particles in a size class. This is largely owing to the Poisson statistics of
counting. The standard deviation of the number of particles in a certain
volume is, for complete random distribution, equal to the square root of the
average. This means that the relative standard deviation of the number of
particles in a size classiis equal to or larger than 1=HNi,ifNiis the number
actually counted (i.e., before any multiplication with a dilution factor, etc.).
Counting just one particle thus leads to an uncertainty (standard deviation)
of over 100%. This becomes especially manifest for very large particles, as
can be derived from Figure 9.10. It is seen that very small numbers of large
particles can give rise to a large proportion of particle volume (or mass),
which means that the large-particle end of a volume distribution is often
subject to large errors. Especially if the size distribution is wide (high value
ofc 2 ), tens of thousands of particles may have to be counted to obtain
reliable results.
Question 1
Butter and margarine are water-in-oil emulsions. These products are inevitably
contaminated with some microorganisms, and especially yeasts can cause spoilage.
The organisms can only proliferate in the aqueous phase, and cannot move from one
drop to another. If the number of yeasts present is thus very much smaller than the
number of drops, the fraction of the aqueous phase that is contaminated may be too
small to give perceptible spoilage.
In a certain margarine, the volume fraction of aqueous phase is 0.2. The
number of water drops is counted by microscopy and is estimated at 10^9 per ml of
product. The aqueous phase of the freshly made product is separated (by melting and
centrifuging) and the count of yeasts is determined at 5 6105 per ml, which means
105 per ml product. It is concluded that 10^5 /10^9 or 0.01%of the aqueous phase is
contaminated, which would give negligible spoilage. Is this conclusion correct?
Answer
No. The reasoning followed above would imply that the fraction of the aqueous
phase that is contaminated is proportional to the number average volume of the
drops:S 3 =S 0 ¼ðd 30 Þ^3. However, the chance that a droplet is contaminated with a
yeast cell is proportional to its volume. Consequently, the drop volume distribution
should be weighted withd^3 , which then means that the volume fraction contaminated
is proportional toS 6 /S 3 ,orto(d 63 )^3. Since the droplet size distribution of the
products concerned tends to be very wide, the difference will be large. ForS 0 ¼
109 cm^3 andj¼ 0 :2, we derived 30 ¼ 7 : 3 mm. We have no way of determiningd 63 ,
but assuming a size distribution similar to the one depicted in Figure 9.10, we see that
d 63 would be about 10 timesd 30. The volume ratio would then be 10^3 , implying that