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

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

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.

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 = ± 0.2 inches typically one significant digit number and its uncertainty agree in decimal place

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.

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

Determining the Uncertainty in a Measurement Direct determination of mass g g Each reading has an uncertainty taresample What is the uncertainty in the mass of the sample? ± g± g± g

Determining the Uncertainty in a Measurement Determination of mass by difference g g Each reading has an uncertainty taresample & tray What is the uncertainty in the mass of the sample? ± g± g± g g weighing tray ± g

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 g g Each reading has an uncertainty tareproduct What is the uncertainty in the lost mass? ± g± g± g g reactant g tare ± g

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

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

Functions Estimate of Uncertainty Temp (°C)VP (torr) If the temperature of water is 25.4 ± 0.2 °C, what is the vapor pressure and its uncertainty?

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

6/27/2015