1.  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/

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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