Two Experiments Illustrating the Importance of Sampling in a Quantitative
Chemical Analysis
D. Harvey, J. Chem. Ed., (2002), 79, 360-363
Creating a Sampling Plan for A Mixture of Solids
The quality of an analysis is no better than its weakest link. A typical analysis consists of three
common steps -- a sample is collected, the sample is prepared for analysis, and the prepared
sample is analyzed. Since variances are additive, the total variance for the analysis, σ2total, is a
simple summation of the variances for each step in the analysis; thus,
σ2total = σ2samp + σ2prep + σ2meas
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where σ prep is the variance for the preparation of the samples, σ2meas is the variance for the
measurements, and σ2samp is the variance for obtaining samples.
There are two goal for this experiment. First, you will evaluate each of the three sources of
variance for the analysis of a solid mixture, verifying that sampling is the most significant source
of variance. Second, you will develop a sampling plan that provides a satisfactory uncertainty for
the analysis.
Composition of the Mixture
Erythrosin B, whose structure is shown here, is a dye that has several analytical uses, including a
biological stain for bacteria in soils, a plasma stain for nerve cells (when used in conjunction with
methylene blue), as a phosphorescent probe for studying the diffusion of membrane proteins, and
in the quantitative determination of phospholipids. It also is an acid-base indicator whose color is
orange in strongly acidic solutions and red in aqueous solutions of pH greater than 3. And, of
course, erythrosin B, which also is known as Acid Red 51 and FD&C red dye no. 3, has been
used as a food coloring in maraschino cherries. The sample to be analyzed is a mixture of
I
I
NaO
O
O
I
I
COONa
erythrosin B (the analyte) and NaCl (the matrix). The erythrosin B adheres to the surface of the
salt crystals, imparting a pink color to the NaCl (in fact, this is a common way to dispense
indicators that are unstable in solution).
Run a scan from 500 nm – 550 nm of one of your samples. Measure the absorbance, A, for each
sample at a wavelength of 526 nm using distilled water as the reference (do not include more
than one sample spectrum in your report). Calculate the concentration of erythrosin B in each
Level IV sample using Beer’s law
A = abC
-1
-1
where a is erythrosin B’s absorptivity, which has a value of 0.0916 cm ppm , b is the
pathlength, which is 1.00 cm, and C is the concentration of erythrosin B in ppm. Convert these
values to % w/w erythrosin B by accounting for the sample’s mass and its dilution.
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Experiment 1. Finding the Weakest Link Using a Nested Design
For this first part of the experiment, you will be provided with approximately 50 g of this mixture,
which represents your gross sample. Pour your gross sample onto a piece of paper, shape it
into a flattened cone, and divide it into quarters. Obtain an approximately 1-g sample from each
quarter; these are your four Level I samples.
Transfer each Level I sample into a clean glass or agate mortar and pestle. Grind each sample
for several minutes to reduce the particle size and further homogenize the sample. Divide each
of these processed samples in half and obtain an approximately 0.25-g sample from each half.
Quantitatively transfer each sample to a 50-mL volumetric flask and dilute to volume with distilled
water. Be sure to thoroughly mix each solution. These are your eight Level II samples.
Divide each of your Level II samples approximately in half by transferring into separate 20-mL
scintillation vials or testtubes (the remaining Level II sample can be discarded). These are your
16 Level III samples.
Adjust the spectrometer to a wavelength of 526 nm. Using distilled water as a reference,
measure the absorbance, A, of each Level III sample twice, without removing the sample cell
from the spectrometer between measurements. These 32 absorbances are the results for Level
IV. Calculate the concentration of erythrosin B in each Level IV sample. Convert these values to
% w/w erythrosin B by accounting for the sample’s mass and its dilution. To determine the % w/w
erythrosin B for the Level III samples, average the associated results for Level IV. For example,
the % w/w erythrosin B for sample IA1 is the average result for samples IA1a and IA1b. The
results for the Level II and Level I samples are found in a similar manner; thus, the % w/w
erythrosin B for sample IA is the average result for samples IA1 and IA2, and the result for
sample I is the average result for IA and IB.
Report
Summarize your data using four tables. In the first table reports results for the Level IV samples
using the following headings: sample ID, mass of sample, absorbance, and % w/w erythrosin B.
The remaining tables reports results for the Level III, Level II, and Level I samples and include the
following headings: sample ID and % w/w erythrosin B. Using your data, calculate values for
σ2samp, σ2prep, σ2pos, σ2spect, and σ2total. In addition, briefly answer the following questions
Are the differences between σ2samp, σ2prep, σ2pos, and σ2spect, statistically significant?
Based on your results, which step is the weakest link in this analysis?
How might you go about improving the overall standard deviation for this analysis?
Explain why the difference between the results for samples IA1a and IA1b are influenced only
by indeterminate errors. Is the same true for the difference between samples IA1 and IA2?
How about for samples IA and IB, or samples I and II?
5. The data from your four-level nested design also can be used to evaluate the accuracy of
your analysis. The best experimental estimate of the % w/w erythrosin B is the average
result for your eight Level II samples (why is this?). Using the total variance for your analysis,
determine the 95% confidence interval for the % w/w erythrosin B and compare to the
expected value provided by your instructor. For this calculation there are eight samples and
seven degrees of freedom.
6. When sampling is the weakest link the sampling plan should be designed to minimize its
contribution to the overall variance. One approach is to find a way to increase the gross
sample’s homogeneity before collecting individual samples. For example, you could grind the
gross sample to decrease the average particle size. Another approach is to make a
composite sample by collecting several portions of the gross sample and mixing them
together before they are analyzed. If the sampling variance for a particular sample size is
1.
2.
3.
4.
2
σ2samp, and σ2meth is the variance due to the analysis, then the total variance for the analysis of
one sample is
σ2total = σ2samp + σ2meth
If we collect and analyze n separate samples of the same size, then the total variance is
σ2total = σ2samp + σ2meth
n
n
If, however, we form a composite sample by mixing together these n samples and removing k
identical samples for analysis, the total variance is
σ2total = σ2samp + σ2meth
nk
n
Using your results for σ2samp and σ2meth (which is σ2total - σ2samp), determine the total variance
for (a) the analysis of four separate samples taken from the gross sample, and (b) the
analysis of two portions of a composite sample formed by collecting and mixing four samples
from the gross sample. Finally, find a combination of n and k that will give you a total
variance that is less than that in (b), but that requires the analysis of only sample.
Experiment 2. Evaluating the Sampling Constant
For this second part of the experiment you will be provided with approximately 100 g of this
mixture, which represents your gross sample.
Obtaining and Analyzing Individual Samples
One method for determining the sampling variance is to analyze samples of different size. The
table below shows the nominal masses to use as samples (ideally, each sample should be within
±10% of the nominal mass) and the solution volume to which you should dilute your samples
(using volumetric flasks). Each sample will need to be prepared six times, giving a total of 30
samples.
nominal mass (g)
0.10
0.25
0.50
1.00
2.50
volumetric flask (mL)
10
25
50
100
250
Measure the absorbance, A, for each sample at a wavelength of 526 nm using distilled water as
the reference (do not include more than one sample spectrum in your report). Calculate the
concentration of erythrosin B in each Level IV sample using Beer’s law.
Convert these values to % w/w erythrosin B by accounting for the sample’s mass and its dilution.
For your last replicate sample of nominal mass 2.50 g, measure the absorbance 10 additional
times (using different aliquots of solution) and calculate the % w/w erythrosin B for each trial.
Determining the Total Variance
For each nominal mass, calculate the variance using the associated six replicate samples (do not
include the additional 10 measurements on your last replicate sample of nominal mass 2.50 g).
These variances are the total variances for the analysis of a given nominal mass, and include the
variances in sampling, the variances in preparing samples for analysis and the variances in
measuring absorbance.
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Determining the Variance Due to Measuring Absorbance
Calculate the variance for your 10 absorbance measurements for the last replicate sample of
nominal mass 2.50 g. This variance is that due to the measurement of absorbance and is the
same for all nominal masses.
Determining the Variance Due to Sample Preparation.
For each nominal mass, use a propagation of error to estimate the variance due to sample
preparation. Use the average sample size and the average % w/w erythrosin B for each nominal
mass.
Determining the Variance Due to Sampling
Now that you have values for the total variances, the variance due to the measurement of
absorbance, and the variances due to sample preparation, you can calculate the variance due to
sampling for each nominal mass.
Analysis of Data
Your report should include several neatly prepared tables summarizing your data and your
results. Give careful consideration as to how you can present this information in a manner that is
clear. Include a plot of % w/w erythrosin B as a function of the actual mass taken (this graph will
have 30 data points) and comment on its significance. The sampling statistics for many well
mixed materials are consistent with the following equation
2
mR = Ks
where m is the sample’s mass, R is the percent relative standard deviation and Ks is Ingamells’
sampling constant. Report Ks for each nominal mass and the average value for Ks. In addition,
briefly answer the following questions.
1. Based on your results, how useful is Ingamells’ sampling constant for predicting the sampling
variance?
2. Using your average value for Ks, add boundary lines to your plot of % w/w erythrosin B as a
function of sample mass representing ±1 standard deviations about your overall average %
w/w erythrosin B. What is the significance of these lines?
3. Based on your results, how large of a sample do you need if you wish to have a percent relative
standard deviation of ±2%? Does this make for a reasonable analysis? Explain. If not, then
discuss how you might further process your sample so that the analysis is more feasible.
Organizational Tip
Experiment 1 requires the preparation of 16 samples and 32 absorbance measurements, and
experiment 2 requires 30 samples and 40 absorbance measurements. Although this seems like
an excessive number of samples, the experiment can be done fairly quickly if the team is
organized. It is suggested that one or two team members be responsible for weighing the
samples and transferring the sample to a volumetric flask; then another team member takes
responsibility for dissolving the sample, diluting to volume, and thoroughly mixing the sample; and
another team member can make absorbance measurements.
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