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Uncertainty and Error in Measurement ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com http://www.atr.com.my/store/index.php?main_page=product_info&cPath=1_186_261&products_id=429 http://wattsupwiththat.com/2009/06/10/quote-of-the-week-9-negative-thermometers/
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Measurements Science often involves quantifying measurements to a standard. A problem that often arises is that when two different instruments are used to take the same measurement, rarely do they give the exact same reading. So how so we deal with this? ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com
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Useful Definitions First we need to identify a few things to assist us with our understanding of this problem. Precision is the closeness of the experimental results to each other. Accuracy represents closeness to the actual value. Repeatable refers to close measurements which have been taken by one person. Reproducible refers to the case where several similar readings were taken by different people. Systematic Error arises when we use faulty equipment or poor technique when taking a measurement. This type of error tends to accumulate. ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com
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Significant Figures To determine the correct number of sig figs when doing calculations, follow these rules: 1. Multiplication & Division: give as many sig figs in the answer as there are in the measurement with the least number of sig figs. 2. Addition & Subtraction: give the same number decimal places in the answer as there are in the measurement with the least number of decimal places. ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com
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Uncertainty When making a single measurement, absolute uncertainty and percentage uncertainty can be easily calculated. For instance, if a 25.0 cm 3 measuring pipette measures to ±0.1 cm 3, then: Absolute uncertainty is simply 0.1 cm 3, And the percentage uncertainty is 0.1/25.0 x 100% = 0.4% A 50.0 cm 3 measuring device with the same tolerance would have a percentage uncertainty of 0.1/50.0 = 0.2% ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com http://www.cardinal.com/us/en/distributedproducts/images/P/P4675-125.jpg
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Uncertainty Now, if we use this 50.0 cm 3 measuring device to measure a smaller quantity, say 20.0 cm 3, the tolerance doesn’t change, but the percentage uncertainty will. We still have an absolute uncertainty of ±0.1 cm 3, But our percentage uncertainty will be 0.1/20.0 = 0.5% ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com http://www.cardinal.com/us/en/distributedproducts/images/P/P4675-125.jpg
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Uncertainty When we add and subtract two measurements, then we add the uncertainties. Again, using a 25.0 cm 3 ±0.1 cm 3 measuring device, we might actually measure 24.9 cm 3 and 24.9 cm 3 to get a total volume of 49.8 cm 3. Alterntatively, we might measure them to be 25.1 cm 3 and 25.1 cm 3 whereby we’ll get a volume of 50.2 cm 3. So, our measurement would be somewhere between 49.8 cm 3 and 50.2 cm 3. In other words, 50.0 cm 3 ±0.2 cm 3. ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com http://www.cardinal.com/us/en/distributedproducts/images/P/P4675-125.jpg
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Uncertainty When multiplying, dividing, or using powers, then percentage uncertainties should be used in the calculations and then converted back into absolute uncertainty when the final result is reported. For example, let’s say we are performing a titration of an unknown acid to determine its molar mass. ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com http://www.dartmouth.edu/~chemlab/chem3-5/ionx2/overview/procedure.html
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Uncertainty We begin by dissolving 2.500 g of an unknown acid in dH 2 O in a volumetric flask with a final volume of 250 cm 3. Next we use 25.0 cm 3 of a standardized base for each titration. We pipette this volume into a conical flask. Into the burette, we place 50 cm 3 of the acid and perform our titrations. ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com http://www.dartmouth.edu/~chemlab/chem3-5/ionx2/overview/procedure.html
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Uncertainty Now consider we’ve used a digital scale, volumetric flask, a pipette, and a burette for the titration— each one of which contributes to the overall uncertainty of our experiment. The conical flask doesn’t contribute to the error, because it is just a collecting jar. ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com
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Accounting for Uncertainty The balance weighs to ±0.001 g, so the uncertainty is 0.001/2.500 x 100% = 0.04% The pipette measures 25.00 cm 3 ±0.10cm 3, so the uncertainty is 0.10/25.00 x 100% = 0.40% The volumetric flask measures 250.00 cm 3 ±0.15 cm 3, so the uncertainty is 0.15/250.00 x 100% = 0.060% The burette measures 50.00 cm 3 ±0.10 cm 3, so the uncertainty is 0.10/50.00 x 100% = 0.20% Thus the overall uncertainty is: 0.04% + 0.40% + 0.060% + 0.20% ≈ 0.70% ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com
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Accounting for Uncertainty So, if the answer for the molar mass is determined to be 129 g/mol, the uncertainty is 0.70%. Now we must convert it back to absolute uncertainty: 0.007 x 129 = 0.903 g/mol. Thus, the answer should be reported as 129 ± 1 g/mol ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com
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Percent Error Let’s say that the actual value of the molar mass of the acid is 126 g/mol. To find our percentage error, we use the following equation: % Error = |(Observed – Expected)| x 100% Expected = |(129 – 126)| x 100% = 2.4% 126 ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com
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Other Uncertainties There are other sources of uncertainty such as the tolerance of the equipment used to prepare the standard solution as well as the end-point reading. These should be mentioned in your evaluation of the data, but do not have to be included in your calculations. ©2010 Travis Multhaupt, M.S., www.travismulthaupt.com
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