Coordination and Pauling's Rules

The arrangement of atoms in a crystal structure not only 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 will be surrounded by other atoms or ions.  The number of  ions or atoms that immediately surround an atom or ion of interest is called the coordination number,  - C.N.  As we shall see, 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. Generally atoms with higher numbers of electrons have larger radii than those with smaller numbers of electrons.  Thus ions will have radii different from the atoms because ions will have 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 be 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. As the atomic 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 will vary somewhat as a function of the environment in which the atoms or ions are found. A  list of ionic radii of the more common elements for various coordination numbers is shown on page 67 of your text.  First, let's examine the table looking at one column of the Periodic Table to see how the radii are affected by increasing atomic number.

Here we see the effect 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 downward in table.  Ionic radius also increases with increasing coordination number, the electron cloud is drawn out by the presence of more surrounding ions.

Ion R(Å) C.N. = 6 R (Å) C.N. = 8 Li+1 0.74 0.92 Na+1 1.02 1.18 K+1 1.38 1.51 Rb+1 1.52 1.61 Cs+1 1.67 1.74

Next, we examine one row of the Periodic Table to see how the radii are affected by charge of the ion. Here we see 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 large because there are more electrons in their outermost shells.

Ion R(Å) C.N. = 4 R (Å) C.N. = 6 Na+1 0.99 1.02 Mg+2 0.57 0.72 Al+3 0.39 0.48 Si+4 0.26 0.40 P+5 0.17 0.38 S+6 0.12 0.29 S-2 1.84 Cl-1 1.81

Reference: http://www.tulane.edu/~sanelson/eens211/paulingsrules.htm