Page 205 - Modern Analytical Chemistry

1400-CH07 9/8/99 4:03 PM Page 188

188 Modern Analytical Chemistry

13
A sample containing 10 particles can be fairly large. Suppose this is equivalent
to a mass of 80 g. Working with a sample this large is not practical; but does this
mean we must work with a smaller sample and accept a larger relative sampling
variance? Fortunately the answer is no. An important feature of equation 7.4 is that
the relative sampling variance is a function of the number of particles but not their
combined mass. We can reduce the needed mass by crushing and grinding the par-
13
ticles to make them smaller. Our sample must still contain 10 particles, but since
each particle is smaller their combined mass also is smaller. If we assume that a par-
ticle is spherical, then its mass is proportional to the cube of its radius.
Mass µ r 3
Decreasing a particle’s radius by a factor of 2, for example, decreases its mass by a fac-
3
13
tor of 2 , or 8. Instead of an 80-g sample, a 10-g sample will now contain 10 particles.
7 5
EXAMPLE .
13
Assume that the sample of 10 particles from Example 7.4 weighs 80 g. By how
much must you reduce the radius of the particles if you wish to work with a
sample of 0.6 g?
SOLUTION
To reduce the sample from 80 g to 0.6 g you must change its mass by a factor of

80
= 133 times
06.
This can be accomplished by decreasing the radius of the particles by a factor of
3
x = 133
x = 5.1
Decreasing the radius by a factor of approximately 5 allows you to decrease the
sample’s mass from 80 g to 0.6 g.

Treating a population as though it contains only two types of particles is a use-
ful exercise because it shows us that the relative sampling variance can be improved
by collecting more particles of sample. Furthermore, we learned that the mass of
sample needed can be reduced by decreasing particle size without affecting the rela-
tive sampling variance. Both are important conclusions.
Few populations, however, meet the conditions for a true binomial distribu-
tion. Real populations normally contain more than two types of particles, with the
analyte present at several levels of concentration. Nevertheless, many well-mixed
populations, in which the population’s composition is homogeneous on the scale at
which we sample, approximate binomial sampling statistics. Under these conditions
the following relationship between the mass of a randomly collected grab sample,
m, and the percent relative standard deviation for sampling, R, is often valid. 6
2
mR = K s 7.5
where K s is a sampling constant equal to the mass of sample producing a percent
relative standard deviation for sampling of ±1%.* The sampling constant is evalu-
*Problem 8 in the end-of-chapter problem set asks you to consider the relationship between equations 7.4 and 7.5.