wavelengths, the resulting color can roughly be predicted. Furthermore, the
rules of thumb on whiteness given above can be of some use to explain
whiteness. But otherwise, we have to rely on empiricism.
9.3 PARTICLE SIZE DISTRIBUTIONS
The particles in a dispersion are hardly ever of the same size. Nature may
often succeed in making rather monodisperse systems, like protein
molecules, cells, or wheat kernels, although wider distributions also occur,
e.g., starch granules. Most man-made dispersions have a fairly wide range of
sizes and are thus polydisperse or heterodisperse; examples are emulsion
droplets, particles obtained by grinding (flour, etc.), and spray-dried milk.
Since many properties of a dispersion depend on particle size, as we have
seen in the previous section, such properties may also depend on the
distribution of sizes: how many particles of each possible size are present?
This is the subject of this section.
9.3.1 Description
We start by defining a size variablex. Various definitions can be chosen:x
may be particle diameter, molar mass, number of molecules in a particle,
particle volume, etc. Thecumulative number distribution F(x) is now defined
as the number of particles with a size smaller thanx. ConsequentlyF(0)¼ 0
and Fð?Þ¼N¼the total number of particles in the dispersion. The
dimension ofF(x) generally is [L^3 ] (where L stands for length), i.e., number
per unit volume, but other definitions can also be taken, for instance number
per unit mass. Often, a cumulative distribution is recalculated to a
percentage ofN, hence puttingFð?Þat 100%.
Thefrequency distributionof the number is now defined as
fðxÞ¼F^0 ðxÞ¼dFðxÞ
dxð 9 : 2 ÞThe frequency distribution is thus a differential quantity, and it is given in
number per unit volume per unit ofx.Ifxis expressed in units of length,
e.g., particle diameter, the dimension off(x)is[L^4 ].
So far we have assumed the distribution to be a continuous one, which
is nearly always a good approximation, because of the very large number of
particles in most dispersions. In practice, a distribution is often split into size