework where a Linear Combination of Atomic Orbitals (LCAO) is invoked ± N   ± B  • According to the Born interpretation, the probability electron density is proportional to    N 2   ± B   bonding • The probability electron density   of  bond is thus proportional to 1. The probability density of the electron confined to atom A,  2. The probability density of the electron confined to atom B, B2 3. The additional probability of finding the electron intermediate between A and B, AB   N 2  B2 AB  • The additional contribution from AB is known as the overlap density and represents an enhancement of finding the electron in the internuclear distance. • Thus electrons accumulate in regions of constructive orbital overlap between bonding nuclei and lowering the overall energy.  antibonding • • The probability electron density   of the  antibonding orbital is thus proportional to 1. The probability density of the electron confined to atom A,  2 2. The probability density of the electron confined to atom B, B B2 The probability of finding the electron intermediate between A and B is now AB due to destructive interference in the internuclear region resulting in a node   N  2   B2  AB  • Occupation of the  antibonding orbital thus decrease electron density in the internuclear region thus pushing the bonding nuclei apart, weakening the bond and raising the energy of the system. system • In fact the following relationship holds true |E  EA| > |E  EA| thus the antibonding orbital is more antibonding than the bonding orbital is bonding. MOs from 1s Orbitals ‐ Homonuclear Diatomic Molecules • Positive combination causes a build‐up of electron density between the nuclei, resulting in a sigma i b di MO (σ). bonding ( ) • Negative combination results in a nodal plane between the nuclei, which works against bonding, resulting in a sigma antibonding MO (σ*). • In general, number of MOs formed as LCAOs equals number of AOs used. bonding MO  = a b    anti‐bonding MO  = a  b   MO Energy Level Scheme First Period Homonuclear Diatomic Cases • Scheme is filled in the usual aufbau manner, following the Pauli exclusion principle and Hund's rule of maximum multiplicity (for the ground state). • Bond order is defined as follows: bond order = ½ ((bondingg e's – antibonding g e's)) H2 and He2 • H2 bond order = ½(2 – 0) = 1 • He2 bond order = ½(2 – 2) = 0 (He2 does not exist.) H2+ and He2+ ions • H2+ bond order = ½(1 – 0) = ½ • He2+ bond order = ½(2 – 1) = ½ MOs for Second Period Homonuclear Diatomic Molecules • • In general, atomic orbitals that are most similar in energy interact most effectively in forming MOs. MO MOs for second period diatomic molecules are combinations of the type 2s ± 2s and 2p ± 2p. 2s bonding MO 2s anti‐bonding MO MO Energy Level Scheme for Li2, Be2, and Related Ions Li2 bond order = ½(4 – 2) = 1 Be2 bond order = ½(4 – 4) = 0 2p ± 2p MOs ( and ) • Singly degenerate 2p and 2p bonding and antibonding MOs from 2pz ± 2pz • Two degenerate π2p bonding MOs, one from 2px + 2px and one from 2py + 2py • Two degenerate π*2p antibonding MOs, one from 2px – 2px and one from 2py – 2py 2p bonding MO 2p bonding MO 2p antibonding MO 2p anti‐bonding MO MO Scheme for O2 through Ne2 and Related Ions • The core configuration levels σ1s and σ*1s are omitted. MO Scheme for B2 through N2 and Related Ions • In diatomic molecules of the lighter elements B2, C2, and N2, mixing between sigma‐type MOs causes the σ*2s level to move down and the σ2p level to move up in energy. • The σ2p level rises above that of π2p. • This results in the following scheme, in which the relative ordering π2p < σ2p occurs: MO Description of O2 and Its Ions • MO model predicts the observed paramagnetism, which VB theory cannot explain easily. Second Period Heteronuclear Diatomic Molecules • MO scheme for homonuclear diatomic molecules can be adapted to describe bonding in heteronuclear diatomic molecules. • The two atoms do not contribute equally to each MO. •  More electronegative element has lower energy AOs and makes a greater contribution to bonding MOs.  Less electronegative atom makes a greater contribution to antibonding MOs. Ordering of MOs energies in heteronuclear cases tends to be like lighter homonuclear cases: i.e., π2p < σ2p Heteronuclear Diatomic Molecules Examples • CO, CN–, NO+ (10 valence electrons, like N2) bond order = 3 • NO (11‐electron free radical, like O2+) bond order = 2.5 MO Description of Pi‐Delocalized Systems • MO theory has no difficulty explaining the π‐delocalized electron density implied by resonance forms, which VB theory cannot easily represent. π electron density of O3 • • The three 2p orbitals perpendicular to the molecular plane combine to form three π MOs: 1. π – bonding MO 2. πn – nonbonding MO 3. π* – antibonding MO π A non‐bonding MO neither builds nor destroys bonding between the oxygen atoms. Pi MO Energy Level Scheme for O3 π* – antibonding MO • The π system has two pairs of electrons, one pair in the π MO, and the other in the πn non‐bonding MO. • The configuration (π)2 adds a bond order of 1 across the two O–O bonds (i.e., 0.5 to each bond). • When this is added to the sigma bond between each oxygen pair, the O–O bond order becomes 1.5. • The configuration (πn)2 neither adds nor subtracts from the overall strength of the bonds. πn – nonbonding MO π – bonding MO Pi MOs of Benzene, C6H6 • C6H6, has three pairs of electrons delocalized in a π system extending around the hexagonal ring. π electron density of C6H6 • The six 2p orbitals perpendicular to the ring on the six carbon atoms combine to form three bonding (π1, π2, π3) and three antibonding (π4*, π5*, π6*) MOs. Pi MO Energy Level Scheme for C6H6 π* 6 π* 4 • • Three pairs in bonding MOs add a total of three bond orders over six C–C linkages, or 0.5 for each. π* 5 π2 π3 When this is added to the sigma bond between each carbon pair, the C–C bond order becomes 1.5. π1 Wave Equations of Molecules • Approximate wave functions are sought by combining atomic wave functions for the bonded atoms. • Several different approaches have been taken to constructing trial wave functions.  V l Valence B d (VB) ‐ Heitler Bond H itl & London L d (1927) bond = a b Hybrid orbitals ‐ Pauling (1920s)  Molecular Orbital (MO) ‐ Hund, Mulliken (1920s) MO = a A + b B LCAOs ‐ Lennard‐Jones (1929) ( ) Normalization Requirement • From the Copenhagen interpretation we associate the probability of finding the electron in a vanishingly small volume element (“a point”) as P   • The electron has unit probability over all space: P = ∫  d = 1 • TTo ensure that th t any trial t i l wave ffunction, ti R R, meets t thi this criterion it i we generally ll mustt multiply lti l b by a normalization constant, N, such that P = N2 ∫  d = 1 • If  is a solution to the Schrödinger equation, then so too is N , and E in ,H( = E( is unaffected. Normalization Requirement • When two orbitals on different atoms A and B overlap, the nature and effectiveness of their interaction is given by the Slater overlap integral S = ∫ A B d which can be evaluated precisely only for H2 . • • For our purposes we only need to recognize when certain interactions of orbitals have S > 0, S = 0, or S < 0  S > 0 – bonding interaction  S = 0 – nonbonding interaction  S < 0 – antibonding interaction p p pictures of interactingg orbitals. The sense of S is evident from simple  S > 0 – reinforcing overlap between nuclei  S = 0 – equally reinforcing and cancelling overlaps  S < 0 – cancelling overlap between nuclei Examples of S > 0, S = 0, and S < 0 AO Shape and Symmetry • All AOs with the same value of l have the same symmetry, regardless of n. • As isolated orbitals, each type can be identified with a particular point group.  s = R3  p = C4v  d = D2h ("cloverleaf") or D4h (dz2) • Orbitals that have bonding combinations usually have the same kind of symmetry with respect to the internuclear axis. • If a bonding combination can be identified for any pair of orbitals, there must also be an antibonding combination. • Nonbonding orbitals are oriented relative to each other such that they do not have the same kind of symmetry relative to the internuclear axis. Symmetry Definition of Sigma, Pi, and Delta Interactions • Sigma  interactions are symmetrical relative to a C2 axis collinear with the internuclear axis. • Pi  interactions are anti‐symmetric ( changes sign) with respect to C2 collinear with the internuclear axis and with respect to v coplanar with the shared nodal plane. • Delta  interactions are symmetric to C2 and anti anti‐symmetric symmetric to C4 . • Bonding and antibonding interactions of the same type have the same symmetry with respect to the internuclear axis.  bonding  anti‐bonding  bonding  anti‐bonding Z 3p p px py pz dx2-dy2 dxy Y X 4s s 3d dz2 dxz dyz Metal Carbonyl C Y Z X Metal dz2 Z Z sp C bond carbonyl O Y X dy2 Metal dx2-dy sp C O bond carbonyl Y X Metal dyz * C -back-donation O carbonyl Metal Alkene Y Z X Metal dz2 Z CR2 CR2 complex alkene Y X Metal dx2-y2 Z CR2 CR2 complex alkene *CR 2 Y X Metal dyz back-donation CR2 alkene Fischer carbene Y Z X Metal dz2 C sp2 bond Z Z carbene sp2 R R Y X dy2 Metal dx2-dy C sp2 bond carbene sp2 R Y X R Metal dyz back-donation C R R carbene p