Chapter 3 - Experimental error › 2017 › 06 › chapter-3... · / v } µ v ] v P } ( ( v µ u U...

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• Reading assignment Chapter 0: The analytical process Chapter 1: Chemical measurements Chapter 2: Tools of the trade • Problem set #1 Chapter 0: Problems 0-1, 0-2, 0-4 Chapter 1: Problems 1-12, 1-13, 1-15, 1-17, 1-18, 1-19, 1-20, 1-21, 1-24, 1-28, 1-32 Chapter 2: Problems 2-1, 2-5, 2-14, 2-15

Transcript of Chapter 3 - Experimental error › 2017 › 06 › chapter-3... · / v } µ v ] v P } ( ( v µ u U...

Page 1: Chapter 3 - Experimental error › 2017 › 06 › chapter-3... · / v } µ v ] v P } ( ( v µ u U o } } l Z o ( u } ] P ] = v r W Z } µ v } ( ( } Z o v µ u } ( ] u o

• Reading assignment Chapter 0: The analytical process Chapter 1: Chemical measurements Chapter 2: Tools of the trade

• Problem set #1 Chapter 0: Problems 0-1, 0-2, 0-4 Chapter 1: Problems 1-12, 1-13, 1-15, 1-17, 1-18, 1-19, 1-20, 1-21, 1-24, 1-28, 1-32 Chapter 2: Problems 2-1, 2-5, 2-14, 2-15

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Significant figures

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Significant figures refers to the minimum number of digits needed to express a value without loss of precision.

Brown, , E. LeMay, and B. Bursten. 2000. Chemistry: The Central Science. 8th ed. Phils: Pearson Education Asia Pte. Ltd.Silberberg, M. 2010. Principles of General Chemistry. 2nd ed. New York: McGraw-Hill.

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There are several rules of significant figures.

• Any non-zero digit is significant 5.35 g (3 sf)

• Zeros between nonzero digits are significant 607 K (3 sf)

• Zeros to the left of the first nonzero digit are not significant 3.870 moles (4 sf)

• For numbers with a decimal point, zeros after the last non-zero digit are significant 0.00080 L (2 sf)

Chang, R. 2002. Chemistry 7th ed. Singapore: McGraw-Hill.

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There are several rules of significant figures.

• For numbers without a decimal point, zeros after the last non-zero digit may or may not be significant Use scientific notation to remove the ambiguity

• Counting numbers and conversion factors have unlimited number of significant figures 3 trials (infinite number of sf) 1 L = 1000 mL (infinite number of sf)

Chang, R. 2002. Chemistry 7th ed. Singapore: McGraw-Hill.

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Significant figures in arithmetic

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In rounding off numbers, look at the leftmost digit to be dropped.• If the leftmost digit to be removed…

• less than 5: the preceding number is left unchanged• 5 or greater, the preceding number is increased by 1

• For multi-step calculations, leave some insignificant digits to avoid rounding-off errors

Example: 7.248 gRounded to two significant figures: 7.2 gRounded to three significant figures: 7.25 g

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+ and -: Round off to the least number of decimal places.• For example, the molecular weight of KrF2

• Rounded to the correct number of significant figures: 121.795 g/mol

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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+ and – of numbers in scientific notation:1) Express numbers with the same exponent2) Round off to the least number of decimal places• For example,

• Rounded to the correct number of significant figures: 11.51 x 105 or 1.151 x 106

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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x and /: Round off to the least number of significant figures.• For example,

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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Log: The number of digits in the mantissa of log nshould equal the number of significant figures in n.

• Where n is the antilog of a

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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Antilog: The number of digits in the antilog should equal the number of digits in the mantissa.

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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For graphs that should represent quantitative data: Provide the appropriate amount of tick marks on both x and y axis.

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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Types of experimental error

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Systematic error is consistent and caused by a flaw in equipment or experimental design.

• Examples pH reading from a pH meter that

was incorrectly standardized volume reading from an

uncalibrated buret or pipet

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.https://www.e-education.psu.edu/natureofgeoinfo/c5_p5.html

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There are ways to detect a systematic error.

• Analyze a certified reference material• Analyze a sample that does not contain the analyte being measured

(aka blank sample)• Use a different analytical technique to measure the same quantity• Different people analyze identical samples by the same or different

method

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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Random error is caused by uncontrolled variables that are always present in an experiment. It cannot be corrected.

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.https://www.e-education.psu.edu/natureofgeoinfo/c5_p5.html

• Examples volume reading from a buret or

pipet measurement reading from an

instrument that is subject to electrical noise

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Terms related to experimental errorPrecision and accuracy

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Precision is a measure of how closely individual measurements agree with one another.

Brown, , E. LeMay, and B. Bursten. 2000. Chemistry: The Central Science. 8th ed. Phils: Pearson Education Asia Pte. Ltd.

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Accuracy refers to how close a measured value is to the “true” value.

Brown, , E. LeMay, and B. Bursten. 2000. Chemistry: The Central Science. 8th ed. Phils: Pearson Education Asia Pte. Ltd.

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A precise measurement is not necessarily accurate, and an accurate measurement is not necessarily precise.

Brown, , E. LeMay, and B. Bursten. 2000. Chemistry: The Central Science. 8th ed. Phils: Pearson Education Asia Pte. Ltd.

Ideal case

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Terms related to experimental errorAbsolute and relative uncertainty

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Absolute uncertainty expresses the margin of uncertainty associated with a measurement.• For example, the volume reading from a calibrated buret is

12.35 ± 0.02 mL

• True value is anywhere in the range of 12.33 to 12.37 mL

Absoluteuncertainty

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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Relative uncertainty compares the size of the absolute uncertainty with the size of its associated measurement.

• For example, the volume reading from a calibrated buret is 12.35 ± 0.02 mL

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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Percent relative uncertainty is relative uncertainty x 100.

• For example, the volume reading from a calibrated buret is 12.35 ± 0.02 mL

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company.

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Propagation of uncertainty from random error

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+ and -: uncertainty is based on the absolute uncertainties individual terms.

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

Final value: 3.06 (± 0.041)

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The volume delivered by a buret is the difference between the final and initial readings. Suppose that the initial reading is 0.05 (± 0.02) mL and the final reading is 17.88 (± 0.02) mL, what is the uncertainty in the volume delivered?

Final value: 17.83 (± 0.03)

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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x and /: first step is to convert all uncertainties into percent relative uncertainties.

Percent relative uncertainty e1 = * 100

= 1.7%

0.03

1.76

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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x and /: first step is to convert all uncertainties into percent relative uncertainties.

Percent relative uncertainty e2 = * 100

= 1.1%

0.02

1.89

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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x and /: first step is to convert all uncertainties into percent relative uncertainties.

Percent relative uncertainty e3 = * 100

= 3.4%

0.02

0.59

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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x and /: first step is to convert all uncertainties into percent relative uncertainties.

absolute uncertainty

percent relative uncertainty

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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x and /: next step is to calculate the error of the product or quotient.

Final value: 5.64 (± 4.0%)

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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It is possible to convert percent relative uncertainty into absolute uncertainty.

4.0%

absolute uncertainty

percent relative uncertainty

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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Mixed operations: split them into + - and x /.

Step 1: Work on the difference in the numerator using absolute uncertainties

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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Mixed operations: split them into + - and x /.

Step 2: Work on the quotient using relative uncertainties

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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Mixed operations: split them into + - and x /.

absolute uncertainty

percent relative uncertainty

absolute uncertainty

percent relative uncertainty

Rounded off,

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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Real rule for significant figures: The first digit of the absolute uncertainty is the last significant digit in the answer.

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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You prepared a 0.250 M NH3 solution by diluting 8.45 (± 0.04) mL of 28.0 (± 0.5) wt% NH3[density = 0.899 (± 0.003) g/mL] up to 500.0 (± 0.2) mL.Find the uncertainty in 0.250 M.The molar mass of NH3 is 17.0305 g/mol, whose uncertainty is negligible relative to other uncertainties in this problem.

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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Propagation of uncertainty from systematic error

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Atomic mass of the same atom from different sources approximates a rectangular distribution.

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

Where x = averagea = range on each side

of the average

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Uncertainties in the masses of the atoms of a molecule are obtained by multiplying the standard uncertainties by the number of atoms of each type.

• For example, C2H4

Standard uncertainty in the masses of individual atoms

Uncertainty in the masses of individual atoms

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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Uncertainties in the masses of the atoms of a molecule are obtained by multiplying the standard uncertainties by the number of atoms of each type.

• For example, C2H4

Uncertainty in the mass of the atoms combined

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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Multiple deliveries from a single pipet or a single volumetric flask approximates a triangular distribution.

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

Where x = averagea = range on each side

of the average

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The uncertainty in the volume of multiple deliveries from a single pipet is obtained by multiplying the standard uncertainty by the number of volume deliveries.

• For example, using a single 25 mL pipet (± 0.03) four times to deliver a total of 100 mLStandard uncertainty in the volume of a single delivery

Uncertainty in the volume of four deliveries

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

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Calibration improves certainty by removing systematic error.• For example, using a single uncalibrated 25 mL pipet (± 0.03)

four times to deliver a total of 100 mL

Standard uncertainty in the volume of a single delivery

Uncertainty in the volume of four deliveries

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

Uncertainty due to systematic error

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Calibration improves certainty by removing systematic error.• For example, a 25 mL pipet was calibrated and found to deliver

24.991 ± 0.006 mL

Total volume of four deliveries

Uncertainty in the volume of four deliveries

Harris, Daniel 2010. Quantitative Chemical Analysis 8th ed. New York: W.H. Freeman and Company

Uncertainty due to random error

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Problem set #1 (continued)

• Chapter 3 Problems 3-1, 3-2, 3-5 a-f, 3-9, 3-10, 3-12, 3-15 a-c


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