Pauling

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Pauling's Rules

Linus Pauling studied crystal structures and the types of bonding and coordination that occurs

within them. His studies found that crystal structures obey the following rules, now known as

Pauling's Rules.

Rule 1

Around every cation, a coordination polyhedron of anions forms, in which the cation-anion

distance is determined by the radius sums and the coordination number is determined by the

radius ratio.

This rule simply sets out what we have discussed above, stating that the different types of

coordination polyhedra are determined by the radius ratio, R!R", of the cation to the anion.

Rule 2, The Electrostatic Valency Principle

An ionic structure will be stable to the extent that the sum of the strengths of the electrostatic

bonds that reach an ion equal the charge on that ion.

#n order to understand this rule we must first define electrostatic valency, e.v.

e.v $ %harge on the ion!%.&.

or eample, in &a%l each &a( is surrounded by ) %l * ions. The

&a is thus in ) fold coordination and %.&. $ ). Thus e.v. $ +!).

o +!) of a negative charge reaches the &a ion from each %l. othe (+ charge on the &a ion is balanced by )-+!) $+ negative

charge from the ) %l ions.

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imilarly, in the %a structure, each %a( ion is surrounded by /

* ions in cubic or /*fold coordination. The e.v. reaching the %a

ion from each of the ions is thus +!0. ince there are / ions,the total charge reaching the %a ion is /-+!0 or . o, again the

charge is balanced.

&otice that in &a%l, each %l ion is also

surrounded by ) &a ions in octahedral

coordination. o, again, the +!) of a positivecharge from each &a reaches the %l ion and

thus the %l ion sees )-+!) $ + positive charge,

which eactly balances the *+ charge on the%l.

#n the case of &a%l the charge is eactly

balanced on both the cations and anions. #n

such a case, we say that the bonds are of e1ualstrength from all directions. 2hen this occurs

the bonds are said to be isodesmic.

This is not the case for %(0 ion in triangular coordination with 3*

. Here, e.v. $ 0!4 5% has a charge of (0 and is coordinated by 4

oygens6. Thus, the 4 3ygens each contribute 0!4 charge to

the %arbon ion, and the charge on the carbon is balanced. 7ut,each 3ygen still has !4 of a charge that it has not used. Thus,

a carbonate structural group is formed * %34*.

#n cases like this, where the electrostatic valency is greater than

+! the charge on the anion 50!4 8 +!-6, the anion will be morestrongly bound to the central coordinating cation than it can be

bonded to other structural groups. 2hen this occurs the bonding

is said to be anisodesmic.

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9 third case arises when the e.v. reaching the cation is eactly +!

the charge on the anion. This is the case for i(0 in tetrahedralcoordination with 3*. Here, the e.v. reaching the i is 0!0 $+. This

leaves each 3ygen with a *+ charge that it has not shared. ince

this *+ is eactly +! the original charge on 3 *, the 3ygens in the

i30

*0

group can be :ust as tightly bound to ions outside the group asto the centrally coordinated i. #n this case the bonding is said to

be mesodesmic.

The i30*0 group is the basic building block of the most common

minerals in the ;arth's crust, the silicates.

Rule 3

Shared edges, and particularly faces of two anion polyhedra in a crystal structure decreases

its stability.

The reason for this is that

sharing of only corners of

polyhedra places the

positively charged cations atthe greatest distance from

each other. #n the eample

shown here, for tetrahedral

coordination, if the distance between the cations in the

polyhedrons that sharecorners is taken as +, then

sharing edges reduces the

distance to <.=/, and sharing

of faces reduces the distanceto <.4/.

Rule 4

In a crystal structure containing several cations, those of high valency and small

coordination number tend not to share polyhedral elements.

haring of polyhedral elements for cations of high charge will place cations close enough

together that they may repel one another. Thus, if they do not share polyhedral elements they

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can be better shielded from the effects of other positive charges in the crystal structure.

Rules + through 0 maimi"e the cation * anion attractions and minimi"e the anion*anion and

cation*cation repulsions.

Rule 5, The Principle of Parsimony

The number of different inds of constituents in a crystal tends to be small.

This means that there are only a few different types of cation and anion sites in a crystal. ;ven

though a crystal may have tetrahedral sites, octahedral sites, and cubic sites, most crystals will

be limited to this small number of sites, although different elements may occupy similar sites.

&et time we will see how these principles apply to the silicate minerals.

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