Chapter 3 - Experimental error › 2017 › 06 › chapter-3... · / v } µ v ] v P } ( ( v µ u U...
Transcript of Chapter 3 - Experimental error › 2017 › 06 › chapter-3... · / v } µ v ] v P } ( ( v µ u U...
• 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
Significant figures
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.
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.
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.
Significant figures in arithmetic
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
+ 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.
+ 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.
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.
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.
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.
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.
Types of experimental error
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
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.
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
Terms related to experimental errorPrecision and accuracy
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.
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.
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
Terms related to experimental errorAbsolute and relative uncertainty
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.
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.
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.
Propagation of uncertainty from random error
+ 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)
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
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
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
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
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
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
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
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
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
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
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
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
Propagation of uncertainty from systematic error
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
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
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
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
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
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
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
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