What Determines The Size Of The Charge Of An Ion
| EENS 2110 | Mineralogy |
| Tulane University | Prof. Stephen A. Nelson |
| Coordination and Pauling's Rules | |
The arrangement of atoms in a crystal structure not just depends on the charge on the ion and type of bonding between atoms, but also on the size of the atoms or ions. In any given molecule or crystal structure each atom or ion volition exist surrounded by other atoms or ions. The number of ions or atoms that immediately surround an atom or ion of interest is chosen the coordination number, - C.N. As we shall encounter, the coordination number depends on the relative size of the atoms or ions. So, we must first discuss their sizes.
Atomic and Ionic Radii
The size of an atom or ion depends on the size of the nucleus and the number of electrons. More often than not atoms with higher numbers of electrons have larger radii than those with smaller numbers of electrons. Thus ions volition have radii dissimilar from the atoms considering ions will take either gained or lost electrons. The number of positive charges in the nucleus determines both the number of electrons that surround an atom and the number of electrons that can be lost or gained to form ions.
- Thus as the charge on the ion becomes more positive, there will exist less electrons and the ion will have a smaller radius.
- As the charge on the ion becomes more negative, there will be more electrons and the ion will have a larger radius.
- Equally the diminutive number increases in any given column of the Periodic Table, the number of protons and electrons increases and thus the size of the atom or ion increases.
Atomic and ionic radii also depend on the type of bonding that takes place between the constituents, and on the coordination number. Thus, atomic and ionic radii volition vary somewhat as a function of the environment in which the atoms or ions are constitute.
A list of ionic radii of the more mutual elements for various coordination numbers is shown on folio 67 of your text. First, let's examine the table looking at one column of the Periodic Table to come across how the radii are affected past increasing atomic number.
| Here we run into the upshot of increasing the atomic number (and total number of electrons) for ions of equal charge and the effect of changing the coordination number. The radii increase with increasing total number of electrons downwards in table. Ionic radius as well increases with increasing coordination number, the electron cloud is drawn out by the presence of more than surrounding ions. |
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| Next, we examine ane row of the Periodic Table to see how the radii are afflicted by charge of the ion. Here we run across that as the charge becomes more positive the radius of the cation decreases. This is because there are fewer electrons in the outer shells of the ions. The sizes of anions are relatively big considering there are more electrons in their outermost shells. |
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Coordination of Ions
Coordination number, C.Due north. depends on the relative size of the ions. If all of the atoms in a crystal are the same size, so there are two means to pack the atoms to form a crystal structure. In this case, the maximum number of atoms that exist coordinated effectually any individual is 12. Nosotros call this 12-fold coordination. There are two ways that atoms tin be packed in 12-fold coordination.
| First, examine a single layer of atoms of equal size. Notation that there are two kinds of voids between the atoms, those that have a sort of triangular shape with the triangles pointing up we'll call B voids, and those with the triangles pointing downward nosotros'll call C voids. If nosotros add the next layer of atoms and so that they occupy the space above the B voids, so add together the next layer higher up the A atoms, this will result in a stacking sequence that runs AB AB AB ....etc. This type of closest packing is referred to as hexagonal closest packing. It results in a hexagonal lattice with the c-centrality oriented perpendicular to the AB AB layers. If after adding the layer of B atoms we place the next layer so that the atoms occupy positions over the C voids in the A layer, and go along the process upwards, we get a stacking sequence that runs ABC ABC ABC.... etc. This type of packing is referred to as cubic closest packing. It results in a cubic or isometric lattice with the |  |
centrality perpendicular to the layers.If we subtract the size of the cation in such an organization, still allowing for the surrounding anions to impact each other and bear on the cation, with decreasing size of the cation, the coordination will first issue in 8 anions surrounding the cation.
| This is chosen 8-fold coordination or cubic coordination because the shape of the object synthetic by drawing lines through the centers of the larger ions is a cube. If the size of the coordinated cation becomes smaller, it will get as well small to touch on the surrounding anions. Thus, there is limiting radius ratio that volition occur when the Rx/Rz becomes also minor. To see what this limit is, nosotros must await at the vertical plane running through the centers anions labeled A and B. |  |
In this structure we tin determine the radius ratio for the limiting condition, often called the "no rattle limit" considering if the radius ratio becomes smaller than this the cation volition "rattle" in its site. Using the Pythagorean theorem we can write:
Giving Rx/Rz = 0.732 |  |
Thus, for Rx/Rz < 0.732 the cation will be too small-scale or will rattle in its site and the structure will have to change to 6-fold coordination.
| Six-fold coordination is also called octahedral coordination because the shape defined by drawing planes through the center of the larger ions is an octahedron. Octahedral coordination is stable when Rx/Rz , 0.732, but decreasing the radius of cation, Rx, will eventually reach a limit where once again the smaller ion volition rattle in its site. |  |
The no rattle limit can be determined by looking at the horizontal plane running through the ions labeled C and D. In this example we tin write:
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| For Rx/Rz < 0.414 the construction goes into 4-fold coordination. Planes through the centers of the larger atoms in this case will class a tetrahedron, then 4-fold coordination is as well called tetrahedral coordination. |  |
The calculation to determine the "no rattle" limit for tetrahedral coordination is circuitous (meet text past Klein and Dutrow, page 70). The result shows that the limit is reached when Rx/Rz = 0.225. As the radius ratio becomes smaller than this, triangular coordination becomes the stable configuration.
| For triangular coordination, the coordination number is 3, that is three anions surround the smaller cation. The "no rattle" limit is reached for triangular coordination when Rx/Rz becomes less than 0.155. |  |
| At values of Rx/Rz < 0.155 the only way the smaller ion can be coordinated by the larger ions is to have 2 of the larger ions on either side. This two-fold coordination is termed linear coordination. |  |
| The table here summarizes the cation to anion radius ratios, Rx/Rz, for various coordination numbers and gives the name of the coordination polyhedron for each coordination number. |
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In describing the construction of crystals and the locations of various ions or atoms within crystals, reference is often made to a crystallographic site on which an atom resides. Such sites are unremarkably referred to in terms of the coordination number or coordination polyhedron that surrounds the ion. For example, in the silicate minerals Si is surrounded past 4 oxygens in tetrahedral coordination. Thus Si is often said to occupy the tetrahedral sites in silicate minerals. Nosotros volition explore these concepts in more than detail latter in the course.
For the elements that occur in common minerals in the World's crust, the most common coordinating anion is Oxygen. The following table gives the ionic radius and coordination of these common metal cations coordinated with oxygen.
Ion C.N.
(with Oxygen)Coord. Polyhedron Ionic Radius, � G+ 8 - 12 cubic to closest 1.51 (8) - 1.64 (12) Na+ 8 - 6 cubic to octahedral 1.xviii (8) - 1.02 (6) Ca+2 8 - half dozen one.12 (eight) - 1.00 (vi) Mn+ii 6 Octahedral 0.83 Fe+ii half dozen 0.78 Mg+2 6 0.72 Fe+3 6 0.65 Ti+4 half dozen 0.61 Al+iii 6 0.54 Al+iii iv Tetrahedral 0.39 Si+4 4 0.26 P+5 four 0.17 S+6 iv 0.12 C+4 3 Triangular 0.08
Pauling'due south Rules
Linus Pauling studied crystal structures and the types of bonding and coordination that occurs within them. His studies plant that crystal structures obey the following rules, now known as Pauling's Rules.
Rule 1
Effectually 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 past 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, Rx/Rz, of the cation to the anion.
Rule 2, The Electrostatic Valency Principle
An ionic structure volition be stable to the extent that the sum of the strengths of the electrostatic bonds that reach an ion equal the accuse on that ion.
In club to understand this dominion we must beginning define electrostatic valency, e.v.
eastward.five = Charge on the ion/C.North.
| For example, in NaCl each Na+ is surrounded by 6 Cl- ions. The Na is thus in 6 fold coordination and C.Due north. = vi. Thus eastward.v. = ane/6. So one/half dozen of a negative accuse reaches the Na ion from each Cl. So the +1 charge on the Na ion is counterbalanced by 6*1/6 =1 negative charge from the half-dozen Cl ions. |  |
| Similarly, in the CaF2 structure, each Ca+2 ion is surrounded past 8 F- ions in cubic or 8-fold coordination. The eastward.5. reaching the Ca ion from each of the F ions is thus one/four. Since in that location are 8 F ions, the total charge reaching the Ca ion is 8*ane/4 or two. And so, again the charge is balanced. |  |
| Find that in NaCl, each Cl ion is also surrounded by half-dozen Na ions in octahedral coordination. And so, over again, the one/half dozen of a positive accuse from each Na reaches the Cl ion and thus the Cl ion sees six*1/6 = 1 positive charge, which exactly balances the -1 charge on the Cl. In the case of NaCl the accuse is exactly balanced on both the cations and anions. In such a case, we say that the bonds are of equal strength from all directions. When this occurs the bonds are said to be isodesmic . |  |
| This is not the instance for C+4 ion in triangular coordination with O-2. Here, e.v. = four/3 (C has a charge of +iv and is coordinated by 3 oxygens). Thus, the 3 Oxygens each contribute four/iii charge to the Carbon ion, and the charge on the carbon is balanced. But, each Oxygen still has ii/iii of a charge that it has not used. Thus, a carbonate structural group is formed - CO3 -2. In cases like this, where the electrostatic valency is greater than 1/2 the charge on the anion (4/3 > 1/ii*two), the anion will be more strongly bound to the central coordinating cation than it can be bonded to other structural groups. When this occurs the bonding is said to be anisodesmic . |  |
| A third case arises when the e.5. reaching the cation is exactly 1/ii the charge on the anion. This is the example for Si+iv in tetrahedral coordination with O-2. Here, the eastward.v. reaching the Si is 4/iv =1. This leaves each Oxygen with a -1 charge that it has non shared. Since this -ane is exactly 1/2 the original accuse on O-2, the Oxygens in the SiOfour -4 group can exist simply as tightly spring to ions outside the grouping as to the centrally coordinated Si. In this example the bonding is said to exist mesodesmic . The SiO4 -4 group is the basic building block of the most common minerals in the Earth'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 at the greatest distance from each other. In the example shown here, for tetrahedral coordination, if the distance between the cations in the polyhedrons that share corners is taken equally ane, and then sharing edges reduces the distance to 0.58, and sharing of faces reduces the distance to 0.38. |  |
Rule 4
In a crystal construction containing several cations, those of high valency and small-scale coordination number tend not to share polyhedral elements.
Sharing 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 tin be ameliorate shielded from the effects of other positive charges in the crystal structure.
Rules 1 through 4 maximize the cation - anion attractions and minimize the anion-anion and cation-cation repulsions.
Rule five, The Principle of Parsimony
The number of different kinds of constituents in a crystal tends to be small.
This ways that there are only a few unlike types of cation and anion sites in a crystal. Even though a crystal may have tetrahedral sites, octahedral sites, and cubic sites, most crystals volition be express to this pocket-sized number of sites, although different elements may occupy similar sites.
Next fourth dimension we volition run into how these principles apply to the silicate minerals.
Examples of questions on this material that could exist asked on an exam
- Define the post-obit: (a) coordination number, (b) octahedral coordination, (c) tetrahedral coordination, (d) tringular coordination, (east) electrostatic valence, (f) isodesmic, (g) anisodesmic, (h) mesodesmic.
- Explain what factors control the size of ions and why.
- Why are anions in general larger than cations?
- Expalin Pauling'due south five rules in uncomplicated terms and explain why they are important.
Render to EENS 2110 Folio
What Determines The Size Of The Charge Of An Ion,
Source: https://www.tulane.edu/~sanelson/eens211/paulingsrules.htm
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