1. Measurements and Uncertainties Edward A. Mottel Department of Chemistry Rose-Hulman Institute of Technology

2. Resources Available  Zumdahl: Appendix A1.5-A1.6  Exp. DD: Uncertainties in Measurements and Error Analysis

3. Measurements  Measurement refers to the process of obtaining a physical value by comparing it to an accepted standard. Example: determining the length of a room with a tape measure.

4. Uncertainty  Every measurement is accurate to a degree of uncertainty. tolerance sign, ± It doesn’t refer to a mistake, but rather a recognition that when a measurement is made, it isn’t 100% perfect. Room length = 147.5 ± 0.2 inches typically one significant digit number and its uncertainty agree in decimal place

5. Measurements Without Uncertainty  Some values are not measured but defined. 1 inch = 2.54 cm 1 atm = 760 torr 1 min = 60 seconds Conversion factors of defined values have no uncertainty.  Some relationship values are exact. Two hydrogen atoms per oxygen in water.

6. Determining the Uncertainty in a Measurement  Repetitive measurements standard deviation  Estimation smallest gradation limiting digit  Uncertainties add together buret 22 23 ± 0.03 mL

7. Determining the Uncertainty in a Measurement Direct determination of mass 0.000 g17.345 g Each reading has an uncertainty taresample What is the uncertainty in the mass of the sample? ± 0.000 g± 0.001 g± 0.002 g

8. Determining the Uncertainty in a Measurement Determination of mass by difference 0.000 g17.345 g Each reading has an uncertainty taresample & tray What is the uncertainty in the mass of the sample? ± 0.001 g± 0.002 g± 0.003 g 1.000 g weighing tray ± 0.002 g

9. Determining the Uncertainty in a Measurement The balance is tared, a reactant is weighed, a reaction is performed, the balance is tared, the product is weighed. 0.000 g17.000 g Each reading has an uncertainty tareproduct What is the uncertainty in the lost mass? ± 0.002 g± 0.003 g± 0.004 g 17.345 g reactant 0.000 g tare ± 0.004 g

10. Propagation of Error “Error Analysis”  The uncertainties in several measurements lead to a cumulative uncertainty in a calculated value. Partial differential estimate of uncertainty Sum of percentage uncertainty values Worst case: Maximum-minimum method

11. Maximum-Minimum Estimate of Uncertainty values which are added or subtracted have their uncertainty values added together numbers in the numerator are maximized numbers in the denominator are minimized n = (P atm - VP) V RT

12. Functions Estimate of Uncertainty 25.023.76 25.224.04 25.424.33 25.624.62 25.824.91 26.025.21 Temp (°C)VP (torr) If the temperature of water is 25.4 ± 0.2 °C, what is the vapor pressure and its uncertainty?

13. If 235 ± 3 mL of gas is trapped by water displacement at 25.4 ± 0.2 °C, and the barometric pressure is 732.4 ± 0.2 torr, then how many of moles of gas were trapped and what is the uncertainty in this value? n = 0.008939 n max = 0.009066 digit of uncertainty n = 0.0089 ± 0.0001 mol

15. 6/27/2015