Precision and Uncertainties in Common Lab Equipment PDF

Precision and Uncertainties for Common Lab Equipment When you record a scientific measurement, the last digit that you record is understood to have some uncertainty, and to be your best estimate. When reading non-electronic devices such as rulers, thermometers, and glassware, the general rule of th...

Precision and Uncertainties for Common Lab Equipment When you record a scientific measurement, the last digit that you record is understood to have some uncertainty, and to be your best estimate. When reading non-electronic devices such as rulers, thermometers, and glassware, the general rule of thumb is to "read between the lines"! This means that you can estimate one more digit or decimal place than the device is marked. But this rule does NOT APPLY to electronic equipment (such as a balance or electronic thermometer) which gives you a direct digital readout. For these digital devices, your teacher will provide you the precision of the instrument. The following uncertainties apply to careful measurements made by a trained observer: Length (common metric rulers): +/- 0.01 cm (or 0.1 mm) Mass (electronic balances): always +/- one unit in the last digit. This means that a common centigram balance is +/- 0.01 grams; a milligram balance +/- 0.001 grams. Volumetric Glassware  10 mL graduated cylinder: +/- 0.02 mL (always record to 2 decimal places)  25 mL graduated cylinder: +/- 0.1 mL (always record to 1 demical places)  100 mL graduated cylinder: +/- 0.5 mL (always record to 1 decimal place)  500 mL graduated cylinder: +/- 5 mL  50 mL buret: +/- 0.02 mL (always record to 2 decimal places)  10 mL graduated pipet: +/- 0.01 mL (always record to 2 decimal places)  Fixed volume pipets (glass): +/- 0.2 % of the capacity (Ex: 25 mL = +/- 0.05 mL) Beakers and Flasks: Approximately 5% of the capacity. (But of course, you would never use one of these to measure a precise amount of liquid, would you?) Thermometer  (alcohol or mercury): +/- 0.2 oC  TI CBL temperature probe: +/- 0.1 oC pH Measurements  pH paper: +/- 1 pH unit (pH paper gives a "quick and dirty" estimate)  TI CBL pH probe: +/- 0.1 pH units (even though it reads out to 0.01). pressure  TI CBL pressure probe: +/- 2 kPa (even though it may read out to decimal places) Treatment of Error and Uncertainty in Chemistry Measurements How to do an Error Analysis—a self-teaching guide 1. In science, we are always seeking to better understand the world around us. We do this by designing and carrying out experiments according to the scientific method. And in these experiments, we are often concerned with measuring something—coming up with a numerical value for some property or behavior we observe. Eventually, if we can interpret and make sense of all these measurements, we might propose a scientific law that allows us to predict the values in future cases without actually having to conduct the experiment each time. 2. But it takes work and effort, often trial and error, to design good experiments. And even in a well-designed experiment, it is impossible to make absolutely perfect measurements. There are two types of error or uncertainty that will always limit the precision and the accuracy of our results. The two types are called random error and systematic error.  Random error comes from the measuring device itself and depends upon its precision. All measuring devices produce some uncertainty in the last measured digit. We cannot eliminate random error totally. But we can minimize it by using good measuring devices and more importantly, reading them carefully and skillfully to as many significant digits as they allow.  Systematic error refers to errors or limitations that can be avoided. They might be due to an improperly calibrated instrument. Or perhaps we are not reading the instrument correctly. Finally (and most often), perhaps our experimental method was flawed and can be improved by more careful experimental design. 3. How to Deal with Error and Uncertainty—follow these four steps: a. When recording your data, also record the precision (+/-) for all measurements due to random error, depending on the measuring device (See previous page). You can do this either by writing the +/- value after each measurement or by including the +/- value in the heading of a data table column. b. Do your calculations to obtain your experimental result. c. Using the uncertainties in each data element, calculate the percent uncertainty in your result that is due to random error alone. (See para 4 below). d. Now take the literature value of the result and calculate the percent error between your value and the literature value. e. Compare the results of steps (c) and (d) to decide whether random error alone can account for how far you were off the literature value, or whether systematic error also affected your results. 4. Calculating the Uncertainty of a Numerical Result  When you add or subtract data, the uncertainty in the result is the sum of the individual uncertainties. Convert this sum to a percentage. Example 1: Mass of crucible + product: 74.10 g +/- 0.01 g Mass of empty crucible: - 72.35 g +/- 0.01 g Mass of product 1.75 g +/- 0.02 g The individual uncertainties are added to give +/- 0.02 g for the result. Converting to a percentage, (0.02 g / 1.75 g) x 100 = 1 %. This is the percent uncertainty due to random error. Note that the random error introduced by a centigram balance is very tiny when you are weighing quantities of about 5 grams or more. However, the random error of the balance begins to contribute a large uncertainty if you try to weigh a very tiny quantity.  For multiplication and division of data, the percent uncertainty in the result is the sum of the percent errors of each measurement. Example 2: A student did an experiment to measure the density of a liquid. He weighed an empty graduated cylinder, placed a volume of liquid in the cylinder, and then weighed it again.. Her data is shown here: Mass of empty graduated cylinder: 25.64 g +/- 0.01 g Mass of grad cylinder with liquid: 28.02 g +/- 0.01 g Volume of liquid: 3.00 mL +/- 0.05 mL Since density = mass/volume, the student calculates the experimental density value: 2.38 g / 3.00 mL = 0.793 g/mL Suppose the literature value for this density is 0.809 g/mL. Then the percent error between the student’s result and the literature value is [(0.809 - 0.793) / 0.809] x 100 = 1.98 %. Can random error alone account for this difference? To find out, we must calculate the percent uncertainty that is due to random error. The mass of the liquid is (28.02 g - 25.64 g) = 2.38 g +/- 0.02 g (Notice that the mass uncertainty is now +/- 0.02 g because we had to subtract two mass values) What % uncertainty in the mass is this? 0.02 g / 2.38 g x 100 = 1 % Similarly, the % uncertainty in the volume measurement is 0.05 mL / 3.00 mL x 100 = 2 % To find the overall uncertainty of this density value, we simply add 1% (the mass uncertainty) to 2% (the volume uncertainty), for a total uncertainty due to random error of 3%. This percent uncertainty is larger than the student’s overall percent error! This means that random error alone can account for the difference between the student’s value and the literature value. We can say that the student got the same value as the literature value, within the limitations of random error. Systematic error, if present, did not appear to affect the result. Example 3: A student performs a calorimetry experiment to determine the amount of heat transferred in an experiment. He takes the following measurements: Mass of water: 100.00 g +/- 0.01 g (negligible uncertainty) Initial temperature of water: 23.6 +/- 0.2 oC Final temperature of water: 27.4 +/- 0.2 oC Change in temp (T) 3.8 +/- 0.4 oC (11% uncertainty) His calculation would be: Q = (mass) (T) (Cp of water) Q = (100.00 g) (3.8 oC) (4.184 J /g oC) = 1589.92 J = 1.6 kJ How should the student report the result? What is his uncertainty? The balance uncertainty is negligible, and there is no uncertainty shown in the Cp value for water. The biggest source of random error is the thermometer. Since the T value can be plus or minus 11%, the overall result must also be plus or minus 11%, which is the sum of the three uncertainties in the heat transfer equation. Since 1.6 kJ x 11 % = 0.18 kJ So the student should report his result as 1.6 kJ +/- 11%, or 0.18 kJ. Suppose the literature value is 1.7 kJ. Looking at the student’s value, the discrepancy can be explained purely by random error. The student’s result is the same as the literature value within the limitations of his measuring device, since his value can be plus or minus 11%. But what if the literature value was 2.4 kJ?. In this case, random error alone cannot account for the discrepancy. Therefore, some systematic error must have occurred. It is this error that the student should seek to identify and make some suggestions for eliminating it next time.