9 Coordination

8/12/2019 9 Coordination

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Lecture 9:Lecture 9:Coordination andCoordination and PaulingPaulingss RulesRules

To understand theTo understand the atomic structure of mineralsatomic structure of minerals, we typically think in, we typically think interms ofterms of sphericalsphericalcationscations and anions held together predominantlyand anions held together predominantlybyby ionic bondsionic bonds

One of the properties of these ions that we would like to know is theirOne of the properties of these ions that we would like to know is their

RADIUSRADIUS, I.e., their size, I.e., their size

This sounds easy, but remember, ions are really formed from nucleiThis sounds easy, but remember, ions are really formed from nucleiwith probability density clouds of electrons moving around them.with probability density clouds of electrons moving around them.Thus, we refer to:Thus, we refer to:

Effective Electrostatic Radii:Effective Electrostatic Radii: imaginaryimaginaryfixed radii of an ionfixed radii of an ion

Why imaginary?Why imaginary?Because electrons are inBecause electrons are inprobability densityprobability densitycloudscloudsand becauseand because radiiradiiactuallyactually changechangesomewhatsomewhat depending ondepending onbonding environmentbonding environment


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Given that the effective electrostatic radii is imaginary, howGiven that the effective electrostatic radii is imaginary, howdo wedo we measuremeasureit???it???

!!Look atLook at anion-cationanion-cation bond lengthsbond lengthsin structures with predominantlyin structures with predominantlyionic bondingionic bonding

!!Once you know theOnce you know the total bond distancetotal bond distanceandand one ion radiusone ion radius, you can, you can

calculatecalculatethethe otherother

!!Shannon and Prewitt (1969); Shannon (1976)Shannon and Prewitt (1969); Shannon (1976) used many oxideused many oxidestructuresstructuresto regressto regress averageaverageeffective electrostatic radii ofeffective electrostatic radii of cationscations

Example:Example:

PericlasePericlase ((MgOMgO))

Mg-O length 2.11 according to XRDMg-O length 2.11 according to XRD

OO--22effeff= 1.32 (by definition)= 1.32 (by definition)

So, MgSo, Mg++22effeff = 2.11 - 1.32 = 0.79= 2.11 - 1.32 = 0.79


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Effective Electrostatic RadiiEffective Electrostatic Radii

CationCation Radii ?? Elemental Radii (> or


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Some General Rules Regarding Effective Electrostatic Radii:Some General Rules Regarding Effective Electrostatic Radii:

!!LargerLargerwith Z within groups (going down columns)with Z within groups (going down columns)

!! Larger with increasing numberLarger with increasing number

Note: AnNote: An exceptionexceptionis the so-calledis the so-called Lanthanide ContractionLanthanide Contraction;;trivalenttrivalentlanthanides (Lalanthanides (La+3+3to Luto Lu+3+3) decrease in radius with increasing Z) decrease in radius with increasing Z

Why? Inner electronWhy? Inner electron orbitalsorbitals build before outerbuild before outer orbitalsorbitals are added;are added;higher nuclear charge combined with relatively weak shielding drawshigher nuclear charge combined with relatively weak shielding drawselectrons into the nucleuselectrons into the nucleus

!! ForFor cationscations withwithsame electronic structure,same electronic structure, radiiradiidecrease withdecrease withincreasing chargeincreasing charge

i.e.i.e. RReffeffPP+5+5


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COORDINATION NUMBER (C.N.):COORDINATION NUMBER (C.N.): in packed ionic structures,in packed ionic structures, thethenumber of nearest neighborsnumber of nearest neighbors(1st coordination shell); i.e. the(1st coordination shell); i.e. the

number of anions surrounding anumber of anions surrounding a cationcation (or number of(or number of cationscationssurrounding an anion)surrounding an anion)

Example: KExample: K++

6-fold coordination -> 1.386-fold coordination -> 1.38

8-fold coordination -> 1.518-fold coordination -> 1.5110-fold coordination -> 1.5910-fold coordination -> 1.59

This increase in radius with coordination number reflects expansion of theThis increase in radius with coordination number reflects expansion of thecationcation into more available space between larger number of anionsinto more available space between larger number of anions

Regular CoordinationRegular Coordination PolyhedraPolyhedra: all: all cation-anioncation-anion distances assumed todistances assumed tobe equalbe equal


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Is it possible toIs it possible to predictpredictcoordination environments?coordination environments?


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PaulingPaulingss Rule #1 (Rule #1 (Radius Ratio PrincipalRadius Ratio Principal))::

!! The distance betweenThe distance between cationscations and anions can be calculatedand anions can be calculatedfrom their effective electrostatic radii;from their effective electrostatic radii; coordination numbercoordination numberdepends on the relative radii ofdepends on the relative radii of cationscations and surroundingand surroundinganionsanions

!! Greater radius = greater coordination numberGreater radius = greater coordination number

Note: ONLY strictly true for ionic bonding with undistortedNote: ONLY strictly true for ionic bonding with undistorted polyhedrapolyhedra,,pretending that ions are spheres (which they arenpretending that ions are spheres (which they arent ) and they aret ) and they areunpolarizedunpolarized

PolarizationPolarization: distortion of ion shape; large,: distortion of ion shape; large, monovalentmonovalent ions areions aremost easily polarizedmost easily polarized


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Radius Ratio (R.R.) =Radius Ratio (R.R.) = RRcc/R/Raa

RRcc== cationcation radiusradiusRRaa= anion radius= anion radius

2-fold: R.R. < 0.1552-fold: R.R. < 0.155

3-fold: R.R. 0.155 - 0.23-fold: R.R. 0.155 - 0.22255

4-fold: R.R. 0.24-fold: R.R. 0.2225 - 0.4145 - 0.414

6-fold: R.R. 0.414 - 0.7326-fold: R.R. 0.414 - 0.7328-fold: R.R. 0.732 - 1.0008-fold: R.R. 0.732 - 1.000


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Example:Example:

RReffeffNaNa++= 1.10= 1.10

RReffeffClCl--= 1.72= 1.72

C.N. of Na inC.N. of Na in NaClNaCl??

R.R. = 1.10/1.72 = 0.64R.R. = 1.10/1.72 = 0.64

6-Fold6-Fold

In 1929In 1929 Linus PaulingLinus Pauling came up with some other useful rulescame up with some other useful rules


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PaulingPaulingss Rule #2 (Rule #2 (ElectrostaticElectrostatic ValencyValency PrincipalPrincipal))::

!! The strength of a bond (electrostatic valence) equals the ionicThe strength of a bond (electrostatic valence) equals the ionic

valence (charge) divided by the coordination numbervalence (charge) divided by the coordination number

!! Sum of bond valences = ionic valenceSum of bond valences = ionic valence

E.V. = Z./C.N.E.V. = Z./C.N.

Example: SiOExample: SiO44

Si+4

O-2

O-2O-2

O-2

Si+4

Each:Each: Si-OSi-Obondbond

E.V. = +4/4 = 1E.V. = +4/4 = 1

Each:Each: O-SiO-Si bondbond

E.V. = -2/2 = -1E.V. = -2/2 = -1


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Example:Example: NaClNaCl

Each:Each: Na-ClNa-Cl bondbond

E.V. = +1/6 =E.V. = +1/6 =+1/6+1/6

Each:Each:Cl-NaCl-Na bondbondE.V. = -1/6 = -1/6E.V. = -1/6 = -1/6

Na

+

Cl-

Cl-Cl-

Cl-

Cl-

Cl-

Note: some problems with this approach -- irregular coordinationNote: some problems with this approach -- irregular coordinationpolyhedrapolyhedra, mixed bonding types, mixed bonding types


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IsodemicIsodemic:: compounds with allcompounds with all bonds of equal strengthbonds of equal strength

Ex.Ex. SpinelSpinel ABAB22OO44 A = +2; B = +3A = +2; B = +3

By XRD, we know that AIV, BVIBy XRD, we know that AIV, BVISo,So, A = +2/4 = +1/2A = +2/4 = +1/2

B = +3/6 = +1/2B = +3/6 = +1/2

AnisodesmicAnisodesmic:: bonds of unequal strength;bonds of unequal strength; common in compounds withcommon in compounds withanionic complexes; electrostatic valence within the anionicanionic complexes; electrostatic valence within the anioniccomplex is greater than half the anion chargecomplex is greater than half the anion charge

Ex. CarbonateEx. Carbonate COCO33-2-2 CC+4+4is 3-fold with respect to Ois 3-fold with respect to O-2-2

E.V. = +4/3 = 1 1/3E.V. = +4/3 = 1 1/3(most strength within complex)(most strength within complex)

MesodesmicMesodesmic:: bond strength is exactly half the anion chargebond strength is exactly half the anion charge

Ex.Ex. Si-OSi-O O in silicatesO in silicates SiSi+4+4is 4-fold with respect to Ois 4-fold with respect to O-2-2

E.V. = +4/4 = +1E.V. = +4/4 = +1


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PaulingPaulingss Rule #3Rule #3!! Vertex-sharingVertex-sharing(2(2 cationscations share 1 anion) betweenshare 1 anion) between tetrahedratetrahedra

oror octahedraoctahedra isis energetically stableenergetically stable

!! Edge-sharingEdge-sharing (2(2 cationscations share 2 anions) betweenshare 2 anions) between polyhedrapolyhedra isisless stableless stable; rare for; rare for tetrahedratetrahedra, more common for, more common for octahedraoctahedra

!! Face-sharingFace-sharing(2(2 cationscations share 3 anions) betweenshare 3 anions) between polyhedrapolyhedra isisunstableunstable; never occurs for; never occurs for tetrahedratetrahedra; rare for; rare for octahedraoctahedra

Why? Electrostatic RepulsionWhy? Electrostatic RepulsionLess of a problem forLess of a problem for octahedraoctahedra becausebecause cation-cationcation-cation distances aredistances arelonger andlonger and cationscations are typically of lower chargeare typically of lower charge


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PaulingPaulingss Rule #4Rule #4

!! CationsCations of high charge and small coordination number tend not toof high charge and small coordination number tend not to

share anions with othershare anions with other cationscations

!! Why? Repulsion betweenWhy? Repulsion between cationscations

PaulingPaulingss Rule #5 (Rule #5 (Principal of ParsimonyPrincipal of Parsimony))

!! The number of different components in a crystal tends to be small;The number of different components in a crystal tends to be small;if lots of ions are present, they tend to occupy the same structuralif lots of ions are present, they tend to occupy the same structural

position (position (sitessites))


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Coordination of CommonCoordination of Common CationsCations and Anions:and Anions:

Most common element in crust by weight? OMost common element in crust by weight? OTherefore, many minerals contain OTherefore, many minerals contain OOO-2-2effective electrostatic radius:effective electrostatic radius:1.27-1.341.27-1.34

SiSi+4+4: 4-fold: 4-foldAlAl+3+3: 4-fold or 6-fold: 4-fold or 6-foldFeFe+3+3: 6-fold (4-fold): 6-fold (4-fold)MgMg+2+2: 6-fold: 6-foldor 8-foldor 8-foldFeFe+2+2: 6-fold: 6-foldMnMn+2+2: 6-fold or 8-fold: 6-fold or 8-foldNaNa++: 8-fold: 8-fold

CaCa+2+2

: 8-fold: 8-foldKK++: 12-fold: 12-fold

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9 Coordination