# External Calibration and Propagation of Error

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## External Calibration and Propagation of Error

Learning Objectives

After completing this exercise, students will be able to:

• Identify a single-point external calibration
• Determine the amount of unknown in a sample for a single-point external calibration
• Use propagation of error to determine the relative error in a measurement.
1. A 12.3 mM solution of analyte yields a signal of 489. A solution containing an unknown concentration of analyte yields a signal of 318.  Determine the concentration of analyte in the unknown. (answer = 8.00 mM)
2. A standard solution was prepared by dissolving 25.7 ± 0.1 mg of dye to a volume of 100.00 ± 0.08 mL in a volumetric flask. Calculate the concentration of the standard solution in ppm (mg/L). (answer = 257 ± 2 ppm)
3. The absorbance of the dye standard solution in part (2) was measured 7 times in a 1.00 ± 0.01 cm cuvette, yielding an average absorbance of 0.749 ± 0.002. Determine the absorptivity of the dye in ppm-1 cm-1.  (A = abc, where A is absorbance, a is absorptivity, b is path length, and c is concentration) (answer = 2.91 (± 0.04) x 10-3 ppm-1 cm-1)
4. A solution with an unknown concentration of dye yielded an absorbance of 0.884 ± 0.003 in the same cuvette for 3 replicate measurements. Calculate the concentration of dye in the unknown.  (answer = 304 ± 5 ppm)

## Methods of Calibration

Learning Objectives

After completing this exercise, students will be able to:

• Identify a quantitation as an external calibration or standard addition.
• Identify when an internal standard is used in a quantitation.
• Describe under what circumstances an external calibration, standard addition, and internal standard are used in a quantitation.
• Generate a calibration curve based on a set of external standards.
• Use a calibration curve to determine the amount of unknown analyte in a sample.
• Compare an experimental value to a reported value.

Part 1.  For each of the following situations, identify what type of calibration is being performed.1  Choose between the following:

• external calibration or standard addition
• single- or multi-point
• with or without an internal standard.

Justify your answer by describing the characteristics of the situation that lead to your determination.  (You can use these situations as practice, but you are not required to solve the problem for this assignment.  Answers are provided.)

1. A spectrophotometric method for the quantitative analysis of Pb2+ in blood uses Cu2+ as an internal standard. A standard that is 1.75 ppb Pb2+ and 2.25 ppb Cu2+ yields a ratio of (SA/SIS)std of 2.37. A sample of blood spiked with the same concentration of Cu2+ gives a signal ratio, (SA/SIS)samp, of 1.80. What is the concentration of Pb2+ in the sample of blood? (answer = 1.33 ppb)
2. A spectrophotometric method for the quantitative analysis of Pb2+ in blood yields an Ssamp of 2.80. A calibration curve prepared from standard Pb2+ solutions has the equation S = (2.11 ppb-1) × ppb Pb2+ - 0.006.  What is the concentration of Pb2+ in the original sample of blood? (answer = 1.33 ppb)
3. A spectrophotometric method for the quantitative analysis of Pb2+ in blood yields an Ssamp of 0.712 for a 5.00 mL sample of blood. After spiking the 5.00 mL blood sample with 5.00 µL of a 1560 ppb Pb2+ standard, an Sspike of 1.546 is measured. What is the concentration of Pb2+ in the original sample of blood? (answer = 1.33 ppb)
4. A spectrophotometric method for the quantitative analysis of Pb2+ in blood was designed wherein 1.00 mL of the original blood sample and varying volumes of a 1560 ppb Pb2+ standard were combined in 5.00 mL volumetric flasks. All samples were diluted to volume before measuring the signal. A plot of Sspike(V/V0) versus ppb Pb2+ has the equation Sspike(V/V0) = (1.55 ppb-1) × ppb Pb2+ + 2.06. What is the concentration of Pb2+ in the original sample of blood? (answer = 1.33 ppb)

Part 2.  An experiment was performed to determine the milligrams Cu and Fe in a vitamin tablet.  The following procedure was performed:

1. A vitamin tablet was dissolved in acid and diluted to 100.00 mL with water.
2. A 2.00 mL aliquot of the solution from part 1 is transferred by volumetric pipette to a 100.00 mL volumetric flask and diluted to volume.
3. The solution from part 2 is analyzed with an atomic absorption spectrometer (AA).  The AA instrument reports the absorbance of the element of interest.  Recall that Beer’s Law tells us the absorbance of a sample is directly proportional to its concentration.
4. A series of calibration standards are prepared from Cu and Fe metal standards.  These calibration standards are also analyzed with the AA.

The following data were obtained:

Vitamin Tablet Samples

Sample

AbsCu

AbsFe

1

0.2955

0.6927

2

0.2766

0.6518

3

0.2519

0.6223

4

0.2590

0.6015

Cu calibration data

[Cu] (ppm)

Abs

10.0

0.0635

25.0

0.2772

50.0

0.4723

75.0

0.6912

100.0

0.8814

Fe calibration data

[Fe] (ppm)

Abs

50.0

0.2413

100.0

0.4586

150.0

0.6615

200.0

0.9284

250.0

1.1312

Your goal is to determine the mg of Cu and Fe in the original tablet and compare the experimental values to the manufacturer’s reported values of 150 mg of Cu and 750 mg of Fe.

Learning Objectives

After completing this exercise, students will be able to:

• Identify a quantitation as an external calibration or standard addition.
• Use a single-point standard addition to calculate an analyte concentration.
• Generate a standard addition plot.
• Use a standard addition plot to determine the amount of unknown analyte in a sample.
• Compare an experimental value to a reported value.

Part 1.  Identify what type of calibration is being performed.2  Choose between the following:

• external calibration or standard addition
• single- or multi-point
• with or without an internal standard.

Justify your answer by describing the characteristics of the situation that lead to your determination.  (You can use these situations as practice, but you are not required to solve the problem for this assignment.  Answers are provided.)

1. A tonic water sample is diluted 50-fold. A 3.00 mL aliquot of the diluted tonic water yields a fluorescence intensity of 302.56 a.u.  The 3.00 mL diluted tonic water aliquot is spiked with 35 µL of a 52.3 ppm quinine standard solutions.  The spiked solution yields a fluorescence intensity of 434.05 a.u..  Calculate the concentration of quinine in the tonic water.  (answer = 67.6 ppm)
2. One method for analyzing volatile compounds is called headspace analysis. The headspace is the vapor above a liquid or solid sample in a closed container.  An example is the determination of formaldehyde in pharmaceutical excipients.3  Formaldehyde is a residual in hydroxypropyl methylcellulose, a binding agent.  A 500 mg sample of hydroxypropyl methylcellulose is placed into a 20 mL headspace vial and dissolved in 5 mL of 1% p-toluenesulfonic acid in ethanol. The vial is heated at 60 °C for 15 min under continuous agitation in the headspace autosampler.  Injection of 1.0 mL of the headspace into the GC-MS resulted in a formaldehyde peak with an area of 27843.  Another 500 mg sample of hydroxypropyl methylcellulose is analyzed under the same conditions, except the sample is spiked with 100 µL of a 50.0 µg/mL formaldehyde standard solution prior to agitation.  The peak area of formaldehyde from the spiked sample was 51045.  Determine the concentration of formaldehyde in the hydroxypropyl methylcellulose sample in ppm. Note – MS is a mass-sensitive detector, not concentration-sensitive.  (answer = 12.0 ppm)
3. The concentration of gold in serum was determined. A serum sample (1.00 mL) was transferred to a polypropylene test tube. Saturated KMnO4 (1 mL) was added and the solution mixed with a Vortex mixer. HCl (2 mL, 6 M) was added and the solution was allowed to sit for 10 minutes. The solution was placed in a water bath at approximately 50°C. When the vigor of the chlorine evolution diminished, the temperature of the bath was increased and the solution boiled for 10 minutes. The colorless suspension was cooled to room temperature and 2.00 mL of methyl isobutyl ketone (MIBK) was added. The vial was capped and vigorously shaken for 2 minutes. The MIBK layer, which contained the gold, was separated from the aqueous layer and analyzed with graphite furnace atomic absorption spectroscopy.  The absorbance of the solution of 0.415.  The procedure was repeated using 1.00 mL of serum spiked with 100 µL of a 25.0 µg/L Au standard solution.  The absorbance of the resulting MIBK layer was 0.528.  Calculate the concentration of Au in the serum sample in µg/L. (answer = 9.18 µg/L)

Part 2.  The following procedure was used to quantify the amount of quinine in commercial tonic water.

Procedure

1. Prepare three tonic water samples by transferring 2.00 mL of tonic water by pipet to each of three 100-mL volumetric flasks and dilute each to the mark with 0.10 M H2SO4. Mix well.
2. Transfer 3.00 mL of a tonic water sample by pipet into a fluorescence cuvette and place it in the spectrometer. Acquire the fluorescence spectrum and record the fluorescence intensity at the peak maximum.
3. Add 0.035 mL of a 52.3 ppm quinine stock solution to the sample cuvette. Mix the sample solution with a disposable pipette by drawing in the solution and expelling it.  Acquire the fluorescence spectrum and record the fluorescence intensity.
5. Remove the cuvette from the spectrometer and thoroughly rinse it with DI water.
6. Repeat with the other two tonic water samples.

Data

The following data were acquired:

mL quinine

Intensity

Trial 1

Trial 2

Trial 3

0.000

302.56

298.42

294.74

0.035

434.05

421.77

418.86

0.070

549.70

542.78

529.83

0.105

665.39

649.24

646.64

0.140

768.66

757.93

753.56

0.175

870.26

858.85

858.85

Data Analysis

1. Calculate the ppm of quinine in tonic water.  Report the average, standard deviation and 95% CI. (answer = 75.8 ± 1.5 ppm, 75.8 ± 3.8 ppm)
2. The Food and Drug Administration limits the amount of quinine in tonic water to 83 ppm.  Is the commercial tonic water below the limit?

Learning Objectives

After completing this exercise, students will be able to:

• Generate a standard addition plot.
• Use a standard addition plot to determine the amount of unknown analyte in a sample.
• Describe when an internal standard is needed in a quantitative analysis.
• Determine the amount of unknown in a sample in the presence of an internal standard.
1. To determine the concentration of analyte in a sample, a standard addition was performed. A 5.00-mL portion of sample was analyzed and then successive 0.10-mL spikes of a 600.0-mg/L standard of the analyte were added, analyzing after each spike. The following table shows the results of this analysis.
 Vspike (mL): Stotal (arbitrary units): 0 0.1 0.2 0.3 0.119 0.231 0.339 0.442

Construct an appropriate standard addition plot and use a linear regression analysis to determine the concentration of analyte in the original sample.4 (answer = 12.3 ppm)

2. Mercury(II) can be determined by measuring the absorbance of the complex between Hg2+ and the complexing agent TTC. The Hg2+ in a 5.12 g soil sample was extracted into organic solvent containing excess TTC, and the resulting solution was diluted to 100.0 mL in a volumetric flask.  Five-mL aliquots of the analyte solution were transferred to six 25-mL volumetric flasks.  Various volume of a 5.00 x 10-6 M Hg2+ standard solution was added to each flask and diluted to volume.  The absorbance of each solution was measured at 255 nm in 1.00 cm quartz cells.  Calculate the ppm of Hg2+ in the soil sample.5 (answer = 41.3 ppm)

Volume of Standard Solution (mL)

Absorbance at 255 nm

0.00

0.582

2.00

0.689

4.00

0.767

6.00

0.869

8.00

1.009

10.00

1.127

1. In the flame emission determination of sodium, lithium is often added as an internal standard. The following emission intensity data were obtained for solutions containing Na+ and 100 ppm Li+

[Na+] (ppm)

[Li+] (ppm)

Int, Na

Int, Li

0.1

100

0.11

86

0.5

100

0.52

80

1

100

1.8

128

5

100

5.9

91

10

100

9.5

73

A water sample is spiked with 100 ppm Li.  Flame emission spectroscopy yields a Na intensity of 4.4 and a Li intensity of 95.  Determine the concentration of Na+ in the water sample.6 (answer = 3.5 ppm)

## References

[1] Questions taken from Harvey, David.  “Chapter 3.”  Analytical Chemistry 2.1. http://dpuadweb.depauw.edu/harvey_we...rsion_2.1.html

[2] Questions taken from Harvey, David.  “Chapter 3.”  Analytical Chemistry 2.1. http://dpuadweb.depauw.edu/harvey_we...rsion_2.1.html

[3] Mary-Anne del Barrio, Jack Hu, Pengzu Zhou, Nina Cauchon.  Simultaneous determination of formic acid and formaldehyde in pharmaceutical excipients using headspace GC/MS. Journal of Pharmaceutical and Biomedical Analysis, Volume 41, Issue 3, 2006, Pages 738-743.