Page 27 - Modern physical chemistry

16 Structure in Solids

rearrange

and solve for the desired radius ratio

r A + =.J3 -1 = 0.732.
r B -

1.12 Avogadro's Number

From the density of a well formed crystal and from its internal structure, one can
determine the mass of a formula unit. Knowing the mass of a mole of the substance, one
can then find the number of units in a mole.
By chemical analysis, a person can determine the amount of a pure substance that
reacts with, or is chemically equivalent to, a given amount of another substance. The
equivalent mass is defined as the mass that is equivalent to 3.0000 g tetravalent 12C. Mul-
tiplying this mass by the pertinent valence, or the pertinent valence change, yields the
mass of one mole. Thus, one mole of 12C weighs 12.0000 g.
In a given discussion, the elementary unit may be taken as an atom, or a molecule,
an ion, an electron, a proton, ... , or even a specified group of such particles. By definition,
a mole is the amount of substance that contains the same number of elementary units
as there are atoms in 12.0000 g pure 12C. The number of elementary units in a mole is
called the Avogadro number N A •
By X-ray methods, a person can determine the nature and the size of a unit cell in a
given crystal. From how each particle is shared with neighboring unit cells, one deter-
mines the net number n of molecules or formula units in the cell. If m is the mass of a
formula unit, we have
pV
m=-, [1.8]
n
where p is the density and V the volume of the cell. If there are only negligible defects
in the crystal, the conventionally determined whole-crystal density can be used for p.
The volume of the cell is obtainable from the X-ray data
Avogadro's number NA equals the mass per mole M divided by the mass per formula
unitm.
M
N A =-· [1.9]
m
In practice, very few crystalline materials yield good data; for, even a crystal that
appears to be perfect to the eye may contain many faults. There may be lattice defects,
particles missing from lattice points, and particles that are not associated with lattice
points, but with interstitial positions. Extensive dislocations of various kinds may occur
with respect to inner surfaces. The surfaces may encompass small volumes so that the
crystal is broken up into a mosaic of crystallites. The defects tend to decrease the sharp-
ness of diffraction patterns and decrease the over-all density of the crystal.
A solid metal usually exhibits a mosaic structure. An alkali chloride prepared by freez-
ing its melt is generally defective. But when it is slowly crystallized out of a solution at
low temperatures, suitable crystals may form. Some naturally occurring crystals of calcite,
quartz, and diamond are also suitable.