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INORGANIC CHEMISTRY

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CONTENTS

Units Page No.

1. Stereo chemistry and bonding of compounds of main groups 1

2. Nuclear Chemistry 38

3. Chemistry of f-Block Elements 80

4. Molecular Symmetry 103

5. Group Theory 151

Notes

Stereo chemistry and bonding of compounds of

main groups

1Self Learning Material

UNIT–1

Stereo Chemistry and Bonding of Compounds of Main Groups

(Structure) 1.1 Learning Objectives 1.2 Introduction 1.3 VSEPR theory 1.3.1 Gillespie laws 1.3.2 Applications of gillespie laws 1.3.3 Comparison of ch4, nh3, h2o and h3o

+

1.3.4 Comparison of pf5, sf4, clf3 and [icl2-]

1.3.4 Limitations of vsepr theory 1.4 Walsh diagrams 1.4.1 Application to triatomic molecules 1.4.2 Application to penta atomic molecules 1.5 Dp- pp bonds. 1.5.1 Dp- pp bonding in phosphorous group elements 1.5.2 Dp- pp bonding in nitrogen, oxygen and sulphur compounds 1.6 Bent's rule and energetics of hybridisation: bent rule 1.6.1 Third and fourth groups halides 1.6.2 Fifth and sixth groups hydrides & halides 1.6.3 Isovalent hybridisation 1.6.4 Apicophilicity 1.7 Some simple reactions of covalently bonded molecules 1.7.1 Atomic inversion 1.7.2 Berry pseudo rotation 1.7.3 Nucleophillic substitution 1.7.4 Free radical mechanism 1.8 Boranes and Higher Boranes. 1.9 Carboranes. 1.10 Let us sum up 1.11 Review questions 1.12 Further readings

Notes

Inorganic Chemistry

Self Learning Material2

1.1 ObjectivesAfter studying the chapter, students will be able to:

zz Discuss absolute shapes of various molecules;zz Explain VSPER Theory;

zz Describe Boran and carboranes.

1.2 Introduction The concept of molecular shape is of the greatest importance in inorganic chemistry for not only does it affect the physical properties of the molecule, but it provides hints about how some reactions might occur. Pauling and Slater (1931) in their Valence Bond Theory (VBT) proposed the use of hybrid orbital, by the central atom of the molecule, during bond formation. Thus, with the knowledge of the hybridisation used by the central atom of the molecule, one can predict the shape and also the angles between the bonds of a molecule. However, since then more advanced theories have come into existence. In this chapter we explore some of the consequences of molecular shape in terms of VSEPR theory and refine that concept into the powerful concept of molecular symmetry and the language of group theory, using Walsh diagrams and Bent's Theory, towards the end of the unit. You may recall what you have already studied about the directional property of a covalent bond and the concept of hybridisation of orbital to predict molecular geometry.

1.3 VSEPR THEORY

In order to predict the geometry of covalent molecules, Valence Shell Electron Pair

Repulsion Theory is used. This theory was given by Gillespie and Nyholm. According

to this theory the geometry of a molecule depends upon the number of bonding and

non-bonding electron pairs in the central atom. These arrange themselves in such a way

that there is a minimum repulsion between them so that the molecule has minimum

energy (i.e. maximum stability).

1.3.1 Gillespie Laws

The following rules have been reported by Gillespie to explain the shape of some

covalent molecules:

1. If the central atom of a molecule is surrounded only by bonding electron pairs and not by non-bonding electron pairs (lone pairs), the geometry of the molecule will be regular. In other words we can say that the shape of covalent

Notes


main groups


molecule will be linear for 2 bonding electron pairs, triangular for 3 bonding electron pairs. tetrahedral for 4 bonding electron pairs, trigonal bipyramidal for 5 bonding electron pairs:

2. When the central atom in a molecule is surrounded by both, bonding electron pairs as well as by lone pairs, then molecule will not have a regular shape. The geometry of the molecule will be disturbed. This alteration or distortion in shape is due to the alteration in bond angles which arises due to the presence of lone pairs on the central atom. How the presence of lone pairs causes an alteration in bond angles can be explained as follows:

At a fixed angle the closer the electric-pairs to the central atom, the greater is the repulsion between them. Since the lone-pair electrons are under the influence of only one positive centre (i.e. nucleus), they are expected to have a greater electron density than the bond-pair electrons which are under the influence of two positive centres. Thus lone pair is much closer to the central atom than the bond pair. Hence it is believed that lone pair will exert more repulsion on any adjacent electron pair than a bond pair will do on the same adjacent electron pair.

(lp - lp) > (lp - bp) ....................(i)

(lp = lone pair and bp = bond pair)

If the adjacent electron pair is a bond pair, then repulsive force between lone pair and bond pair will be greater than repulsive force between two bond pairs.

(lp - bp) > (bp - bp) ....................(ii)

On combining relations (i) and (ii) we get

(lp - lp) > (lp - bp) > (bp - bp)

Thus the repulsion between two lone pairs is maximum in magnitude, that between a bp and lp is intermediate while that between two bond pairs is the minimum.

Notes

Inorganic Chemistry


The more the numbers of lone pairs on a central metal atom, the greater is the contraction caused in the angle between the bonding pairs. This fact is clear when we compare the bond angles in CH4, NH3 and H2O molecules. (Table 1.1)

Table 1.1

3. B-A-B bond angle decreases with the increase in electro negativity of atom B in AB2 molecule where A is the central atom.

Example: Pl3 (102o) > P Br3 (101.5o) > PCl3 (100o)

4. Bond angles involving multiple bonds are generally larger than those involving only single bonds. However, the multiple bonds do not affect the geometry of the molecule.

5. Repulsion between electron pairs in filled shells are larger than the repulsion between electron pairs in incompletely filled shells.

Examples: H2O (105.5o) < H2S (92.2o)

1.3.2 Applications of Gillespie Laws

Let us take some examples in support of these laws:

(a) AX2 molecule, which has only two bond-pairs, will be linear:

X----A-----X

Examples in this groups will be BeCl2, CaCl2, CO2 etc.

(b) If the molecule is AX3 (I) or AX2 with a lone pair of electrons on the central atom A, i.e. AX2E (II), then the molecule will be triangular (Fig 1.1):

Fig 1.1 : (I) = BCl3, BF3 etc. (II) = SO2, SnCl2 etc.

Notes


main groups


(c) If the molecule is AX4 (III) or AX3E (IV) or AX2E2, then AX4 will be tetrahedral; AX3E will be pyramidal and AX2E2 will be angular. (Fig. 1.2):

Fig 1.2 : (III) = CCl4, CH4, SiCl4, GeCl4 etc.

(IV) = NH3, PCl3, As2O3 etc.

(V) = H2O, SeCl2, etc.

(d) If the molecule is AX5 (VI) or AX4E (VII) or AX3E2 (VIII) or AX2E3 (IX) then AX5 will be triangular bi pyramidal; AX4E will irregular tetrahedral; AX3E2 will be T-shaped,; and AX2E3 will be linear. (Fig. 1.3)

(VI) (IX)

(VIII) (IX)

Fig 1.3 : (VI) = PCl5; (VII) SF4, TeCl4 etc. (VIII) = ClF3, BrF3 etc. (IX) = XeF2, ICl2

- , or I3- etc.

(e) If the molecule is AX6 (X) or AX5E (XI) or AX4E2 (XII) then AX6 will be octahedral, AX5E will be square pyramidal; and AX4E2 will be square planar. (Fig. 1.4)

(X) (XI) (XII) Fig 1.4 : (X) = SF6, WF6, etc. (XI) = BrF5, IF5 etc. (XII) = ICl4,

XeF2- etc.

Notes

Inorganic Chemistry


1.3.3 Comparison of CH4, NH3, H2O and H3O+

In table 1.1 bond angles in CH4, NH3 and H2O molecules are given. In all these

molecules, the central atom (C, N and O respectively) is sp3 hybridised. But they differ

in the number of lone pair (s) present on the central atom, being zero in CH4, one in

NH3 and two in case of H2O. Thus the repulsive force between electron pairs gradually

increases in these molecules from CH4 to H2O, resulting in the change of geometry and

the bond angles. This CH4 (Four bond pairs) is tetrahedral with the characteristic bond

angle of 109.5. NH3 is pyramidal (Three bond pairs and one lone pair) and has a bond

angle of 107o. While H2O is angular (Two bond-pairs and two lone-pairs) and has bond

angle of 105o. The increasing lp-lp repulsion decreases the bond angles from 109.5o to

~107o in NH3 and ~105.5o in H2O.

On comparing H3O+ with these molecules, we notice that, it resembles with NH3

molecule. As the central atom of H3O + on (Oxygen) is also sp3 hybridised and has 3bp+1 lp. Thus H3O

+ will also be pyramidal with a bond angle of ~107o. (Fig 1.5)

Fig 1.5 :A- CH4, Tetrahedral, bond angle 109.5o

B- NH4, Pyramidal, bond angle ~107o

C- H2O, Angular, bond angle ~105o

D- H3O+, Pyramidal, bond angle ~107o

1.3.4 Comparison of PF5, SF4, ClF3 and [ICl2-]

comparison of PF5, SF4, ClF3 and [ICl2]- species clearly indicates that each of these molecules have 10 electrons in the valence shell of the central atom, being P, S, Cl and I respectively. In addition to this, the central atom in each case is sp3d hybridised, and has 0, 1, 2 and 3 lone pairs (lp) respectively. In PF5, all the five electron pairs (= 10 electrons) are bond pairs and are housed in the five sp3d hybrid orbital; resulting

Notes


main groups


in the trigonal bipyramidal geometry of the molecule. (Fig 1.6 (a)). SF4 molecule has 4 bond-pairs and one lone pair on the central S atom. The lp in this molecule has two options- it can sit in a axial or in an equatorial orbital. In the axial position (Fig. 1.6 (b))i)) it has three bps at 90o and one bp directly opposite to itself. While in equatorial position (Fig. 1.6 (b)(ii)) it has two bps at 90o and two bps at 120o. As in the equatorial position lp-bp repulsion is less and expansion is easy the lp prefers the equatorial position and the molecule is therefore irregular tetrahedral.

In ClF3., the two lps may be axial-axial (Fig 1.6(c)(i)), or axial-equatorial (Fig 1.6(c)(ii)), or equatorial-equatorial (Fig 1.6(c)(iii)) positions. As the axial position will result in maximum repulsion hence the axial position for the lp is ruled out. Thus the molecule will have T. shaped geometry, according to the Fig. 1.6(c)(ii).

Similarly, due to reduced lp-lp repulsions and a larger volume that a lp can occupy at equatorial positions, the [ICl2]- ion will be linear. The three lps occupy the three equatorial positions, leaving the axial positions for the Cl atoms (Fig. 1.6(d)).

(a) PF5, Trigonal bipyamidal

(I) (II)

(b) SF4, Irregular tetrahedral

(c) ClF3, T-Shaped

Notes

Inorganic Chemistry


(d) [ICl2]-, Linear

Fig. 1.6 : Structures of PF5, SF4, ClF3 and [ICl2]-

Limitations of VSEPR Theory

1. This theory is not able to predict the shapes of certain transition element complexes.

2. This theory is unable to explain the shapes of certain molecules with an inert pair of electrons.

3. This theory is unable to explain the shapes of molecules having extensive delocalised p-electron system.

4. This theory can not explain the shapes of molecules which have highly polar bonds.

1.4 WALSH DIAGRAMS

We know all the systems want to be in a stable state, and the stable state is one in which

it has the minimum possible energy. Same is true for the stereochemistry or the geometry

of a molecule. The VSEPR theory considered that the most stable configuration of a

molecule is one in which repulsive forces between the valence electron-pairs is minimum.

In contrast, the molecular orbital theory (MOT) considers that the stable geometry of a

molecule can be determined on the basis of the energy of molecular orbitals formed as

a result of linear combination of atomic orbitals (LCAO). In 1953 A.D.Walsh proposed

a simple pictoral-approach to determine the geometry of a molecule considering and

calculating the energies of molecular orbitals of the molecule.

The basic approach is to calculate the energies of molecular orbitals for two limiting

structures, say linear or bent to 90o for an AB2 molecule, and draw a diagram showing

how the orbitals of one configuration correlate with those of the other. Then depending

on which orbitals are occupied, one or the other structure can be seen to be preferred.

By means of approximate MO Theory implemented by digital computers, this approach

has been extended and generalized in recent years. Walsh's approach to the discussion

Notes


main groups


of the shape of an AB2 triatomic molecule (such as BeH2 and H2O) is illustrated in Fig.

1.8. The illustration shows an example of a Walsh diagram, a graph of the dependence

of orbital energy on molecular geometry. A Walsh diagram for an B2A or AB2 molecule

is constructed by considering how the composition and energy of each molecular orbital

changes as the bond angle changes from 90o to 180o. The diagram is in fact just a more

elaborate version of the correlation diagram.

1.4.1 Application to Triatomic Molecules

The coordinate system for the AB2 molecule is shown in Figure 1.8. The AB2 molecule

has C2v symmetry when it is bent and, when linear D2h symmetry. To simplify notations,

however, the linear configuration is considered to be simply an extremum of the C2v

symmetry. Therefore the labels given to the orbitals through the range 90o ≤ θ < 180o

are retained even when θ = 180o. The symbols used to label the orbitals are derived

from the orbital symmetry properties in a systematic way, but a detailed explanation

is not given here. For present purposes, these designations may be treated simply as

labels. (Fig. 1.7).

Fig. 1.7

The A atom of AB2 molecule will be assumed to have only s, px, py and pz orbitals in

its valence shell, whereas each of the B atoms is allowed only a single orbital oriented

to form a σ bond to A. In the linear configuration PAX and PAZ are equivalent non-

bonding orbitals labelled 2a, and b1 respectively. The orbitals SA and PAY interact with

σ1B and σ2B, orbitals on the B atoms, to form one very strongly bonding orbital, 1a, one

less strongly bonding orbital, 1b2, one less strongly bonding 3a1 and 3b2. The ordering

of these orbitals and in more detail, the approximate values of their energies can be

estimated by an MO calculation. Similarly, for the bent molecule the MO energies may

be estimated. Here only pzA is non bonding, spacing and even the order of the other

orbitals is function of the angle of bending θ.

The complete pattern of orbital energies, over a range of θ, is obtained with typical input

parameters. This is shown in the figure 1.8. Calculations in the Huckel approximation

are simple to perform and give the correct general features of the diagram but for certain

Notes

Inorganic Chemistry


cases (e.g. AB2E2) very exact computations are needed for an unambiguous prediction

of structure.

(Angle θ)Figure 1.8 Orbital Correlation Diagram For AB2 Triatomic Molecules Where A

uses only s and p orbitals

Form the approximate diagram (Fig. 1.8) it is seen that an AB2 molecule (one with no lone pairs) is more stable when linear then when bent. The 1b2 orbital drops steadily in energy form θ = 90o to 180o; while the energy of the 1a1 orbitals is fairly insensitive to angle.

For an AB2E molecule the results are ambiguous, because the trend in the energy of the

2a1 orbital approximately offsets that of the 1b2 orbiral.

For AB2E2 molecules, the result should be the same as for AB2E. Since the energy of

b1 orbirtal is independent of the angle. Thus it is not clear in this approach that AB2E2

molecules should necessarily be bent, but all known ones are.

The H2O Molecule:

Because of its unique importance, this molecule has been subjected to more detailed

study than any other AB2E2 molecule. A correlation diagram calculated specially for H2O

is shown in the figure. Although it differs in detail for the general AB2E2 shown in the

figure it is encouraging to see that the important qualitative features are the same. The

general purpose diagram pertains to a situation in which there is only a small energy

difference between the ns and np orbitals of the central atom. As stated in discussing that

general purpose diagram, it is not clear whether an AB2E2 molecule ought necessarily

to be bent.

In the diagram calculated expressly for H2O the lowest level's is practically pure 2s and its

energy is essentially constant for all angles. It can be determined from this diagram that

the energy is minimized at an angle of 106o, essentially in accord with the experimental

Notes


main groups


value of 104.5o. (Fig. 1.9).

Fig. 1.9

BeH2 - Molecule

The simplest AB2 molecule in Period 2 is the transient gas-phase BeH2 molecule

(BeH2 normally exists as a polymeric solid), in which there are four valence electrons.

These four electrons occupy the lowest two molecular orbitals. If the lowest energy is

achieved with the molecule angular, then that will be its shape. We can decide whether

the molecule is likely to be angular by accommodating the electrons in the lowest two

orbitals corresponding to an arbitrary bond angle in Fig. 1.7. We then note that the

HOMO decreases in energy on going to the right of the diagram and that the lowest

total energy is obtained when the molecules is linear. Hence, BeH2 is predicted to be

linear and to have configuration 1sg2 2su

2 . In CH2, which has two more electrons than

BEH2, three of the molecular orbital must be occupied. In this case, the lowest energy

is achieved if the molecule is angular and has configuration .

The principal feature that determines whether or not the molecule is angular is whether

the 2a1 orbital is occupied. This is the orbital that has considerable A-2s character in

the angular molecule but not in the linear molecule.

1.4.2 Application to Penta Atomic Molecules

For peta-atomic molecules examples of CH4 and SF4 may be taken for consideration.

For these molecules, two geometries are possible: one a symmetrical tetrahedral and

the other, a distorted tetrahedral geometry (or a tetragonal geometry of relatively lower

symmetry).

CH4 - Molecule

Methane, CH4, has eight valence electrons. During bonding, for orbitals [a1g (2s)

and t1u (2p)] of carbon, and one [a1g (1s)] orbital of each four hydrogen atoms take

part. Overlapping of these eight orbitals, eight molecular orbitals are formed, the four

Notes

Inorganic Chemistry


bonding (2σg or 2a1 and 2tσ or 2t1) and the four antibonding (2σg* and 2t*σ). The eight

valency electrons of CH4 molecule are distributed in the four bonding molecular orbitals

(Electronic Configuration, 2a12 2t1

b ). In the tetrahedral geometry due to overlapping

with the orbitals of hydrogen atom the energy of 2a1g and 2t1u orbitals is considerably

reduced. In contrast, in the distorted geometry, comparatively less overlapping of t1u

orbitals with the hydrogen orbitals (as compared to that in the tetrahedral geometry)

the energy of 2t1 molecular orbitals increases. Thus the geometry of CH4 molecule is

symmetrical tetrahedral, rather than a distorted tetrahedral.

SF4 - MoleculeIn the valence shall of sulphur atom, in SF4 molecule, in addition to 3s (α1g) and 3p

(t1u) orbitals, 3d (t2g and eg) orbitals are also present. During the bonding, 2pz orbital of each of the four fluoring atoms take part. As a result for bonding (2a1 and 2t1) and four antibonding (2a1

* 2t1* ) molecular orbitals formed; while the d-orbitals [α1(dz2), b1(dx2

- y2) and t2 (dxy, dxz, dyz)] are present as non-bonding molecular orbitals. Ten valency electrons of SF4 molecule remain distributed in the four bonding and one non-bonding molecular orbitals, resulting in . 3a1

2 configuration. As in the distorted geometry, overlapping of 2t1 orbitals is comparatively greater (thus reducing their energy) and the filling in 3α1 orbital considerably reduce the energy of the system as compared to that in the regular tetrahedral structure. Hence SF4 molecule has a distorted tetrahedral geometry, rather than a regular tetrahedral structure. Thus we can say 'Walsh Diagrams' are complementary to the VSEPR concept.

1.5 dp- pp BONDS.

There are several structural phenomena that have traditionally been attributed to the

formation of dp- pp Bonds. Recent work has raised some doubts. The phenomena in

questions are exemplified by:

The fact that for amines such as (R3Si)2 NCH3, (R3Si)3 N and (H3Ge)3 N, the

central NSi2C, NSi3 and NGe3 skeletons are planar. Many tetrahedral species such as

SiO4-4, PO4

-3, SO4-2 and ClO-4 have bond lengths shorter than those predicted from

conventional tables of single bond radii. In silicates the Si-O-Si units also show what

were considered to be Si-O distance that are "too short" for single bonds.

Recent re-examinations of these phenomena by both theoretical and experimental

methods together with earlier arguments now suggest that the dp- pp contributions to

these effects are at best small. Thus, in reading literature written prior to 1985, where

such interactions are often accorded great importance, one should now be sceptical of

all but the facts themselves.

Notes


main groups


This is not to say that dp- pp bonding in main group compounds is never important.

Probably in the case of - S - N = S units, and in F3F ≡ N, where the S - N distances are

very short indeed. However, it is always dangerous to attribute all structural effects to

simple orbital overlaps, even if the explanation seems to fit, and the rise and fall of the

dp- pp overlap hypothesis is a case in point. In a multiple bonded molecule having

bond pairs + lone pairs = 4, 5 or 6, the p bonds will be dp- pp bond. In this case the

central atom uses all its p-orbitals for hybridisations and has only d-orbitals available

to overlap with p-orbitals of the adjacent atom to give dp- pp bond.

The formation of dp- pp bond is common for all the second period elements and is

not important for the elements of third and higher periods. The dp- pp bonding is

more favourable than the dp- pp bonding for higher atoms of third and higher periods.

1.5.1 dp- pp Bonding in Phosphorous Group Elements

Phosphine Oxide, R3P = 0, presents an important example of the participation of d-atomic

orbitals of nonmetallic elements in p bonding. Presence of p bonding is defected with

the help of the evidences, such as reduction in the bond length, increase in the bond

strength and the stabilisation of charge distribution. On these grounds, compared to

ammine oxide, phosphine oxide presents a strong evidence of the presence of dp- pp

bond, in a very high stability of P = O.

Similarly the fact that almost all t-phosphines are readily oxidised into R3P = 0, also

indicates that dp- pp bond is present in the P = O bonding. This is supported by the

lower dipole moment of triethyl phosphine oxide (1.4 x 10-3 Cm. cf 16.7 x 10-3 Cm.

of trimethyl amine oxide), higher dissociation energy of P = O bond (500-600 KJ, cf

200-300 KJ of N→O bond) and the smaller P-O bond lengths in phosphoryl compounds.

1.5.2 dp- pp Bonding in Nitrogen, Oxygen and Sulphur Compounds

There are number of examples, which show dp- pp bonding in nitrogen, oxygen and

sulphur compounds:

(i) Mobile p bonding in trisilyl amine results in the resonance in the molecule :

(2) The bond angles in disiloxane, H3SiOSiH3, and silyl isothiocyanate, H3Si-N=C=S indicates dp- pp back-bonding (in comparison to ether and methyl isothiocyanate)

Notes

Inorganic Chemistry


The argument given against the use of d atomic orbitals in bonding by non-metallic elements is that a very high excitation energy is required for the same. Hence it may be concluded that the use of d-atomic orbitals in bonding by non-metallic elements will be possible only in their higher oxidation states and when they are linked with strong electronegative elements, eg. PF5,SF6,OPX3 etc.:

(i) The N - S bond length in N ≡ SF3 indicates. The bond order = 2.7 indicating dp - pp bonding:

Thiazytrifluoride

N - S bond length = 141.6 pm; Bond order = 2.7

(cf N - S = 174 pm, b.0 = 1, N = S = 154 pm; b.0 = 2)

In S4N4F4 also there is indication of dp - pp bonding (compare with S4H4N4):

Tetra Sulphur tetramide Tetra sulphur tetramide fluoride

(Isoelectronic and Isomorphous with S8) (Alternate S = N bond)

Notes


main groups


(ii) In diphenyl phosphonitrilic fluoride, there is evidence of ה bonding in the ring:

1.6 BENT'S RULE AND ENERGETICS OF HYBRIDISATION: BENT RULE

Bent rule may be stated as follows: "More electronegative constituents 'prefer' hybrid

orbitals having less s character and more electropositive substituents 'prefer' hybrid orbitals

having more s character." An example of Bent rule is provided by the fluoromethanes. In

CH2F2, the F-C-F bond angle is less than 109.5o, indicating less than 25% s character,

but the H-C-H bond angle is larger and C-H bond has more s character. The bond angle

in the other fluoromethanes yield similar results. The tendency of more electronegative

substituents to seek out the low electronegative pxdx2 apical orbital in TBP structures

is often termed "apicophilicity". It is well illustrated in a series of oxysulfuranes of the

type-prepared by Martin and Co-workers. These, as well as related phosphoranes provide

interesting insight into certain molecular rearrangements.

Bent's rule is also consistent with and may provide alternative rationalization for

Gillespie's VSEPR model. Thus the Bent's rule prediction that highly electronegative

constituents will 'attract' p character and reduce bond angles is compatible with the

reduction in regular volume of the bonding pair when held tightly by an electronegative

substituents. Strong, s-rich covalent bonds require a larger volume in space to bond. Thus

double bonded oxygen, despite the high electro negativity of oxygen, seeks s-rich orbitals

because of the shortness and better overlap of the double bond. Again, the explanation,

whether in purely s-character terms (bent's rule) or in larger angular volume for a double

bond (VSEPR), predicts the correct structure.

The mechanism operating behind Bent's rule is not completely clear. One factor favouring

increased p character in electronegative substituents is the decreased bond angles of p

orbitals and the decreased steric requirements of electronegative substituents. There

Notes

Inorganic Chemistry


may also be an optimum strategy of bonding for a molecule in which the character

is concentrated in those bond in which the electronegativity difference is small and

covalent bonding is important. The p character, if any, is then directed towards bonds

to electronegative groups. The latter will result in greater ionic bonding in a situation in

which covalent bonding would be low anyway because of electronegativity difference.

Some light may be thrown on the workings of Bent's rule by observations of apparent

exceptions to it. The rate exceptions to broadly useful rules are unfortunate with respect

to the universal applications of those rules. They also have the annoying tendency to

be confusing to someone who is encountering the rule for the first time. On the other

hand, any such exception or apparent exception is a boon to the research since it almost

always provides insight into the mechanism operating behind the rule. Consider the

cyclic bromophosphate ester.

The phosphorus atom is in an approximately tetrahedral environment using four σ bonds

of approximately sp3 character. We should expect the more electronegative oxygen

atoms to bond to s-poor orbitals on the phosphorus and the two oxygen atoms in the

ring do attract hybridizations of about 20%s. The most electropositive constituent on

the phosphorus is the bromine atom and Bent's rule would predict an s-rich orbital,

but instead it draws another s-poor orbital on the phosphorus atom is that involved in

σ bond to the exocyclic oxygen. This orbital has nearly 40% s-character. The oxygen

atom ought to be about as electronegative as the other two, so why the difference? The

answer probably lies in the overlap aspect.

The large bromine atom has diffuse orbitals that overlap poorly with the relatively

small phosphorus atom. Thus, even though the bromine is less electronegative than the

oxygen, it probably does not form as strong a covalent bond.

The presence of a p bond shortens the exocyclic double bond and increases the

overlap of the σ orbitals. If molecules respond to increase in overlap by rehybridization

in order to profit from it, the increased s-character then becomes reasonable. From this

point of view, Bent's rule might be rewarded. The p character tends to concentrate in

orbitals with weak covalently (from either electro negativity or overlap considerations),

and s-character tends to concentrate in orbitals with strong covalently matched electro

Notes


main groups


negativities and good overlap. Some quantitative support for the above qualitative

arguments comes from average bond energies of phosphours, bromine and oxygen.

P-Br 264 KJ Mol-1

P-0 335 KJ Mol-1

P=0 544 KJ Mol-1

Bent's rule is a useful tool in inorganic and organic chemistry. For example, it has been

used to supplement the VSPER interpretation of the structures of various non-metal

fluorides, and should be applicable to a wide range of question on molecular structure.

Energetics of Hybridisation:According to hybridisation model, bond directions are determined by a set of

hybrid orbitals on the central atom which are used to form bonds to the ligand atoms and to hold unshared pairs. Thus AB2 molecules are linear due to the use of linear sp hybrid orbitals. AB2 molecule should be equilaterally triangular, while AB2E molecule should be angular, due to use of trigonal sp2 hybrids. AB4, AB3E and AB2E2 molecule should be tetrahedral, pyramidal and angular, respectively, because here sp3 hybrid orbitals are used.

These cases are, of course, very familiar and involve no more than an octet of electrons.

For the AB5, AB4E, AB3E2 and AB2E3 molecules the hybrid must now include orbitals in

their formation. The hybrid orbitals used must be of the sp3dz2 leading to TBP geometry

and Sp3dx2-y2 leading to SP geometry. There is no way to predict with certainty which

set is preferred, and doubtless that difference between them connot be great, since we

know experimentally that AB5 molecules nearly all have TBP structures, the same

arrangement is assumed for the AB4E cases, and so on. Even this adhoc assumption

does not solve all difficulties, since the position preferred by lone pairs must be decided

and there is no simple physical model here (as there was in the VSEPR approach) to

guide us. A preference by lone pairs for equatorial positions has to be assumed. With

these assumptions, a consistent correlation of all structures in this five-electron-pair

class is possible.

For AB6 molecule, octahedral sp3d2 hybrids are used. For AB4E2 molecule; there is

nothing in the directed valence theory itself to show whether the lone pairs should be

cis or trans. The assumption that they must be trans leads to consistent results.

The most fundamental problem with the hybridisation model is that in all cases in which

there are more than four electron pairs in the valence shell of the central atom, it is

necessary to postulate that at least one d orbital becomes fully involved in the bonding.

Notes

Inorganic Chemistry


There are both experimental and theoretical reasons for believing that this is too drastic

an assumpiton. Some recent MO calculations and other theoretical considerations suggest

that although the valence shell d orbitals make a significant contribution to the bonding

in many cases, they never play as full a part as do the valence shell p orbitals. Fairly

directed experimental evidence in the form of nuclear quadruple resonance studies of

the I Cl-2 and ICl-34 ions shows that in these species, d-orbitals participation is very

small. This participations is probably greater in species with more electronegative ligand

atoms such as PF5, SF6 and Te (OH)6 but not of equal importance with the contribution

of the s and p orbitals.

Perhaps it is surprising that by going to the opposite extreme, namely by omitting all

consideration of d orbitals, but still adhering to the concept of directed orbitals it is again

possible to rationalize many of the principal features of the structures of main group.

Es+p3 = [2(-1806)]+[3(-981)] = 6555 KJ mol-1

For tetrahedral hybridised phosphours.

(3te22te13te1) the energy will be:

te = 5x (-1187) = -5935 KJ mol-1

In this case the hybridisation has cast 620 KJ mol-1 of energy or roughly two bonds worth of energy. This is shown graphically in the figure 1.10.

Figure 1.10

The energy difference between the hybridised and unhybridised atom represents the increase in energy of the two electrons in the filled 3s orbital and the decrease in energy of the electrons in the half-filled 3p orbitals.

The energetics of hybridisation, together with the principle of good overlap, are important in determining the electronic structure of molecules.

1.6.1 Third and Fourth Groups Halides

For promoting an atom to hybridised excited state energy is required. But when the hybrid orbitals give very strong bonds, the energy gained from bonding may be used for excitation. For example, when carbon forms four covalent bonds, although there is promotion energy from Is22s22p2 ls22s12p3, this is independent of the hybridization to the valence state :

Notes


main groups


Figure 1.11 Hybridisation energy of carbon

This energy of hybridisation is the order of bonding energy. Its important use is to determine structure of molecules. For example, the stability of the halides of the III A (Gr. 13) and IV A (Gr. 14) viz BCl3, AlCl3, GaCl3, InCl3, TiCl3 and CCl4, SiCl4, GeCl4, SnCl2, PbCl2, can be explained on the basis of hybridisation energy. The heaviest elements of these groups (Thallium and, Tin and Lead) the stable oxidation-states are two unit less than the maximum oxidation states (3 and 4 respectively) i.e. One in Thallium, TlCl and two in Tin and Lead, SnCl2 and PbCl2. Although, in these elements (Compared to the light elements of the group) excitation is easy. It is because, in the heavier elements orbitals are more diffused (Fig 1.12), hence in a volume region, values of Ψ Tl or Pb are less, compared toΨ B or Ψ Si. This results in lesser overlapping of the central atom orbitals with orbitals of chlorine atoms. Hence, in the compounds of heavy elements bonds are weaker, compared to that of the light elements. Because of this less effective overlapping, the heavy elements utilise pure p-orbitals in bonding and letting the lone-pair 'sink' into a pure s-orbital. Hence, the stable chlorides of thallium and lead are TlCl and PbCl2.

Figure 1.12 Effect of orbital size on overlapping

1.6.2 Fifth and Sixth groups Hydrides & Halides

As the energy of hybridization is of the order of magnitude of bond energies and can thus be important in determining the structure of molecules. It is responsible for the

Notes

Inorganic Chemistry


tendency of some lone pairs to occupy spherical, nonstereochemically active s-orbitals rather than stereochemically active hybrid orbitals. For example, the hydrides of the Group VA (15) and VIA (16) elements are found to have bond angles considerably reduced as one progresses fro m the first element in each group to those that follow (table 1.2 ).

Table 1.2: Bond angles in the hydrides of Groups VA (15) and VIA (16)

NH3 = 107.2o PH3 = 93.8o AsH3 = 91.8o SbH3 = 91.3o

OH2 = 104.5o SH2= 92o SeH2 = 91o TeH2 = 89.5o

An energy factor that favours reduction in bond angle in these compounds is the hybridization discussed above. It costs about 600 KJ mol-1 to hybridize the central phosphorus atom. From the standpoint of this energy factor alone the most stable arrangement would be utilizing pure p-orbital in bonding and letting the lone pair "sink" into a pure s-orbital. Opposing this tendency is the repulsion of electrons, both bonding and nonbonding (VSEPR). This favours an approximately tetrahedral arrangement. In the case of the elements N and O the steric effects are most pronounced because of the small size of atoms of these elements. In the larger atoms, such as those of P, As, Sb, S, Se and Te, these effects are somewhat relaxed, allowing the reduced hybridization energy of more p character in the bonding orbitals to come into play. The molecule is thus forced to choose between higher promotion energies and better overlap for an-s-rich hybrid, or lower promotion energies and poorer overlap for an s-poor hybrid. (s-character: sp3 (25%)<sp2(-33%)<sp(50%)); s-rich means>25%; s-poor means<25%)

1.6.3 Isovalent Hybridisation

In many tetravalent molecules the bond angle is seen slightly distorted, than the ideal

109o28' for example in CH3Cl, H-C-H bond is 110o20'. This deviation may be explained

in terms of 'Isovalent Hybridisation". Consider an imaginary molecule A-M-B. If in this

molecule, B is replaced by a strong electronegative element C, then M is rehybridised

in such a way that the orbital used for bonding C has more p character, than the orbital

used for bonding B. Hence in A-M-C molecule, M-A bond will have more s-character,

compared to that in A-M-B molecule. For example, as compared to CH3NH2, the carbon

atom in CH3OH uses a hybrid orbital having more s-character to link methyl hydrogen.

As a result the C-O bond has more p character as compared to the C-N bond. This is

an example of Bent's rule.

1.6.4 Apicophilicity

Good example of the effect of the differences in hybrid bond strengths are shown by the

bond lengths in MXn Molecules with both equatorial and axial constituents. (Table 1.3)

Table 1.3

req(pm) rax(pm)

Notes


main groups


PF5 153.4 157.7PCl5 202 214SbCl5 231 243SF4 154 164ClF3 159.8 169.8BrF3 172.1 181.0

An sp3d hybrid orbital set may be considered to be a combination of pzdz2 hybrids

and spxpy hybrids. The former make two linear hybrid orbitals bonding axially and

the latter form the trigonal, equatorial bonds. The sp2 hybrid orbitals are capable of

forming stronger bonds, and they are shorter than the weaker axial bonds. When the

electronegativities of the substituents on the phosphorus atom differ, as in the mixed

chlorofluorieds, PClxF5-x, and the alkylphosphorus fluorides, RxPF5-x it is experimentally

observed that the more electronegative substituent occupies the axial position and the

less electronegative substituent is equatorially situated. This is an example of Bent's

rule which states: More electronegative substituents 'prefer' hybrid orbitals having less

s-character, and more electropositive substituents 'prefer' hybrid orbitals having more

s-character. A second example of Bent's rule discussed earlier is that of the fluoromethanes.

In CH2F2 the F-C-F bond angle is less than 109.5o indicating less than 5% s character,

but the H-C-H bond angle is larger and the C-H bond has more s character. The bond

angles in the other fluoromethanes yield similar results.

The tendency of more electronegative substituents to seek out the low electronegativity

pzdz2 apical orbital in TBP structure is often termed "apicophilicity". It is well illustrated

in a series of oxysulfuranes.

1.7 SOME SIMPLE REACTIONS OF COVALENTLY BONDED MOLECULES

One of the major differences between organic and inorganic chemistry is the relative

emphasis placed on structure and reactivity. Structural organic chemistry is relatively

simple, as it is based on diagonal, trigonal or tetrahedral carbon. Thus organic chemistry

has turned to the various mechanisms of reaction as one of the more exciting aspects

of the subject, to contrast, inorganic chemistry has a wide variety of structural types to

consider, and even for a given element there are many factors to consider. Inorganic

chemistry has been, and to a large extent still is more concerned with the static structure

of reactants or products than with the way in which they interconvert. This has also been

largely a result of the paucity of unambiguous data on reaction mechanisms. However,

Notes

Inorganic Chemistry


this situation is changing. Interest is increasingly centring on how inorganic molecules

change and react. Most of this work has been done on coordinate chemistry, and much

of it will be considered later on, but a few simple reactions of covalent molecules will

be discussed here.

1.7.1 Atomic Inversion

The simplest reaction is seen in a molecule of ammonia. This can undergo the simple

inversion of the hydrogen atoms about the nitrogen atom. This is analogous to the

inversion of an umbrella in a high wind.

One might argue that above equation does not represent a reaction because the product

is identical to the reactant and no bonds were formed or broken in the process. Leaving

aside, the process illustrated above is of chemical interest and worthy of chemical study.Consider the trisubstituted amines and phophines shown in the figure below.

(Chiral amines and phophines)

Because these molecules are non superimposable upon their mirror images (i.e. they

are chiral) they are potentially optically active, and separation of the enantiomers is

at least theoretically possible. Racemization of the optically active material can take

place as shown in mechanism of NH3. It is of interest to note that the energy barrier to

inversion is strongly dependent on the nature of the central atom and that of subsequent.

For example, the barrier to inversion of methyl propyl phenylphosphine is about 120 MJ

Mol-1. This is sufficient to allow the separation of optical isomers, and their racemization

may be followed by classical techniques. In contrast, the barrier to inversion in most

amines is low (-40 KJ mol) with such low barriers to inversion, optical isomers cannot

be separated because racemization takes place faster than the resolution can be affected.

Since traditional chemical separations cannot effect the resolution of the racemic

mixture, the chemist must turn to spectroscopy to study the rate of interconversion of

the enantiomers.

Notes


main groups


1.7.2 Berry Pseudo Rotation

In PF5 the fluorine atoms are indistinguishable by means of NMR of F. This means that

they are exchanging with each other faster than the NMR instrument can distinguish

them. The mechanism for this exchange is related to the inversion reaction we have seen

for amines and phosphines. The mechanism for this exchange is believed to take place

through conversion of the ground state trigonal bipyramidal into a square pyramidal

transition state and back to a new trigonal bipyramidal structure.

This process results in complete scrambling of the fluorine atoms at the equatorial and

axial positions in phosphorus pentafluoride. If it occurs faster than the time scale of

NMR experiment, all the fluorine atoms appear to be identical. Because it was first

suggested by Berry, and because, if all of the substituents are the same as in PF5, the

two triogonal bipyramidal arrangements are related to each other by simple rotation,

the entire process is called a Berry pseudorotation. Note that the process can take place

very readily because of the similarity in energy between trigonal bypyramidal and square

pyramidal structures.

(Berry pseudorotation in Pentavalent Phosphorus Compound)

In fact the series of 5-coordinated structures collected by Muetterties and Guggenberger,

which are intermediate between trigonal bipyramidal and square pyramidal geometrically

effectively provides a reaction coordinate between the extreme structures in the Berry

pseudorotation.

1.7.3 Nucleophillic Substitution

The simplest reaction path for neucleophilic displacement may be illustrated by solvolysis

of a chlorodialkylphophine oxide.

Notes

Inorganic Chemistry


We would expect the reaction to proceed with inversion of configuration of the phosphorus

atom. This is generally observed especially when the entering and leaving groups are

highly electronegative and is thus favorably disposed at the axial positions, and when

the leaving group is one that is easily displaced. In contrast in some cases when the

leaving group is a poor one, it appears as though front side attack takes place because

there is retention of configuration. In either case, the common inversion or the less

common retention, there is a contrast with the loss of stereochemistry associated with

a carbonium ion mechanism.

1.7.4 Free Radical Mechanism

In the atmosphere there are many free radical reactions initiated by sunlight. One of the

most important and controversial sets of atmospheric reactions at present is that revolving

around stratospheric ozone. The important of ozone and the effect of ultraviolet radiation

on life will be discussed later, but we may note briefly that only a small portion of the

sun's spectrum reaches the surface of the earth and that parts of UV portion that are

largely screened can cause various ill effects to living systems. The earth is screened from extremely high energy UV radiation cleaves the oxygen molecule to form two free radicals of oxygen atoms.

O2 + hv (below 242 mm) O. + O.

The oxygen atoms can then attack oxygen molecules to form ozone.

O. + O2 + m m + O3

The neutral body m carries off some of the kinetic energy of the oxygen atoms. This

reduces the energy of the system and allows the bond to form to make ozone. The net

reaction is therefore:

3O2 + hv 2O3

The process protects the earth from the very energetic, short wavelength UV radiation

and at the same time produces ozone, which absorbs somewhat longer wavelength

radiation by similar process:

O3 + hv (220-230 mm) O2 + O.

Thus the process is repeated.

1.8 BORANES AND HIGHER BORANES.

Boron hydrides are known as Boranes. These are named boranes in analogy with

alkanes. These are gaseous substance at ordinary temperatures. It is expected that boron

would form the hydride BH3, but this compound is unstable at the room temperature.

Notes


main groups


However, higher hydrides like B2H6(diborance). B4H12 (tetraborane), B6H10(hexaborane),

B10H14(decaborane) etc. are known. The general formula of boranes are BnHn + 4 and

BnHn + 6 (Proposed by stock). In addition to these is one, recently discovered series of

closed polyhedral structures with the formula [BnHn]2-. Higher boranes have different

shapes, some resemble with nests, some with butterfly and some with spider's web.

The modern explanation of the structure of boranes is due to C.L.Higgins, who proposed

the concept of three centred two electron bond ( -bond) Fig. He also proposed the concept

of completely delocalised molecular orbitals to explain structures of boron polyhedrons.

He established icosahedral structure of [B12H12] Fig given below.

Fig.: 3C, 2e bond in B2H6

Fig.: B12H12 Icosahedron

In higher boranes, in addition to two centred two electron (2c, 2e) and the three

centred two electron bond (3c, 2e bond) present in diborance, B-B 2C, 2e and B-B-B

(3c, 2e) bonds are also important. In B-B-B bonds, three atoms of boron with their sp3

hybridisation are placed at the corners of a equilateral triangle .

Fig. : B-B-B bond

Notes

Inorganic Chemistry


1.8.1 Wade's Rule

In 1970 K. Wade gave a rule relating the number of electrons in the higher

borane molecules with their formulae and shapes. Using these rules one can predict the

general shapes of the molecules from their formulae. These rules are also applicable

on carboranes and other polyhderal molecules called 'Deltahedral's Deltahedrons are so

called, as they are composed of delta, ∆ , shaped triangular faces.

According to Wade's rule, the building blocks of deltahedrons are BH units, which are

formed by sp-hybridisation of boron atom. Out of the two sp hybrids one is used in the

formation of 2c, 2e B-H exo bond of the deltahedron and the other sp hybrid is directed

inside as a radial orbital. Remaining two unhybridised p orbitals of each boron atoms are

placed perpendicular to the radial orbitals and are known as tangential orbitals. These

radial and tangential orbitals combine by linear combination method to form skeleton

or framework of the deltahedron. To fill all bonding molecular orbitals of the skeleton,

necessary number of electrons are obtained form the radial orbitals of BH units and s

orbitals of the extra hydrogen atoms.

These electrons are called Skeletal electrons. For example in B4H10, four BH units

contribute 8 electrons (4 x 2 = 8) and six extra hydrogens give six electrons thus B4H10

has total 14 skeletal electrons figure given below shows the molecular energy diagram

of [B6H6]2-. This molecule has seven pairs of skeletal electrons (six boron atoms and

one pair from two negative charges). These are used to saturate seven skeletal molecular

orbitals (a1g, t1u and t2g).

Fig. : Skeletal molecular energy diagram of [B6H6]2-

Notes


main groups


Classification:

On the basis of structures, molecular formula and skeletal electrons higher

boranes are classified into Closo, Nido, Arachno and Hypo :

Table 1.3

Name Formula Skeletal

Electron Pair

Examples

Closo [BnHn]2- n+1 [B5H5]

2- to [B12H12]2-

Nido [BnHn+4] n+2 B2H6 , B5H9, B6H19Arachno [BnHn+6] n+3 B4H10 , B5H11Hypo [BnHn+8] n+4 Only derivatives are known.

1.8.2 Closo Boranes

These are closed structured (Closo, Greak, meaning cage) boranes with the

molecular formula [BnHn]2- and skeletal electrons = n+1 pairs (= 2n+2 electrons). In

this structure, there is one boron atom placed at each apex and there are no B-H-B

bonds present in the molecule. All the member of the series from n=5 to 12 are known.

[B5H5]2- is trigonal bipyramidal, [B6H6]2- is octahedral and [B12H12]2- is icosahedral.

All are stable on heating and are quite inert.

1.8.3 Nido-Boranes

These boranes have nest (Nido, Latin, meaning Nest) like structure. Their general

formula is BnHn+4 and have (n+2) pairs = 2n+4 skeletal electrons on removing one

boron atom from an apex of closo structure, nido structure is obtained. Because, of the

lost boron atom, these boranes have extra hydrogens for completing the valency. The

polyhedra in this series have B-H-B bridge bonds in addition to B-B bonds. They are

comparatively less stable than 'Closo', but more than 'Arachno' on heating.

1.8.4 Arachno-Boranes

These boranes have the general formula (BnHn +6) and skeletal electrons = (n+3)

pairs = 2n+6 = electrons. These molecules are obtained by removing two boron atoms

from two apexes of the closo structure and have spider-web like structure. They have

B-H-B bridge-bonds in their structures and are very reactive and unstable on heating.

1.8.5 Structural Inter-relation

The structural interrelation between closo, nido arachno species is shown in Fig. given

below

Notes

Inorganic Chemistry


Fig : Arachno B4H10

This is based on the observation that the structures having same number of

skeletal electrons are related with one another by the removal of BH unit one by one

and the addition of suitable number of electrons and hydrogen atoms, e.g. by removing

one BH unit and two electrons from octahedral closo. [B6H6]2- ion and adding four

hydrogens, we get square pyramidal nido- B5H9 borane.

On repeating same process on nido B5H9 (i.e. removing one BH unit and adding two

hydrogen's), we get butterfly shaped arachno. B4H10. Each of these three boranes have

14 skeletal electrons, but due to removal of BH unit, the resulting structure becomes

more open gradually.

The most symmetrical closo structure has (n+1) skeletal molecular orbital, which requrie

2n+2 electrons. Similarly, nido-boranes have (n+2) molecular-orbitals and need 2n+4

skeletal electrons; while for (n+3) molecular orbital, arachno boranes require 2n+6

skeletal electrons.

1.8.6 Synthesis

The simplest method for synthesis of higher boranes is the controlled pyrolysis

of diborance, B2H6 it is a gas phase reaction, BH3 formed in the first step reacts with

borane to give higher boranes:

Notes


main groups


Closp Nido Arachno

Fig: Interrelation between closo, nido and arachno-boranes

1.8.7 Reactions The important reactions of higher boranes are with Lewis bases, which involve removal of BH2 or BHn from the cluster, growth of the cluster or removal of one or more number of protons:

Notes

Inorganic Chemistry


1. Decomposition by Lewis-bases:

B4H10 + 2NH3 → [BH2(NH3)2] + [B3H8]

The reaction is analogous to the reaction of diborane with ammonia.

2. Deprotonation :

Higher boranes give deprotonation reaction easily rather than decomposition:

B4H10 + N(CH3)3→ [HN(NH3)3] + [B10H13] -

This deprotonation takes place from 3c, 2e BHB-bond. The bronsted acidity of boranes

increases with their size:

B4H10 < B5H9 < B10H14

For deprotonation of B5H9 strong-base like Li4(CH3)4 is required:

B5H9 + Li(CH3) → Li+[B5H8]- + CH4

3. Cluster Building:

Reactions of borane with borohydride are important with respect to synthesis of higher

boranes:

5K[B9H14] + 2 B5H9 →5K[B11H14] + 9 H2

4. Electrophilic displacement of proton:

Electrophilic displacement of proton by the catalytic activity of Lewis acids like AlCl3

is the basis of alkylation and halogenation of boranes:

B5H9 + CH3Cl [CH3B5H8] + HCl

1.9 CARBORANES

Carboranes are mixed hydrides of carbon and boron, having both carbon and boron

atoms in an electron - deficient; skeletal framework. There are two types of carboranes:

1. Closo-Carboranes: These have closed cage structrues in which hydrogen bridges are

structurally analogous to the Bn Hn-2 anions with B- replaced by isoelectronic carbon.

These carboranes have the general formula. C2Bn+2 (n=3) to 12. The important member

is C2B10H12 . Which is isoelectronic with [B12H12]2- similarly B4C2H6 is isoelectronic

with [B6H6]2-.

(A) 31, 2, C2 B10 H12 (B) C2B4H6

Notes


main groups


2. Nido Carboranes: They are having an open case structure in which some framework

members are attached likely by hydrogen bridges. These are derived formally from one

or other of several borones.

These contain one to four carbon atoms in the skeleton.

In addition to the above types of carboranes, there are a number of carboranes with an

additional heteroatom such as phosphorus built into the basic structure and a family

of metallo carboranes, some of which are similar to ferrocene. One peculiar feature

common to all carboranes is that to date no compound has been synthesized with either

carbon bridging two boron atoms in a three centre two electron bond or acting as one

end off a hydride bridge.

First carborane was obtained in 1953 when mixtures of diborane and acetylene were

ignited with a hot wire. Since that time, many new carboranes have been isolated.

Nomenclature:

Rules for naming carboranes are as follows:

1. First of all, give the positions and number of carbon atoms, then the type of carborane

(either closo or nido) and finally the name of the borane from which the carborane is

formally derived and the number of hydrogen atoms shown in bracket. For example CB5H9

is name as monocarbonido hexaborane (9). Similarly, the three isomers of C2B10H12

are named as 1, 2; 1, 7 and 1, 12 dicarbo-closo-dodecaborane (12).

2. Number of atoms in these structure are counted by starting the numbering from that

in the apical position and proceeding through successive rings in a clockwise direction.

This rule is important in naming the isomers.

Closo-Carboranes or Closed Cage Carboranes

These carboranes are having general formula C2BnHn+2 (n=3 to 10) in which the

constituents are only terminal. These are isoelectronic with the corresponding [BnHn]2-

ions and have the same closed polyhedral structures, with one hydrogen atom bonded to

each carbon and boron. No bridging hydrogen atoms are present in the C2Bn skeleton.

They are considered in three groups.

a. small, n = 3 - 5

b. large, n = 6-10 and

c. dicarbo-closo-dodecaborone

Preparation:

I(a) The Small Closo Carboranes (C2BnHn+2 where n = 3 to 5)

Notes

Inorganic Chemistry


B5H9 + C2H2 1,5 - C2B3H5 + 1,6 - C2B4H6 +2,4 - C2B5H7

Example - The closo hexaborane isomers, C2BnH6,

(b) The Large Closo Carboranes (C2B2Hn+2 where n = 6 to 9)

The first three members of this group of carboranes are obtained by the thermolysis of

1,3 - C2B7H13 and 1,3 - C2B2H12.

Example : C2B6H8 is made from hexaborane (10) and dimethylacetylene. The structure

of 1,7 - Me2C2B6H6 is based on the bicapped triangular prism. The carbon atoms are

present one on the prism and the other above the face opposite.

(c) Dicarobo-closo-dodecaborone:

Preparation: The orthocarborane is the only isomer which can be synthesized directly.

However, it is synthesized by the base catalysed reaction of acetylenes with decarborane

(14) or via B10H12L2.

B10H14 + 2L B10H12L2 R2L2B10H10 + H2 + 2L

Example: C2B10H12 gives three isomeric structure - 1,2 (ortho), 1-7 (meta) and 1, 12 (para)

(II) Nido-Carboranes or Open Cage Carboranes

These structures are derived formally from one or other of several boranes and contain

from one to four carbon atoms in the skeleton.

Examples: CB5H9, C2B4H8, C3B3H7, C4B2H6 etc.

Preparation: The smaller nido-carboranes are generally prepared by reacting a borane

with acetylene under mild conditions.

Example: B5H9and C2H2 undergo reaction in the gas phase at 215oC to give mainly the

nidocarborane 2,3 - C2B4H8 together with methyl derivatives of CB5H9.

The preparation method described above does not yield a single product

but a mixture of several products whose separation is not an easy task. However some

smaller nidocaroranes are prepared by the following specific methods:

i. Mono carbo-nido-hexaborane (7) CB5H7 is formed by passing silent electric

discharge through 1-methyl pentaborane (9).

ii. The only example isoelectronic with B5H9 is 1,2-dicarbonido - pentaborane(7),

C2B3H7, which is prepared as follows:

iii. Monocarbonidohexaborane (9), CB5H9 is formed from ethyldifluoroborane and

lithium. The nido-carboranes are formally related to B6H10. All are having eight pairs

of electrons which are bonding the six cage atoms together.

Notes


main groups


Large Nido-Carborane:

Dicarbo-nido-undecaborane, C2B9H13, is the second member of the class of nido-

carboranes C2BnHn+4 (n =4 or 9),. The parent carborane and its substituted derivatives can

be prepared by the base degradation of ortho-carborane (1,2-dicarbocloso-dodecaborane

(C2B10H12).

When C2B9H13 is heated, the closo-undeca-Borone (11) cage is formed.

Properties

Properties of carboranes resemble with that of the corresponding boranes closely.

Thus, 1.2 dicarbo closo-dodecarborane-12 is stable in both air and heat. On heating in

inert atmosphere at 500oC, it is converted into 1, 7 isomer i.e. meta or neo isomer; while

at 700oC it is concerted to 1, 12 isomer i.e. para-isomer.

Fig. : (a) C2B10H12 (b) 1,7 C2B10H12 (c) 1,12 C2B10H12

Analogous to boranes, carboranes are also classified into closo, nido and arachno structure.

The chemical reactions, in so far as they are known, are very similar to those of

C2B10H12, which are described below. Various substitution reactions have been studied

and the hydrogen atoms bonded to carbon are weakly acidic.

All three of the icosahedral isomers are stable both to heat and to chemical

attack, and much more so than decaborane (14). They are white crystalline solids which

resist both strong oxidizing agents and strong reducing agents and are also stable to

hydrolysis. This is important because it allows reactions to be carried out on substitutions,

often under quite drastic conditions, without destroying the cage structure, rather as the

chemistry of derivatives of an aromatic ring such as benzene can be developed without

destroying the ring.

Most chemical studies have been concerned with substituents on the two carbon

atoms. These may be introduced in the first place by employing substituted acetylene in

Notes

Inorganic Chemistry


the carborane syntheses. Such groups as C-alkyl, -haloalkyl, -aryl, -alkaenyl and -alkenyl

may be introduced into the structure in this way. Further reactions on the subsequents

groups may then be carried out by the usual synthetic methods of organic chemistry to

give, for example, carboxylic acid, ester, alcohol, ketone, amine or unsaturated groups

in the side chain.

The nido-carborane 2.3-C2B4H8 is converted to the closo-carboranes C2B3H5,

C2B4H6 and C2B5H7 on pyrolysis or ultraviolet irradiation.

Largely because of preparative difficulties, relatively little is known about the

reactions of the smaller nido-carboranes. They are only moderately stable to heat and

are less resistant to hydrolysis and oxidation in air than the closo species. Halogen

substitutions have been observed, as has the formation of anions; for example,

Similarly with LiC4H9, Lithium derivative is former:

B10C2H12 + 2LiC4H9 →B10C2H10Li2 + 2C4H10

The Sodium derivative with FeCl3 gives Fe-derivative:

2Na2[B9C2H11] + FeCl3→2NaCl + Na2[Fe(B9B2C11)2]

Structures

Structural studies of carboranes have been done using X-ray analysis and nmr studies.

The C2B3, C2B4 and C2B5 closo-carboranes, for example, have trigonal bipyramidal,

octahedral and pentagonal bipyramidal skeletal structrues respectively, and positional

isomers have been identified.

The icosahedra structure is similar to that of B12H122- (Fig. 9.8) and is electron-

deficient, with electron delocalization extending over the whole framework. It is thus

in effect a three-dimensional aromatic molecule, with marked electron withdrawing

character, the most important result of which is to render the two hydrogen atoms bonded

to carbon acidic. All the C-H and B-H bonds are of the normal two-electron type and

the electron deficiency is associated with the framework, in which there are multicentre

bonds.

The Structure of nido C3B3H7 is shown in Fig. In the diagram hydrogen bridges

are shown by curved lines, but terminal B-H and C-H bonds are ommitted. It can be

seen that the introductions of successive carbon atoms to the framework involves the

elimination of one bridge hydrogen atom and one B-H (i.e. the replacement of BH2 by

an isoelectronic CH unit). Like all the carboranes these compounds are electron-deficient,

with multicentered bonds and delocalization extending over the entire framework. In

Notes


main groups


much the same way, C2B3H7 has a square pyramidal structure that is formally derived

from that of B5H9, with two BH2 replaced by 2CH.

1.10 LET US SUM UP

After going through this unit, you would have achieved the objectives stated earlier in

this unit. Let us recall what we have discussed so for:

VSEPR theory considers that the geometry of a molecule depends upon the number of

bonding and nonbonding electron pairs in the central atom. These arrange themselves

in such a way that there is a minimum repulsion between them so that the molecule has

minimum energy (i.e. maximum stability).

The repulsive force between the electron pairs very as:

(lp - lp) > (lp - bp) > (bp - bp)

Thus, the more that number of lone pair on a central metal atom, the greater is the

contraction caused in the angle between the bonding pair. Hence, the bond-angle in

CH4 (4bp + Olp) is 109o28', in NH3 (3bp + lp) is 107o and H2O (2bp + 2lp) is 105o.

Similarly. BCl3 is triangular, but SO2 is angular; PCl5 is trigonal bipyramidal, but SF4

is irregular tetrahedral; ClF3 is T-shaped, while XeF4 is square planar.

Walsh diagrams propose a simple pictoral approach to determine the geometry

of a molecule considering and calculating the energies of molecular orbitals of the

molecule. The molecule will have that geometry in which the energies of the molecular

orbitals used are minimum. Using this concept we can understand why H2O is angular

and BeH2 is linear; or why CH4 is tetrahedral and SF4 is distorted tetrahedral.

Number of compounds of non-metallic elements of group VA and VIA (N, P,

O, S etc.) use d p -pp bond.

The formation of dp -pp bond is common for all the second period elements and is not

important for the elements of third and higher periods. The pp -dp bonding is more

favorable than the dp-pp bonding for higher atoms i.e. atoms of third and higher periods.

Bent's rule states that more electronegative substituents prefer hybrid orbitals

having less s-character and more electropositive substituents prefer hybrid orbitals having

more s-character. Thus in CH2F2, the F-C-F bond angle is less than 109.5o indicating

Notes

Inorganic Chemistry


less then 25% s-character in C-F bond, but the H-C-H bond angle is larger indicating

C-H bond has more than 25% s-character. The energetics of hybridization also explains

the geometry of a molecule. The molecule will have the geometry which involves the

hybridization of lower energy. Thus while the lighter elements of groups IIIA, IVA,

VA and VIA use sp2, sp3, sp3d and sp3d2 hybridizations respectively for their hydride

formation, the haviour elements of these groups use their pure p-orbital, leaving a lp-

sink into a pure s-orbital. Hence in the heaviest elements of groups IIIA and IVA (Tl,

Sn and Pb) the stable oxidation states are two unit less (1 and 2 respectively) then the

maximum oxidation states (3 and 4 respectively) i.e. TlCl, SnCl2 and PbCl2. Similarly,

the hydrides of group VA and VIA elements are found to have bond-angles considerably

reduced as one moves down in these groups.

As the more electronegative atoms use those hybrid orbitals, which have higher

p-character and less electronegative atoms use hybrid orbitals with more s-character the

bond angle in many tetravalent molecules is seen slightly distorted than 109o28', e.g. in

CH3Cl, HCH bond is 110o20'. This is known as Isovalent hybridization. Apicophilicity

is the tendency of more electronegative substituents, in trigonal bipyramidal molecules

(e.g. PCl4F or PCl3F2) to seek out the apical orbitals, e.g. in oxysulfuranes. The common

reaction of covalently bonded molecules are atomic inversion, Berrypseudorotation;

nucleophilic displacement and free radical mechanism. Atomic inversion is the simple

inversion of atoms about the central atom of the molecule. This is analogous to the

inversion of umbrella in a high wind. The energy barrier to inversion is strongly dependent

on the nature of the central atom and that of substituents. Hence the separation of optical

isomers and their racemization is possible only in such case which have values of energy

barrier sufficiently high (e.g. methylpropylphenyl phosphine, 120 KJ mol-1). Berry pseudorotation involves scrambling of atoms at the equatorial and axial position in a trigonal bipyramid geometry (e.g. in PF5). This is believed to take place through conversion of the ground state trigonal bipyramidal into a square pyramidal transition state back to a new trigonal bipyramidal structure.

Nucleophilic displacement, e.g. the solvolysis (Alkoholysis) of a chlorodialkyl phosphine oxide proceeds with inversion of configuration of the phosphorus atom. The highly electronegative entering and the leaving groups are favourably disposed at the axial positions.

1.11 Review Questions

1. What are the postualtes of VSEPR theory ? 2. Discuss the limitations of VSEPR theory?

Notes


main groups


3. Explain the concept of Walsh diagrams 4. What is Dp- pp bonds? 5. Explain Dp- pp bonding in phosphorous group elements 6. Describe the Dp- pp bonding in nitrogen, oxygen and sulphur compounds 7. Discuss Bent's rule and energetics of hybridisation 8. What are Boranes and Higher Boranes. 9. Explain the Wades Rule.

10. Describe the structure of carboranes.

1.12 Further Readings

zz Advanced inorganic chemistry, F.A. Cotton and G.Wilkison , John Wiley.

zz Inorganic Cghemistry , J.E.Huheey Harper and Row.

zz Chemistry of Elements, N.N. Greenwood and A. Earnshaw, Pergamor

Notes

Inorganic Chemistry


UNIT–2

Nuclear Chemistry

(Structure) 2.1 Learning Objectives 2.2 Introduction 2.3 Radioactive rays 2.4 Theory of Radioactive Decay 2.5 The liquid-drop model 2.6 The shell model 2.7 Nuclear Reactor 2.8 Rates of Radioactive Transitions. 2.9 Nuclear Reaction Cross section 2.10 Isotopes and their application 2.11 Let us sum up 2.12 Review questions 2.12 Further readings


zz Discuss Nuclear Reactons and Nuclear reaction cross section;zz Explain Shell model and liquid drop model;

zz Describe application of radio isotopes.

2.2 Introduction The Nuclear chemistry is the study of the chemical and physical properties of elements as influenced by changes in the structure of the atomic nucleus. Modern nuclear chemistry, sometimes referred to as radiochemistry, has become very interdisciplinary in its applications, ranging from the study of the formation of the elements in the universe to the design of radioactive drugs for diagnostic medicine. In fact, the chemical techniques pioneered by nuclear chemists have become so important that biologists, geologists, and physicists use nuclear chemistry as ordinary tools of their disciplines. While the common perception is that nuclear chemistry involves only the study of radioactive

Notes

Nuclear Chemistry


nuclei, advances in modern mass spectrometry instrumentation has made chemical studies using stable, nonradioactive isotopes increasingly important.In 1896 Henri Becquerel was using naturally fluorescent minerals to study the properties of x-rays, which had been discovered in 1895 by Wilhelm Roentgen. He exposed potassium uranyl sulfate to sunlight and then placed it on photographic plates wrapped in black paper, believing that the uranium absorbed the sun’s energy and then emitted it as x-rays. This hypothesis was disproved on the 26th-27th of February, when his experiment "failed" because it was overcast in Paris. For some reason, Becquerel decided to develop his photographic plates anyway. To his surprise, the images were strong and clear, proving that the uranium emitted radiation without an external source of energy such as the sun. Becquerel had discovered radioactivity.Becquerel used an apparatus similar to that displayed below to show that the radiation he discovered could not be x-rays. X-rays are neutral and cannot be bent in a magnetic field. The new radiation was bent by the magnetic field so that the radiation must be charged and different than x-rays. When different radioactive substances were put in the magnetic field, they deflected in different directions or not at all, showing that there were three classes of radioactivity: negative, positive, and electrically neutral.The term radioactivity was actually coined by Marie Curie, who together with her husband Pierre, began investigating the phenomenon recently discovered by Becquerel. The Curies extracted uranium from ore and to their surprise, found that the leftover ore showed more activity than the pure uranium. They concluded that the ore contained other radioactive elements. This led to the discoveries of the elements polonium and radium. It took four more years of processing tons of ore to isolate enough of each element to determine their chemical properties.Ernest Rutherford, who did many experiments studying the properties of radioactive decay, named these alpha, beta, and gamma particles, and classified them by their ability to penetrate matter. Rutherford used an apparatus similar to that depicted in Fig. 3-7. When the air from the chamber was removed, the alpha source made a spot on the photographic plate. When air was added, the spot disappeared. Thus, only a few centimeters of air were enough to stop the alpha radiation. Because alpha particles carry more electric charge, are more massive, and move slowly compared to beta and gamma particles, they interact much more easily with matter. Beta particles are much less massive and move faster, but are still electrically charged. A sheet of aluminum one millimeter thick or several meters of air will stop these electrons and positrons. Because gamma rays carry no electric charge, they can penetrate large distances through materials before interacting–several centimeters of lead or a meter of concrete is needed to stop most gamma rays.

2.3 Radioactive raysThe radioactive substance usually gives rise to three types of radiation known as alpha beta and gamma rays squirrel another fort and William billiard started their characteristics property.

Notes

Inorganic Chemistry


A radium source was kept inside a small hole in a lead lead block and subjected to the action of magnetic field directed to the place of the paper . The Alpha particles was slightly deflected to the left, the beta particle were sharply deviated to the right and gamma rays went Straight through.The Alpha particles are positively charged particles having +2 units and mass 4 unit and so they are identical to helium nuclei. This was proved by the fact that a strong Alpha source is a sealed glass to produced helium gas, which was identified by the characteristic atomic spectrum. They were ejected with velocity ranging 1.5 - 2.00 x 109 cm/second. They have greater Momentum due to their large mass, but they can be stopped by Aluminium foil less than 0.1 mm thickness. The absorption value of a metal foil depends not only on its thickness but also upon the atomic weight of the foil. The higher the atomic weight of the foil. The higher the atomic weight greater is the absorption. The alpha rays ionize molecules present in the air. The range of particles through a gas depends upon the nature of a gas and inversely proportional to its pressure and directly proportional to absolute Temperature and pass through a few cm of air, where as Alpha particle can penetrate four times further in H2 than in O2 . The elements with rapid decay periods emit Alpha particles with great ranges. For example at 15 degree Celsius and 1 atmospheric pressure.

Th232(t1/2 = 1.39 x 1010 yrs) = 2.8 cmand P 2/12 (t1/2 = 3 x 10-7 sec) = 8.6 cm

The beta is a negatively charged and their composition is that of electron the mass is that of electrons.The mass is 0.00055 amu and they travel a the velocity of light. They produce less number of ions on passing through the molecules of air. Neutrons decompose to produce a proton and electron ( b-particles) and meutrino.

10n → 11H + b + g-

The mechanism of beta decay is based on nutron-proton transformation with simultaneous emission of neutrino. These b-ray are also having ionizing power.The gamma rays are shortwave radiation and undeflected by magnetic field. They are identical to X-ray accept that their wavelength is much shorter. These rays have no mass and they are not considered as particles. Instead they may be described as form of electromagnetic magnetic radiations. They have greater penetrating power due to their short wavelength and greater energy. It may penetrate 25 cm iron, and up to 8 cm lead. Thus lead is about eight times effective than A1 and 4 times better than Cu.

Notes

Nuclear Chemistry


In radioactive processes, particles or electromagnetic radiation are emitted from the nucleus. The most common forms of radiation emitted have been traditionally classified as alpha (a), beta (b), and gamma (g) radiation. Nuclear radiation occurs in other forms, including the emission of protons or neutrons or spontaneous fission of a massive nucleus.Of the nuclei found on Earth, the vast majority are stable. This is so because almost all short-lived radioactive nuclei have decayed during the history of the Earth. There are approximately 270 stable isotopes and 50 naturally occurring radioisotopes (radioactive isotopes). Thousands of other radioisotopes have been made in the laboratory.Radioactive decay will change one nucleus to another if the product nucleus has a greater nuclear binding energy than the initial decaying nucleus. The difference in binding energy (comparing the before and after states) determines which decays are energetically possible and which are not. The excess binding energy appears as kinetic energy or rest mass energy of the decay products.The Chart of the Nuclides, part of which is shown above is a plot of nuclei as a function of proton number, Z, and neutron number, N. All stable nuclei and known radioactive nuclei, both naturally occurring and manmade, are shown on this chart, along with their decay properties. Nuclei with an excess of protons or neutrons in comparison with the stable nuclei will decay toward the stable nuclei by changing protons into neutrons or neutrons into protons, or else by shedding neutrons or protons either singly or in combination. Nuclei are also unstable if they are excited, that is, not in their lowest energy states. In this case the nucleus can decay by getting rid of its excess energy without changing Z or N by emitting a gamma ray.Nuclear decay processes must satisfy several conservation laws, meaning that the value of the conserved quantity after the decay, taking into account all the decay products, must equal the same quantity evaluated for the nucleus before the decay. Conserved quantities include total energy (including mass), electric charge, linear and angular momentum, number of nucleons, and lepton number (sum of the number of electrons, neutrinos, positrons and antineutrinos–with antiparticles counting as -1).137Ba decay data, counting numbers of decays observed in 30-second intervals. The best-fit exponential curve is shown. The points do not fall exactly because of statistical counting fluctuations.The probability that a particular nucleus will undergo radioactive decay during a fixed length of time does not depend on the age of the nucleus or how it was created. Although the exact lifetime of one particular nucleus cannot be predicted, the mean (or average) lifetime of a sample containing many nuclei of the same isotope can be predicted and measured. A convenient way of determining the lifetime of an isotope is to measure how long it takes for one-half of the nuclei in a sample to decay–this quantity is called the half-life, t1/2. Of the original nuclei that did not decay, half will decay if we wait another half-life, leaving one-quarter of the original sample after a total time of two half-lives. After three half-lives, one-eighth of the original sample will remain and so on. Measured half-lives vary from tiny fractions of seconds to billions of years, depending on the isotope.

2.4 Theory of Radioactive Decay

Notes

Inorganic Chemistry


The number of nuclei in a sample that will decay in a given interval of time is proportional to the number of nuclei in the sample. This condition leads to radioactive decay showing itself as an exponential process, as shown above. The number, N, of the original nuclei remaining after a time t from an original sample of N0 nuclei is

N = N0e-(t/T)where T is the mean lifetime of the parent nuclei. From this relation, it can be shown that t1/2 = 0.693T.In alpha decay, the nucleus emits a 4He nucleus, an alpha particle. Alpha decay occurs most often in massive nuclei that have too large a proton to neutron ratio. An alpha particle, with its two protons and two neutrons, is a very stable configuration of particles. Alpha radiation reduces the ratio of protons to neutrons in the parent nucleus, bringing it to a more stable configuration. Many nuclei more massive than lead decay by this method.Consider the example of 210Po decaying by the emission of an alpha particle. The reaction can be written 210Po Æ 206Pb + 4He. This polonium nucleus has 84 protons and 126 neutrons. The ratio of protons to neutrons is Z/N = 84/126, or 0.667. A 206Pb nucleus has 82 protons and 124 neutrons, which gives a ratio of 82/124, or 0.661. This small change in the Z/N ratio is enough to put the nucleus into a more stable state, brings the "daughter" nucleus (decay product) into the region of stable nuclei in the Chart of the Nuclides.In alpha decay, the atomic number changes, so the original (or parent) atoms and the decay-product (or daughter) atoms are different elements and therefore have different chemical properties. In the alpha decay of a nucleus, the change in binding energy appears as the kinetic energy of the alpha particle and the daughter nucleus. Because this energy must be shared between these two particles, and because the alpha particle and daughter nucleus must have equal and opposite momenta, the emitted alpha particle and recoiling nucleus will each have a well-defined energy after the decay. Because of its smaller mass, most of the kinetic energy goes to the alpha particle.Beta particles are electrons or positrons (electrons with positive electric charge, or antielectrons). Beta decay occurs when, in a nucleus with too many protons or too many neutrons, one of the protons or neutrons is transformed into the other. In beta minus decay, a neutron decays into a proton, an electron, and an antineutrino: n Æ p + e - +. In beta plus decay, a proton decays into a neutron, a positron, and a neutrino: p Æ n + e+ +n. Both reactions occur because in different regions of the Chart of the Nuclides, one or the other will move the product closer to the region of stability. These particular reactions take place because conservation laws are obeyed. Electric charge conservation requires that if an electrically neutral neutron becomes a positively charged proton, an electrically negative particle (in this case, an electron) must also be produced. Similarly, conservation of lepton number requires that if a neutron (lepton number = 0) decays into a proton (lepton number = 0) and an electron (lepton number = 1), a particle with a lepton number of -1 (in this case an antineutrino) must also be produced. The leptons emitted in beta decay did not exist in the nucleus before the decay–they are created at the instant of the decay.

Notes

Nuclear Chemistry


To the best of our knowledge, an isolated proton, a hydrogen nucleus with or without an electron, does not decay. However within a nucleus, the beta decay process can change a proton to a neutron. An isolated neutron is unstable and will decay with a half-life of 10.5 minutes. A neutron in a nucleus will decay if a more stable nucleus results; the half-life of the decay depends on the isotope. If it leads to a more stable nucleus, a proton in a nucleus may capture an electron from the atom (electron capture), and change into a neutron and a neutrino.Proton decay, neutron decay, and electron capture are three ways in which protons can be changed into neutrons or vice-versa; in each decay there is a change in the atomic number, so that the parent and daughter atoms are different elements. In all three processes, the number A of nucleons remains the same, while both proton number, Z, and neutron number, N, increase or decrease by 1.In beta decay the change in binding energy appears as the mass energy and kinetic energy of the beta particle, the energy of the neutrino, and the kinetic energy of the recoiling daughter nucleus. The energy of an emitted beta particle from a particular decay can take on a range of values because the energy can be shared in many ways among the three particles while still obeying energy and momentum conservation.In gamma decay, depicted in Fig. 3-6, a nucleus changes from a higher energy state to a lower energy state through the emission of electromagnetic radiation (photons). The number of protons (and neutrons) in the nucleus does not change in this process, so the parent and daughter atoms are the same chemical element. In the gamma decay of a nucleus, the emitted photon and recoiling nucleus each have a well-defined energy after the decay. The characteristic energy is divided between only two particles.

2.5 The liquid-drop modelThe average behaviour of the nuclear binding energy can be understood with the model of a charged liquid drop. In this model, the aggregate of nucleons has the same properties of a liquid drop, such as surface tension, cohesion, and deformation. There is a dominant attractive-binding-energy term proportional to the number of nucleons A. From this must be subtracted a surface-energy term proportional to surface area and a coulombic repulsion energy proportional to the square of the number of protons and inversely proportional to the nuclear radius. Furthermore, there is a symmetry-energy term of quantum-mechanical origin favouring equal numbers of protons and neutrons. Finally, there is a pairing term that gives slight extra binding to nuclei with even numbers of neutrons or protons.The pairing-energy term accounts for the great rarity of odd–odd nuclei (the terms odd–odd, even–even, even–odd, and odd–even refer to the evenness or oddness of proton number, Z, and neutron number, N, respectively) that are stable against beta decay. The sole examples are deuterium, lithium-6, boron-10, and nitrogen-14. A few other odd–odd nuclei, such as potassium-40, occur in nature, but they are unstable with respect to beta decay. Furthermore, the pairing-energy term makes for the larger number of stable

Notes

Inorganic Chemistry


isotopes of even-Z elements, compared to odd-Z, and for the lack of stable isotopes altogether in element 43, technetium, and element 61, promethium.The beta-decay energies of so-called mirror nuclei afford one means of estimating nuclear sizes. For example, the neon and fluorine nuclei, 19/10Ne9 and 19/9F10, are mirror nuclei because the proton and neutron numbers of one of them equal the respective neutron and proton numbers of the other. Thus, all binding-energy terms are the same in each except for the coulombic term, which is inversely proportional to the nuclear radius. Such calculations along with more direct determinations by high-energy electron scattering and energy measurements of X-rays from muonic atoms (hydrogen atoms in which the electrons are replaced by negative muons) establish the nuclear charge as roughly uniformly distributed in a sphere of radius 1.2 A1⁄3 × 10−13 centimetre. That the radius is proportional to the cube root of the mass number has the great significance that the average density of all nuclei is nearly constant. Careful examination of nuclear-binding energies reveals periodic deviations from the smooth average behaviour of the charged-liquid-drop model. An extra binding energy arises in the neighbourhood of certain numbers of neutrons or protons, the so-called magic numbers (2, 8, 20, 28, 50, 82, and 126). Nuclei such as 4/2He2,

16/8O8, 40/20Ca20,

48/20Ca28, and 208/82 Pb126 are especially stable species, doubly magic, in view of their having both proton and neutron numbers magic.

The liquid drop model was proposed by N. Bohr and Kalcker which provided the

reasonable explanation of many nuclear phenomena, not explained on the basis of other

nuclear models. Some of these important phenomena are given below.

Phenomena explained by liquid drop model:

(1) The constant density of nuclei with radius, just as density of liquid drop is independent of the size of liquid drop.

(2) Systematic dependence of neutron excess (N-Z) on A5/3 for stable nuclei.

(3) The approximate constant value of binding energy per nucleon (B/A) which is analogues to latent heat of vaporization

(4) Fission by thermal neutrons of U235 and other odd N nuclides

(5) Systematic variation of decay energies with N and Z

Bohr suggested that the properties of the nucleus can very well be compared with that

of liquid in which the molecules of liquid drop correspond to nucleons in the nucleus.

The similarities between liquid drop and nucleus are given below.

Similarities between nucleus and liquid drop:

(1) The density of liquid is almost independent of its size, so that radius R of the liquid is proportional to cube root of number A of molecules in the drop.

Notes

Nuclear Chemistry


(2) Both the molecules of the liquid drop and nucleons in the nucleus interact only with their immediate neighbours.

(3) The energy necessary to completely evaporate the liquid drop into its molecules is approximately proportional to mass number A just as the binding energy of nucleus is proportional to mass number A.

Attempts were made to describe the nuclear dynamics in terms of motion of liquid drop.

The most important motions are surface vibrations. A deformation of spherical liquid

drop gives rise to periodic oscillations of the surface.

Surface vibrations of liquid drop:

Let there exists a spherical drop of radius R. Any deformation of its surface can be

described by a function ) which is the distance of deformed surface from centre.

The difference of the two radii is given by

(1)

can be expressed in terms of spherical harmonics as

Taking the cylindrically symmetric deformation for which m = 0, so that

The restoring force is supplied by the surface tension which opposes the deformation

of surface. The surface energy is

where α is coefficient of surface tension. In Weizsacker-Bethe mass formula, the surface

energy term is given as

Equating the two values of Es, we get

Rayleigh gave the formula for frequency as

where μ is the mass of liquid drop. Taking μ = MA, where M is the mass of single

nucleon and putting the value of α, we get

Notes

Inorganic Chemistry


The corresponding energy is

This gives too high value of excitation energy. The frequency ω is reduced by Coulomb effect

as

where γ is the ratio between the coulomb energy and surface energy

. This equation leads to somewhat smaller frequencies for heavier

nuclei. However even this correction is not able to explain the low lying energy levels.

The liquid drop model is however able to explain the stability of nuclei against the

breakup into two fragments i.e. nuclear fission, explained later.

Energy release in symmetric fission

If a nucleus (A,Z) breaks into two equal halves (A/2,Z/2), then the fission is

called symmetric fission. The energy release in this symmetric fission is

From the semi empirical mass formula, binding energy for nucleus (A,Z) is

and binding energy for two fission fragments (A/2,Z/2) is

From the mass formula

and using equation (7), (8) and (9) we finally get

Pairing energy difference is small and can be neglected. Taking =13MeV and =0.6 MeV

Notes

Nuclear Chemistry


From this equation Q > 0 for

Thus for all nuclei having Z > 35, A > 80, fission is possible and will release energy.

However the slow neutron fission does not take place even with many of heavy nuclei.

This discrepancy was explained by Bohr and Wheeler by considering the Coulomb

potential barrier of the two fragments at the instant of separation.

Potential barrier for symmetric fission:

The existence of potential barrier prevents the immediate breaking of two fission

fragments. Let the height of potential barrier is Ec then the nucleus will be unstable

and will break in two parts if Q > Ec. The barrier height of coulomb potential between

the two symmetric fragments when they just touch each other, is

Thus the condition for stability is

may have value 50 for a nucleus of mass number 250, hence nuclei (A>250)

would be too unstable to exist.

Nuclear fission and deformation of liquid drop:

The fission process can be explained with the help of liquid drop model as shown in fig.

1. A highly energetic compound nucleus is formed when an incident neutron combines

with the nucleus. This energy starts a series of rapid oscillations in the drop which

distorts the spherical shape of liquid drop and drop becomes ellipsoidal in shape. The

surface tension force makes the drop to return to its original shape while the excitation

energy tends to distort the shape further. For sufficiently large excitation energy, the drop

may acquire the dumb bell shape. Again if oscillations are violent enough to produce

the critical stage (stage IV) then nucleus cannot return to first stage and finally fission

takes place. The energy required to produce fourth stage is called threshold energy or

critical energy.

Notes

Inorganic Chemistry


The potential energy curve for fission is plotted in fig 2. On X axis, is the distance r

which is the separation of centers of two fission fragments. The curve is divided in

three regions. In region I fragments are completely separated and their potential energy

is the electrostatic Coulomb energy resulting from their mutual repulsion. When

distance r = 2R, the drops just touch each other. The energy E at this point is less than the

corresponding Coulomb energy by an amount equal to CD. In region II when we reach

the critical distance rc, the potential energy curve has its maximum value corresponding

to the barrier height and explains why fission does not take place when Q>0.

A critical energy called threshold or activation energy given by Ea =Ec- Q is required

to overcome the potential barrier and fission to take place. In region III, the fragments

coalesce and short range attractive forces become dominant.

For the mathematical analysis, drop is considered incompressible, the volume of sphere

of radius R is same as that of ellipsoid of semi major axis 'a' and semi minor axis 'b' i.e.

If denotes the eccentricity then,

Then surface energy of ellipsoid is, Es= (area of ellipsoid) × surface tension

Notes

Nuclear Chemistry


And the Coulomb energy of ellipsoid is

Taking the terms up to ε2 only, the stability condition against spontaneous fission is

This equation gives much more experimentally realistic stability limit against

the spontaneous fission.

2.6 The shell modelIn the preceding section, the overall trends of nuclear binding energies were described in terms of a charged-liquid-drop model. Yet there were noted periodic binding-energy irregularities at the magic numbers. The liquid drop model only discusses the properties of nuclear matter but does not tell anything about single nucleon. For certain numbers of neutrons and protons called magic numbers, nuclei exhibit special stability. This stability is not explained by liquid drop model. Also the other properties of nucleons like spin, magnetic moment and quadrupole moments are unexplained in liquid drop model.

The periodic occurrence of magic numbers of extra stability is strongly analogous to the extra electronic stabilities occurring at the atomic numbers of the noble-gas atoms. The explanations of these stabilities are quite analogous in atomic and nuclear cases as arising from filling of particles into quantized orbitals of motion. The completion of filling of a shell of orbitals is accompanied by an extra stability.

The nuclear model accounting for the magic numbers is, as previously noted, the shell model. In its simplest form, this model can account for the occurrence of spin zero for all even–even nuclear ground states; the nucleons fill pairwise into orbitals with angular

Notes

Inorganic Chemistry


momenta canceling.

The shell model also readily accounts for the observed nuclear spins of the odd-mass nuclei adjacent to doubly magic nuclei, such as 208/82Pb. Here, the spins of 1/2 for neighbouring 207/81Tl and 207/82Pb are accounted for by having all nucleons fill pairwise into the lowest energy orbits and putting the odd nucleon into the last available orbital before reaching the doubly magic configuration (the Pauli exclusion principle dictates that no more than two nucleons may occupy a given orbital, and their spins must be oppositely directed); calculations show the last available orbitals below lead-208 to have angular momentum 1/2. Likewise, the spins of 9/2 for 209/82 Pb and 209/83 Bi are understandable because spin-9/2 orbitals are the next available orbitals beyond doubly magic lead-208. Even the associated magnetization, as expressed by the magnetic dipole moment, is rather well explained by the simple spherical-shell model.

The orbitals of the spherical-shell model are labeled in a notation close to that for electronic orbitals in atoms. The orbital configuration of calcium-40 has protons and neutrons filling the following orbitals: 1s1/2, 1p3/2, 1p1/2, 1d5/2, and 1d3/2. The letter denotes the orbital angular momentum in usual spectroscopic notation, in which the letters s, p, d, f, g, h, i, etc., represent integer values of l running from zero for s (not to be confused with spins) through six for i. The fractional subscript gives the total angular momentum j with values of l + 1/2 and l − 1/2 allowed, as the intrinsic spin of a nucleon is 1/2. The first integer is a radial quantum number taking successive values 1, 2, 3, etc., for successively higher energy values of an orbital of given l and j. Each orbital can accommodate a maximum of 2j + 1 nucleons. The exact order of various orbitals within a shell differs somewhat for neutrons and protons (see table for the orbitals comprising each shell). The parity associated with an orbital is even (+) if l is even (s, d, g, i) and odd (−) if l is odd (p, f, h).

An example of a spherical-shell-model interpretation is provided by the beta-decay scheme of 2.2-minute thallium-209 shown below, in which spin and parity are given

Notes

Nuclear Chemistry


for each state. The ground and lowest excited states of lead-209 are to be associated with occupation by the 127th neutron of the lowest available orbitals above the closed shell of 126.

From the last line of the table, it is to be noted that there are available g9/2, d5/2, and s1/2 orbitals with which to explain the ground and first two excited states. Low-lying states associated with the i11/2 and j15/2 orbitals are known from nuclear-reaction studies, but they are not populated in the beta decay.The 2.13-MeV state that receives the primary beta decay is not so simply interpreted as the other states. It is to be associated with the promotion of a neutron from the 3p1/2 orbital below the 126 shell closure. The density (number per MeV) of states increases rapidly above this excitation, and the interpretations become more complex and less certain.By suitable refinements, the spherical-shell model can be extended further from the doubly magic region. Primarily, it is necessary to drop the approximation that nucleons move independently in orbitals and to invoke a residual force, mainly short-range and attractive, between the nucleons. The spherical-shell model augmented by residual interactions can explain and correlate around the magic regions a large amount of data on binding energies, spins, magnetic moments, and the spectra of excited states.

2.7 Nuclear ReactorNuclear reactors created not only large amounts of plutonium needed for the weapons programs, but a variety of other interesting and useful radioisotopes. They produced 60Co, in which the non-conservation of parity was first discovered, and a number of transuranic isotopes that are used to study the limits of the periodic table. Reactors also produce isotopes for commercial and medical purposes:241Am–used in smoke detectors,60Co–used in industry to inspect weld quality, also used in cancer therapy,99mTc–used for medical diagnosis, and137Cs–also used for medical therapy.Reactor neutrons have been used for material studies that involve their scattering from the crystal planes.

Notes

Inorganic Chemistry


2.8 Rates of Radioactive Transitions here is a vast range of the rates of radioactive decay, from undetectably slow to unmeasurably short. Before considering the factors governing particular decay rates in detail, it seems appropriate to review the mathematical equations governing radioactive decay and the general methods of rate measurement in different ranges of half-life.Exponential-decay lawRadioactive decay occurs as a statistical exponential rate process. That is to say, the number of atoms likely to decay in a given infinitesimal time interval (dN/dt) is proportional to the number (N) of atoms present. The proportionality constant, symbolized by the Greek letter lambda, λ, is called the decay constant. Mathematically, this statement is expressed by the first-order differential equation,

This equation is readily integrated to give

in which N0 is the number of atoms present when time equals zero. From the above two equations it may be seen that a disintegration rate, as well as the number of parent nuclei, falls exponentially with time. An equivalent expression in terms of half-life t1⁄2 is

It can readily be shown that the decay constant λ and half-life (t1⁄2) are related as follows: λ = loge2/t1⁄2 = 0.693/t1⁄2. The reciprocal of the decay constant λ is the mean life, symbolized by the Greek letter tau, τ.For a radioactive nucleus such as potassium-40 that decays by more than one process (89 percent β−, 11 percent electron capture), the total decay constant is the sum of partial decay constants for each decay mode. (The partial half-life for a particular mode is the reciprocal of the partial decay constant times 0.693.) It is helpful to consider a radioactive chain in which the parent (generation 1) of decay constant λ1 decays into a radioactive daughter (generation 2) of decay constant λ2. The case in which none of the daughter isotope (2) is originally present yields an initial growth of daughter nuclei followed by its decay. The equation giving the number (N2) of daughter nuclei existing at time t in terms of the number N1(0) of parent nuclei present when time equals zero is

in which e represents the logarithmic constant 2.71828.The general equation for a chain of n generations with only the parent initially present (when time equals zero) is as follows:

in which e represents the logarithmic constant 2.71828.

Notes

Nuclear Chemistry


These equations can readily be modified to the case of production of isotopes in the steady neutron flux of a reactor or in a star. In such cases, the chain of transformations might be mixed with some steps occurring by neutron capture and some by radioactive decay. The neutron-capture probability for a nucleus is expressed in terms of an effective cross-sectional area. If one imagines the nuclei replaced by spheres of the same cross-sectional area, the rate of reaction in a neutron flux would be given by the rate at which neutrons strike the spheres. The cross section is usually symbolized by the Greek letter sigma, σ, with the units of barns (10−24 cm2) or millibarns (10−3 b) or microbarns (10−6 b). Neutron flux is often symbolized by the letters nv (neutron density, n, or number per cubic centimetre, times average speed, v) and given in neutrons per square centimetre per second.The modification of the transformation equations merely involves substituting the product nvσi in place of λi for any step involving neutron capture rather than radioactive decay. Reactor fluxes nv even higher than 1015 neutrons per square centimetre per second are available in several research reactors, but usual fluxes are somewhat lower by a factor of 1,000 or so. Tables of neutron-capture cross sections of the naturally occurring nuclei and some radioactive nuclei can be used for calculation of isotope production rates in reactors.

Measurement of half-lifeThe measurement of half-lives of radioactivity in the range of seconds to a few years commonly involves measuring the intensity of radiation at successive times over a time range comparable to the half-life. The logarithm of the decay rate is plotted against time, and a straight line is fitted to the points. The time interval for this straight-line decay curve to fall by a factor of 2 is read from the graph as the half-life, by virtue of equations (1) and (2). If there is more than one activity present in the sample, the decay curve will not be a straight line over its entire length, but it should be resolvable graphically (or by more sophisticated statistical analysis) into sums and differences of straight-line exponential terms. The general equations (4) for chain decays show a time dependence given by sums and differences of exponential terms, though special modified equations are required in the unlikely case that two or more decay constants are identically equal.For half-lives longer than several years it is often not feasible to measure accurately the decrease in counting rate over a reasonable length of time. In such cases, a measurement of specific activity may be resorted to; i.e., a carefully weighed amount of the radioactive isotope is taken for counting measurements to determine the disintegration rate, D. Then by equation (1) the decay constant λi may be calculated. Alternately, it may be possible to produce the activity of interest in such a way that the number of nuclei, N, is known, and again with a measurement of D equation (1) may be used. The number of nuclei,

Notes

Inorganic Chemistry


N, might be known from counting the decay of a parent activity or from knowledge of the production rate by a nuclear reaction in a reactor or accelerator beam.Half-lives from 100 microseconds to one nanosecond are measured electronically in coincidence experiments. The radiation yielding the species of interest is detected to provide a start pulse for an electronic clock, and the radiation by which the species decays is detected in another device to provide a stop pulse. The distribution of these time intervals is plotted semi-logarithmically, as discussed for the decay-rate treatment, and the half-life is determined from the slope of the straight line.Half-lives in the range of 100 microseconds to one second must often be determined by special techniques. For example, the activities produced may be deposited on rapidly rotating drums or moving tapes, with detectors positioned along the travel path. The activity may be produced so as to travel through a vacuum at a known velocity and the disintegration rate measured as a function of distance; however, this method usually applies to shorter half-lives in or beyond the range of the electronic circuit.Species with half-lives shorter than the electronic measurement limit are not considered as separate radioactivities, and the various techniques of determining their half-lives will hence not be cited here.Decay-rate considerations for various types of radioactivity are given here in the same order as listed above in Types of radioactivity.Alpha decayAlpha decay, the emission of helium ions, exhibits sharp line spectra when spectroscopic measurements of the alpha-particle energies are made. For even–even alpha emitters the most intense alpha group or line is always that leading to the ground state of the daughter. Weaker lines of lower energy go to excited states, and there are frequently numerous lines observable.The main decay group of even–even alpha emitters exhibits a highly regular dependence on the atomic number, Z, and the energy release, Qα. (Total alpha energy release, Qα, is equal to alpha-particle energy, Eα, plus daughter recoil energy needed for conservation of momentum; Erecoil = (mα/[mα + Md])Eα, with mα equal to the mass of the alpha particle and Md the mass of the daughter product.) As early as 1911 the German physicist Johannes Wilhelm Geiger, together with the British physicist John Mitchell Nuttall, noted the regularities of rates for even–even nuclei and proposed a remarkably successful equation for the decay constant, log λ = a + b log r, in which r is the range in air, b is a constant, and a is given different values for the different radioactive series. The decay constants of odd alpha emitters (odd A or odd Z or both) are not quite so regular and may be much smaller. The values of the constant b that were used by Geiger and Nuttall implied a roughly 90th-power dependence of λ on Qα. There is a tremendous range of known half-lives from the 2 × 1015 years of 144/60Nd (neodymium) with its 1.83-MeV alpha-particle energy (Eα) to the 0.3 microsecond of 212/84 Po (polonium) with Eα = 8.78 MeV.The theoretical basis for the Geiger–Nuttall empirical rate law remained unknown until the formulation of wave mechanics. A dramatic early success of wave mechanics was

Notes

Nuclear Chemistry


the quantitative theory of alpha-decay rates. One curious feature of wave mechanics is that particles may have a nonvanishing probability of being in regions of negative kinetic energy. In classical mechanics a ball that is tossed to roll up a hill will slow down until its gravitational potential energy equals its total energy, and then it will roll back toward its starting point. In quantum mechanics the ball has a certain probability of tunneling through the hill and popping out on the other side. For objects large enough to be visible to the eye, the probability of tunneling through energetically forbidden regions is unobservably small. For submicroscopic objects such as alpha particles, nucleons, or electrons, however, quantum mechanical tunneling can be an important process—as in alpha decay.The logarithm of tunneling probability on a single collision with an energy barrier of height B and thickness D is a negative number proportional to thickness D, to the square root of the product of B and particle mass m. The size of the proportionality constant will depend on the shape of the barrier and will depend inversely on Planck’s constant h.In the case of alpha decay, the electrostatic repulsive potential between alpha particle and nucleus generates an energetically forbidden region, or potential barrier, from the nuclear radius out to several times this distance. The maximum height (B) of this alpha barrier is given approximately by the expression B = 2Ze2/R, in which Z is the charge of the daughter nucleus, e is the elementary charge in electrostatic units, and R is the nuclear radius. Numerically, B is roughly equal to 2Z/A1⁄3, with A the mass number and B in energy units of MeV. Thus, although the height of the potential barrier for 212/84Po decay is nearly 28 MeV, the total energy released is Qα = 8.95 MeV. The thickness of the barrier (i.e., distance of the alpha particle from the centre of the nucleus at the moment of recoil) is about twice the nuclear radius of 8.8 × 10−13 centimetre.

The tunneling calculation for the transition probability (P) through the barrier gives approximately in which M is the mass of the alpha particle and ℏ is Planck’s constant h divided by 2π. By making simple assumptions about the frequency of the alpha particle striking the barrier, the penetration formula (5) can be used to calculate an effective nuclear radius for alpha decay. This method was one of the early ways of estimating nuclear sizes. In more sophisticated modern techniques the radius value is taken from other experiments, and alpha-decay data and penetrabilities are used to calculate the frequency factor.The form of equation (5) suggests the correlation of decay rates by an empirical expression relating the half-life (t1⁄2) of decay in seconds to the release energy (Qα) in MeV:

Values of the constants a and b that give best fits to experimental rates of even–even nuclei with neutron number greater than 126 are given in the table. The nuclei with

Notes

Inorganic Chemistry


126 or fewer neutrons decay more slowly than the heavier nuclei, and constants a and b must be readjusted to fit their decay rates.

Semiempirical constants* a b98 californium (Cf) 152.86 −52.950696 curium (Cm) 152.44 −53.682594 plutonium (Pu) 146.23 −52.089992 uranium (U) 147.49 −53.656590 thorium (Th) 144.19 −53.264488 radium (Ra) 139.17 −52.147686 radon (Rn) 137.46 −52.459784 polonium (Po) 129.35 −49.9229

*From correlation of ground-state decay rates of even-even nuclei with N > 126.The alpha-decay rates to excited states of even–even nuclei and to ground and excited states of nuclei with odd numbers of neutrons, protons, or both may exhibit retardations from equation (6) rates ranging to factors of thousands or more. The factor by which the rate is slower than the rate formula (6) is the hindrance factor. The existence of uranium-235 in nature rests on the fact that alpha decay to the ground and low excited states exhibits hindrance factors of over 1,000. Thus the uranium-235 half-life is lengthened to 7 × 108 years, a time barely long enough compared to the age of the elements in the solar system for uranium-235 to exist in nature today.The alpha hindrance factors are fairly well understood in terms of the orbital motion of the individual protons and neutrons that make up the emitted alpha particle. The alpha-emitting nuclei heavier than radium are considered to be cigar-shaped, and alpha hindrance factor data have been used to infer the most probable zones of emission on the nuclear surface—whether polar, equatorial, or intermediate latitudes.Beta decayThe processes separately introduced at the beginning of this section as beta-minus decay, beta-plus decay, and orbital electron capture can be appropriately treated together. They all are processes whereby neutrons and protons may transform to one another by weak interaction. In striking contrast to alpha decay, the electrons (minus or plus charged) emitted in beta-minus and beta-plus decay do not exhibit sharp, discrete energy spectra but have distributions of electron energies ranging from zero up to the maximum energy release, Qβ . Furthermore, measurements of heat released by beta emitters (most radiation stopped in surrounding material is converted into heat energy) show a substantial fraction of the energy, Qβ , is missing. These observations, along with other considerations involving the spins or angular momenta of nuclei and electrons, led Wolfgang Pauli to postulate the simultaneous emission of the neutrino (1931). The neutrino, as a light and uncharged particle with nearly no interaction with matter, was supposed to carry off the missing heat energy. Today, neutrino theory is well accepted with the elaboration that there are six kinds of neutrinos, the electron neutrino, mu neutrino, and tau neutrino and corresponding antineutrinos of each. The electron neutrinos are involved in nuclear beta-decay transformations, the mu neutrinos are encountered in decay of muons to electrons,

Notes

Nuclear Chemistry


and the tau neutrinos are produced when a massive lepton called a tau breaks down.Although in general the more energetic the beta decay the shorter is its half-life, the rate relationships do not show the clear regularities of the alpha-decay dependence on energy and atomic number. The first quantitative rate theory of beta decay was given by Enrico Fermi in 1934, and the essentials of this theory form the basis of modern theory. As an example, in the simplest beta-decay process, a free neutron decays into a proton, a negative electron, and an antineutrino: n → p + e− + ν. The weak interaction responsible for this process, in which there is a change of species (n to p) by a nucleon with creation of electron and antineutrino, is characterized in Fermi theory by a universal constant, g. The sharing of energy between electron and antineutrino is governed by statistical probability laws giving a probability factor for each particle proportional to the square of its linear momentum (defined by mass times velocity for speeds much less than the speed of light and by a more complicated, relativistic relation for faster speeds). The overall probability law from Fermi theory gives the probability per unit time per unit electron energy interval, P(W), as follows:

in which W is the electron energy in relativistic units (W = 1 + E/m0c2) and W0 is the

maximum (W0 = 1 + Qβ/m0c2), m0 the rest mass of the electron, c the speed of light,

and h Planck’s constant. This rate law expresses the neutron beta-decay spectrum in good agreement with experiment, the spectrum falling to zero at lowest energies by the factor W and falling to zero at the maximum energy by virtue of the factor (W0 − W)2.In Fermi’s original formulation, the spins of an emitted beta and neutrino are opposing and so cancel to zero. Later work showed that neutron beta decay partly proceeds with the 1/2 ℏ spins of beta and neutrino adding to one unit of ℏ. The former process is known as Fermi decay (F) and the latter Gamow–Teller (GT) decay, after George Gamow and Edward Teller, the physicists who first proposed it. The interaction constants are determined to be in the ratio gGT

2/gF2 = 1.4. Thus, g2 in equation (7) should be replaced

by (gF2 + gGT

2).The scientific world was shaken in 1957 by the measurement in beta decay of maximum violation of the law of conservation of parity. The meaning of this nonconservation in the case of neutron beta decay considered above is that the preferred direction of electron emission is opposite to the direction of the neutron spin. By means of a magnetic field and low temperature it is possible to cause neutrons in cobalt-60 and other nuclei, or free neutrons, to have their spins set preferentially in the up direction perpendicular to the plane of the coil generating the magnetic field. The fact that beta decay prefers the down direction for spin means that the reflection of the experiment as seen in a mirror parallel to the coil represents an unphysical situation: conservation of parity, obeyed by most physical processes, demands that experiments with positions reversed by mirror reflection should also occur. Further consequences of parity violation in beta decay are that spins of emitted neutrinos and electrons are directed along the direction of flight, totally so for neutrinos and partially so by the ratio of electron speed to the speed of light for electrons.

Notes

Inorganic Chemistry


The overall half-life for beta decay of the free neutron, measured as 12 minutes, may be related to the interaction constants g2 (equal to gF

2 + gGT2) by integrating (summing)

probability expression (7) over all possible electron energies from zero to the maximum. The result for the decay constant is in which W0 is the maximum beta-particle energy in relativistic units (W0 = 1 + Qβ/m0c

2), with m0 the rest mass of the electron, c the speed of light, and h Planck’s constant. The best g value from decay rates is approximately 10−49 erg per cubic centimetre. As may be noted from equation (8), there is a limiting fifth-power energy dependence for highest decay energies.

In the case of a decaying neutron not free but bound within a nucleus, the above formulas must be modified. First, as the nuclear charge Z increases, the relative probability of low-energy electron emission increases by virtue of the coulombic attraction. For positron emission, which is energetically impossible for free protons but can occur for bound protons in proton-rich nuclei, the nuclear coulomb charge suppresses lower energy positrons from the shape given by equation (7). This equation can be corrected by a factor F(Z,W) depending on the daughter atomic number Z and electron energy W. The factor can be calculated quantum mechanically. The coulomb charge also affects the overall rate expression (8) such that it can no longer be expressed as an algebraic function, but tables are available for analysis of beta decay rates. The rates are analyzed in terms of a function f(Z,Qβ) calculated by integration of equation (7) with correction factor F(Z,W).Approximate expressions for the f functions usable for decay energies Q between 0.1 MeV and 10 MeV, in which Q is measured in MeV, and Z is the atomic number of the daughter nucleus, are as follows (the symbol ≈ means approximately equal to):

For electron capture, a much weaker dependence on energy is found:

The basic beta decay rate expression obeyed by the class of so-called superallowed transitions, including decay of the neutron and several light nuclei is

Like the ground-to-ground alpha transitions of even–even nuclei, the superallowed beta transitions obey the basic rate law, but most beta transitions go much more slowly. The extra retardation is explained in terms of mismatched orbitals of neutrons and protons involved in the transition. For the superallowed transitions the orbitals in initial and final states are almost the same. Most of them occur between mirror nuclei, with one more or less neutron than protons; i.e., beta-minus decay of hydrogen-3, electron capture of beryllium-7 and positron emission of carbon-11, oxygen-15, neon-19, . . . titanium-43.The nuclear retardation of beta decay rates below those of the superallowed class may

Notes

Nuclear Chemistry


be expressed in a fundamental way by multiplying the right side of equation (9) by the square of a nuclear matrix element (a quantity of quantum mechanics), which may range from unity down to zero depending on the degree of mismatch of initial and final nuclear states of internal motion. A more usual way of expressing the nuclear factor of the beta rate is the log ft value, in which f refers to the function f(Z,Qβ). Because the half-life is inversely proportional to the decay constant λ, the product fβt1⁄2 will be a measure of (inversely proportional to) the square of the nuclear matrix element. For the log ft value, the beta half-life is taken in seconds, and the ordinary logarithm to the base 10 is used. The superallowed transitions have log ft values in the range of 3 to 3.5. Beta log ft values are known up to as large as ∼ 23 in the case of indium-115. There is some correlation of log ft values with spin changes between parent and daughter nucleons, the indium-115 decay involving a spin change of four, whereas the superallowed transitions all have spin changes of zero or one.Gamma transitionThe nuclear gamma transitions belong to the large class of electromagnetic transitions encompassing radio-frequency emission by antennas or rotating molecules, infrared emission by vibrating molecules or hot filaments, visible light, ultraviolet light, and X-ray emission by electronic jumps in atoms or molecules. The usual relations apply for connecting frequency ν, wavelength λ, and photon quantum energy E with speed of light c and Planck’s constant h; namely, λ = c/ν and E = hv. It is sometimes necessary to consider the momentum (p) of the photon given by p = E/c.Classically, radiation accompanies any acceleration of electric charge. Quantum mechanically there is a probability of photon emission from higher to lower energy nuclear states, in which the internal state of motion involves acceleration of charge in the transition. Therefore, purely neutron orbital acceleration would carry no radiative contribution.A great simplification in nuclear gamma transition rate theory is brought about by the circumstance that the nuclear diameters are always much smaller than the shortest wavelengths of gamma radiation in radioactivity—i.e., the nucleus is too small to be a good antenna for the radiation. The simplification is that nuclear gamma transitions can be classified according to multipolarity, or amount of spin angular momentum carried off by the radiation. One unit of angular momentum in the radiation is associated with dipole transitions (a dipole consists of two separated equal charges, plus and minus). If there is a change of nuclear parity, the transition is designated electric dipole (E1) and is analogous to the radiation of a linear half-wave dipole radio antenna. If there is no parity change, the transition is magnetic dipole (M1) and is analogous to the radiation of a full-wave loop antenna. With two units of angular momentum change, the transition is electric quadrupole (E2), analogous to a full-wave linear antenna of two dipoles out-of-phase, and magnetic quadrupole (M2), analogous to coaxial loop antennas driven out-of-phase. Higher multipolarity radiation also frequently occurs with radioactivity.Transition rates are usually compared to the single-proton theoretical rate, or Weisskopf formula, named after the American physicist Victor Frederick Weisskopf, who developed

Notes

Inorganic Chemistry


it. The table gives the theoretical reference rate formulas in their dependence on nuclear mass number A and gamma-ray energy Eγ (in MeV).

Gamma transition rates*

transition type partial half-life tγ (seconds)

illustrative tγ values for A = 125, E = 0.1 MeV (seconds)

E1 5.7 × 10−15 E−3 A−2/3 2 × 10−13

E2 6.7 × 10−9 E−5 A−4/3 1 × 10−6

E3 1.2 × 10−2 E−7 A−2 8E4 3.4 × 104 E −9 A−8/3 9 × 107

E5 1.3 × 1011 E−11 A−10/3 1 × 1015

M1 2.2 × 10−14 E −3 2 × 10−11

M2 2.6 × 10−8 E −5 A−2/3 1 × 10−4

M3 4.9 × 10−2 E−7 A−4/3 8 × 102

M4 1.3 × 105 E−9 A−2 8 × 109

M5 5.0 × 1011 E−11 A−8/3 1 × 1017

*The energies E are expressed in MeV. The nuclear radius parameter r0 has been taken as 1.3 fermis. It is to be noted that tγ is the partial half-life for γ emission only; the occurrence of internal conversion will always shorten the measured half-life.It is seen for the illustrative case of gamma energy 0.1 MeV and mass number 125 that there occurs an additional factor of 107 retardation with each higher multipole order. For a given multipole, magnetic radiation should be a factor of 100 or so slower than electric. These rate factors ensure that nuclear gamma transitions are nearly purely one multipole, the lowest permitted by the nuclear spin change. There are many exceptions, however; mixed M1–E2 transitions are common, because E2 transitions are often much faster than the Weisskopf formula gives and M1 transitions are generally slower. All E1 transitions encountered in radioactivity are much slower than the Weisskopf formula. The other higher multipolarities show some scatter in rates, ranging from agreement to considerable retardation. In most cases the retardations are well understood in terms of nuclear model calculations. Though not literally a gamma transition, electric monopole (E0) transitions may appropriately be mentioned here. These may occur when there is no angular momentum change between initial and final nuclear states and no parity change. For spin-zero to spin-zero transitions, single gamma emission is strictly forbidden. The electric monopole transition occurs largely by the ejection of electrons from the orbital cloud in heavier elements and by positron–electron pair creation in the lighter elements.

2.9 Nuclear reaction and cross sectionWhen a nucleus is bombarded with different projectiles, then either elastic or inelastic scattering may take place or one or more particles which are all together different may be knocked out of the nucleus. When the mass number and/or atomic number of target nuclei changes, we say a nuclear reaction takes place. Typically a nuclear reaction is written as

Notes

Nuclear Chemistry


where x is incident projectile, X is target nucleus, Y is new nucleus and y is outgoing particle. The above reaction can also be written is short form as . Nuclear reactions are classified on the basis of projectiles used, particles detected and residual nucleus. There are two types of nuclear reactions:(A) Direct reaction: In this type of reaction, projectile nucleons enter or leave the target nucleus without disturbing the other nucleons. These are further classified as-

(i) Scattering: In the scattering reaction, the projectile and outgoing particles are same. The scattering is elastic when residual nucleus is left in ground state. The scattering is called inelastic when residual nucleus is in excited state.

(ii) Pickup reactions: When the projectile gains nucleons from the target, the nuclear reaction is called pick up reaction. i.e.

(iii) Stripping reaction: In this type of nuclear reaction, the projectile loses nucleons to the target nucleus viz.

(B) Compound nuclear reaction: Here projectile and target form a compound nucleus which has a life time of 10-16 sec. The compound nucleus does not "remember" how it was formed as life span of 10-16 sec is very large in comparison to nuclear time of 10-22 sec. The same compound nucleus can be formed by number of nuclear reactions and can decay in a number of ways or channels.

Transmutations

Notes

Inorganic Chemistry


Transmutation by neutrons

Notes

Nuclear Chemistry


Notes

Inorganic Chemistry


Notes

Nuclear Chemistry


Fission Reaction

Notes

Inorganic Chemistry


Notes

Nuclear Chemistry


Product Distribution

Notes

Inorganic Chemistry


Notes

Nuclear Chemistry


2.10 Isotopes and their application

Isotopes are variants of a particular chemical element which differ in neutron number, although all isotopes of a given element have the same number of protons in each atom. The term isotope is formed from the Greek roots isos ( "equal") and topos ("place"), meaning "the same place". The number of protons within the atom’s nucleus is called the atomic number. Each atomic number identifies a specific element, but not the isotope; an atom of a given element may have a wide range in its number of neutrons. The number of both protons and neutrons in the nucleus is the atom's mass number, and each isotope of a given element has a different mass number.

Stable isotopes are generally defined as non-radioactive isotopic elements that do not decay over time. Radioactive isotopes may also be classified as stable isotopes when their half-lives are too long to be measured. These elements can often be found to occur in nature and include isotopes of carbon, nitrogen, hydrogen, oxygen, noble gases and metals.

For example, there are a lot of carbon atoms in the universe. The normal ones are carbon-12. Those atoms have 6 neutrons. There are a few straggler atoms that don't have 6. Those odd ones may have 7 or even 8 neutrons. Carbon-14 actually has 8 neutrons.

Notes

Inorganic Chemistry


C-14 is considered an isotope of the element carbon.The applications of radioisotopes have played a significant role in improving the quality of life of human beings.

1. Radiotracer (radioisotopes)

Radiotracers are widely used in medicine, agriculture, industry, and fundamental research. Radiotracer is a radioactive isotope; it adds to nonradioactive element or compound to study the dynamical behavior of various physical, chemical, and biological changes of system to be traced by the radiation that it emits. The tracer principle was introduced

by George de Hevesy in 1940 for which he was awarded the Nobel prize.

Radioisotope production

The sustainability of radioisotope production is one of the critical areas that receive great attention. There are more than 160 different radioisotopes that are used regularly in different fields; these isotopes are produced either in a medium or in high-flux research reactors or particle accelerators (low or medium energy) . Some of the radioisotopes

produced by the reactor and particle accelerators and their applications are given in Table

Reactor radioisotope Half-life ApplicationsRadioisotopes produced by reactorsBismuth-213 45.59 min It is an alpha emitter (8.4 MeV). Used for

cancer treatment, e.g., in the targeted alpha therapy (TAT)

Cesium-131 9.7 days It emits photon radiation in the X-ray range (29.5–33.5 keV). Used in brachytherapy of malignant tumors

Cesium-137 30 years Used in medical devices (sterilization) and gauges (661.64 keV)

Chromium-51 28 days Used in Diagnosis of gastrointestinal bleeding and to label platelets (320 keV)

Cobalt-60 5.27 years Used for controlling the cancerous growth of cells (1173.2 keV)

Dysprosium-165 2 h Used for synovectomy treatment of arthritis (95 keV)

Erbium-169 9.4 days Used for relieving arthritis pain in synovial joints (8 keV)

Holmium-166 26 h Diagnosis and treatment of liver tumors (81 keV)

Iodine-125 60 days U s e d i n c a n c e r b r a c h y t h e r a p y a n d radioimmunoassay (35 keV)

Notes

Nuclear Chemistry


Iodine-131 8 days Widely used in treating thyroid cancer and in imaging the thyroid, diagnosis, and renal blood flows (284 keV)

Iridium-192 74 days Used as an internal radiotherapy source for cancer treatment. Strong beta emitter for high-dose rate brachytherapy (317 keV)

Iron-59 46 days Used in studies of iron metabolism in the spleen (1095 keV)

Lead-212 10.6 h Used in TAT for cancers (239 keV)Molybdenum-99 66 h Used as the parent in a generator to produce

technetium-99 m (740 keV)Palladium-103 17 days Used to make brachytherapy permanent implant

seeds for early-stage prostate cancer. Emits soft X-rays (362 keV)

Potassium-42 12.36 h Used for potassium distribution in bodily fluids and to locate brain tumors (1524 keV)

Radium-223 11.4 days Used to treat prostate cancers that have spread to the bones

Rhenium-186 3.71 days Used for therapeutic purpose to relief pain in bone cancer. Beta emitter with weak gamma for imaging (137 keV)

Samarium-153 47 h Effective in relieving the pain of secondary cancers lodged in the bone, sold as Quadra met. Beta emitter (103 keV)

Selenium-75 120 days Used to study the production of digestive enzymes (265 keV)

Sodium-24 15 h Used for studies of electrolytes within the body (2754 keV)

Ytterbium-169 32 days Used for cerebrospinal fluid studies in the brain (63 keV)

Radioisotopes produced by acceleratorsCobalt-57 272 days Used as a marker to estimate organ size and for

in vitro diagnostic kits (122 keV)Copper-64 13 h Used for PET imaging studies of tumors and

also cancer therapy (511 keV)Copper-67 2.6 days Beta emitter, used in therapyFluorine-18 110 min Used as fluorothymidine (FLT)Gallium-67 78 h Used for tumor imaging and locating

inflammatory lesions (infections)

Notes

Inorganic Chemistry


Indium-111 2.8 days Brain studies, infection, and colon transit studiesIodine-123 13 h Used for diagnosis of thyroid functionRubidium-82 1.26 min Convenient PET agent in myocardial perfusion

imagingStrontium-82 25 days Used as the parent in a generator to produce

Rb-82Thallium-201 73 h Used for location of low-grade lymphomas

Some of the radioisotopes produced by the reactor and particle accelerators and their

applications.

2. Solubility of sparingly soluble salt

In determining the solubility of lead sulphate, Hevsey and paneth, 1913, added a quantity

of Pb212 radioisotope which radioactive, viz radioactive pb(NO3)2 with pbcl2 and lead

chloride with Pb212 was precipitated. The activity of resulting mixture was measured,

so that a relationship was established was to the activity of material per mg of lead.

Pb(NO3)2 + H2SO4 (or) Na2SO4 →PbSO4

Pb(NO3)2 + K2CrO4 →PbCrO4

The soluble mixture of lead Ion was created as lead sulphate and held at constant

temperature until a saturated solution of fairy insoluble material and reached equilibrium.

After ascertaining the total volume of supranatant liquid, a sample of definite volume

was withdrawn and evaporated to the dryness. The activity of minute residue was

determined in the same fashion as upon the initial mixture before precipitated of lead

sulphate from the activity of the residue it was possible to establish the amount of lead

sulphate. From the activity of the residue it was possible to establish the amount of lead

sulphate dissolved in the known quantity of solution by using the relationship;

Amount = moles of lead sulphate per litre

Where S1 = Initial activity , S2 = Final activity M = Molecular weight of lead sulphate

2. Isotopic dilution method

It is used in determining the volume of blood in an animal or human being. The

procedure is that in a donor, a tracer is injected into and allowing sufficient time for

incorporation (sometimes weeks) of a tracer into the circulating red cell. A tracer used

for this purpose is 59Fe. A sample is withdrawn from the donor and activity per millitre

of blood is determined and then an aliquot portion of the sample introduced into the

recipient subject. After proper elapse of time, a sample from recipient is assayed for

Notes

Nuclear Chemistry


radioactivity. From the dilution of the activity it is possible to calculate the volume of

blood in the recipient.

The solubility of benzene or other hydrocarbons is too small to be measured by ordinary

physical or chemical method. If however, a definite quantity of 3H1 tritium in the form of

tritium oxide is added to a given amount of water, the resulting mixture has an activity

that can be measured in the minute amounts that may dissolved organic solvents. Thus

using dilution methods one has to know the amount of labeled tracer added, the weight

of the mixture and the labeled content of a purified sample from the combined mixture .

3. Neutron activation analysis

When a sample is subjected to nuclear bombardments and then analysed for its radioactive

content is called "activation analysis". Most elements give rise to a radioactive isotope

with characteristic radiations, when bombarded with neutrons form a reactor or neutron

generator. This permits qualitative identification and also quantitative analysis by

comparative method.

The sample to be analysed is exposed to thermal neutrons from a reactor for a sufficiently

long time to produce measurable amount of desire radioisotope. The capture rate is

proportional to the neutron flux f (neutrons cm-2) sec-1) and to the number of target

nuclei N available and the induced activity is represented as;

A = f no (1-e0.693t/T1/2)

Where A = Induced activity o

F = neutron flux

N = number of target nuclei

= weight of target material in multiply 6.023 x 1023 /atomic weight

O = nuclear cross section in barns

t = period of irradiation in seconds

T1/2= half line, seconds

This is the most sensitive method to determine the trace elements like gallium in iron,

copper in nickel and halfnium in Zirconium.

Hair accumulates arsenic at the bottom end of the hair and hair grows normally 0.5

mm per day. if a person is dead, the arsenic in hair can be neutron irradiated rated

the distribution of 76As along the length compared, the pattern and the schedule of

Arsenic poisoning may be determined. In normal hair 76As content is lower and remains

almost constant all along the length except towards the tip, while a pattern in case of

Notes

Inorganic Chemistry


Arsenic poisoning reveals distinct peaks corresponding to days of poisoning each day

corresponding to 0.5 mm length of the hair. The technique was used in 1962 in the

examination of Napoleon's hair and revealed abnormal amount of arsenic. However

this evidence was not considered conclusively.

4. Structure of Thiosulphate ion One of the applications of tracer chemistry to structural problem was the establishment

of thiosulphate ion. If sulphur is heated with labelled sulphide ion, the thiosulphate ion

is produced as follows as

When the thiosulphate containing a labelled sulphur atom is broken down in acid

solution, the product are

After decomposition of thiosulphate the label sulphur is linked to the oxygen atom

in the sulphate ion. This clearly indicates that the the labelled sulphur is not affected

in the synthesis and decomposition of the thiosulphate and also it shows that both S

atoms in the thiosulphate are not equivalent. Otherwise the label isotope would have

been distributed between the product of decomposition. This lead to the conclusion that

thiosulphate ion has the structure.

5. Reaction mechanism

The course of the certain organic reactions can be traced with labelled atoms. In ester

hydrolysis since oxygen does not have a radioisotope sufficiently long lived for tracer

studies; it was necessary to use O18 the stable isotope as a tracer. The hydrolysis of an

ester by water enriched with heavy oxygen is indicated as

The fact that the labelled oxygen is in the acid proves that OR' group has been substituted by the O*H in the hydrolytic reaction. The acyl-oxygen bond breaks.

6. Medicine

Nowadays radiotracer has become an indispensable and sophisticated diagnostic tool in medicine and radiotherapy purposes.

Notes

Nuclear Chemistry


Diagnostic purpose

The most common radioactivity isotope used in radioactive tracer is technetium (99Tc).

Tumors in the brain are located by injecting intravenously 99Tc and then scanning the head

with suitable scanners. 131I and most recently 132I and 123I are used to study malfunctioning

thyroid glands. Kidney function is also studied using compound containing 131I. 33P is

used in DNA sequencing. Tritium (3H) is frequently used as a tracer in biochemical

studies. 14C has been used extensively to trace the progress of organic molecule through

metabolic pathways.

A most recent development is positron emission tomography (PET), which is a more

precise and accurate technique for locating tumors in the body. A positron emitting

radionuclide (e.g., 13N, 15O, 18F, etc.) is injected to the patient, and it accumulates in

the target tissue. As it emits positron which promptly combines with nearby electrons,

it results in the simultaneous emission of two γ-rays in opposite directions. These

γ-rays are detected by a PET camera and give precise indication of their origin, that is,

depth also. This technique is also used in cardiac and brain imaging. Compound X-ray

tomography or CT scans. The radioactive tracer produces gamma rays or single photons

that a gamma camera detects. Emissions come from different angles, and a computer

uses them to produce an image. CT scan targets specific area of the body, like the neck

or chest, or a specific organ, like the thyroid .

Therapeutic

The most common therapeutic use of radioisotopes is 60Co, used in treatment of cancer.

Sometimes wires or sealed needles containing radioactive isotope such as 192Ir or 125I

are directly placed into the cancerous tissue. The radiations from the radioisotopes

attack the tumor as long as needle/wire is in place. When the treatment is complete,

these are removed. This technique is frequently used to treat mouth, breast, lung, and

uterine cancer. 131I is used to treat thyroid for cancers and other abnormal conditions

of thyroid. 32P is used to treat excess of red blood cells produced in the bone marrow.

7. Agricultural research

Development of high yielding varieties of plants, oil seeds, and other economically important crops and protection of plant against the insects are the thrust area of

agricultural research.

New varieties of crops

The irradiated seeds of wheat, rice, maize, cotton, etc., are undergoing profound genetic changes in order to improve crop varieties and mutation breeding. These varieties of

Notes

Inorganic Chemistry


crops are more disease resistant and have high yields. Several countries all over the

world produce new variety of crops from radiation-induced mutants.

Eradication of insect and pests

The best technique for the control of insects and pests is sterile insect technique (SIT). Irradiation is used to sterilize mass-reared insects so that, while they remain sexually competitive, they cannot produce offspring. As a result, it enhances the crop production

and preservation of natural resources.

8. Food preservation and sterilization

As per WHO reports, about 25–35% of world food production is susceptible to the attack by pests, insects, bacteria, and fungi causing a great loss of the economy of the country. Food irradiation has more advantages than conventional methods. All types of radiations are not recommended for food irradiation; only three types of radiation are recommended by CODEX general standard for food irradiation which are 60Co or 137Cs, X-rays, or electron beams from particle accelerators. The food products are exposed to γ-radiations from the intense controlled sources to kills pests, bacteria, insects, and parasites and extends shelf-life but also reduces the food’s nutritional value somewhat by destroying vitamins A, B1 (thiamin), C, and E. No radiation remains in the food after treatment.

Depending on the radiation dose and its application, radiations are classified into three categories: they are low dose (<1 kGy), medium dose (1–10 kGy), and high dose (>10

kGy) .

Low-dose applications

Sprout inhibition in bulbs and tubers: Irradiated potato can be stored at higher temperature

of around 15°C. This not only conserves energy but also prevents sweetening of potato,

commonly occurring at low temperatures. It gives advantage to the manufacturers of

chips as low-sugar potato gives desired lighter color to fries and chips.

Delayed ripening of fruits: Irradiated fruits (all kinds of mangoes) of these at hard

mature pre-climacteric stage at 0.25–0.75 kGy delay the ripening process by about 7

days, thus improving shelf-life. These doses are also effective in destroying quarantine

pests. Irradiated fresh fruits can be stored for longer duration, sometimes up to 30 days

at 12–14°C and in modified atmospheres.

Medium-dose applications

Under ice, sea food such as fish and prawns, fish-like Bombay duck, pomfret, Indian

salmon, mackerel, and shrimp can be stored for about 7–10 days. Studies have

demonstrated that irradiation at 1–3 kGy followed by storage at melting ice temperatures

increases its shelf-life nearly threefold.

Notes

Nuclear Chemistry


Meat and meat products including poultry have a shelf-life of about a week at 0–3°C,

which could be extended up to 4 weeks by applying a dose of 2–5 kGy, which inactivates

spoilage bacteria. Radiation treatment has been employed to enhance the shelf-life of

intermediate moisture meat products.

High-dose applications

While transporting the spices, due to inadequate handling and processing conditions,

spices get contaminated with insect eggs and microbial pathogens. When incorporated

into semi-processed or processed foods, particularly, after cooking, the microbes, both

spoilers and pathogens, in spices can outgrow causing spoilage and posing risk to

consumers. Many of the spices develop insect infestation during storage, and unscrupulous

traders convert them into spice powders. A dose of 10 kGy brings about near sterility or

commercial sterility while retaining the natural characteristics of spices.

Irradiation at higher doses can also be employed for total sterilization of diets for

immunocompromised patients, adventure sports, military, and astronauts.

9. Industry and civil engineering

Radioisotopes are commonly used in industry for checking blocked water pipes and

detecting leakage in oil pipes. For example, small quantity of radioactive 24Na is placed

in a small enclosed ball and is allowed to move in pipe with water. The moving ball

containing radioisotope is monitored with a detector. If the movement of ball stops, it

indicates the blocked pipe. Similarly, radioisotope 24Na is mixed with oil flowing in an

underground pipe. With radiation detector, the radioactivity over the pipe is monitored.

If there is a leakage place, the radiation detector will show large activity at that particular

place. Radioisotopes are also used to monitor fluid flow and filtration, detect leaks, and

gauge engine wear and corrosion of process in equipment.

Radioactive materials are used to inspect metal parts and the integrity of welds across a

range of industries. The titanium capsule is a radioactive isotope which is placed on one

side of the object being screened, and some photographic film is placed on the other side.

The gamma rays pass through the object and create an image on the film. Gamma rays

show flaws in metal castings or welded joints. The technique allows critical components

to be inspected for internal defects without damage. Radiotracer is also used to inspect

for internal defect without damage.

In industries, the production methods need to be constantly monitored in order to check

the quality of products and to control the production process. The monitoring is carried

out by quality control devices using the unique properties of radiation; such devices are

Notes

Inorganic Chemistry


called nuclear gauges. They are more useful in extreme temperature, harmful chemical

process, molten glass, and metals. The gauges are also used to measure the thickness of

sheet materials, including metals, textiles, paper, and plastic production.

2.11 LET US SUM UPIn 1896 Henri Becquerel was using naturally fluorescent minerals to study the properties of x-rays, which had been discovered in 1895 by Wilhelm Roentgen. He exposed potassium uranyl sulfate to sunlight and then placed it on photographic plates wrapped in black paper, believing that the uranium absorbed the sun’s energy and then emitted it as x-rays. This hypothesis was disproved on the 26th-27th of February, when his experiment "failed" because it was overcast in Paris. For some reason, Becquerel decided to develop his photographic plates anyway. To his surprise, the images were strong and clear, proving that the uranium emitted radiation without an external source of energy such as the sun. Becquerel had discovered radioactivity.Becquerel used an apparatus similar to that displayed below to show that the radiation he discovered could not be x-rays. X-rays are neutral and cannot be bent in a magnetic field. The new radiation was bent by the magnetic field so that the radiation must be charged and different than x-rays. When different radioactive substances were put in the magnetic field, they deflected in different directions or not at all, showing that there were three classes of radioactivity: negative, positive, and electrically neutral.The term radioactivity was actually coined by Marie Curie, who together with her husband Pierre, began investigating the phenomenon recently discovered by Becquerel. The Curies extracted uranium from ore and to their surprise, found that the leftover ore showed more activity than the pure uranium. They concluded that the ore contained other radioactive elements. This led to the discoveries of the elements polonium and radium. It took four more years of processing tons of ore to isolate enough of each element to determine their chemical properties.

In liquid drop model, the molecules of liquid drop correspond to nucleons in the

nucleus.This liquid drop model explains the stability of nuclei against the breakup into

two fragments i.e. nuclear fission.For certain numbers of neutrons and protons called

magic numbers, nuclei exhibit special stability. This stability is not explained by liquid

drop model. Also the other properties of nucleons like spin, magnetic moment and

quadrupole moments are unexplained in liquid drop model.The magic numbers and

other properties of nuclei are explained by nuclear shell model.

In shell model, all magic numbers are reproduced when spin orbit interaction

term is included in simple harmonic oscillator potential.

Radiotracers are widely used in medicine, agriculture, industry, and fundamental research. Radiotracer is a radioactive isotope; it adds to nonradioactive element or compound to study the dynamical behavior of various physical, chemical, and biological changes of

Notes

Nuclear Chemistry


system to be traced by the radiation that it emits. The tracer principle was introduced

by George de Hevesy in 1940 for which he was awarded the Nobel prize.

Radioisotope production

The sustainability of radioisotope production is one of the critical areas that receive great attention. There are more than 160 different radioisotopes that are used regularly in different fields; these isotopes are produced either in a medium or in high-flux research

reactors or particle accelerators (low or medium energy) .


1. What do you mean by nuclear chemistry?

2. Explain the concept of Theory of radio decay?

3. What are types of nuclear reaction?

4. Describe the nuclear shell model .

5. What do you mean by liquid drop model?

6. Define nuclear reactor.

7. Write a short note on Isotopes Dilution method .

8. Discuss the use of isotopes in medicine





Notes

Inorganic Chemistry


UNIT–3

Chemistry of f-Block Elements

(Structure) 3.1 Learning Objectives 3.2 Introduction 3.3 Electronic structure 3.4 Oxidation states and ionic radii 3.5 Lanthanide contration 3.6 Complex formation 3.7 Occurrence and isolation 3.8 Lanthanide compounds 3.9 Chemistry of actinides 3.10 Separation of Np, Pu and Am from U 3.11 Similarities between the latter actinides and the latter lanthanides 3.12 Let us sum up 3.13 Review questions 3.14 Further readings


zz Discuss the chemistry of Lanthanide compounds;

zz Explain Chemistry of actinides;

zz Describe Separation of Np, Pu and Am from U.

3.2 Introduction

In order Lanthanides, also sometimes called lanthanons, are the elements which constitute a distinct series of fourteen elements from cerium (Ce, Z = 58) to lutetium (Lu, Z = 71). They are so called because these elements succeed lanthanum (La, Z =57), the element of Group 3 and lie between it and hafnium (Hf, Z = 72), the element of Group 4, both belonging to third transition series (or 5d-transition series). That is why they are also known as inner transition elements. These fourteen elements belong to f-block and are the members of 4f-series because the last or differentiating electron in the atoms of these elements enters 4f-subshell, i.e., the f-subshell of ante-penultimate shell (here n=6).

Notes



Thus, f-block elements have partly filled f-subshells of (n-2)th shell in the elementary or ionic state.

The first and the last elements of the 4f-series have been mentioned above (i.e., Ce and Lu). The rest twelve elements along with their symbols and atomic numbers are given below:

praseodymium (Pr, Z = 59), neodymium (Nd, Z = 60), promethium (Pm, Z = 61), samarium (Sm, Z = 62), europium (Eu, Z = 63), gadolinium (Gd, Z = 64), terbium (Tb, Z = 65), dysprosium (Dy, Z = 66), holmium (Ho, Z = 67), erbium (Er, Z = 68), thulium (Tm, Z =69) and ytterbium (Yb, Z = 70). In analogy with various series of d-block elements, these elements are called first inner transition series elements. All the fourteen elements of the series resemble closely in their electronic configuration and chemical properties with one another as well as with lanthanum which may be called a prototype of lanthanide elements. All these elements said to be the members of Group 3 i.e., Sc group and have been allotted one single position in the periodic table.

These elements were originally called rare earth elements because the elements then known occurred as oxides (earths) and were available scarcely. All the elements have similar physical and chemical properties hence are considered together for their study.

It is interesting to note that the elements with even atomic numbers are relatively more abundant and also have a larger number of isotopes but those with odd atomic numbers are less abundant and do not have more than two isotopes. Promethium (Z = 61) has been made artificially only and does not occur in nature.

3.3 ELECTRONIC STRUCTURE OF LANTHANIDESThe electronic configurations of the lanthanides have been derived from the

electronic spectra of the atoms of these elements. These spectra have great complexity thereby inferring to some doubt about the configurations. In the following table, the expected and alternative probable configurations of the elements have been listed. The expected configurations have been derived by taking into account the electronic configuration of lanthanum and supposing that in all the lanthanide elements succeeding lanthanum, additional electrons are filled in 4f-subshell successively from cerium to lutetium. Thus the 4f-electrons are embedded in the interior while 5d and 6s electrons are exposed to the surroundings. The electronic configurations of lanthanide elements are given in Table 3.1.

Table 3.1 Electronic structures of lanthanum and lanthanides

Element (Z) A t . number

Expected electronic configuration

Alternative probable configuration

Lanthanum (La) 57 [Xe]5d16s2 -Cerium (Ce) 58 [Xe]4f15d16s2 [Xe]4f26s2Praseodymium (Pr) 59 [Xe]4f25d16s2 [Xe]4f36s2Neodymium (Nd) 60 [Xe]4f35d16s2 [Xe]4f46s2Promethium (Pm) 61 [Xe]4f45d16s2 [Xe]4f56s2

Notes

Inorganic Chemistry


Samarium (Sm) 62 [Xe]4f55d16s2 [Xe]4f66s2Europium (Eu) 63 [Xe]4f65d16s2 [Xe]4f76s2Gadolinium (Gd) 64 [Xe]4f75d16s2 [Xe]4f75d16s2Terbium (Tb) 65 [Xe]4f85d16s2 [Xe]4f96s2Dysprosium (Dy) 66 [Xe]4f95d16s2 [Xe]4f106s2Holmium (Ho) 67 [Xe]4f105d16s2 [Xe]4f116s2Erbium (Er) 68 [Xe]4f115d16s2 [Xe]4f126s2Thulium (Tm) 69 [Xe]4f125d16s2 [Xe]4f136s2Ytterbium (Yb) 70 [Xe]4f135d16s2 [Xe]4f146s2Lutetium (Lu) 71 [Xe]4f145d16s2

According to the latter view, as is evident from the table, the solitary 5d-electron shifts into the 4f-subshell in all the cases except in Gd (64) and Lu (71) because it is favourable energetically to move the single 5d-electron into 4f-orbital in most of the elements and give more appropriate electronic configuration which in widely accepted. In Gd such a shift would have destroyed the symmetry of a half filled f-subshell and the resulting configuration would have been less stable than the probable configuration i.e., 4f7 is more stable than 4f8 configuration. In lutetium, the f-subshell is already completely filled and cannot accommodate any additional electron.

From the above, it may be concluded that the general electronic configuration of lanthanide elements could be written as [Xe](n-2)f1-14(n-1)d0,1ns2 where n is the principal quantum number, i.e., 6. The three subshells, viz., (n-2)f, (n-1)d and ns together form the valence shell of these elements, i.e., 4f.5d.6s = valence shell.

3.4 OXIDATION STATESIt has been shown that the lanthanide elements are highly electropositive and form

essentially ionic compounds. It is observed for these elements that +3 (i.e. formation of tripositive ions, Ln3+) is the principal or common oxidation state exhibited by all of them. This is said to be the most stable oxidation state of the lanthanides. Some of these elements also show + 2 and +4 oxidation states but except a few such ions, they have the tendency to get converted to +3 state. For example, Sm and Ce form Sm2+ and Ce4+ ions but are easily converted to +3 states. That is why Sm2+ is a good reducing agent while Ce4+ is a good oxidising agent, i.e.,

Sm2+ → Sm3+ + e (electron provider, a reductant)

Ce4+ + e → Ce3+ (electron acceptor, an oxidant)

It means Ln2+ and Ln4+ ions are less frequent than Ln3+ ions among the lanthanides. +2 and +4 oxidation states are shown by the elements particularly when they lead to:

(a) Noble gas electronic configuration, e.g., Ce4+ (4f0),

Notes



(b) Half-filled f-orbital, e.g., Eu2+ and Tb4+ (4f7), and

(c) Completely filled f-orbital, e.g., Yb2+ (4f14) in the valence shell.

Among the above, +2 and +4 oxidation states, which exist only in aqueous solutions, are exemplified by Sm2+, Eu2+, Yb2+ and Ce4+.

There are some exceptions also, i.e., sometimes +2 and +4 oxidation states are also shown by the elements which are close to f0, f7 and f14 states, e.g., the valence shell configurations of the ions given below are 4f1, 4f2, 4f3, 4f6 and 4f8, etc.:

Ce3+: 4f1; Ce2+: 4f2 ; Sm2+: 4f6 ; Pr4+: 4f1; Pr3+: 4f2; Dy2: 4f8 ; Nd4+: 4f2; Tm2+: 4f13.

No satisfactory explanation for these exceptions has yet been given. These oxidation states have only been explained on the basis of thermodynamic and kinetic factors, that too arbitrarily. Due to the only one stable oxidation state (i.e., +3), lanthanide elements resemble each other much more than do the transition (or d-block) elements. It has also been observed that the higher oxidation states of the lanthanides are stabilized by fluoride or oxide ions, while the lower oxidation sates are favoured by bromide or iodide ions. Among the lanthanides, in addition to +3 states, +2 states is shown by Nd, Sm, Eu, Tm, and Yb only whereas +4 state is exhibited by Ce, Pr, Nd, Tb and Dy elements. Rest five elements show only +3 states.

3.5 IONIC RADII AND LANTHANIDE CONTRACTIONHere, ionic radii of tripositive ions (i.e., Ln3+) have only been considered because

+3 is the most stable and common oxidation state of all the lanthanides, in general. It has been observed that the atomic as well as the ionic radii of lanthanides decrease steadily as we move from Ce to Lu. The ionic radii have been listed below (for Ln3+ ions):

From the above list it is clear that the ionic radii decrease steadily all along the series amounting in all to 18 pm. This shrinking in the ionic size of the Ln3+ ions with increasing atomic number is called lanthanide contraction. The term steadily decrease means the values decrease regularly and with a very small difference though the nuclear charge increases by +14 units from the first to the last element.

The atomic radii of these elements also decrease from Ce to Lu (Ce : 165 pm, Lu : 156 pm) but the overall shift is only of 165-156 = 9 pm. These values do not decrease regularly like ionic radii rather there are some irregularities at Eu and Yb which have abnormally high atomic radii. (Eu : 185 pm, Yb : 170 pm).

Notes

Inorganic Chemistry


The atomic radii for the metats are actually the metallic radii which are recorded for the metal atoms surrounded by 8 or 12 nearest neighbors (in bulk). Various metal atoms in metal crystal are bonded together by metallic bonding. In Eu and Yb, only two 6s-electrons participate in metallic bonding, 4f-subshells being stable. For other lanthanides, three electrons are generally available for this purpose. This results in larger atomic volumes for Eu and Yb because of weaker bonding among atoms. The larger values ultimately give rise to the larger size to the atoms of the elements.

Cause of Lanthanide Contraction

As we move along the lanthanide series from Ce to Lu, the addition of electrons takes place to the 4f-orbitals, one at each step. The mutual shielding effect of f-electrons is very little, being even smaller than that of d-electrons, due to the scattered or diffused shape of these orbitals. However, the nuclear charge (i.e. atomic number) goes on increasing by one unit at each step (i.e., each next element). Thus, the attraction between the nucleus and the outermost shell electrons also goes on increasing gradually at each step. The 4f-electrons are not able to shield effectively the attraction of the nucleus (i.e. inward pull) for the electrons in the outer most shell as the atomic number of lanthanide elements increases.This results in the increased inward pull of the outer most electrons by the nucleus, finally causing the reduction in the atomic or ionic size of these elements. The sum of the successive reductions gives the total lanthanide contraction.

It may be concluded that the lanthanide contraction among the 4f-sereies elements and their ions takes place due to the poor shielding effect of 4f-electrons and gradual increase in the nuclear charge.

Consequences of Lanthanide Contraction

Lanthanide contraction plays an important role in determining the chemistry of lanthanides and heavier transition series elements. Some important consequences of lanthanide contraction are discussed below:

(a) Basic character of lanthanide hydroxides, Ln(OH)3

Because the size of tripositive lanthanide ions (Ln3+) decreases regularly with increasing atomic number (or nuclear charge), the process being called lanthanide contraction, therefore, the covalent character between Ln3+ ion and OH- ions increases from La(OH)3 to Lu(OH)3 (Fajans’ rules). As a result, the beasic character of the hydroxides decreases with increasing atomic number. Consequently, La(OH)3 is the most basic while Lu(OH)3 is the least basic.

(b) Resemblance between the atomic radii of the second and third transition series elements

Notes



The lanthanide contraction is an important factor in allowing the separation of lanthanides from one another. Also it has significant effect on the relative properties of the elements which precede and succeed the lanthanides. Normally in the same group, the atomic (or covalent) radii increase as the value of n (principal quantum number) increases due to increased distance between the nucleus and the outermost shell of the electrons which counterbalances the increased nuclear charge. This fact is evident when the values of atomic radii are compared for the elements of first and second transition series. On the same analogy, the atomic radii of the elements of third transition series should be greater than those of the second transition series elements. This statement is valid only for the elements of Group 3, i.e. Sc, Y and La in terms of their atomic radii (see the table given below). But, when these values are compared for the elements of 4d and 5d series in the next group, viz. Group 4, 5……………..12, it is observed that the values are unexpectedly almost equal. The similarity in the values of atomic radii for the elements of second and third transition series is attributed to the inclusion of 14 lanthanides between La (Z = 57, Group 3) and Hf (Z = 72, Group 4) of third transition series which due to “lanthanide contraction” cancel the increase in the values of a tomic radii. Due to the similarity in the size of the elements of the two series (i.e., second and third), the elements of a particular Group resemble each other more closely in their properties than do the elements of first and second transition series. The examples of the pairs of elements which show similar properties and hence are difficult to separate are Zr- Hf, Nb-Ta, Mo-W, Ru-Os, Rh-Ir, Pd-Pt and Ag-Au. The atomic radii of the elements of the three transition series are given below to justify the above statement.

From this table, it is evident that due to lanthanide contraction, the atomic and ionic radii of second and third transition series elements do not differ much though they are appreciably higher than those of first transition series elements. It is also observed

Notes

Inorganic Chemistry


that the atomic radii of the elements falling immediately after the lanthanide series are closer to those of their 4d-congeners and the effect slowly decreases along the series of 5d-elements as we move away from it.

(a) Densities of the elements of the three transition series

The density and atomic volume are inversely proportional to each other. All the transition metals have low values for their atomic volumes and hence their densities are high. In a given transition series, the atomic volumes of the elements first decrease on proceeding from left to right and generally attain a minimum value for the elements of group VIII (i.e., Groups 8,9,10). They then start increasing further up to Group 12 elements. Accordingly, the density of the elements increases from left to right up to the elements of Group VIII (Groups 8,9,10) and then decreases up to Group 12. Down the group, the densities of the elements increase regularly. But the striking feature that has been observed on moving from the first element to the last element in every group is that the densityes of the elements belonging to second transition series are only slightly higher than those of the corresponding elements of 3d-series while the values for the elements from Hf (Z =72) to Hg (Z =80) (i.e., 5d-sereis) are almost double of those for the elements from Zr (Z=40) to Cd (Z=48), respectively (4d-sereis). This analogy does not apply to Y(Z=39) and La (Z=57). This can be explained as follows:

Because of lanthanide contraction, the atomic sizes of the elements of third transition series after La (Z=57) i.e. form Hf onwards become very small and as a result, the packing of atoms in their metallic crystals becomes much compact which results in high densities. Also, there is only a small difference in the atomic sizes of the elements of the two series, viz., 4d-and 5d-series but the atomic masses of the elements of 5d-series are almost double to the corresponding elements of 4d-series. This makes the densities of 5d-series elements almost double to those of the elements of 4d-series.

(b) Similarities among lanthanides

There is very small change in the radii of lanthanides and hence their chemical properties are quite similar. This makes the separation of these elements using the usual physical and chemical methods difficult. Consequently new methods like ion exchange technique, solvent extraction etc. have now been used for their separation which are based on slight difference in the properties like hydration, complex ion formation, etc.

Notes



3.6 COMPLEX FORMATION BY LANTHANIDESThe lanthanides have low charge density due to their larger size in spite of having

high charge (+3). Hence, they do not cause much polarization of the ligands and have a weak tendency for complex formation. This reluctance for complex formation may be attributed mainly to:

(i) The unfavourable electronic configuration on the lanthanide ions.

(ii) The larger size which leads to little attraction for electron rich species.

Because of the above reasons, only the high energy 5d, 6s and 6p-orbitals are available for coordination, the 4f-orbitals being screened, so that only strong (usually chelating) coordinating groups can interact. Thus, only a few complexes with unidentate ligands are formed but stable complexes are formed by Ln3+ ions with chelating ligands such as (i) oxygen containing, viz., EDTA, β-diketones, citric acid, oxalic acid, acetyl acetone, oximes, (ii) nitrogen containing, viz., ethylene diamine, NCS, etc. The Ln3+ ions do not form complexes with π-bonding ligands such as CO, NO, CNR, etc., at all. The complex forming tendency and the stability of the complexes increases with increasing atomic number. This fact is taken as a basis to take advantage in their separation from one another. Ce(IV) complexes are relatively common, an example of high oxidation state ion seeking stabilization through complexation.

The most important class of lanthanide complexes are the anionic type. Complexation with hydroxycarboxylic acids such as citric and tartaric acid is used in the separation procedure of lanthanides. The EDTA complexes have achieved importance in the recent years. The coordination number of lanthanide complexes is usually six.

3.7 OCCURRENCE AND ISOLATION OF LANTHANIDE ELEMENTS

Except promethium which is unstable and occurs only in traces, all the lanthanides occur in nature to a considerable extent, cerium being the most abundant of all the elements. There are more than hundred minerals known to contain lanthanides but very few are of commercial importance. Monazite sand is the best known and most important mineral of lanthanide elements which is essentially a mixture of orthophosphates, LnPO4 containing upto 12% thorium, the element of 5f-series, small amounts of Zr, Fe and Ti as silicates, lanthanum and about 3% yttrium. Among lanthanides contained in monazite, the bulk is of Ce, Nd, Pr and others occur in minute quantities.

Extraction of lanthanide metals

After conventional mineral dressing which gives minerals of more than 90 percent purity, the mineral is broken down by either acidic or alkaline attack. By making use

Notes

Inorganic Chemistry


of different solubilites of double salts: Ln2(SO4)3.Na2SO4.xH2O for light and heavy lanthanides and low solubility of hydrated oxide of thorium, the lanthanide fractions and thorium containing portions are separated in acidic medium.

Monazite is treated with hot conc. H2SO4 when thorium, lanthanum and lanthanons dissolve as sulphates and are separated from insoluble material (impurities). On partial neutralisaion by NH4OH, thorium is precipitated as ThO2. Then Na2SO4 is added to the solution. Lanthanum and light lanthanides are precipitated as sulphates leaving behind the heavy lanthanides in solution. To the precipitate obtained as above, is added hot conc. NaOH. The resulting hydroxides of light lanthanides are dried in air at 1000C to convert the hydroxides to oxides. The oxide mixture is treated with dil. HNO3. This brings CeO2 as precipitate and other lanthanides in solution. From the solutions obtained as above for heavy and light lanthanides, individual members of lanthanide series are isolated by the following methods:

Isolation of Individual Lanthanide Elements:

All the lanthanides have the same size and charge (of +3 unit). The chemical properties of these elements which depend on the size and charge are, therefore, almost identical. Hence, their isolation from one another is quite difficult. However, the following methods have been used to separate them from one another.

1. Fractional Crystallization Method:

This method is based on the difference in solubility of the salts such as nitrates, sulphates, oxalates, bromates, perchlorates, carbonates and double salts of lanthanide nitrates with magnesium nitrate which crystallize well and form crystals. Since, the solubility of these simple and double salts decreases from La to Lu, the salts of Lu will crystallize first followed by those of lighter members. The separation can be achieved by repeating crystallization process a number of times. A non-aqueous solvent, viz., diethyl ether has been used to separate Nd(NO3)3 and Pr(NO3)3.

2. Fractional Precipitation Method:

This method is also based on the difference is solubility of the precipitate formed, which is formed on addition of the precipitant, i.e. Precipitating agent. If a little amount of precipitant is added, the salt with lowest solubility is precipitated most readily and rapidly. For example, when NaOH is added to a solution of Ln(NO3)3, Lu-hydroxide being the weakest base and having the lowest solubility product is precipitated first while La-hydroxide which is the strongest base and has the highest solubility product is precipitated last. By

Notes



dissolving the precipitate in HNO3 and reprecipitating the hydroxides a number of times, it is possible to get the complete separation of lanthanide elements.

3. Valency change Method:

This method is based on the change of chemical properties by changing the oxidation state of the lanthanide elements. The most important application of this method is made in the separation of cerium and europium elements from mixture of lanthanides.

(i) The mixture containing Ln3+ ions if treated with a strong oxidising agent such as alkaline KMnO4, only Ce3+ ion is oxidized to Ce4+ while other Ln3+ ions remain unaffected. To this solution alkali is added to precipitate Ce(OH)4 only, which can be filtered off from the solution.

(ii) Eu2+ can be separated almost completely from Ln3+ ions from a solution by reducing it with zinc-amalgam and then precipitating as EuSO4 on adding H2SO4 which is insoluble in water and hence can be separated. The sulphates of other Ln3+ ions are soluble and remain in solution.

4. Complex Formation Method:

This method is generally employed to separate heavier lanthanide elements from the lighter ones by taking the advantage of stronger complexing tendency of smaller cations with complexing agents. When EDTA is added to Ln3+ ion solution, lanthanides form strong complexes. If oxalate ions are added to the solution containing EDTA and Ln3+ ions, no precipitate of oxalates is obtained.

However, on adding small amount of acid, the least stable complexes of lighter lanthanides are dissociated and precipitated as oxalates, but the heavier lanthanides remain in solution as EDTA complexes.

5. Solvent Extraction Method:

This method is based on the difference in the values of partition coefficient of lighter and heavier lanthanides between two solvents, e.g., water and tri-butyl phosphate (TBP). Heavier lanthanides are more soluble in TBP than lighter ones whereas reverse trend of solubility is found in water and other ionic solvents. La(NO3)3 and Gd(NO3)3 have been separated by this method because the partition coefficient of Gd-nitrate in water and TBP is different from that of La-nitrate. Thus, Gd-nitrate can be separated from La-nitrate by continuous extraction with water from a solution of these salts in TBP in kerosene oil or by using a continuous counter-current apparatus which gives a large number of partitions automatically.

Notes

Inorganic Chemistry


6. Modern Ion-Exchange Method:

This is the most rapid and most effective method for the isolation of individual lanthanide elements from the mixture. An aqueous solution of the mixture of lanthanide ions (Ln3+aq) is introduced into a column containing a synthetic cation exchange resin such as DOWAX-50 [abbreviated as HR (solid)]. The resin is the sulphonated polystyrene containing-SO3H as the functional group. As the solution of mixture moves through the column, Ln3+aq ions replace H+ ions of the resin and get themselves fixed on it:

Ln3+aq + 3H(resin) → Ln(resin)3 + 3H+aq

The H+aq ions are washed through the column. The Ln3+aq. ions are fixed at different positions on the column. Since, Lu3+aq. is largest (Lu3+ anhyd. is smallest and is hydrated to the maximum extent) and Ce3+aq. is the smallest, Lu3+aq. ion is attached to the column with minimum firmness remaining at the bottom and Ce3+aq. ion with maximum firmness remaining at the top of the resin column. In order to move these Ln3+aq. ions down the column and recover them, a solution of anionic ligand such as citrate or 2-hydroxy butyrate is passed slowly through the column (called elution). The anionic ligands form complexes with the lanthanides which possess lower positive charge than the initial Ln3+aq ions. These ions are thus displaced from the resin and moved to the surrounding solutions as eluant- Ln complexes.

For example, if the citrate solution (a mixture of citric acid and ammonium citrate) is used as the eluant, during elution process, NH4+ ions are attached to the resins replacing Ln3+aq. ions which form Ln-citrate complexes:

Ln (resin)3 + 3NH4+ → 3NH4- resin + Ln3+ aq

Ln3+aq + citrate ions → Ln-citrate complex

As the citrate solution (buffer) runs down the coloumn, the metal ions get attached alternately with the resin and citrate ions (in solution) many times and travel gradually down the column and finally pass out of the bottom of the column as the citrate complex. The Ln3+aq cations with the largest size are, eluted first (heavier Ln3+aq ions) because they are held with minimum firmness and lie at the bottom of the column. The lighter Ln3+aq ions with smaller size are held at the top of the column (with maximum firmness) and are eluted at last. The process is repeated several times by careful control of concentration of citrate buffer in actual practice.

3.8 LANTHANIDE COMPOUNDSThe lanthanides are very electropositive and reactive metals, the reactivity depends

on the size. Europium with the largest size is most reactive. All the lanthanides generally give normal and complex compounds.

Notes



Oxides:

If lanthanide elements are ignited in air or O2, they readily form the oxides of Ln2O3 type except Ce which gives a dioxide, CeO2. The oxides are ionic and basic. The basic nature of oxides decreases along the series with decreasing ionic size

Ytterbium resists the action of air even at 1000oC due to the formation of a protective layer of oxide on its surface.

Hydroxides:

The lanthanides react slowly with cold water but readily with hot water:

2 Ln + 6 H2O → 2 Ln(OH)3 + 3H2

On adding aqueous ammonia to this aqueous solution, hydroxides are precipitated as gelatinous precipitate. These hydroxides are also ionic and basic, the basic nature decreasing with increasing atomic number. La(OH)3 is most basic and Lu(OH)3 is least basic. Their basic character is more than that of Al(OH)3 but less than that of Ca(OH)2.

Oxo-salts:

Lanthanides form oxo-salts such as nitrates, sulpates, perchlorates and salts of oxo-acids which are soluble in water but carbonates and oxalates are insoluble. The difference in basicity is responsible for the difference in thermal stability of the oxo-salts which decreases along the series. Thus, La(NO3)3 is more stable than Lu(NO3)3.

Halides and Hydrides:

The lanthanides also burn in halogens to produce LnX3 type halides and combine with H2 at high temperature to give stable MH2 or MH3 type hydrides. Among halides, fluorides are insoluble but other halides are soluble in water.

Ln also form complexes with chelating ligands, the detailed account has been given earlier.

3.9 CHEMISTRY OF ACTINIDEThe group of fourteen elements from thorium (Th, Z = 90) to lawrencium (Lr,

Z=103) are called actinides, actinoids or actinons. These are named so because these elements succeed the element actinium (Ac, Z = 89). These elements are also known as inner-transition elements as they lie between actinium and rutherfordium (Rf, Z =104), i.e., the elements of fourth transition series. Thus, they constitute the second inner-transiton series of which actinium is the prototype.

Notes

Inorganic Chemistry


In these elements 5f-subshell of the antepenultimate shell (n=7) is successively filled by the additional or differentiating electrons, one at a time in each step, which are embedded in the interior while 6d- and 7s-electons are exposed to the surroundings.

In the outermost and penultimate shell of these elements, the number of electrons remains almost the same. That is why the actinide elements resemble one another very closely. The actinides lying beyond uranium, i.e., the elements with Z = 93 to 103 are called transuranium elements. The first and the last elements of 5f-series have been mentioned above with their names and symbols, i.e., Th and Lr. The remaining twelve elements are listed below:

protactinium (Pa, Z = 91), uranium (U, Z = 92), neptunium (Np, Z = 93), plutonium (Pu, Z = 94), americium (Am, Z = 95), curium (Cm, Z = 96), berkelium (Bk, Z = 97), californium (Cf, Z= 98), einsteinium (Es, Z = 99), fermium (Fm, Z= 100), mendelevium (Md, Z = 101) and nobelium (No, Z= 102).

These elements of 5f-series are also said to belong to Group 3 and Period 7 and have been allotted a position below those of 4f-series in the periodic table and thus the elements of both f-series have been placed separately below the main body of table to avoid unnecessary expansion of the periodic table.

General features of the actinides

(1) Occurrence and Nature

Only the first four elements, viz., Ac, Th, Pa and U occur in nature in uranium minerals that too only Th and U occur to any useful extent. All the remaining actinides, i.e. trans uranium elements are unstable and are made artificially. The elements above Fm (Z =100) exist as short lived species, some of them existing only for a few seconds. All the actinides are radioactive in nature.

(2) Electronic structure (or Configuration)

The electronic configuration of actinium (Z = 89) which is followed by fourteen actinides is [Rn]5f06d17s2, the last electron entering the 6d-subshell. In the next element, Th, the first member of the actinide series, the additional electron must enter 5f-subshell and the filling of 5f-subshell must continue progressively till the last element, Lr. Thus, 6d-subshell in all the elements must remain singly filled thereby giving the expected valence shell configuration of 5f1-146d17s2 for these elements. Since, the energies of 6d- and 5f- subshells are almost the same and the atomic spectra of the elements are very complex, it is difficult to identify the orbital in terms of quantum numbers as well as to write down the configuration. For chemical behaviour, the valence shell electronic

Notes



configuration of the elements is of great importance and the competition between 5fn6d07s2 and 5fn-16d17s2 is of interest. It has been observed that the electronic configuration of actinides does not follow the simple pattern as is observed for the lanthanides. For the first four actinde elements, viz., Th, Pa U and Np, due to almost equal energies of 5f and 6d, the electrons may occupy the 5f or 6d subshells or sometimes both. From Pu (Z=94) onwards, 6d1 electron gets shifted to 5f-subshell except for Cm (Z=96) and Lr (Z=103) in which 6d1 electron does not shift to 5f due to stable 5f7 and 5f14 configurations. In view of the above considerations, the general valence shell electronic configuration of the actinide elements may be written as: 5f0-146d0-27s2. For individual elements the observed or actual valence shell configurations are listed below:

From the above valence shell configurations of the actinide elements, it is clear that Th does not have any f-electron though this element belongs to 5f-series (i.e., actinides). For Pa, U, Np, Cm and Lr, both the expected and observed (actual) configurations are same. For the rest of the actinides, 6d-subshell does not contain any d-electron.

(3) Oxidation States

The important oxidation states exhibited by actinides are compiled below in the tabular form. Some of them are stable but most of these oxidation states are unstable. It may be seen from these oxidation states that the +2 state is shown by Th and Am only in the few compounds like ThBr2, ThI2, ThS, etc. The +3 oxidation state is exhibited by all the elements and it becomes more and more stable as the atomic number increases. The +4 oxidation state is shown by the elements from Th to Bk, the +5 oxidation state by Th to Am, the +6 state by the elements from U to Am and the +7 state is exhibited by only two elements, viz ., Np and Pu. Np in the +7 state acts as an oxidising agent.

Notes

Inorganic Chemistry


The principal cations given by actinide elements are M3+, M4+ and oxo-cations such as MO2+ (oxidation state of M = + 5) and MO22+ (oxidation state of M = +6). The examples of oxo-cations are UO2+, PuO2+, UO22+ and PuO22+ which are stable in acid and aqueous solutions. Most of the M3+ ions are more or less stable in aqueous solution. Np3+ and Pu3+ ions in solution are oxidized to Np4+ and Pu4+ by air. The latter ions are further oxidized slowly to UO22+ and PuO22+ by air. Various oxidation states of the actinides are listed below:

The lighter elements up to Am show variable oxidation states, the maximum being for Np, Pu and Am, but the heavier elements show constant oxidation state of +3.

(4) Atomic and Ionic Radii

The atomic (metallic) and ionic radii of cations in common oxidation states (i.e. M3+ and M4+ cations) of some of the actinide elements have been evaluated. A look into the values of the atomic radii reveals that the metallic radii first decrease from Th to Np and then increase gradually up to Bk. For the higher actinides the values are not known. The values of ionic radii for both types of ions go on decreasing. This steady fall in the ionic radii along the actinide series is called actinide contraction which is analogous to lanthanide contraction found in lanthanides. The cause of actinide contraction is the same as has been discussed for the lanthanides. Here also, increasing nuclear change and poor shielding effect of 5f-electrons play an important role. The atomic and ionic radii are listed below:

Notes



It is clear from above table that there is only a small variation in the atomic and ionic radii of the actinide elements; hence they show similar chemical properties.

(5) Magnetic and Spectral Properties

It has already been mentioned that the paramagnetic nature of the substances is due to the presence of unpaired electrons. The actinide elements like lanthanides show paramagnetism in the elemental and ionic states. Tetravalent thorium (Th4+) and hexavalent uranium (U6+) ions are diamagnetic due to the absence of unpaired electrons. Th4+ = U6+ = Rn (Z= 86) structure (diamagnetic, paired electrons). Since, actinides constitute second f-series, it is natural to expect similarities with lanthanides (the first f series) in their magnetic and spectroscopic properties. But, there are some differences between the lanthanides and actinides. Spin-orbit coupling is strong (2000-4000cm-1) in the actinides as happens in the lanthanides but because of the greater exposure of the 5f-electrons, crystal field splitting is now of comparable magnitude and J is no longer such a good quantum number. It is also noted that 5f-and 6d-subshalls are sufficiently close in energy for the lighter actinides to make 6d- levels accessible. As a result each actinide compound has to be considered individually. This must allow the mixing of J levels obtained from Russel-Saunders coupling and population of thermally available excited levels.

Accordingly, the expression µ = ( + 1) is less applicable than for the lanthanides and magnetic moment values obtained at room temperature are usually lower and are much more temperature dependent than those obtained for compounds of corresponding lanthanides.

The electronic spectra of actinide compounds arise from the following three types of

electronic transitions:

(a) f-f transitions: These are Laporte (orbitally) forbidden but the selection rule in relaxed partially by the action of crystal field in distorting the symmetry of

Notes

Inorganic Chemistry


the metal ion. Because the actinides show greater field, hence the bands are more intense. These bands are narrow and more complex, are observed in the visible and UV regions and produce the colours in aqueous solutions of simple actinide salts.

(b) 5f-6d transitions: These are Laporte and spin allowed transitions and give rise to much more intense bands which are broader. They occur at lower energies and are normally confined to the UV region hence do not affect the colours of the ions.

(c) Metal to ligand charge transfer: These transitions are also fully allowed and produce broad, intense absorptions usually found in UV region, sometimes trailing in the visible region. They produce the intense colours which are characteristic of the actinide complexes.

The spectra of actinide ions are sensitive to the crystal field effects and may change from one compound to another. It is not possible to deduce the stereochemistry of actinide compounds due to complexity of the spectra. Most of the actinide cations and salts are coloured due mainly to f-f transitions. Those with f0, f7 and f14 configurations are colourless. The colours of some of the compounds in different oxidation states are given below:

NpBr3 : green; NpI3 : brown; NpCl4 : red-brown; NpF6 : brown

PuF3: purple; PuBr3 : green; PuF4 : brown; PuF6 : red brown

AmF3: pink; AmI3 : yellow; AmF4: dark tan.

The coordination chemistry of actinides is more concerned with aqueous solutions. Because of the wider range of oxidation states available in actinides, their coordination chemistry is more varied. Most of the actinide halides form complex compounds with alkali metal halides.

For example, ThCl4 with KCl forms complexes such as K[ThCl5] and K2[ThCl6], etc. ThCl4 and ThBr4 also form complexes with pyridine, e.g. ThCl4.py Chelates are formed by the actinides with multidentate organic reagents such as oxine, EDTA, acetyl acetone, etc.

The actinides with small size and high charge have the greatest tendency to form complexes. The degree of complex formation for the various ions decreases in the order: M4+ > MO22+ > M3+ > MO2+. The complexing power of different anions with the above cations is in the order:

Monovalent anions : F- > NO2- > Cl-

Bivalent anions : CO32- > C2O4

2- > SO42-

Notes



Chemistry of actinides

The actinide elements are highly electropositive and reactive. They show similar properties to those of lanthanides. But these elements have much higher tendency to form complexes. They react with water and tarnish in air forming oxide coating. They react readily with HCl but slowly with other acids. The metals show basic nature and do not react with NaOH but they react with halogens, oxygen and hydrogen to form halides, oxides and hydrides. Some of the compounds of the actinides are discussed below:

(i) Oxides: The metals on reacting with air or oxygen give various oxides under different experimental conditions. Uranium is one of the reactive elements and gives the oxides: UO, UO2, U3O8 and UO3:

Similarly, the oxides of Np, Pu and Am are: NpO, NpO2, Np3O8, PuO, Pu2O3, Pu2O7, PuO2, AmO and AmO2.

(ii) Hydrides, nitrides and carbides: U and Pu form hydrides UH3 and PuH3, by the direct union of the elements. U reacts with H2 even at room temperature but the reaction is faster at 250 0C as compared to room temperature:

2U + 3H2 → 2UH3

This compound is reactive and is hydrolysed by water:

2UH3 + 4H2O → 2UO2 + 7H2

This also reacts with Cl2, HF and HCl as follows:

2UH3 + 4Cl2 → 2UCl4 + 3H2

2UH3 + 8HF → 2UF4 + 7H2

UH3 + 3HCl → UCl3 + 3H2

When treated with ammonia, the metals give nitrides of the type :

UN, U2N3, UN2 and PuN.

These metals also give carbides: UC and PuC.

All the MX type compounds where M = U, Np, Pu or Am and X = O, C or N have the rook-salt (or NaCl) structure.

(iii) Halides: Actinide elements on reacting with halogens or hydrogen halides form halides, the most common being those of U and Np. Trihalides of MX3 type are formed by the actinides which are isomorphous with one another. Some of

Notes

Inorganic Chemistry


the actinides also form the tetra, penta and hexa halides as well. For example, U fluorides are obtained as is shown by the following reactions:

Np, Pu and Am are also reactive similar to U and give the analogous reactions and products. Some examples of halides are : UF6, UF5, UF4, UF3, UCl4, UCl3, UBr4, UBr3, NpF6, NpF4, NpF3, NpCl3, NpI3, etc.

3.10 CHEMISTRY OF SEPARATION OF NP, PU AND AM FROM U

Although several isotopes of Np, Pu and Am elements are known yet only a few are obtained. But Np237 and Pu239 are found in the uranium fuel elements of nuclear reactors from which Pu is isolated on a kilogram scale. Np237 is also found in substantial amounts and is recovered primarily for conversion by neutron irradiation of NpO2 into Pu238 which is used as a power source for satellites. Am produced from intense neutron irradiation of pure plutonium. The main problem involved in the extraction of these elements includes the recovery of the expensive signatory material and the removal of hazardous fission products that are formed simultaneously in amounts comparable to the amount of the synthetic elements themselves. There are various methods available for the separation of Np, Pu and Am which are based on precipitation, solvent extraction, differential volatility of compounds and ion exchange. The chemistry of the most important methods of separation is given below:

(a) Method based on stabilities of oxidation states

The stabilities of major ions of these elements involved are UO22+ > NpO22+ > PuO22+ > AmO22+ and Am3+ > Pu3+ >> Np3+ > U4+. By choosing a suitable oxidising or reducing agent, it is possible to obtain a solution containing the elements in different oxidation states. The elements can then be separated by precipitation or solvent extraction method. For example, Pu can be oxidized to PuO22+ whereas Am remains as Am3+. Thus PO22+ can be easily removed by solvent extraction or Am3+ by precipitation as AmF3.

Notes



(b) Method based on extraction by using organic regents

It is a well known fact that MO22+ ions can be extracted from nitrate solutions

into organic solvents. The M4+ ions can be extracted into tributyl phosphate in kerosene from 6M-HNO3 solution. Similarly M3+ ions can be separated from 10-16M HNO3. Thus, the actinides close to each other can be separated by changing the conditions.

(c) Method based on precipitation

The actinide ions in M3+ or M4+ state only give insoluble fluorides or phosphates in acid solution. In the higher oxidation states these elements are either soluble or can be prevented to get precipitated by complex formation with sulphate or other ions.

(d) Method based on ion-exchange

This method has been found suitable for small amount of material. In this method, both cationic and anionic ion ex`changers can be used to separate the actinide ions. The separation of the actinides by ion-exchange methods is given below:

1. Isobutyl methyl ketone method: In this method the following scheme is used

The two layers are separated and collected. Here Ac4+ represents the actinide ions such as Np4+, Pu4+ and Am4+.

2. Tributyl phosphate method: This method is dependent on the difference in extraction coefficients from 6N-HNO3 into 30% tributyl phosphate in kerosene. The order of extraction is :

Pu4+ > PuO22+, Np4+ > Np+O2 >> Pu3+ and UO2

2+ > NpO22+ > PuO2

2+.

The Ac3+ ions have very low extraction coefficients in 6M-HNO3 but in 12M-HCl or 16M-HNO3 the extraction increases. The order of extraction is: Np < Pu< Am < Cm, etc. In this method the scheme used is as follows:

Notes

Inorganic Chemistry


Repeat extraction to get Ac4+. Here also, Ac = Actinide particularly Np and Pu.

3. Method Based on Lanthanum fluoride cycle: This method was developed for the isolation of Np but has been found to be of great utility in the separation of Pu from U. The scheme used for the separation is given below

3.11 SIMILARITIES BETWEEN LANTHANIDES AND ACTINIDES

(i) In the atoms of the elements of both the series, three outermost shells are partially filled and remaining inner shells are completely filled but the additional or differentiating electron enters (n-2) f-subshell.

(ii) The elements of both series exhibit +3 oxidation state which is prominent and predominant state.

(iii) Like Lanthanide contraction found in the lanthanide elements, there occurs contraction in size in the actinide elements called actinide contration. Both the contractions are due to poor shielding effect produced by f-electron with increasing nuclear charge.

(iv) The elements of both the series are quite reactive and are electropositive.

(v) The electronic absorption bands of the elements of both the series are sharp and appear like lines. These bands are produced due to f-f transitions within (n-2)f-subshell though such transitions are orbital forbidden.

(vi) Most of the lanthanide and actinide cations are paramagnetic.

(vii) The nitrates, perchlorates and sulphates of trivalent lanthanide and actinide elements are soluble while the hydroxides, fluorides and carbonates of these elements are insoluble.

(viii) The lanthanide and actinide elements show similarity in properties among their

Notes



series though the lanthanides are closer among them sieves in properties as compared to actinides.

3.12 LET US SUM UPThe present unit covers all the important and interesting aspects of lanthanides such

as their electronic structure, oxidation states, ionic radii and corresponding lanthanide contraction, consequences of lanthanide contraction, complex formation by lanthanides, their occurrence, extraction and various methods employed for the isolation of the elements and a brief account of the lanthanide compounds. The readers can understand well all these aspects after going through the unit text. The text material of this unit contains the introductory part which is quite interesting and important from the view point of the readers and a detailed account of the general features of the actinides such as their occurrence, electronic structure, oxidation states which have greater variability than those of lanthanides, atomic and ionic radii-the actinide contraction, their magnetic and spectral properties along with exhibition of colour, formation of complexes, etc has been given.

The chemistry of actinides including the formation of various compounds, e.g., oxides, hydrides, nitrides, carbides and halides in various oxidation states has also been discussed. The unit also contains a detailed account of the chemistry of separation of Np, Pu and Am from U including the method used for separation. At last the points of similarities between the lanthanides and actinides have been mentioned.


1. Why are the lanthanides grouped together?

2. Why the f-block elements are also called the inner transition elements?

3. Give a brief account of lanthanide contraction.

4. The number of f-electrons in Eu2+ and Yb2+ ions is

(a) 7 and 14

(b) 7 and 13

(c) 6 and 14

(d) 6 and 13

5. Tb4+ ions are stable, explain.

6. Discuss the ion exchange method of isolation of lanthanides. Which one is more basic La(OH)3 or Lu(OH)3 and why?

7. Why do actinides show higher oxidation states than lanthanides?

Notes

Inorganic Chemistry


8. Oxocations MO22+ are formed by U, Np, Pu and Am only wheras heavier actinides do not form such ions, why?

9. The elements beyond atomic number 102 are unstable, Explain. 10. Write the electronic configuration of Th, Cm and No. 11. Name the actinides along with their symbols and atomic numbers. 12. Which actinide ion in +3 oxidation state has just half-filled 5f-subshell? (a) Pu3+

(b) Am3+

(c) Cm3+

(d) Bk3+

13. The first member of the post actinide transition series is: (a) Rutherfordium (b) Seaborgium (c) Meitnerium (d) Hahnium





Notes

Molecular Symmetry


(Structure) 4.1 Objectives

4.2 Introduction

4.3 Concept of symmetry

4.4

4.5 Rotation axis or proper axis of rotation or symmetry axis

4.6

4.7 Alternate axis of symmetry or improper axis of rotation, or rotation

4.8

4.9 Total symmetry elements and total operations generated

4.10

4.11

4.12 Products of symmetry operations

4.13

4.14

4.15

Let us sum up

Special point groups

Classification of molecules in point group 4.16

Review questions 4.17

4.18 4.19 Further readings

UNIT–4

Molecular Symmetry


zz Know the concept of symmetry

zz Define symmetry elements

zz Know the types of symmetry elements

zz Define symmetry operations

zz Identify symmetry elements in molecules

zz Perform symmetry operations on molecules

zz Know/state the characteristics of a group

Notes

Inorganic Chemistry


4.2 Introduction Symmetry is an important concept in chemistry. Firm understanding of this concept of the symmetry and group theory is extremely useful in dealing with molecular structures, stereochemistry, spectroscopy and reaction mechanisms.

4.3 Concept of SymmetryThe concept of symmetry is important to almost every aspect of life in our

universe. The “principle of symmetry” plays an important role in modern science. The idea of symmetry principle originally resulted from the study of geometrical forms and observations of natural objects. There are countless examples of symmetry in nature. Symmetry principle finds lot of applications in natural science i.e. physics, chemistry where geometrical properties are generally taken into account. The Oxford English dictionary gives the following two definitions for symmetry:

1) Mutual relation of the parts of something in respect of magnitude and position; relative measurement and arrangement of parts; proportion.

2) Due or just proportion, harmony of parts with each other and the whole; fitting, regular or balanced arrangement and relation of parts of elements; the condition or quality of being well proportion or balanced.

The first symmetry definition clearly has more scientific importance. The concept of symmetry sounds simple and familiar, yet symmetry is far more complex and difficult to apply in practice than one might think. Let us introduce the concept of symmetry in molecular systems in more quantitative manner. The symmetry of the molecule is determined by the existence of symmetry elements about which symmetry operations can be performed. To understand the concept of symmetry let us examine the figure1given below

Figure 1Let us examine these objects to understand the concept of symmetry in terms of

symmetry operations (certain movements) that can be performed on these objects. If one performs a symmetry operation on the object then object should look the same before after the symmetry operation is carried out i.e. object before and after the symmetry operation should have equivalent configuration or is indistinguishable. Configurations may not be identical as some parts may have interchanged when symmetry operation is carried out. What are the characteristics of these objects that lead us to say that object in part (x) has no symmetry and but objects in parts (y) and (z) have symmetry? One can even say that the object in part (z) has more symmetry than the object in (y). Dotted lines in parts (y) and (z) indicate that the objects are symmetrical about these lines. Let

Notes

Molecular Symmetry


us explain it further by taking the example of unlabelled square piece of card and rotate this by 0°, 90°, 180°, 270°and 360° successively about dotted line perpendicular to the plane of the card. Figure 2 gives some of these results of rotations.

Figure 2: Rotations of square planar card without labelingAll resulting configurations look the same and cannot be distinguished from each

other. Thus these rotations/movements are the symmetry operations. However if one labels the corners of the square piece of card and then perform similar rotations about the dotted line one gets the result as shown in Figure 3.

Figure 3: Rotations of square planar card with labelingIf we label the four corners, as a,b, c and d then it is possible to observe the effects

of these movements/rotations. If we carry out four successive 90° rotations about dotted line, the configuration V (the final configuration) is now strictly identical to the configuration I (the original configuration). The labels a, b, c, and d help distinguish the effects of rotation. If we have labeled or tagged corners of the square card then the intermediate 90° rotations (configurations II, III and IV) do not lead to equivalent configurations and these can be distinguished from each other. Thus these movements/rotations when the square card corners are labeled are not truly symmetry operations.

Let us now differentiate between closely related terms symmetry elements and symmetry operations. For this let us consider the Figure 4 and label the arms and perform the rotation through 180°.

Figure 4: 180° rotation of the object about the dotted line

Notes

Inorganic Chemistry


Now rotate the figure around the vertical dotted line by 180° (half a complete cycle, a clock wise rotation). Because the hands are identified by letters, the two figures can be differentiated after rotation and thus this rotation/movement is not a symmetry rotation. But if we remove the labeling, the figures will be identical/equivalent/ indistinguishable.

The concept of symmetry is familiar one and generally one speaks of a shape as being, “symmetrical”, “unsymmetrical” or even “more symmetrical than some other shape”. For scientific purposes, one needs to specify concept of symmetry in a more quantitative way. Let us consider two triangles A and B as shown in Figure 5.

Figure 5: Shapes of the triangles A and BFind the triangle which is more symmetrical. Your answer will be that triangle A is

more symmetrical than triangle B, which is correct. Why triangle A is more symmetrical than triangle B? Let us examine it. Label the corners of triangles as x y z. Draw a dotted line in the plane of the paper through vertex x such that it bisects the side yz at p as shown in Figure 6.

Figure 6: The 180° rotation about dotted line in triangle ATriangle A has been divided into two halves which are equivalent i.e. triangle xzp

and triangle xpy are equivalent to each other. Now rotate triangle A through 1800 about dotted line (in plane). Triangle A goes to triangle A′. Triangles A and A′ are distinguishable hence rotation is not a symmetry rotations. If the corners of triangle A are not labeled and 180° rotation is performed about the dotted line you will not be able to distinguish the original triangle A and the triangle obtained after rotation i.e. triangle A′. In that case rotation is a symmetry rotation. Can you carry out similar rotation process in triangle B (Figure 5) i.e. rotate about dotted line through 180° and get the indistinguishable figure B′ ? Your answer will be ‘No’. See the results in Figure 7.

Figure 7: The 180° rotation in triangle B

Notes

Molecular Symmetry


The smaller angle a is pointing in opposite direction. You can distinguish between

triangles B and B′. Thus 1800 rotation on triangle B does not give similar results as

were obtained in case of triangle A.

Hence triangle A is more symmetrical than triangle B about dotted line in plane of the

paper. Let us again take triangle A and pass a dotted line perpendicular to the plane

of the triangle through the midpoint of this triangle A and perform three successive

120° clockwise (negative rotation) rotations about this dotted line. The results of these

rotations are shown in Figure 8.

Figure 8: Effect of successive 120° rotations on triangle A

Configurations I and IV of the triangle are indistinguishable even when the vertices

are labeled. These three successive 120° rotations bring back the triangle A to original

position. If vertices are not labeled then you will not be able to distinguish configurations I,

II, III, IV from each other. In the triangle A, so far, we have given two types of movements

or rotations about dotted line (a line or an axis) in plane and a line perpendicular to plane

of the paper. After these physical movements/rotations (or operations) you find the new

triangle A′. The configurations II, III, IV are equivalent or indistinguishable from A i.e.

from configuration I. In these types of movements/rotations/operations on triangle A we

had imaginary lines in plane and perpendicular to the triangle plane about which these

movements/rotations/operations were performed. Since we are getting indistinguishable

configurations, these imaginary lines and physical movements /rotations / operations

are special.

4.4 Definition of symmetry elements and symmetry operations

4.4.1 Symmetry element

The two very closely related terms-symmetry elements and symmetry operation should

not be confused with each other. The difference between these two terms has been

explained ahead. Symmetry element: It is an imaginary geometric entity -‘a point’ or ‘a

line’ or’ a plane’, (Figure 9) about which symmetric movements (symmetry operations)

can be performed.

Notes

Inorganic Chemistry


Figure 9: Imaginary geometric entities (symmetry elements), a point, a line and a plane

4.4.2 Symmetry operation

It is a physical movement (rotations etc.) of the body of the object around an imaginary

point or a line or a plane in such a manner that after the movement the final configuration

of the object is indistinguishable from the original configuration of the object.

Thus the terms symmetry element and symmetry operation are complimentary to each

other. If a symmetry element exists in a object then there must exist a symmetry operation

about this symmetry element and if there exists a symmetry operation it must be about

already existed symmetry element i.e. you cannot separate these two terms from each

other. Existence one demands the existence of the other. Symmetry elements are imaginary

(a point, a line, a plane) while symmetry operations are the actual movements around

these imaginary geometric entities.

4.4.3 Types of symmetry elements

Discussion shall be restricted to symmetry elements present in isolated molecules i.e.

molecular symmetry. Before discussing the types of symmetry elements let us discuss

their naming system. Commonly there are two types of naming systems and these are:

1. The Schoenflies notation used extensively for molecules and by spectroscopists .

2. The Hermann – Mauguin or international notation used by crystallographers.

In this module Schoenflies notation will be used in dealing with the symmetry of the molecules. The types of symmetry elements that may commonly be present/observed in molecules have been summarized in Table 4.

Table 1: Types of symmetry elements/ symmetry operations No. Types of

Symmetry

element

Schoenflies

notation

Operation/movement

I Identity E C1, rotation by 3600 or does nothing

II Rotation axis or proper axis of rotation

Cn Rotation about an axis

Notes

Molecular Symmetry


III Reflection plane or mirror plane or plane of symmetry

σ Reflection through a plane

IV Centre of symmetry or point of symmetry or inversion centre

i Inversion through a point

V Improper axis of rotation or alternate axis of symmetry or rotation-reflection axis

Sn Rotation followed by reflection

Identity symmetry element (E)Presence of identity symmetry element means that object exists. To perform an

identity operation, rotate the object through 360° about a randomly chosen axis or do nothing to the object. Even though it makes no sense to have such a symmetry element, but the concepts (postulates/conditions) of a group require the existence of a neutral element or identity element. The asymmetric molecules like bromochlorofluoromethane [C(H)(Br)(Cl)(F)] have identity symmetry element only. Rotation through 360°about any chosen axis is represented as a C1 or Cnn rotation (the symbol C stands for cyclic rotation). C1 is Shoenflies symbol for this rotation and I is Hermann-Mauguin symbol (I stands for I-fold rotation). Besides other symmetry elements, every molecule possesses this symmetry element. A single rotation through 360° brings the molecule to original configuration and each and every point on it at its place. Therefore, only one operation is required to reach the original position. Identity is vital in the correct mathematical description of symmetry by group theory. In Cartesian coordinate system identity symmetry operation E on a point (x1, y1, z1) can be shown as given below in Figure 10. E(x1, y1, z1)(x1, y1, z1)

Figure 10: Effect of identity operation E on a point (x1, y1, z1)Reciprocal of identityReciprocal of a symmetry operation is the symmetry operation of the same type

which can bring the object back to its original configuration i.e. the identity situation. Reciprocal of a symmetry operation can be represented by putting an inverse sign over the symbol of symmetry operation. Inverse of E can be written as E-4.What is the reciprocal of E? Since in identity operation we do not do any rotation or simply rotate the object through 360°. Therefore, E-1 is E itself. Because E symmetry operation does not do any change in the orientation of the object. Examples of molecule with identity (E) only.

Some examples of molecules having only identity (E) i.e. with no symmetry elements in them are given in Figure 11.

Notes

Inorganic Chemistry


Figure 11: Molecules belonging to the C1 point group

4.5 Rotation axis or proper axis of rotation or symmetry axis

It is an imaginary line which passes through the body of the molecule and about which

certain rotations through an angle of 2п/n in clock wise direction are carried out in such

a manner that rotation brings back the molecule in an equivalent position. Symmetry

operation is carried out with reference to fixed frame of X, Y and Z axes. If we repeat

the symmetry rotation several times successively molecule finally reaches to original

configuration ie a full cycle has been completed. Symbol for the proper axis of rotation

is Cn (C stands for cyclic rotation in clock wise direction) and n means that rotation has

been performed through an angle of 2п/ө n times to reach to original position.

ie n=2п/ө or ө=2п/n

To give a symmetry rotation to a molecule we need to know (i) centre (centre of gravity)

in the molecule where from symmetry axis is to be passed and (ii) magnitude of angle ө.

By convention, positive rotation go counter clock wise direction and negative rotation go

clock wise direction. The n has integer values ie n=1 or 2 or 3or 4 or 5or 6 or 7………..

up to ∞. For various values of n the magnitude of angle of rotation is summarized as:If n = 1, ө=2π = 3600 i.e. it is identity E. n = 2, ө= π = 1800; n = 3, ө= 2π/3 = 1200 ; n = 4, ө= 2π/4 = 900 n = 5, ө= 2π/5 = 720 ; n = 6, ө= 2π/6 = 600------------------------- and so on.

Operations generated by a symmetry axis:

The number of operations generated by proper/symmetry axis of rotation can be written

as: Cn1, Cn2, Cn3---------------- Cnn . It means for Cn axis you have to rotate the molecule

n time about the axis through an angle of 2п/ө to reach to the original identical position

where n is called the order of the axis, Cn1 means Cn proper rotation has been carried

once and C n2 proper rotation has been carried out two times. Let us explain C2, and C4

axes in more detailed manner. Let us take the example of more common water molecule

which has C2 axis in it and carry out C21 C2

2 and C23 rotations on about C2 axis. The

results are in Figure11

Notes

Molecular Symmetry


Figure 12: Effect of successive C2 operations on H2O moleculeC2

1 operation on water molecule takes it from configuration I to configuration II. Since H1 and H2 are not chemically different so these cannot be differentiated. Configuration I is equivalent to configuration II and C2

2 operation on water molecule takes it from configuration II to equivalent configuration III , as hydrogens are chemically the same and1and 2 have been use to differentiate the stages of operations only. After two successive C2

1 and C21 operations equivalent configuration III is obtained which

is identical to configuration I. After two operations we have reached to the original configuration .Thus the order of the C2 axis is two. Further if one gives C2

3 rotation to Configuration III one gets Configuration II again ie no new configuration is generated. Therefore, C2

3= C24. For C4 axis let us take the example of XeF4. The effects of C4

1, C4

2,,C43 ,C4

4 symmetry operation on this molecule are shown in Figure 2.

Figure 13. Effect of C41, C4

2,,C43 ,C4

4 symmetry operations on XeF4 In case of XeF4 the configurations I to IV are equivalent (since F’s are chemically equivalent and subscript to F is used only to differentiate the different stages of operations)) even if the corners are numbered. So C4

4 is equivalent to identity E. Thus only four operations are generated and order of the C4 axis is four.

Notes

Inorganic Chemistry


Reciprocal of proper axis of rotation:Since the definition of reciprocal demands another symmetry operation of the

same type, which will bring molecule/object back to its original position. Let us find the reciprocal of Cnm. Let the reciprocal be (Cn

m)-1 and it should be a proper axis. Let it be (Cn

p) ie Cn operation carried p times in the same direction.Therefore, (Cn

m)-1= (Cnp).

Reciprocal brings back the molecule to its original position ie E=Cnn

We can express these facts as:(Cn

m)(Cnm)-1= E=Cn

n

(Cnm)(Cn

p) = E=Cnn

Therefore, (Cnp)=(Cn

m)-1= Cnn-m as m times rotation +p time rotation=n time rotation

about the same axis ie gives original configuration. Therefore, m+p=n or p=n-m. Hence reciprocal (Cn

m)-1= (Cnp)= Cnn-m

Let us now find the reciprocals of C31, C4

3, C53, C6

3 symmetry operations.for C3

1 n = 3, m = 1 P = n – m = 2 . Therefore , reciprocal (C3

1)-1 = C32

C43 n = 4, m = 3, --------- p = n - m = 4 - 3 = 1

Therefore, reciprocal of (C43)-1= C4

1

Some examples of molecules having proper axes of rotation: These are summarized in Table.1

Principal axis of rotation: Molecules may have several proper axes of different order. Of the various symmetry axes the axis with highest order is called the principal axis of rotation. Molecule which has several axes of same order then the symmetry axis which lies along z-axis is generally taken as principal axis of rotation. In Figure 3 of the various symmetry axes in XeF4 the C4 axis the principal axis of rotation. There are one C4 axis, 2C2 axes and 2C2’’ in XeF4.

Notes

Molecular Symmetry


Fig.14: Several proper axes of rotation in XeF4

4.6 Reflection plane or symmetry plane:Its symbol is σ. It is an imaginary plane which passes through the body of the

molecule in such a manner that it divides the body into two equal and equivalent halves. Common points that must be taken into consideration for looking for mirror plane in the molecule are (i) plane must lie within the body and cannot be outside the body. (ii) Since σ divides the body of the molecule exactly in two halves, there must be equal number of atoms of the same kind on both side of the mirror i.e. atoms off the plane must exist in even number (reverse may or may not be true). (iii)If molecule has σ and there is one atom of one kind in the molecule (i.e. odd atom), it must lie in the plane itself. And if there are odd number of atoms in the molecule, then the odd atom will lie at the point of intersection of these planes. Reflection operation can not be carried out physically. In Cartesian coordinate system the plane of reflection can be represented as shown in fig.15

Fig .15 Reflection of point (-x1,y1, z1 ) in σxzplane If reflection operation σxz1 exists for point (-x1,y1, z1) , which is at a distance y1

,along a normal to the plane xz ,(reflection plane lies in xz plane), there should be an equivalent point at a distance –y1 . After first reflection σxz1 in σxz plane point (-x1,y1, z1) goes to point (-x1,y1, z1) at equidistance from z axis. Point (-x1,y1, z1) on further σxz2 reflection in σxz plane goes to point (-x1,y1, z1) which is original point ie we have reached to identity situation. Thus by two successive reflection we have reached the original position.

Operations generated by a symmetry plane: In fig.4 we have seen that after carrying out successive operations σxz

1 and σxz2 on the point (-x1,y1, z1) we get the same

point ie identical position. Thus number of reflection operations generated are only two. For reflection plane only two operations are generated always.

Reciprocal of symmetry plane: Reciprocal of a symmetry/reflection plane will be another symmetry/reflection plane which will bring molecule back to its original position. In figure.4 we have seen that first reflection σxz

1 takes the point ((-x1,y1, z1) to (-x1,-y1,

Notes

Inorganic Chemistry


z1) and second reflection σxz2 on new point (-x1,-y1, z1) takes it back to original point (-x1,y1, z1).Therefore, second reflection is the reciprocal of the first ie (σxz

1)-1=( σxz2).

Types of symmetry planes: Let us take the example of XeF4 and explain these. In fig.5 in the structure of XeF4 different reflection planes are shown and these are defined as:

Fig 16: Symmetry elements σv, σd and σh in XeF4

1 Horizontal plane: It is also called as the molecular plane (plane of the paper ) and it contains all the F’s and Xe atom and it is perpendicular to C4 principal axis of rotation ( part I of Figure 16). Its symbol is σh. If there is no Cn axis then molecular axis is taken as the horizontal plane σh .

2 Vertical plane: This is the plane which contains principal axis of rotations C4 (along z axis) and contains x axis or y axis also. Its symbol is σv. There are two σv, s in it. These are designated as σxz or σyz plane. In part I of Figure 16 it is σyz..

3 Dihedral plane: This plane contains C4 principal axis of rotation and lies between X and Y axes. It is designated as σd . In part II of fig17 dihedral planes lie along the diagonals of the square plane of XeF4 molecule.

Examples of reflection planes: These are shown in fig. 17 in various molecules.

Fig .17 Examples of molecules having symmetry planesCentre of symmetry or inversion centre: It is designated by the symbol as i. It lies at

the centre of gravity of the object/molecule. If one starts moving from one point on one side of the object to the other side of the object through the centre and finds a similar point at equidistance from the centre then the centre is called the centre of inversion or centre of symmetry. In cartesian coordinate system it is explained as shown in Figure 18.

Notes

Molecular Symmetry


Fig 18 : Inversion of a point (x1,y1,z1) through origin of cartesian coordinates

On moving from the point (x1,y1,z1) (the arrow head) from one side in quadrant I to other side in quadrant III through the origin of Cartesian coordinates, if one finds a point (-x1, -y1, -z1) at equidistance from the origin then the origin is termed as the centre of inversion. After carrying out the symmetry operation i1 and i2 one goes back to the original position.

Symmetry operations generated by i: In Figure 7, we have seen that after carrying out two successive operations i1 and i2 on the point (x1,y1,z1) we get the same point ie identical position. Thus, the number of inversion operations generated are only two. For inversion centre only two operations are generated always.

Reciprocal of centre of inversion: Reciprocal of a centre of inversion will be another centre of inversion which will bring molecule back to its original position. In figure 18, we have seen that first inversion i1 takes the point (x1,y1,z1) to (-x1,-y1,-z1)and second inversion i2 on new point (-x1,-y1,-z1) takes it back to original point (x1,y1,z1) .Therefore, second inversion i2 is the reciprocal of the first inversion centre i1 ie (i1)-1=( i2)

Some examples of centre of inversion are shown in fig. 19.

i lies at mid point i lies at Pt middle of C6 ringof bond C-C

Fig 19: Some examples of centre of inversion in molecules

4.7 Alternate axis of symmetry or improper axis of rotation, or rotation

Reflection axis: It is designated by the symbol Sn. Improper axis of rotation is performed

in two successive steps. Proper rotation Cn followed by reflection in a plane perpendicular

to Cn axis ie Cn followed by σ ┴ to Cn axis or vice versa i.e. σ followed by Cn ⊥ to σ

(order of performing these symmetry operations here is immaterial).If the combination

of these two operations i.e. σ followed by Cn ⊥ to σ brings the molecule in coincidence

with itself or indistinguishable, then molecule is said to have n-fold alternate axis of

Notes

Inorganic Chemistry


symmetry or improper axis of rotation or rotation-reflection axis of symmetry. Thus

even if Cn and σ ⊥ Cn are not there into the molecule Sn may exist in the molecule. Let

us explain this in more detail by taking the example of CH4 molecule. This molecule

has C2 axis through C and bisecting opposite HCH angles (Fig 20).

Fig 20: S4 axis in CH4 molecule

Take C4 axes along this C2 axis (configuration-I) and perform C41 rotation. This takes

molecule to configuration II. Which is not equivalent to I as above hydrogens are in plane now (i.e. this C4

1is not a symmetry operation). Let us perform σ (⊥ to C4) on configuration II. This takes the molecule to configuration III. Again II and III are not equivalent below hydrogens now come in a plane ie these are distinguishable. Hence, this is not a symmetry operation, but configuration III cannot be distinguished from configuration I. It means combination of operations C4

1 followed by σ (⊥ to C41) bring

back the molecule to indistinguishable position. These two operations when carried out successively gave a resultant configuration which is not different from I ie. S4 operation

exists.

4.8 Definition of group

According to the formal definition of a group, a group is a set /collection of elements,

which are combined with certain operation *, such that:

1. The group contains an identity

2. The group contains inverses

3. The operation is associative

4. The group is closed under the operation or

In mathematical sense a group may be defined as, “collection/set of elements/numbers having certain properties in common i.e. these elements are bound by certain conditions, known as group properties/postulates”. The elements do not need to have some physical significance. My emphasis will be mainly on ‘‘symmetry operations as the collection

of elements of the group”.

Notes

Molecular Symmetry


Basic properties of a group:

There are four basic properties/conditions/postulates/characteristics of a group .The elements belonging to the group should follow/adhere to these conditions in a true sense. Let us state these conditions and explain these by taking suitable examples. Let the elements of the group be [A, B, C, D, E----X-]. The four conditions are:

(I) In the collection (i.e. group) there must an element such when it combines/multiplies with each and every other elements of the group, it leaves them unchanged. This element is the identity element symbolized as ‘E’ and this property can be expressed as:

AE=A, BE=B, CE=C, DE=D and so on. Further this type of combination with E is commutative (order of combination/product is immaterial) i.e. when order of combination is reversed the results, EA=A, EB=B, EC=C, ED=D are also true. Here mode of combination may be multiplication, addition, subtraction in mathematical sense or in symmetry sense one symmetry operation followed by another symmetry operation. There is only one identity element for every group.

(II) Each and every element in the group must have an inverse, say (X) , which is also the member of the group i.e. AX = XA = E . Here, X is the inverse of A and A is the inverse of X i.e. AX =XA=E or A=X-1, A-1=X ie AA-1= E.

Inverses are unique. It is to be noted that there exists only one identity for every single element in the group but each element in the group has a different inverse.

(III) Although the combination of two elements may or may not be commutative, but it must be associative. In group { A,B,C,D----X---} following relations are valid A(BC)=(AB)C=ABC. Provided the order of combination is not changed. This associative property can be extended to any number of elements of the group. This associative result of elements must be an element of the group.

(IV) Closure: Results of combination (multiplication, addition, subtraction) of two or more elements or square of the element (the element is combined with itself) must be equivalent to an element of the group, which is also the member of the group. The group is closed under the given combination.

In group [A, B, C, D---X-] these type of combinations and their results may be AB=C, AC=D, BC= another element of the group only. It is not necessary that combination AB= combination BA i.e. combination may be commutative or non-commutative. Order of combination matters in a group .This order of combination is very significant. This is true in case of symmetry product of symmetry operations.

Notes

Inorganic Chemistry


Examples of group and verification of the characteristics

Example 1Show that a group [0] and mode of combination as addition constitutes a group.

Group contains a single element 0 in it. Property number [I] that there should be an element in the group such that when it combines with each and every element of the group leaves it unchanged. Since group contains only 0 as the element, such element will be 0 itself as 0+0=0. Element 0 is unchanged. Property [II] each element in the group must have its unique inverse. Here inverse of 0 is 0 itself, as 0+0-1=0.Other properties are easy to verify as there is a single element in the group.Example 2Show that the integers [------ -3,-2,-1, 0,+1,+2,+3-----] under the mode of combination addition and as zero as the identity element constitutes a group of infinite order.

Here, the identity element is 0.You add 0 to any element of the group and it will remain unchanged. 3+0=3,5+0=5 ,-6+0=-6 and so on. Addition of 0 to the right or left to the number does not matter, i.e. 4+0=4 or 0+4+4.So property [I] is satisfied. According to property [II] each element in the group has its unique inverse. Let inverse of integer n is n-4.Since the mode of combination is addition in this example the inverse of +n will -n as (n)+(-n)=0(identity). Element when combined with its inverse gives identity. For example -3+3=0, (+4)+(-4)=0 and so on. According to property [III], you can combine any number of elements, the net result will be an element of the group. Order of combination of elements should not be changed. In this example changing of order does not matter. But when we deal with symmetry elements of the group it matters a lot. For example ( -3)+(-2) +(-1) =[-3+(-2)]+(-1) = (-3) + [(-2) +(-1)]= -6 Since this group is of infinite order, property (III) is easily satisfied. Let us take property (IV). Identity is zero. A combination of two or more elements or square of the element (addition of number to itself) must give a number that belongs to group. -2+(-1)=-3, and -3 belongs to group (because it is of infinite order) -2+- 2 = -4 (square of the element),-4 belongs to group -2 +-3 +3 = -2 (more elements combined).-2 belongs to group. Thus property (IV) is satisfied.

This group under the mode of combination as multiplication does not constitute a group as the identity element 0 when combined (here multiplied) with other elements, changes them i.e. 4 x 0=0, thus 4 changes to 0.

The set of natural numbers 0, 1, 2, 3,----- under addition is not a group as there is no inverse of the elements except that of zero .Example 3Show that the group [-1,1] under the mode of multiplication constitutes a group.

First let us find the identity element in the group. There are three possibilities -1 is the inverse,+1 is the inverse or there is no inverse. Let us take the following products:

1x1=1, -1x1=-1, -1x-1=1, number 1 does not change 1or -1 when these are combined

Notes

Molecular Symmetry


(here multiplication) with 4.-1 is not the identity as it change the element when it is combined with other element. Now we need to find the inverse of the element of the group and it should be such that 1x1-1=1, -1x-1-1=4.

Here, 1x1=1 so 1 is the inverse of 1and -1x-1=1 so the inverse of -1 is -1 itself as these products give identity 1 as the element. Since in the group there are only two elements you combine in any way you will get the same result and net result will be an element of the group.

1x1=1, -1x-1=1, -1x1=-1(element is combined with itself or square of the element) or 1x-1=-1 etc.

The group [-1, 1] under the mode of addition does not constitute a group. As 1+-1=0 and 0 is not the element of the group and so on you can verify other properties.

Example 4

Show that the group [1,-1, i, -i ] under mode of combination as multiplication and 1 as the identity element constitutes a group.

Let us prove four properties of the group.

Property (I) AE=A etc. 1x1=1, 1x-1=-1, i x1=i,–i x1=-I, i.e. number are not changed.

So 1 is the identity here Property (II) Let us find the inverses, (-1)-1 = -1,(-i)-1 = i ,(i)-

1= -i, (1)-1 =4.

It can be checked that when element and its inverse is combined identity 1 is obtained

(-1) x (-1)=1 ,(-i) x (i)=-i2=1, (i) x (-i) =-i2 =1, (1) x (1) =1

Property (III) Law of associative multiplication (AB)C=A(BC)-----

(1)x(-1)x(i) ={1x-1}x(i) = 1x{-1x i} = -i.

Property (IV) AB=C----etc

1x-1=-1 ,-1x-1=1, 1x1=1 , i x i=-1 , i x-i=1, i x1=i, i x-1=-i , 1x i x-i=1 etc.

You combine any number of elements you will get the element of the group.

Example 5Show that all powers of [------2-2, 2-1, 20, 21, 22, -----] form an infinite group with mode of combination as multiplication.

Let us first find the identity by taking the following products: 2-2x20=2-2, 2-1x20=2-1 the element is not changed. So 20 is the identity here. Inverse of an element 2n is 2-n as 2n x 2-n =20(identity). Further you combine any number of element in any way here you will get the element of the group only.

2-2x 2-1 x 20 =2-3, 2-1 x22 =21 and so on . Law of closure can also be verified in similar manner.

Example 6

Let us explain the group properties by taking an example of face movements of a

soldier in exercise drill.

Notes

Inorganic Chemistry


L stands for Left Turn command

R stands for Right Turn command

A stands for About Turn command

E means no command or stand at ease

These four face movements L,R,A,E of soldier constitute a group with E (stand at ease) as identity and mode of combination is one face movement on command followed by the other face movement on command i.e. these four face movements on command constitute a group [ L,R,A,E ].Let us now verify the four properties of this group. According to property (I) there must be an element into the group which on combination with other element does not change the element. This is E here ie no command is given to the soldier AE=A , means E movement followed A (about turn)movement i.e. soldier follows the face movements in the order as; stand at ease followed by about turn by 180o. This combined movement is equivalent to A. Also EA=A .Similarly, LE=EL=L and RE=ER=R etc. Here order of face movement matters.

According to property (IV) the product or combination of any two or more elements or square of an element must be an element of the group.

Direction of arrow indicates the order of face movement to be carried i.e. move from right to left and must be followed in all cases.

Theorem of ReciprocalsIn a group the reciprocal of the product of two or more than two elements is equal

to the product of the reciprocals of elements taken in reverse order.Let the group be [A, B, C, D, E, ----------X]. Then according to the theorem

reciprocals(ABCD----------)-1= (D-1C-1B-1A-1--------)Let us prove this . If A, B, C, D,---- are the elements of the group then the product

(ABCD) must be an element of the group (property IV of the group) and (ABCD) product is equivalent another element X of the group such that (ABCD)=X

Right multiply both sides with (D-1C-1B-1A-1-------- )(ABCD)(D-1C-1B-1A-1--------) =X(D-1C-1B-1A-1--------)Taking help of law of associative multiplication we can work out the left hand

side of this relation as;(ABC)(D)(D-1)(C-1B-1A-1--------) =X(D-1C-1B-1A-1 --------)(ABC)(DD-1)(C-1B-1A-1--------) =X(D-1C-1B-1A-1--------)(ABC)(E)(C-1B-1A-1--------) =X (D-1C-1B-1A-1--------)Element and its reciprocal when combined gives identity E and identity E when

combined with element leaves it unchanged

Notes

Molecular Symmetry


(ABC)(E)(C-1B-1A-1-------- ) = X (D-1C-1B-1A-1-------- ) (ABC)(C-1B-1A-1-------- ) = X (D-1C-1B-1A-1-------- ) (AB)(CC-1)(B-1A-1-------- ) = X (D-1C-1B-1A-1-------- ) (AB)(E)(B-1A-1-------- ) = X (D-1C-1B-1A-1-------- ) (AB)( B-1A-1-------- ) = X (D-1C-1B-1A-1-------- ) (A)(B B-1 )( A-1)-------- ) = X (D-1C-1B-1A-1-------- ) (A)(E)( A-1)-------- ) = X (D-1C-1B-1A-1-------- ) (A)(A-1)-------- ) = X (D-1C-1B-1A-1-------- ) ( E ) = X (D-1C-1B-1A-1-------- )

E = X (D-1C-1B-1A-1-------- ) Since X combined with (D-1C-1B-1A-1--------) gives identity E, therefore, X and

(D-1C-1B-1A-1-------- ) are inverse to each other or X-1 = (D-1C-1B-1A-1-------- ) And X=ABCD-----as defined earlier. Putting the value of X one gets (ABCD----)-1= (D-1C-

1B-1A-1--------). Hence the theorem is proved.

Some common types of groupsDuring the discussion of group theory and its applications in chemistry you will

come across some more common types of group. Some basic knowledge of these is must.

Abelian groupIn our previous discussion on group properties the order of combination of elements

was stressed. In case of integers and addition as the mode of combination the order of combination does not matter but in symmetry operations as group elements it matters a lot. An abelian group is that group in which all elements when combine in either way give the same result. In a group [A, B, C, D----] all combination products

AB=BA, AC=CA, BC=CB-------- are true i.e. the elements commute with each other.The integers [------ -3, -2, -1, 0, +1, +2, +3-----] under the mode of combination as

addition and zero as the identity element constitutes an abelian group of infinite order.

Cyclic groupIn this group the elements are generated by taking the power of one group element.

Let the element be X that generates all other group elements. Then the group elements are X1, X2, X3, X4---------Xn =E. There are n elements in the group and thus the order of the group is n. Cyclic group is always an abelian group as powers are additive and their order of combination does not matter.

X1+X2=X2+ X1=X3

Finite and infinite groupsThe group which has infinite number of elements in it is called infinite group.

All positive and negative integers with zero as the identity and addition as the mode of combination constitute an infinite order group ie [------4, 3, 2, 1 0,-1,-1,-3,-4-------] with addition as mode is a n infinite group

In finite group the number of elements is finite. The group (1,-1, i,-i) is a finite group of order four.

Notes

Inorganic Chemistry


4.9 Total symmetry elements and total operations generatedLet us take the example of water molecule and find total symmetry elements in it

and total symmetry operations generated by these symmetry elements. Figure 21 shows the various symmetry elements in structure of water molecule.

Figure 21: Various symmetry elements in H2O molecule

There are four symmetry elements σxz , σyz ,C2 and E in H2O molecule. Total symmetry

elements are four. Let us find the symmetry operations generated by each of these.

(i) E generates only one operation and it is E. (ii) σxz generates σxz

1 and σxz2 =E symmetry operations and these are σxz and E

only. (iii) σyz generates σyz

1 and σyz2 =E symmetry operations and these are σyz and E

only. (iv) C2 generates C2

1 andC22=E symmetry operations and these are C2 and E only.

Thus total symmetry operations generated here: σxz, σyz , C2 and E in H2O i.e. only four symmetry operations are generated by four symmetry elements. In case of H2O molecule number of symmetry elements and symmetry operations are same. It is not same always. These four symmetry operations constitute a group ie [σxz, σyz , C2 , E] is a group of order four. Number of elements in a group is its order. Now let us verify the four group properties.

(I) Here E is the identity, EE=E, EC2=C2E = C2; Eσxz= σxzE = σxz; Eσyz = σyzE = σyz ,thus group element remains unchanged. Figure 22 shows all these results.

Notes

Molecular Symmetry


Figure 22: Combination of E with C2 ,σxz , σyz

(II) Every element has its reciprocal (i.e. inverse)Here (σxz)

-1 is (σxz) ; (σyz )-1 = σyz and (C2)

-1 is C24. Symmetry operation and its

reciprocal when combined give identity i.e. σxz (σxz)-1 = E ; σyz (σyz)

-1 =E; C21( C2

1)-1 =EIn Figure 23, we notice that the symmetry product C2.C2 =E, it means C2 is its

own inverse i.e. (C2)-1=C2. Similarly σxz =E , therefore, σxz is the inverse of itself and

similarly inverse of σyz is σyz.. Thus in the group each element has a unique inverse. Here each and every group element (symmetry operation) is the inverse of itself.

Figure 23: Combination of C2 with C2 , σxz with σxz , σyz with σyz

(III) Law of associative multiplication (order of multiplication should not be changed). (C2

1 σxz ) σyz = C21(σxz σyz) . These products are shown in the Figure 24.

Notes

Inorganic Chemistry


Figure 24: Associative products ( C2 σxz ) σyz = C2

1(σxz σyz) for H2O molecule

(IV) Product of two or more elements or square of the element must be an element of the group. The combinations of various symmetry operations are shown in figure 25.

Notes

Molecular Symmetry


Figure 25: Effect of Combination of Symmetry Operations on H2O MoleculeThus all the four properties of a group [σxz , σyz ,C2 , E ] have been verified. Let

us take another example of C3 axis in an equilateral triangle and see that operations generated by C3 axis constitute a group.C3 axis generates the following symmetry operations: C3

1 ,C32, C3

3 =E i.e. in all three symmetry operations are generated. Various symmetry products are shown in Figure 26.

Similarly other symmetry products can be worked out and all the four properties of the group can be verified. Let us now find the following products or inverses.

C34.C3

2 =? ; (C31)-1=?C3

2.C32 =? C3

4.C31 =? ; ( C3

1)[( C32)( C3

3)] = [( C31) ( C3

2)]( C33)

Inverse of C31)-1=C32 so now show that C34.C32=E.

Figure 26: Various symmetry products of Total symmetry elements present in the given molecule symmetry operations C3

1, C32, C3

3

4.10 Definition of a subgroup

It is defined as a part of the main group which satisfies all the properties of the group ie

within the main group of order’ h’ there may be smaller groups of lower order then ‘h’.

(i) Sub group always contains identity element E.

Notes

Inorganic Chemistry


(ii) E itself constitutes a group of order one and it is trival.

(iii) The sub group of order two also contains identity element E .

(iv) Groups of order two can be E,A ; E,B ; E,C etc in the main group A,B,C,E.

(v) Sub groups always follow the group conditions of the main group.

(vi) Order of the sub group is an integral factor of the order of the main group

(vii) If the order of sub group is’ g’ then h/g is an integer and not in fraction.

(viii) If the order of the main group is six, then sub groups can have orders 1,2,3 only(6/6=1,6/2=3,6/3=2) only and 6/4 or 6/5 are fractions.

Let us now try to find some sub groups in main group by taking suitable examples.[1,-1,I,-I ] is a group of order four under the mode of combination as multiplication.

Since the order of the group is four, the sub groups order can be 4/2=2,4/4=4. There will be no sub group of order 4/3=4.33. Since identity always constitutes a sub group order one. Here, identity is 4. So a sub group of order one is [1] .It satisfies all four group properties. The other sub groups of order two in it can be [1,-1], [I,-I ], [-1,i] [1,I ], [1,-I ], [-1,-I ] . In these groups of order two, only group [1-1] satisfies all four group properties under the mode of multiplication and 1 as the identity. Here we write some products of this sub group as: 1x1=1, 1x-1=-1, -1x-1=4. Multiplication with 1 does not change the number. Number multiplied with itself also gives the element belonging to the sub group. Thus [1,-1] constitutes a sub group of order two and there are only two sub groups [1] of order 1and [1,-1] of order two in the main group [ 1,-1,I,-i] of order four.

Let us take another example of face movements of a soldier during exercise. Here L stands for left turn, R stands for right turn , A stands for about turn and E stands for stand at ease or do nothing or stand still. Thus main group is [L, R, A, E].

L stands for Left Turn command

R stands for Right Turn command

A stands for About Turn command

E means no command or stand at ease

Let us find sub groups in the main group [L, R, A, E] . Here E is trival group of order one and non trival sub groups of order of two can be[ E,A], [E,L], [E,R] ,[L,R],[A,L],[A,R] (because 4/2= 2 is integer) and so on. Out of [E,A ], [E,L], [E,R] sub groups only [E,A ]sub group satisfies all the group condition and this is the only subgroup of order two of the main group (L,R,A,E).

EA=AE=A, AA=E, AAA=A, (A)-1=A and E-1 is E and so on. See Figure 1 for these products.

Notes

Molecular Symmetry


The remaining groups of order of order two [E,L], [E,R],[L,R],[A,L],[A,R] etc do not constitute groups To verify that the [E,L] and [E,R] etc do not constitute sub groups is left an exercise for the reader.All sub groups of order g (h/g=integer) need not be sub groups in true sense and that may not follow the conditions of the group.

Symmetry operations σxz , σyz ,C2 and E in H2O molecules constitute a group. This has been verified in section -5 symmetry operations as group elements. Let us now try to find the sub group in the group [σxz , σyz ,C2 ,E]. E is always a sub group of order one. The other sub groups of order two may be [E,C2] , [E, σyz] , [E, σxz] ,[ σxz , σyz], [C2, σyz ], [C2 , σxz ] and so on. Here [E,C2] ,[E, σyz] and [E, σxz] are the sub groups of order two and [ σxz , σyz ], [C2, σyz ]and [C2, σxz ] are not the sub groups of order two. Let us see some of the products by taking example of water molecule. Results are given in Figure 27 and Figure 28.

Fig. 27 Combination of E with C2, σxz and σyz

Notes

Inorganic Chemistry


Fig. 28: Combination of C2 with C2, σxz, with σxz and σyz and σyz

These results clearly show that E, [E,C2] , [E, σyz] and [E, σxz] are sub groups of the main group [σxz , σyz ,C2 ,E].

4.11 Definition of a Class

This is another way of subdividing a group into smaller collection of elements. It is

defined as the collection of all conjugate elements of the group. It has different bases for

the division of the group. All conjugate elements are related by similarity transformation.

The order of a class is the integral factor of the order of the main group.

Similarity TransformationIf A and X are the the group say it is X-1 element of the group then X-1AX operation

will be equal to some element of B (or it may be same). This transformation is written as:X-1 AX =B

We can express the result by saying that B is the similarity transformation of A by X or we can say that A and B are conjugate elements of the group.

Characteristics of conjugate elements

Conjugate elements have some characteristic properties and these are given as:

(i) Every element is conjugating with itself. It means if we take element A, then it must be possible to find one element, X, in the group such that

(ii) If A is conjugate to B, then B is conjugate to A ie A=X-1BX then there must be another element in the group say Y such that B=Y-1AY holds good.

(iii) If A is conjugate with B and C separately, then B and C are conjugate with each

Notes

Molecular Symmetry


other or if A is conjugate to B and B is conjugate to C, then A will be conjugate with C also.

A=X-1BX (A is conjugate with B) A =Z-1CZ (A is conjugate with C)Then B=Y-1CY (B is conjugate with C)

To find a class in groupLet us take finite group [A, B, C, D, E, and F] of order six with E as the identity

element in this group. We have to find the classes in this group. We take the similarity transformation of each and every element by all the elements of the group including the element itself. Let us first write down all the similarity transformations for E by all the element including E itself and these are: E-1EE=E(element it self) ; A-1EA=E; B-1EB=E ; C-1EC=E ; D-1ED =E ; F-1EF =E ie E transforms into itself by all these transformations. Therefore, E constitutes a separate class itself and number of elements in this class equals to one ie order is one.

Now take element A and perform similar transformations on it and say the results areA-1AE = A B-1AB = C (say) C-1AC= B (say)D-1AD = B(say)F-1AF = C (say)A-1AA = AIn all these transformations A is transformed into A ,B ,C only. Thus A, B, C

constitute a class of order three.Now find similarity transformation for D and the results say are E-1DE = DA-1DA = FE-1DB = FC-1DC = FD-1DD = DF-1DF = DHere D transforms into D and F only. Therefore, D and F constitute a separate

class of order two. Now take the similarity transformations of FE-1FE = F C-1FC = DA-1FA = D D-1FD = FE-1FB =D F-1FF = FF transforms to D or F only so no new class is obtained. Thus the [A, B, C, D, E,

and F] group can be divided into three classes as: (i) E of order one (ii) A,B,C in a class of order three and (iii) D, F in class of order two.

Let us find similarity transformations for PH3 molecule. In Figure 10 top view of PH3 together with various symmetry elements is shown.

Notes

Inorganic Chemistry


Fig. 29: Top view of PH3 moleculeTotal symmetry operations for the molecule PH3 are C3

1, C32, C3

3 = E, σv, σv′ and σv′′ and these constitute a group . Let us see whether σv, σv′ and σv′′ symmetry operations (σ generates only one operation) belong to a class. For this let us find the following transformations:

(i) E σv E = σv (ii) C32 σvC3

1= σv’ (iii) C31 σv C3

2 = σv” (iv) (σv’)-1 σv σv’= σv”(v) (σv)

-1 σv σv = σv (vi) (σv”)-1 σv σv”= σv’During rotations/reflections the already defined symmetry elements and their frame

work do not shift with molecular rotations/reflection ie reference frame remains intact. The results of similarity transformations are shown in figure.14. C32 and C31 are inverse of each other so also each σ is the inverse of itself.

(i) E σv E = σv

(ii) C32σvC3

1= σv’

(iii) C31 σv C32 = σv”

(iv) (σv’)-1 σv σv’= σv”

Notes

Molecular Symmetry


(v) (σv)-1 σv σv = σv

(vi) (σv”)-1σv σv”= σv’

Figure 30: Effect of successive symmetry operations in PH3 moleculeThus all the similarity transformations on σv by E, C3

1 , C32 ,σv, σv’, σv” give

results as σv, σv’, σv” only. Similarly by carrying out similarity transformation on σv’, σv” by E, the results as σv,σv’, σv” only. Thus out of six operations of PH3 3σ’s (σv, σv’, σv”) constitute a separate class. Similarly it can be shown with the help of a diagram that C3

1 and C32 constitute a separate class.

Thus in a group [C31,C3

2, ,σv, ,σv′ σv′′] there are three classes (i) E of order one (ii) σv,σv’, σv” of order three and (iii) C3

1 , C32 of order two. To find classes in this way is

very cumbersome and takes lot of time.For example in case of equilateral triangle there are 12 symmetry operations which

constitute a group. To find classes in this group one has to carry out X-1AX types of similarity transformations 144 times. The procedure can be simplified if certain rules, which are normal, be followed in practice and these are:

Some simple rules for finding classes

(i) The symmetry operations, E, i and σh each are in separate class themselves. (ii) Same (rule ii) is true for Sn

k and Sn−k improper axes of rotation as it is true

for Cnk and Cn

−k axes. (iii) Two reflections σ and σ’ will belong to a same class if there is another symmetry

operation into the group which moves all the points on σ’ into corresponding points on σ.

(iv) σv and σd will belong to different classes.The simplest way of arranging the symmetry operations into classes is to arrange

them into sets of equivalent symmetry operations which have equivalent effects on the molecule.

4.12 Products of symmetry operationsIn a group [X, Y, Z, E] let us take two symmetry operations X and Y and combine

them in the manner YX=Z

Notes

Inorganic Chemistry


X and Y symmetry operations have been carried out in a particular manner and the net product is Z, which is also the element of the group. There is a definite rule/notation that must be followed. First we carry out the symmetry operation which is written in the product

Direction of arrow indicates the order of perforation of the symmetry operation ie X is performed first followed by Y. The net result is Z. If we do not follow the order from right to left we may get different result. If the products XY and YX are same as Z then we say that X and Y commute with each other and if the products XY and YX are not same then XY and YX do not commute with each other .In abelian all group elements commute with each other.

If we apply first symmetry operation X to a point ( x1, y1, z1) it will go to new point ( x2, y2, z2) and on further applying a second symmetry operation Y let ( x2, y2, z2) goes to a point ( x3, y3, z3) i.e. net result of carrying out two successive operations is a point (x3,y3,z3). Now let us look a way i.e. another symmetry operation (third operation) Z which will carry (x1, y1, z1) directly to (x3, y3, z3) .We can represent this process as:

Since Z symmetry operation gives the same result as YX this is called the symmetry

product of YX.

Let us take some general point in Cartesian coordinate system and see the results of

symmetry operationsExample -1 If there are two C2 axes perpendicular to each other in a plane , then

there must be another C2 axis perpendicular to the plane of both the axes i.e. if there is C2x and C2y axes in a molecule, then there will be a C2z axis in the molecule or show that C2y.C2x = C2z

Let us take a point in the Cartesian coordinate system as shown in figure.31 and perform C2x and C2y on it

Fig.31 Effect of C2x and C2y and C2z axes on general point (x1, y1, z1)

Notes

Molecular Symmetry


We operate C2x (clockwise) first (right to left rule) on general point (x1, y1, z1) of an arrow up in first quadrant. It goes to point (x1,-y1,-z1) an arrow down in 2nd quadrant and on further applying C2y (clockwise) it goes to point (-x1, -y1, z1) in 3rd quadrant (arrow up)

Now apply C2z (clockwise) to the original point (arrow) (x1, y1, z1) in 1st quadrant. It will take this point to a point (-x1, -y1, z1) in 3rd quadrant. It is the same result as has been obtained by carrying out C2x and C2y successively on general point in 1st quadrant.

Thus C2y.C2x=C2z

Example-2 Show that if in a molecule C4z and C2y axes exist, then there must be a C2

axis lying in I and III quadrant of XY plane at 450 to C2y.i.e. show thatC4z.C2y = C2 (axis lying in XY plane in 1st and 3rd quadrants)Let us show this in similar manner. Carry out symmetry operations as follows

C4z.C2y on point (x1, y1, z1). First perform C2y on point (x1, y1, z1) in I quadrant. It will take the point to new point (-x1, y1, - z1) (arrow down) in IVth quadrant. On this new point now perform C4z. New point obtained is (x1, y1, - z1) in Ist quadrant. We get the same result by performing C2 operation on (x1, y1, z1) about the axis lying in Ist and III quadrants. The results are shown in fig.32

Fig.32: Results of symmetry product C4z.C2y = C2 on general point group

ThusC4z.C2y = C2 (axis lying XY plane in Ist and IIIrd quadrant)

Example-3 Show that C2z and σxy commute with each other i.e Show that C2z. σxy =

σxy.C2z. Use figure .3 and prove this . Follow the figure and verify the results .

Notes

Inorganic Chemistry


Fig 33: Product of symmetry operations (i) C2z.σxy (ii) σxy .C2z

In fig.3(i) and fig.3(ii) C2z and σxy symmetry operations have been performed in reverse

orders

Thus we see that the final results, of (i) and (ii) routes of taking the products of symmetry operations , are same.

Thus C2z.σxy = σxy .C2z i.e. C2z and σxy commute with each other.

Example-4Let us now see the symmetry product of some symmetry operations of a molecule

trans-dichloroethylene belonging to particular point group.Trans-dichloroethylene has four symmetry elements, E , C2z, σxy, and i which

are shown in figure.34. These four symmetry elements generate only four symmetry operations E , C2z, σxy, and i which constitute a group of order four.

Fig 34: Trans-dichloroethylene and its symmetry elements

Notes

Molecular Symmetry


Let us take symmetry product C2z .i and symmetry product i.C2z symmetry operation’s products in this molecule. On applying C2z. i the configuration I goes to II then goes to III. On applying i.C2z the configuration goes from I to IV and then to III. Therefore, C2z .i = i.C2z. Thus i and C2z are commutative. The results of such operations are shown in Figure.35

Fig. 35: Effect of C2z .i and i.C2z symmetry operations on trans -dichloroethylene

Some general points about relation between symmetry operation and symmetry

elements:

(a) Rotation can be created by combing reflections but reflection cannot be created by proper rotations.

(b) Cn axis and one perpendicular C2 axis requires the existence of nC2 axes perpendicular to Cn axes.

(c) Product of two proper rotations must be a proper rotation.

(d) If there is Cn axis and a plane containing it, there must be n such planes separated by ө=2π/2n angles.

(e) The product of two C2 rotations about axes which intersect at an angle of ө is a rotation by 2 ө about an axis perpendicular to two C2 axes.

4.13 Definition of symmetry equivalent elements

In a group equivalent symmetry elements are those symmetry elements which can be

interchanged by any of the symmetry operation generated by the symmetry element

of the group. Let us explain the definition more clearly by taking the example of MF4

square planar molecule. Example of square planar molecule MF4 In figure. the various

symmetry elements present in square planar MF4 molecule are shown as:

Notes

Inorganic Chemistry


Fig.36: Various symmetry elements present in MF4 moleculeThere are five axes of order two and these are, C2, C2’, C2”, C2’”, C2”” Do these

have same effect when the molecule is rotated about these axes? The C2 lies along C4 axis , C2’, C2”, lie along the diagonals of the square planar plane and C2’”, C2”” lie along Y and X axes bisecting the opposite sides Now let us check whether there is a symmetry operation that can change (i) C2 to C2’, or to C2”, or to C2’”, or to C2”” (ii) C2’ and C2”, to C2’” and C2”” respectively. Let us take C4 axis and perform C4

1 rotation. This type of changes are not possible as C2 does not go to C2’, C2”, C2’”, C2””. The C4

1 rotation only interchanges C2’and C2” with each other. Similarly C2’”and C2”” are interchanged with each other. In no case C2’, C2”( along diagonals) are interchanged with C2’”, C2””( along Y and X axes ). We can also use operations corresponding σv or σd or σh to check whether these interchanges (i) and (ii) are possible or not. Again this is not possible. The σv interchanges C2’and C2” with each other and σd similarly interchanges C2’”and C2”” with each other. In no case C2’, C2”( along diagonals) are interchanged with C2’”, C2””( along X and Y axes ) by σd or σv .Thus of the five symmetry axes of order two C2, C2’, C2”, C2’”, C2”” the set of equivalent symmetry axes are :(i) C2 (ii) C2 ’ is symmetry equivalent to C2” (iii) C2’” is symmetry equivalent to C2”” . Pair of axes (C2’, C2”) is symmetrically nonequivalent to pair of axes (C2’”, C2””).

In MF4 we have four vertical planes and one horizontal plane . Vertical planes can be divided in 2σv and 2σd. Here these two sets of planes cannot be interchanged with each other by any symmetry operation of the molecule. Only 2σv are interchanged amongst them and 2σd are interchanged amongst them. σh cannot be carried into σv or σd by any operation.. So molecule has σh, 2σv and 2σd sets of symmetrically non-equivalent σ symmetry elements.

Example of BF3 : In Figure 2, the various symmetry elements present in trigonal planar BF3 molecule are shown as:

Fig.37: Various symmetry elements in BF3

Notes

Molecular Symmetry


All the 3C2 axes (C2’, C2” and C2.’’’) are symmetrically equivalent. These can be interchanged into one another by C3

1 operation (ө=2π/3). Similarly all three 3σv (containing C3 and one of the C2’, C2” and C2.’’’ axes) are also symmetrically equivalent.

Example of H2O: In fig.37 various symmetry elements are shown in water molecule.

Figure 37: Various symmetry elements in H2O moleculeHere two planes σxz and σyz are not symmetrically equivalent. By C2 symmetry

operation σxz and σyz cannot be interchanged. Example of benzene. In fig.38 various symmetry elements in benzene molecules

are shown as:

Figure 38. Various symmetry axes in benzene moleculeThere is no symmetry operation either C6, or C3,or C2 perpendicular to benzene

ring which can interchange 3C2 and 3C2” with each other . Thus in benzene we have set of 3C2 and 3C2” axes as symmetrically non equivalent symmetry axes.

Definition of symmetry equivalent atomsSymmetry equivalent atoms are those atoms which can be interchanged with each

other by the other symmetry operations generated by symmetry elements of the molecule. These equivalent atoms are chemically same but have different symmetry environment.

Example of MF5 (trigonal bipyramidal). In fig.39 various F’s which are chemically same are numbered to show symmetrically equivalent and symmetrically non equivalent F’s .

Figure 39. Various symmetry elements in MF5 molecule

Notes

Inorganic Chemistry


In MF5, F1 ≡ F2 ≡ F3 are symmetrically equivalent while F4 = F5 are symmetry equivalent to each other. F1 can be changed to F2 or F3 by C3 operation, but F1 or F2 or F3 can not be changed to F4 or F5 or vice versa by C3 or by any of the three C2 axes .These three equatorial fluorine’s F1, F 2, F3 are symmetrically equivalent and two axial fluorine F4 and F5 are symmetrically equivalent to each other (F4 or F5 can be interchanged with each other by any of the C2 axis lying in equatorial plane. Thus in PF5 we have five F’s which are chemically equivalent but symmetrically not equivalent and are divided in two sets (2 axial F and 3 equatorial F). These symmetry non-equivalence of fluorine in MF5 is clearly seen in low temperature NMR spectrum of MF5.

Example of multi ring aromatic molecule: In this molecule (figure.40) we have sets symmetrically equivalent hydrogens

Figure 40. Aromatic ring system with different sets symmetrically non equivalent hydrogen atoms

These sets are (i) (d, d) hydrogen atoms (ii) (a, a, a, a) hydrogen atoms (iii) (b, b, b, b) hydrogen atoms and (iv) (c, c ,c, c) hydrogen atoms. These sets of hydrogen can not be interchanged by the symmetry operations about symmetry axes (dotted lines). All (d) type or (c) type or (b) type or (a) type of hydrogen atoms can be interchanged amongst them by the symmetry operations about these axes(dotted lines).These types of hydrogen atoms are clearly indicated in the 1HNMR spectrum of the molecule.

Example of naphthalene: In case of naphthalene as shown in Figure 7, there are two types hydrogen atoms 4α(1, 4, 5, 8 hydrogen atoms) and 4β ( 2, 3, 6, 7 are β hydrogen atoms). Perform any symmetry operation C2 on the molecule α-hydrogen atoms will be interchanged with α hydrogen atoms and β-hydrogen atoms are interchanged with β-hydrogen atoms only. Thus in naphthalene there are two sets of four equivalent hydrogen atoms i.e. 4α and 4β. Again this symmetry non-equivalence of hydrogen atoms is reflected in 1HNMR spectrum of naphthalene.

Fig.41: Symmetry equivalent sets of hydrogen atoms in naphthalene

Notes

Molecular Symmetry


Some more examples:

Let us take some more examples of (i) mono fluorobenzene and (ii)1-fluoro,3-chloro,5-

bromobenzene and (iii) trans- difluoroethylene as shown in figure 42.

Fig.42: Symmetrically equivalent and non equivalent atoms in (i) mono fluorobenzene, (ii) 1-fluoro,3-chloro,5-bromobenzene and (iii) trans- difluoroethylene molecules.

In mono fluorobenzene (i) there are three sets of different types hydrogen atoms (H1, H2), (H3, H5) and H4. In 1-fluoro, 3-chloro, 5-bromobenzene, all hydrogens are symmetrically non-equivalent. In (iii) trans-difluoroethylene all F’s are symmetrically equivalent and all H’s are symmetrically equivalent. Fig.43 gives some more example of molecule with different sets of equivalent hydrogen atoms.

Fig.43: Various isomers of trichlorobenzene

4.14 Classification of molecules into point group

There are two types of notations for naming the point groups:

(i) Schoenflies Notation: This notation is used by spectroscopists and applied to molecular point groups.

(ii) Hermann-Mauguin notation: This is commonly used by crystallographers and groups are called space groups.

Notes

Inorganic Chemistry


In this module Schoenflies notation will be used for naming the point group. In chemistry sometimes same symbol is used to denote both for a point group and a symmetry element. For example for cyclic Cn point group, the generator symmetry element is also represented by Cn .One should be careful while working with such types of point groups. It is the symmetry operations that constitute a group and not the symmetry elements. But classification of molecules can be described in terms of symmetry elements present in the molecule. Each point group can be generated from a smaller set of symmetry operations known as group generators. Before giving systematic scheme of classification of molecules, let us start looking at small and simple molecule’s symmetry elements and decide the type of group to which the molecule can be placed. It is the combination of symmetry elements inversion and rotation that give molecular point group in which we are interested in this module.

The various point group categories:

Type- I Non axial point groups or molecules with low symmetry.

Type –II Single axial point group or molecules with intermediate symmetry

Type-III Dihedral point groups or molecules with very high symmetry ie molecules having ‘n’ vertical rotational axes perpendicular to principal axis

Type – IV Cubic groups: Molecules having multiple intersecting high order rotational axes.

Before discussing the scheme of classification of the molecules into point groups let us

first take few examples of each category.

Examples of type- I: non axial point groups: Point groups belonging to this category

are C1, Ci, and Cs. These are low order groups where order of the group, h= 1, or 2.

These do not have Cn axis of any order except C1 ( ө=360o).Now we discuss these low

order groups one by one.

(i) C1 point group: Molecules belonging to this point group have only identity E as the symmetry element or we can say that molecules belonging to this category have Cn axis of order 1 ( ө=360o).Few example of molecules belonging to this category are shown in fig.44

Fig .44: Molecules belonging to C1 point group

Notes

Molecular Symmetry


C1 is a trival point group which contains all the molecules that do not have any symmetry elements in them. In above examples there is no, σ, i and Cn axis. Only C1 rotation ө =2π /1=360o is possible.

(ii) Ci point group: molecules belonging to this point group have E (identity) and i (centre of inversion) as the symmetry elements in them. The order of the group is 2 and symmetry elements in these molecules are E and i .Few examples belonging to this point group are shown in fig. 2 . These are (i) 1,2-dichloro-1,2-difluroethane(staggered), (ii)P2F4, (iii)1,3-dichloro-2,4dibromocyclobutane, (iv) meso tartaric acid (v)3,5-dimethylpiperazine-2,5-dione (vi) 1,4-didimethyl,3,6-diflurocyclohexane(chair form).

Fig 45: Few examples of molecules belonging to Ci point group

(iii) Cs point group: Molecules belonging to this point group have mirror plane σ and E as symmetry elements in them and the order of the group is two. Examples of molecules belonging to this point group are shown in fig.46. These are (i) 1,2-dibromo,4-chloro benzene ,(ii) chloroethylene, (iii) 1,2,3-trichlorocyclopropane(iv) PF3Cl2,(v) 1,2- benzpyrene,(vi) formyl chloride ,(vii) F2SO.

Fig.46: Examples of molecules belonging to Cs point group

Notes

Inorganic Chemistry


Examples of type –II: single axial point group: Point groups belonging to this category

are Cn, Cnv, Cnh ,Sn .Let us take these point groups one by one.

(i) Cn point group: Molecules in this point group possess only Cn axis and no other element. The Cn generates Cn

1, Cn2-----Cn

n = E symmetry operations ie n number of symmetry operations only. The order of the group, h = n. It is a cyclic group and hence it is Abelian group. Each operation belongs to a separate class. Few examples of molecules belonging to Cn point group are shown in fig.47. (i)H2O2 nonplanar configuration, (ii) 1,2 –dichloroethane (in the given configuration), (iii)1,1,1-trifluroethane ( neither staggered nor eclipsed,(iv) cis –[Co(en)2Cl2]

+.(v) B(OH)3( hydrogens out of plane of BO3 moiety),(vi) NH2NH2(as shown) (vii) H2S2(HSSH)

Fig. 47: Examples of molecules belonging to Cn point group

(ii) Cnv point group: Molecules belonging to this point group have Cn axis and n σv containing Cn axis. Few examples of molecules belonging to Cnv point group are shown in fig.48: These are (i)BrMn(CO)5,(ii)NH3 ,(iii),PF4Cl (Cl axial) ,(iv) pyrrole

Notes

Molecular Symmetry


Fig.48: Examples of molecules belonging to Cnv point group

(iii) Cnh point group: Molecules belonging to this point group have Cn axis and σh symmetry elements in them. Some examples belonging to this point group are shown in fig.49 and these are(i)trans - butadiene(planar) (ii)B(OH)3 all atoms are in a plane (iii)Trans H2O2(iv)ethylenediamine as shown(v) 1,4-dibromo,2,5-dichlorobenzene(vi) trans-dichloroethene.

Fig.49: Examples of molecules belonging to Cnh point group

(iv) Sn point group: If in a molecule Cn axis exists check for additional S2n

coinciding with Cn axis,if it is there and no other symmetry elements are there then the molecule belongs to S2n point group. Possible examples are S2= i, S4, S6, S8. The S2=i= Ci. We can express this as follows. If in a molecule there is improper axis of even order (in practice S4, S6 and S8 are common)and no plane of symmetry or proper axis of rotation (except colinear one,whose presence is automatically required by improper axis) then the molecule will belong to Sn (n= 4.6.8) group. The point group Sn (n is even) exclusively consists of the operations generated by Sn axis. Thus the molecules which have Sn (n= even) are said to belong to Sn group. The order of the group equals to number of operations generated by Sn axis. Examples of molecules belonging to Sn point group are shown in figure 50.

Notes

Inorganic Chemistry


Fig 50: Examples of molecules belonging to Sn point group

Examples of type-III dihedral point groups (D - groups):In the molecule of type III dihedral point groups there are Cn axis and nC2 axes

perpendicular to Cn axis. Point groups belonging to this category are Dn , Dnh , Dnd . Let us discuss each of these separately.

(i) Dn point groups:Molecules belonging to this type only have Cn axis and nC2 axes perpendicular

to Cn axis and no other symmetry elements are present in them .Few examples of this type are shown in fig.4. These are (i) Cation [Co(en)3]

3+ (ii) Twisted biphenyl (D2) (iii)Twisted ethylene (D2) (iv) perchloro triphenylamine(D3) (v) Twistane (D2) (vi)Gauche form of ethane(D3).

Fig.51 Examples of molecules belonging to Dn point group

(ii) Dnh point groups:Molecules belonging to Dnh point group must have Cn axis, nC2 axes perpendicular

to Cn axis and σh.These symmetry elements generate all symmetry operations which belong to D nh point group. Presence of these symmetry elements automatically requires the existence of nσv. If the Cn axis is of even order its combination with σ h generates ‘i’. Hence the elements of the Dnh group are: E,C2, nC2, σh, i, Sn. If Cn axis is of odd order, then the elements of the group are; E, C2, nC2 ,σh, S2n , nσv. The order of the group is 4n. Some examples of Dnh point group are shown in fig.52 These are:

(i)Ethylene,(ii)Naphthalene,(iii)[PtCl4]

2-,(iv)Trans –[PtCl2Br2]

2-,(v)Cyclopentadienyl

Notes

Molecular Symmetry


Fig.52 Examples of molecules belonging to Dnh point group

(iii) Dnd point groups:Molecules belonging to Dnd point group must have Cn axis, nC2 axes perpendicular

to Cn axis and σd. The dihedral planes lie in Cn axis but bisect the angle made by perpendicular axes. The order of the group is 4n. But order changes with the order of the principal axis of rotation. For n=odd, symmetry elements are; Cn, nC2, nσd , S2n and for n =even the elements are; S2n, nC2, nσd,(symmetry operations of Cn are included in that of S2n ). Examples of molecules belonging to the Dnd point group are given in fig.3. These are: (i) Quashed Td Cs2[CuCl4] (ii) allene,(iii) staggered ethane(iv) S8 molecule.

Fig.53: Examples of molecules belonging to Dnd point group

Examples of type-IV Cubic group or special point groups:The point group discussed so far contained only one high order axis Cn.≥2. Now

we will deal with the molecules which have more than one Cn axis of higher order (n>2). The set of symmetry elements that are obtained from combination of several

Notes

Inorganic Chemistry


higher order axes correspond to the symmetry of regular polyhedra. All the faces of regular polyhedra have same regular polygon shapes. The angle and edges have same dimensions. There are five types regular polyhedra .The five polyhedra together with their characteristics are given in fig.54

Fig.54 Characteristic of five Platonic solidsPoint groups belonging to this category are, T, Th Td , O ,Oh, I, Ih, and so on. Of

these point groups only common point groups Td, Oh and Ih will be discussed.

(i) Td point group:In this point group molecules have four C3 axes along the body of diagonals of

a cube if tetrahedron is inscribed into the cube. The centre of the cube is the origin of coordinate system in which the coordinate axes are parallel to the sides of the cubes as shown below in fig.55

Fig.55 Tetrahedron inscribed in a cubeThe symmetry operations of Td point group are [ E,8C3 3C2,6S4,6σds’]. Three S4

coinciding with three Cartesian coordinates are taken as principal axes. The six 6σds’ bisect the angles between each pair of C3 axes. The order of the group is 24.Example of molecules belonging to Td point group is given in fig.6. These examples are

Fig.56 Examples of molecules belonging to Td point group

Notes

Molecular Symmetry


(ii) Oh point group:These point group’s molecules possess three mutually perpendicular C4 axis which

coincide with cartesian coordinates x,y,z axes and point of their intersection is the origin.. Its symmetry operations are [ E,8C3,6C4,3C2,6C2, 6σv’s,3σh’s ,i,,6S4 and 8S6]. The order of the group is 48. The octahedron inscribed in cube and some symmetry elements/symmetry operations are shown in figure.7

Fig.57 Octahedron inscribed in a cubeExamples of molecules belonging to Oh point group are given in fig.58 and these

are;(i) Mn(CO))6,(ii) [PtCl6]2-,(iii)PF6 ( iv) SF6,(v) [AlF6]

3-.

Fig.58 Examples of molecules belonging to Oh point group

(iii) Ih point group:Molecules belonging to this point group are rare. Molecules with icosahedral (20

triangular faces) and dodecahedral (12 pentgonal faces) belong to Ih point group. The [B12H12]

2- anion is an example of regular icosahedral (fig.59). The symmetry operations in Ih point group are;[E,6C5

1,6C52,6C5

3, 6C54,10C3

1,10C32, 15 C2 ,i ,6S10,6S10

3,6S107,

6S109, 10S6,10S6

5,15 σ] and order of the group is 120, Some more examples for Ih are; dodecahedron, and fullerene(C60) .

Fig.59 Structure of [B12H1

2]2-belongs to Ih point group

4.15 Special point groups

(i) Linear molecules: All linear molecules belong to C∞v or D∞h point groups. Linear molecules with centre of inversion i.e. symmetric linear molecules such

Notes

Inorganic Chemistry


as H2, CO2, C2H2, N2 belong to D∞h point group . These have 2CΦ∞ axes and

∞ C2 axes ┴ to C∞ axes,i ∞ σv and 2SΦ∞ and non symmetric linear molecules

i.e. which lack centre of symmetry such as hetero nuclear diatomic molecules; HCl , CO, NO, HD ,HF, HCN, SiH, KH, HSe, etc belong to C∞v point group. These molecules have CΦ

∞, ∞ σv . Fig.10 gives some symmetrical figures which belong to D∞h point group.

Fig.60 Symmetrical figures belonging to D∞h point group

(ii) Kh group: This point group characterizes the sphere. This geometry is possessed by all free atoms. All possible symmetry elements belong to this point group. Free atoms, H, D, He, Cl., Ca, belong to this point group. Figure 61 shows symmetry of perfect sphere.

Fig.61 Symmetry of a perfect sphere

4.16 Classification of molecules in point groupsThe success of the quickest method of classification of molecules into point groups

depends upon one’s ability to recognize the symmetry elements present in the molecules. The skill can be developed by continuous practice. Before going through the scheme of classification it is very necessary that you are familiar with the shapes of molecules. Table.1 gives some of the shapes/geometries of the molecule according to VSEPR theory.

Notes

Molecular Symmetry


4.17 LET US SUM UPSymmetry is an important concept in chemistry. Firm understanding of this concept of the symmetry and group theory is extremely useful in dealing with molecular structures, stereochemistry, spectroscopy and reaction mechanisms. The concept of symmetry is important to almost every aspect of life in our universe. The “principle of symmetry” plays an important role in modern science. The idea of symmetry principle originally resulted from the study of geometrical forms and observations of natural objects. There

are countless examples of symmetry in nature. According to the formal definition of a

group, a group is a set /collection of elements, which are combined with certain operation *, such that:

1. The group contains an identity

2. The group contains inverses

3. The operation is associative

4. The group is closed under the operation or

In mathematical sense a group may be defined as, “collection/set of elements/numbers having certain properties in common i.e. these elements are bound by certain conditions, known as group properties/postulates”. The elements do not need to have some physical significance. My emphasis will be mainly on ‘‘symmetry operations as the collection of elements of the group”

Subgroup is defined as a part of the main group which satisfies all the properties of

Notes

Inorganic Chemistry


the group ie within the main group of order’ h’ there may be smaller groups of lower

order then ‘h’.

(i) Sub group always contains identity element E.

(ii) E itself constitutes a group of order one and it is trival.

(iii) The sub group of order two also contains identity element E .

(iv) Groups of order two can be E,A ; E,B ; E,C etc in the main group A,B,C,E.

(v) Sub groups always follow the group conditions of the main group.

(vi) Order of the sub group is an integral factor of the order of the main group

(vii) If the order of sub group is’ g’ then h/g is an integer and not in fraction.

(viii) If the order of the main group is six, then sub groups can have orders 1,2,3

only(6/6=1,6/2=3,6/3=2) only and 6/4 or 6/5 are fractions.


1. What do you mean by the concept of symmetry?

2. Define symmetry elements

3. Discuss the types of symmetry elements

4. Define symmetry operations

5. Explain symmetry elements in molecules

6. What do you mean by symmetry operations on molecules?

7. Discuss the characteristics of a group





Notes

Group Theory


UNIT–5

Group Theory

(Structure) 5.1 Objectives 5.2 Introduction 5.3 Definition of group

5.4 Group Multiplication Table

5.5 Subgroup

5.6 Classes

5.7 Representation of group

5.8 The Great Orthogonality Theorem

5.9 Character Table 5.10 Some Examples of Character Tables

5.11 Review questions 5.12 Let us Sum up


zz Discuss basic definition of group theoryzz Explain characteristics of subclasses;

zz Describe Boran and carboranes.

5.2 Introduction

The experimental chemist in his daily work and thought is concerned with observing and, to as great an extent as possible, understanding and interpreting his observations on the nature of chemical compounds. Today, chemistry is a vast subject. In order to do thorough and productive experimental work, one must know so much descriptive chemistry and so much about experimental techniques that there is not time to be also a master of chemical theory. Theoretical work of profound and creative nature, which requires a vast training in mathematics and physics, is now the particular province of specialists. And yet, if one is to do more than merely perform experiments, one must have some theoretical framework for thought. In order to formulate experiments imaginatively

5.13 Further readings

Notes

Inorganic Chemistry


and interpret them correctly, an understanding of the ideas provided by theory as to the behavior of molecules and other arrays of atoms is essential. The problem in educating student chemists-and in educating ourselves is to decide what kind of theory and how much of it is desirable. In other words, to what extent can the experimentalist afford to spend time on theoretical studies and at what point should he say, "Beyond this I have not the time or the inclination to go?" The, answer to this question must of course vary with the special field of experimental work and with the individual. In some areas fairly advanced theory is indispensable. In others relatively little is useful. For the most part, however, it seems fair to say that molecular quantum mechanics, that is, the theory of chemical bonding and molecular dynamics, is of general importance.

The number and kinds of energy levels that an atom or molecule may have are rigorously and precisely determined by the symmetry of the molecule or the environment of the atom. Thus, from symmetry considerations alone, we can always tell what the qualitative features of a problem must be. We shall know, without any quantitative calculations whatever, how many energy states there are and what interactions and transitions between them may be occur. In other words, symmetry considerations alone can give us a complete and rigorous answer to the question “What is possible and what is completely impossible?" Symmetry consideration alone can not , however, tell us how likely it is that the possible things will actually take place. Symmetry can tell us that in principle two Stats of the system must differ in their energy, but only by computation or measurement can we determine how great the difference will be again symmetry can tell us that only certain absorption bands in the electronic or vibration spectrum of a molecule may occur.

By using symmetry considerations alone we may predict the number of vibrational fundamentals, their activities in the infrared and Raman spectra, and the way in which the various bonds and inter bond angles contribute to them for any molecule

possessing some symmetry. The actual magnitudes of the frequencies depend on the interatomic forces in the molecule, and these cannot be predicted from symmetry properties.

In recent years, the use of X-ray crystallography by chemists has improved enormously. No chemist is fully equipped to do research (or read the literature critically) in any field dealing with crystalline compounds, without a general idea of the symmetry conditions that govern the formation of crystalline solids.

An understanding of this approach requires only a superficial knowledge of quantum mechanics.

Notes

Group Theory


In a number of applications of symmetry methods, however, it would be artificial and foolishness to exclude religiously all quantitative considerations. Thus, it is natural to go a few steps beyond the procedure for determining the symmetries of the possible molecular orbitals and explain how the requisite linear combinations of atomic orbitals may be written down and how their energies may be estimated. It has also appeared desirable to introduce some quantitative ideas into the treatment of ligand field theory.

5.3 Definition of group

A group C is a collection of elements which satisfy the following conditions.

1. For any two elements a and b in the group the product a x b is also an element of the group.

In order for this condition to have meaning, we must, of course, have agreed on what we mean by the terms "multiply " and "product. Perhaps we might say combination instead of product in order to avoid pointless incorrect connotations. Let us not yet force ourselves to any particular law of combination but merely say that, if A and B are two elements of a group, we point out that we are combining them by simply writing AB or BA. Now instantaneously the question arises if it makes any difference whether we write AB or BA. In ordinary algebra it does not, and we say that multiplication is commutative, that is xy = yx, or 3 x 6 = 6 x 3. In group theory the commutative law does not in general hold Thus AB may give C and BA may give D where C and D are two more elements in the group. There are some groups, however, in which combination is commutative such groups are called Abelian groups. Because of the truth that multiplication is not in general commutative, it is sometimes convenient to have a means of stating whether an element B is to be multiplied by A in the sense AB or BA. In the first case we can say that B is left-multiplied by A, and in he second case that B is right-multiplied by A

2. There exists an unit element 1 in the group such that 1 x a = a x 1 = a for every element a or in other words one element in a group must commute with all others and leave them unchanged. It is customary to designate this element with the letter E and it is usually called the identity element.

3. There must be an inverse (or reciprocal) element a-1 of each element a such that

a x a-1 = a-1 x a = 5.

or every element must have a reciprocal,which is also an element of the group.

Notes

Inorganic Chemistry


The element R is the reciprocal the element S if RS = SR =E where E is the identity. Obviously, if R is the reciprocal of S, then S is the reciprocal of R. Also, E is its own reciprocal.

At this point we shall prove a small theorem concerning reciprocals which the associative law of multiplication must hold. This is expressed in the following way

A (BC) = (AB)C

In simple words, we may combine B with C in the order BC and then combine this product, S, with A in the order AS, or we may combine A with B in the order AB, obtaining a product, say R, which we then combine with C in the order RC and get the same final product either way. In general, of course, the associative property must hold for the continued product of any number of elements.

Some Examples of Groups

The significance of the above defining rules, we may consider an infinite group and then some finite groups.

As an infinite group we may take all of the integers, both positive, negative, and zero. If we take as our law of combination the ordinary algebraic process of addition, then rule 1 is satisfied. Clearly, any integer may be obtained by adding two others. Note that we have an Abelian group since the order of addition is immaterial. The identity of the group is 0, since 0 + n = n + 0 = n. Also, the associative law of combination holds, since, i.e.,

[(+3) + (-7)] + (+1043) = (+3) + [(-7) + (+1043)]

The reciprocal of any element, n, is ( - n), since ( + n) + ( - n) = 0.

5.4 Group Multiplication Tables

If we have a absolute and non redundant list of the elements of a finite group and we know what all of the possible products (there are h2) are, then the group is absolutely and uniquely defined at least in an abstract sense. The foregoing information can be presented most conveniently in the form of the group multiplication table. This consists of h rows and h columns. Each column is labeled with a group element, and so is each row. The entry in the table under a given column and along a given row is the product of the elements which head that column and that row.

Because multiplication is in general not commutative, we must have an agreed upon and consistent rule for the order of multiplication. Arbitrarily, we shall take factors in the order - (column element) x (row element) thus at the intersection of the column labeled by X and the row labeled by Y we find the element, which is the product XY.

Notes

Group Theory


We now prove an important theorem about group multiplication tables, called the rearrangement theorem.

Dihedral group

The symmetry of a square is the 4-fold dihedral D4 symmetry. To understand a discrete group, we first identify how many elements are in the group which is called the order of the group h. It is obvious that any integer multiples of π/2 rotations would leave the square invariant. There are four in equivalent rotations of this kind. A mirror inversion of the square is also an invariant transformation. Applying inversion to the previous rotations, we get another four rotations Thus, the elements of D4 group are:

D4={I,R,R2,R3,P,PR,PR2,PR3} (1)

That is to say the order of the group

h = 8.

General not commutative, we must have an agreed upon and consistent rule for the order of multiplication. Thus at the intersection of the column labeled by X and the row labeled by Y we find the element, which is the product XY.

We now prove an important theorem about group multiplication tables, called the rearrangement theorem.

Each row and each column in the group multiplication table lists each of the group elements once. Form this it follows that no two rows may be identical nor may any two columns be identical. This each row and each column is a rearranged list of the group elements.

PROOF. Consider the group consist of the h elements E, A2 , A3, ... , Ah. The elements in a given row, say the nth row, are

EAn,A2An,…………….AnAn…………..,AhAn

Since no two group elements, Ai and Aj for instance are the same, no two, products, AiAn and AjAn, can be the same. The h entries in the nth row are all diverse. Since there are only h group elements, each of them must be there once and only once. The argument can obviously be adapted to the columns.

Groups of orders 1, 2, and 3

Consider now systematically examine the possible abstract groups of low order, using their multiplication tables to define them. There is, of course, formally a group of order 1, which consists of the identity element alone. There is only one possible group of order

2 It has the following multiplication table and will be designated G2

Notes

Inorganic Chemistry


For a group of order 3, the multiplication table will have to be, in part, as follows:

There is then only one way to complete the table, either AA = B or AA = E. lf AA = E then BB = E and we would augment the table to give

But then we can get no further, since we would nave to accept BA = A and AB = A in order to complete the last column and the last row, respectively, thus repeating A in both the second column and the second row. The alternative, AA = B, leads unambiguously to the following table:

Cyclic Groups

If G3 is the simplest, nontrivial member of an important set of groups i.e. the cyclic Groups. We note that AA = B while AB (=AAA) = E. Thus we can believe that entire group to be generated by taking the element A and its powers, A2(= B) and A3( = E) In

Notes

Group Theory


general, the cyclic group of order h is defined as an element X and all of its powers Up to Xh = E. We shall presently examine several other cyclic groups. An important property of cyclic, groups is that they are Abelian, that is all multiplications are commutative. This must be so, Since the various group elements are all of the form Xn, Xm, and so on, and, clearly XnXm, Xm Xn, for all m and n.

Groups of Order 4

To carry on, we ask how many groups of order 4 there are and what their multiplication table(s) will be. Obviously, there will be a cyclic group of order 4. Let us employ the relations.

From this we find that the multiplication table, in the usual format, is as follows:

We note that for G4(1) only one element, namely B, is its own Inver be Suppose instead,

we assume that each of two elements , A and B is its own inverse. We shall then have no choice but to also make C its own inverse, since each of the four E's in the table must lie in a different row and column, Thus, we would obtain

A moment's consideration will show that there is only one way to complete this table: ,

Notes

Inorganic Chemistry


It is also clear that there are no other possibilities.Thus, there are two groups of order 4, namely G4

(1) and G4 (2) which may be considered to be defined by their multiplication

tables.Groups of Orders 5 and 6

Similarly, a systematic examination on the possibilities for groups of if we make up to table in which only element (other than E) is its own inverse and let that element be A or C instead of B as in the G4

(1) table given, we are not inventing a different G4 we are only permuting the arbitrary symbols for the group elements. the multiplication table for one of the groups of order 6 is given below:

5.5 Subgroup

Inspection of the multiplication table for the group G6(1) will show that within this group

of order 6 there are smaller groups. The identity E in itself is a group of order 5. This will, of course, be true in any group and is trivial the groups of order 2, namely E,A; B;E,C and the group of order 3namely E, D, F. The last should be known also as the cyclic group G3, since D2 F, D3 DF = FD = E. But to go back to the main point smaller groups that may be found within a larger group are called subgroups. There are, of course, groups that have no Subgroups other than the trivial one of E itself.

Now consider, whether there are any restrictions on the nature of subgroups, restrictions that are logical consequences of the general definition of a group and not of any additional or special characteristics of a particular group. We may note that the orders of the group C6

(1) and its subgroups are 6 and 1, 2, 3; in short, the orders of the subgroups are all factors of the order of the main group. We shall now prove the following theorem:

There order of any subgroup g of a group of order h must be a divisor of h. In other words, h/ g = k where k is an integer.

PROOF. Let us consider that the set of g elements, A1, A2 A3 ... , Ag, forms a subgroup. Now assume another element B in the group which is not a member of this subgroup

Notes

Group Theory


and form all of the g products: BA1 BA2 ... , BAg. No one of these products can be in the subgroup. If, for example,

BA2 = A4then, if we take the reciprocal of A2, perhaps A5, and right-multiply the above equality, we obtain

But this contradicts our assumption that B is not a member of the subgroup A1, A2,

... , Ag, since A4A5 can only be one of the Ai Hence, if all the products BAi in the

large group in addition to the Ai themselves, there are at least 2 members of the group. If h > 2g, we can choose still another element of the group, namely C, which is neither one of the Ai nor one of the BAi and on multiplying the Ai by C we will obtain g more elements, all members of the main group, but none members of the Ai or of the BAi sets.

Thus we now know that h must be at least equal to 3g. However, we must reach the point where there are no more elements by which we can multiply the Ai that are not among the sets Ai, BAi, CAi, and so forth, already obtained. Suppose after having found k such elements, we reach the point where there are no more. Then h = kg, where K is an integer, and h/g =k, which is what we set out to prove.

Even though we have shown that the order of any subgroup, g, must be a divisor of h, we have not proved the converse, namely, that there are subgroups of all orders that are divisors of h, and, indeed, this is not in general true.

Furthermore, as our illustrative group proves, there can be more than one subgroup of a given order.

5.6 Classes

We have studied that in a given group it may be possible to select various smaller sets of elements, each such set including E, however, which are in themselves groups. There is another method in which the elements of a group may be separated into smaller sets, and such sets are called classes. Before defining a class we must consider an operation known as similarity transformation. If A and X are two elements of a group, then X-lAX will be equal to some elements of the group, say B. We have

B = X-l AX

We express this relation in words by saying that B is the similarity transform of A by X. We also say that A and B are conjugate. The following properties of conjugate elements are important.

Notes

Inorganic Chemistry


(i) Every element is conjugate with itself. This means that if we choose any particular element A it must be possible find at least one element X such that

A = X-lAX

If we left-multiply by A-1 we obtain

A-1A = E = A-1X-lAX = (XA)-1(AX)

Which can hold only if A and X commute. Thus the element X may always be E, and it may be any other element that commutes with the chosen element, A.

(ii) If A is conjugate with B, then B is conjugate with A. This means that if

A = X-1 BX

then there must be some element Y in the group such that

B = Y-1 AY

That this must be so is easily proved by carrying out appropriate multiplications, namely,

XAX-1 = XX-l BXX-l = B

Thus, if Y = X-1 (and thus also Y-l = X), we have

B = Y-1AY

and this must be possible, since any element, say X, must have an inverse, say Y.

(iii) If A is conjugate with Band C, then Band C are conjugate with each other. The proof of this is easy to work out from the foregoing discussion and is left as an exercise.

We are now able to define a class of group. A complete set of elements that are conjugate to one another is called a class of the group.

In order to calculate the classes within any particular group we can begin with one element and work out all of its transforms, using all the elements in the group, including itself, then take a second element, which is not one of those found to be conjugate to the first, and determine all its transforms, and so on until all elements in the group have been placed in one class or another.

Let us illustrate this procedure with the group G6. All of the results given below may be verified by using the multiplication table. Let us start with E.

Notes

Group Theory


Thus E must constitute by itself a class, of order 1, since it is not conjugate with any other element. This will, of course, be true in any group. To continue,

Thus the elements A, B, and C are all conjugate and are therefore members of the same class. Continuing we have

It will also be found that every transform of F is either D or F. Hence, D and F constitute a class of order 2.

It will be noted that the classes have orders 1, 2, and 3, which are all factors of the group order, 6. It can be proved, by a method similar to that used in connection with the orders of subgroups, that the following theorem is true: The orders of all classes must be integral factors of the order of the group.

5.7 Representation of groupRepresentation of a group of the type we shall be interested in may be defined as a set of matrices, each corresponding to a single operation in the group, that can be combined among themselves in a manner parallel to the way in which the group elements-in this case, the symmetry operations combine. Thus, if two symmetry operations in a symmetry group, say C2 and σ, combine to give a product C2', then the matrices corresponding to C2 and a must multiply together to give the matrix corresponding to C. But we have already seen that, if the matrices corresponding to all of the operations have been correctly written down, they will naturally have this property.

As an example, let us work out a representation of the group C2, which group consists of the operations E, C2, σv, σv’, a. Let us say that the C2 axis coincides with the z axis

Notes

Inorganic Chemistry


of a Cartesian coordinate system, and let σv, be the xz plane and σv’ be the yz plane. The matrices representing the transformations effected on a general point can easily be seen to be as follows:

now the group multiplication table is:

It can easily be shown that the matrices multiply together in the same fashion. For example,

Again, each element in the group C2 is its own inverse, so the same must be true of the matrices. This is easily shown to be so; for example,

A question that naturally arises at this point is: How many representations can be found for any particular group, say C2v , to continue with that as an example? The answer is: A very large number, limited only by our ingenuity in devising ways to generate them. There are first some very simple ones, obtained by assigning 1 or -1 to each operation, namely,

Notes

Group Theory


Then there are many representations of high order. For example, if we were to assign three small unit vectors directed along the x, y, and z axes to each of the atoms in H20 and write down matrices representing the changes and interchanges of these upon applying the operations, a set of four 9 x 9 matrices constituting a representation of the group would be obtained. Using CH2Cl2 in the same way, we could obtain a representation consisting of 15 x 15 matrices. However, for any group only a limited number of representation are of fundamental significance and we shall now discuss the origin and properties of these.

Suppose that we have a set of matrices, E, A, B, C,…, which form a representation of group. If we make the same similarity transformation on each matrix, we obtain a new set of matrices, namly,

It is easy to prove that the new set of matrices is also a representation of the group. Suppose that

AB = D

then

Clearly, all products in the set of matrices E1 , A1, B1 will run parallel to those in the representation E , A, B hence the primed set also constitutes a representation.

Let us now suppose that, when the matrix A is transformed to A' using D or some other matrix, we find A’ to be a block-factored matrix, namely,

’

for example. If now each of the matrices A’, B’ , E’ , and so forth is blocked out in the same way, then corresponding blocks of each matrix can be multiplied together separately. Thus we can write such equation as:

Notes

Inorganic Chemistry


Therefore the various sets of matrices

are in themselves representations of the group. We then call the set of matrices, E, A, B,E, D, ... , a reducible representation, because it is possible using some matrix, D in this case, to transform each matrix in the set into a new one so that all of the new ones can be taken apart in the same way to give two or more representations of smaller dimension. It if is not possible to find a similarity transformation which will reduce all of the matrices of a given representation in the above manner, the representation is said to be irreducible. It is the irreducible representations of a group that are of fundamental importance, and their main properties will now be described.

5.8 The Great Orthogonality Theorem

All of the properties of group representations and their characters, which are important in dealing with problems in valence theory and molecular dynamics, can be derived from one basic theorem concerning the elements of the matrices which constitute the irreducible representations of a group. In order to state this theorem, which we shall do without proof, some notation must be introduced. The order of a group will, as before, be denoted by h. The dimension of the ith representation, which is the order of each of the matrices which constitute it, will be denoted by li. The various operations in the group will be given the generic symbol R. The element in the mth row and the nth column of the matrix corresponding to an operation R in the ith irreducible representation will be denoted Γ(R)mn. Finally, it is necessary to take the complex conjugate (denoted by *) of one factor on the left- hand side whenever imaginary or complex numbers are involved.

The great orthogonality theorem may then be stated as follows:

……(1)

This means that in the set of metrices constituting any one irreducible representation any set of corresponding metrix elements, one from each metrix, behaves as the component of a vector in h-dimentional space such that all these vectors are matually orthogonal, and each is normalized so that the square of its length equals h/li. This

Notes

Group Theory


interpretation of (1) will perhaps be mare obvious if we take (1) apart into three simpler equations, each of which is contained within it. We shall omit the explicit designation of complex conjugates for simplicity, but it should be remembered that they must be used when complex numbers are involved. The three simpler equations are as follows:

………………………..…..(2)

ΣR Γi(R)mn Γi(R)m’n’ = 0 if m ≠ m’ and/or n ≠ n’…….........(3)

ΣR Γi(R)mn Γi(R)mn = h/li …………………………………………..(4)

Thus, if the vectors differ by being chosen from matrices of different representations, they are orthogonal (2). If they are chosen from the same representation but from different sets of elements in the matrices of this representation, they are orthogonal (3). Finally, (4) expresses the fact that the square of the lenth of any such vector equals h/li.

Five important rules:

1. The sum of the squares of the dimensions of the irreducible representations of a group is equal to the order of the group, that is,

2. The sum of the squares of the characters in any irreducible representation equals h, that is,

3. The vectors whose components are the characters of two different irreducible representations are orthogonal, that is,

4. In a given representation (reducible or irreducible) the characters of all matrices belonging to operations in the same class are identical.

5. The number of irreducible representations of a group is equal to the number of classes in the group.

Illustration of five rules:

Let us now consider the irreducible representations of several typical groups to see how these rules apply. The group C2v consists of four elements, and each is in a separate class. Hence (rule 5) there are four irreducible representations for this group. But it is also required (rule 1) that the sum of the squares of the dimensions of these representations equal h. Thus we are looking for a set of four positive integers, l1, l2, l3, and l4, which satisfy the relation:

Notes

Inorganic Chemistry


Clearly the only solution is

Thus the group C2v has four one-dimensional irreducible representations.

We can actually work out the characters of these four irreducible representations- which are in this case the representations themselves because the dimensions are 1- on the basis of the vector properties of the representations and the rules derived above. One suitable vector in 4-space which has a component of 1 corresponding to E will obviously be

for

thus satisfying rule 2. Now all other representations will have to be such that

which can be true only if each X;(R) = ± 5. Moreover, in order for each of the other representations to be orthogonal to r1 (rule 3 and 4.3-7), there will have to be two + l's and two - 1 's. Thus

(1)( -1) + (1)( -1) + (1)(1) + (1)(1) = 0

Therefore we will have

All of these representations are also orthogonal to one another. For example, taking Γ2

and Γ4, we have

(1)(1) + (-1)(1) + t-1)(-1) + (i)(-l) =0

and so on. These are then the four irreducible representations of the group C2v.

As another example of the working of the rules, let us consider the group C3v. This

Notes

Group Theory


consists of the following elements, listed by classes:

We therefore know at once that there are three irreducible representations. If we denote their dimensions by I}) 12, and 13, we have (rule 1)

The only values of the l, that will satisfy this requirement are 1, 1, and 2. Now once again, and always in any group, there will be a one-dimensional representation whose characters are all equal to 5. Thus we have

We now look for a second vector in 6-space all of whose components are equal to

± 1 which is orthogonal to T l : The components of such a vector must consist of three + l 's and three -1 's. Since X(E) must always be positive and since all elements in the same class must have representations with the same character, the only possibility here is

Now our third representation will be of dimension 2. Hence X3(E) = 2.

In order to find out the values of X3 ( C3) and X3 (σ v) we make use of the orthogonality relationships :

Solving these, we obtain

Notes

Inorganic Chemistry


and

Thus the complete set of characters of the irreducible representations is

We may note that there is still a check on the correctness of r 3: the square of the length of the vector it defines should be equal to h (rule 2), and we see tha t this is so:

Important Practical Relationship

There is a relationship between any reducible representation of a group and the irreducible representations of that group. In terms of practical application of group theory to molecular problems, this relationship is of pivotal importance. We know already that for any reducible representation it is possible to find some similarity transformation which will reduce each matrix to one consisting of blocks along the diagonal, each of which belongs to an irreducible representation of the group. We also know that the character of a matrix is not changed by any similarity transformation. Thus χ(R) , the character of the matrix corresponding to operation in a reducible representation, may be expressed as follows:

……………….(5)

where aj represents the number of times the block constituting the jth irreducible representation will appear along the diagonal when the reducible representation is

completely reduced by the necessary similarity transformation. Now we need not to bother about the difficult question of how to find out what matrix is required to reduce completely the reducible representation in order to find the values of the aj. We can obtain the required relationship by working only with the characters of all representations in the following way. We multiply each side of (5) by χi (R) and then sum each side over all operations, namely,

Notes

Group Theory


Now for each of the terms in the sum over j, we have

since the sets of characters Xj (R) and X i (R) define orthogonal vectors, the squares of whose lengths equal h. Thus, in summing over all j, only the sum over R in which i = j can survive, and in that case we have

which we rearrange to read

Thus we have an explicit expression for the number of times the ith irreducible representation occurs in a reducible representation where we know only the characters of each representation.

Illustrative Examples

Let us take an example. For the group C3v we give below the characters of the irreducible representations, Γ1 , Γ2 , and Γ 3 and the characters of two reducible

representations, Γa and Γ b :

wefind for Γa

Notes

Inorganic Chemistry


and for Γb

The numbers in italics are the numbers of elements in each class. The results obtained above will be found to satisfy, as of course they must. for Γa we have

and for Γb

5.9 Character Table

In group theory, a branch of abstract algebra, a character table is a two-dimensional

table whose rows correspond to irreducible group representations, and whose columns

correspond to conjugacy classes of group elements. The entries consist of characters,

the trace of the matrices representing group elements of the column's class in the given

row's group representation.

In chemistry, crystallography, and spectroscopy, character tables of point groups are

used to classify e.g. molecular vibrations according to their symmetry, and to predict

whether a transition between two states is forbidden for symmetry

reasons.Throughout all of our applications of group theory to molecular symmetry we

Notes

Group Theory


will utilize devices called character tables. In this section the meaning and indicate the

source of the information given in these tables has been discussed. For this purpose we

shall examine in detail a representative character table, one for the group C3v, reproduced

below. The fort main areas of the table have been assigned Roman numerals for reference

in the following dicussion.

In the top row are these entries: In the upper left corner is the Schonflies symbol for the

group. Then, along the top row of the main body of the table, are listed the elements of

the group, gathered into classes.

Area I. In area I of the table are the characters of the irreducible representations of

the group. These have been fully discussed in preceding sections of this chapter and

require no additional comment here.

Area II. We have previously designated the ith representation, or its set of

characters, by the symbol F, in a fairly arbitrary way. Although this practice is still to be

found in some places and is common in older literature, most books and papers-in fact,

virtually all those by English-speaking authors-now use the kind of symbols found in

the C3, table above and all tables in Appendix II. This nomenclature was proposed by

R. S. Mulliken, and the symbols are normally called Mulliken symbols. Their meanings

are as follows:

1. All one-dimensional representations are designated either A or B; two- dimensional representations are designated E; three-dimensional species are designated T (or sometimes F).

2. One-dimensional representations that are symmetric with respect to rotation by 2π/n about the principal Cn axis [symmetric meaning: χ(Cn) = 1] are designated A, while those antisymmetric in this respect [χ(Cn) = -1] are designated B.

3. Subscripts I and 2 are usually attached to A’s and B’s to designate those which are, respectively, symmetric and antisymmetric with respect to a C2 perpendicular to the principal axis or, if such a C2 axis is lacking, to a vertical plane of symmetry.

Notes

Inorganic Chemistry


4. Primes and double primes are attached to all letters, when appropriate, to indicate those which are, respectively, symmetric and antisymmetric with respect to σv.

5. In groups with a center of inversion, the subscript g (from the German gerade,

meaning even) is attached to symbols for representations which are symmetric

with respect to inversion and the subscript u (from the German ungerade,

meaning uneven) is used for those which are antisymmetric to inversion.

6. The use of numerical subscripts for E’s and T’s also follows certain rules, but these cannot be easily stated precisely without some mathematical development. It will be satisfactory here to regard them as arbitrary labels.

Area III. In area III we will always find six symbols: x, y, z, Rx, Ry, Rz. The first three

represent the coordinates x, y, and z, while the R’s stand for rotations about the axes

specified in the subscripts. We shall now show in an illustrative but by no means thorough

way why these symbols are assigned to certain representations in the group C3v, and this

should suffice to indicate the basis for the assignments in other groups.

Any set of algebraic function or vectors may serve as the basis for a representation of

a group. In order to use them for a basis, we consider them to be the components of a

vector and then determine the matrices which show how that vector is transformed by

each symmetry operation. The resulting matrices, naturally, constitute a representation

of the group.

Area IV. In this part of the table are listed all of the squares and binary products of

coordinates according to their transformation properties. These results are quite easy to

work out using the same procedure as for x, y, and z, except that the amount of algebra

generally increases, though not always. For example, the pair of functions xz and

yz must have the same transformation properties as the pair x, y, since z goes into itself

under all symmetry operations in the group. Accordingly, (xz, yz) are found opposite

the E representation.

Properties of characters of representations:

Complex conjugation acts on the character table: since the complex conjugate of a

representation is again a representation, the same is true for characters, and thus a

character that takes on non-trivial complex values has a conjugate character.

Some properties of the group G can be deduced from its character table:

The order of G is given by the sum of the squares of the entries of the first column (the

degrees of the irreducible characters). Moreover , the sum of the squares of the absolute

values of the entries in any column gives the order of the centralizer of an element of

Notes

Group Theory


the corresponding conjugacy class. All normal subgroups of G (and thus whether or not

G is simple) can be recognised from its character table. The kernel of a character χ is

the set of elements g in G for which χ(g) = χ(1); this is a normal subgroup of G. Each

and every normal subgroup of G is the intersection of the kernels of some of

the irreducible characters of G.

The derived subgroup of G is the intersection of the kernels of the linear characters of

G. In particular, G is Abelian if and only if all its irreducible characters are linear.

It follows that the prime divisors of the orders of the elements of each conjugacy class

of a finite group can be deduced from its character table (an observation of Graham

Higman).

The character table does not in general determine the group up to isomorphism: e.g., the

quaternion group Q and the dihedral group of 8 elements (D4) have the same character

table. Brauer asked whether the character table, together with the knowledge of how

the powers of elements of its conjugacy classes are distributed, determines a finite

group up to isomorphism. The linear characters form a character group, which has

important number theoretic connections.

5.10 Some examples of character tables

1. C2 character table.



Notes

Inorganic Chemistry


4. C2v character table.




Notes

Group Theory



9. C∞v character table.

10. D2 character table.

11. D3 character table.

12. D2h character table.

Notes

Inorganic Chemistry





5.11 LET US SUM UP

Today, chemistry is a vast subject. In order to do through and productive experimental

work, one must know so much descriptive chemistry and so much about experimental

techniques that there is not time to be also a master of chemical theory. Theoretical

work of profound and creative nature, which requires a vast training in mathematics

and physics, is now the particular province of specialists. And yet, if one is to do

more than merely perform experiments, one must have some theoretical framework for

Notes

Group Theory


thought. In order to formulate experiments imaginatively and interpret them correctly, an

understanding of the ideas provided by theory as to the behavior of molecules and other

arrays of atoms is essential. The problem in educating student chemists-and in educating

ourselves is to decide what kind of theory and how much of it is desirable. In other

words, to what extent can the experimentalist afford to spend time on theoretical studies

and at what point should he say, "Beyond this I have not the time or the inclination to

go?" The, answer to this question must of course vary with the special field of experimental

work and with the individual. In some areas fairly advanced theory is indispensable.

In others relatively little is useful. For the most part, however, it seems fair to say that

molecular quantum mechanics, that is, the theory of chemical bonding and molecular

dynamics, is of general importance. A group is a collection of elements which satisfy

the some conditions.

If we have a absolute and non redundant list of the elements of a finite group and we

know what all of the possible products (there are h2) are, then the group is absolutely

and uniquely defined at least in an abstract sense. The foregoing information can be

presented most conveniently in the form of the group multiplication table.

The main point smaller groups that may be found within a larger group are called

subgroups. We have studied that in a given group it may be possible to select various

smaller sets of elements, each such set including E, however, which are in themselves

groups. There is another method in which the elements of a group may be

separated into smaller sets, and such sets are called classes.

Inspection of the multiplication table for the group G6(1) will show that within this group

of order 6 there are smaller groups. The identity E in itself is a group of order 5. This will, of course, be true in any group and is trivial the groups of order 2, namely E,A; B;E,C and the group of order 3namely E, D, F. The last should be known also as the cyclic group G3, since D2 F, D3 DF = FD = E. But to go back to the main point smaller groups that may be found within a larger group are called subgroups. There are, of course, groups that have no Subgroups other than the trivial one of E itself.Representation of a group of the type we shall be interested in may be defined as a set of matrices, each corresponding to a single operation in the group, that can be combined among themselves in a manner parallel to the way in which the group elements-in this case, the symmetry operations combine. Thus, if two symmetry operations in a symmetry group, say C2 and σ, combine to give a product C2', then the matrices corresponding to C2 and a must multiply together to give the matrix corresponding to C. But we have already seen that, if the matrices corresponding to all of the operations have been

Notes

Inorganic Chemistry


correctly written down, they will naturally have this property.

As an example, let us work out a representation of the group C2, which group consists of the operations E, C2, σv, σv’, a. Let us say that the C2 axis coincides with the z axis

of a Cartesian coordinate system, and let σv, be the xz plane and σv’ be the yz plane.


1. What Discuss symmetry operations.

2. What do you mean by representation of group? discuss it in detailed.

3. Write detailed notes on the great orthogonality theorem.

4. What do you mean by character table? Discuss various areas of character table.

5. Write short note on properties of character table.

6. Draw the character table for the C2v and C3v point group.





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