Concise Physical Chemistry

c11 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

174 LIQUIDS AND SOLIDS

FIGURE 11.10 A face-centered cubic unit cell. The pointer indicates the face-centered Cl−

ion in the front face of the cell.

the cube may be distorted to a rhomb, tetragon, or other geometry. Let’s stick to the

simple cube to illustrate the principle.

From the angles within the unit cell and its orthogonal distancesa,b, andc, one

can find its volumeVcell. A measurement of the densityρ=m/Vof the crystalline

solid gives the mass of atoms in the cell. Knowing the mass of the atoms in the cell

and theirmolar mass, we can calculate how many of them there are—for example,

four in the case of Na+and Cl−.Theface-centeredunit cell for NaCl resembles two

interpenetrating simple cubic cells (Fig. 11.10). This presents a cell with Na+and

Cl−ions alternating along each edge. The edge of the NaCl cell is 564 pm, but it has

Na+−Cl−−Na+along each edge so the Na+−Cl−distance is half that dimension,

564 / 2 =282 pm. From the relative electronic structures and the fact that Na loses an

electron and Cl−gains one in the ionic bond, we can guess that the Cl−ion will be

about twice as large as Na+. This estimate leads to an approximate ionic radius of

Na+,rNa+=(282/3)=94 pm, leaving 282− 94 =188 pm forrCl−.

11.4.2 The Packing Fraction

One of the things we want to know about a crystal is how efficiently the atoms,

molecules, or ions are packed. This is measured by the packing fraction, the space

in the unit cell that is occupied by atoms relative to the total space in the cell. The

larger the packing fraction, the greater the space occupied by atoms and the less

space “wasted” in the interstices. This can be found by straightforward geometric

calculation.

The idea of a unit cell packing fraction can be illustrated by arranging discs on

a table top. One way is similar to the one shown for the marbles in Fig. 11.8. The

repeating planar unit is a square (Fig. 11.11). A table top covered with many discs

packed in this waycan be thought of as simply a repeating pattern of square unit

cells. Once we know the length of the edge of one unit cell and we know that it

is square, we know the geometry of the entire table top full of discs. Chemists are

not usually interested in table tops covered with discs, but we are very interested in