Problem Set #2

For each column, the ionic radius increases with increasing row number. The reason is that as the row number increases, the number of electrons increases. There is a general trend for ionic radius to decrease as the column number increases along a row. For the first 5 columns, all ions in a row have the same number of electrons, that is they are iso-electronic. The only exceptions to this rule are larger ions of column V which have multiple valence states (As, Sb, Bi). As the column number increases, the nuclear charge increases which attracts the electrons more strongly inward, tending to shrink the ion.

A atom in blue, and B atom in red.

b) The coordination polyhedron is an octahedron (indicated in bold bonds).

c) Polyhedra share edges; a second polyhedron is drawn (dashed bonds); the shared edge contains the two B atoms common to both polyhedra.

d) NaCl structure.

Extra Credit

Coordination polyhedra are octahedral, the central one is shown complete. The upper and lower edge are shared with neighboring octahedral. The other vertices are corner shared with two other neighboring octahedral. This is the rutile structure.

3. We relate the vector joining cation and anion (length R_C+R_A) to half the vector joining anion to anion (length R_A), through the complementary angle, q:

(3.1)

(3.2)

(3.3)

q is half the Bond Angle, the angle between neighboring cation-anion bonds, and an important quantity in crystal chemistry (see drawings, following page).

The value of q for 3- 6- and 8-fold coordination can be found from examination of the drawings. For 3-fold coordination, we have q=60^o. In the case of 6-fold coordination, we consider the plane normal to the 4-fold rotation axis, and find that q=45^o. In the case of 8-fold coordination, we consider the plane normal to a 2-fold rotation axis. The rectangle formed by the anions in this plane has aspect ratio sqrt(2), and the angle q=tan^-1[1/sqrt(2)]=35.26^o.

For four-fold coordination, we find the value of q by inscribing the tetrahedron in a cube, of length 2, centered at the origin. With cation at the center, anions are located at (1,-1,1) (-1,1,1) (1,1,-1) (-1,-1,-1). The cosine of the bond angle is just the dot-product of two cation-anion vectors, e.g. cos(bond angle) = (1,-1,1)•(-1,1,1)/3 = -1/3. So the tetrahedral bond angle is 109.47^o, and q=54.74^o.

Now that we have found the value of the bond angle and q for each coordination polyhedron, we substitute these values into (3.3) to find the radius ratio:

Coordination Bond Angle q Smallest Possible R_C/R_A

2 180 90 0

3 120 60 0.1547

4 109.47 54.74 0.2247

6 90 45 0.4142

8 70.53 35.26 0.7321

3-fold

4-fold

4. Recalculating the chemical analyses, we find the following compositions in mole fraction of the oxides.