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WORKSHOP 3: UNCERTAINTY IS
AN UNAVOIDABLE PART OF LIFE
PART I: Statistical Uncertainty
Recall that radioactive decay is a random process, so in a given minute, identical amounts of a given sample could undergo slightly different amounts of decay. The average value of many trials, therefore, is a good way to report how many counts per minute one can expect to get from a typical sample. In last week’s lab, the class data was pooled to arrive at average values for the number of counts per minute emitted by each radioisotope. The average value by itself, however, does not provide any information about how much fluctuation there was from trial to trial. For example, suppose your class data looked like this for the counts per minute recorded from a particular sample: Trial # cpm 1 1034 2 1033 3 1035 4 1034
Now suppose another lab’s data looked like this: Trial # cpm 1 1950 2 460 3 1000 4 726
In both sets of data, the average value is 1034 cpm. What is different between the two data sets? Which set of data reflects statistical consistency? Which demonstrates large statistical fluctuation? About which average value are you more “certain”?
The standard deviation is a measure of how widely spread out the data are around the average value. Your lab instructor will explain the mathematical expression for standard deviation. The standard deviation is the root-mean-square deviation from the average (mean) value.
- For each individual value, subtract the average value
- square that result so that you are not working with negative numbers
- Find the average of the squared deviations (variance)
- Take the square root of this number Note that Excel uses a slightly different, also common method of calculating standard deviation, which we’ll mention in class.
Question for thought: Which set of data would you expect to have a higher standard deviation – the height of every person in the United States, or the height of every NBA player? Would the average values of these two data sets be the same?
PART B : How to properly report uncertainty: Your lab instructor will review how uncertain values should be reported, and WHY. Together, you will properly report the uncertain values in your results from last week’s lab.
Questions for discussion:
- Why is uncertainty rounded to one sig fig?
- Why do we round the average value to the same precision (i.e. the same place value) as the uncertainty?
PART II: Uncertainty in measurements
Uncertainty can also occur due to the inherent imprecision of measuring instruments. What would you do if a recipe called for 1.453 tablespoons of sugar? Do you have any cooking gadgets that can measure something that precisely? Probably not, so that measurement will be less precise (more uncertain) than the recipe requires! Similarly, instruments in chemistry laboratory have differing levels of precision with which they can measure substances, and the level of precision you need influences which piece of glassware you use. The last digit that you read off of an instrument or piece of glassware is uncertain, but all other digits are precise. For example, if your glassware has markings every 0.5 ml, you would be estimating the final number in between the ½ ml markings.
Questions for discussion: Would you be able to measure 1.43 mL with a 50 mL beaker? Why or why not? Would you be able to measure 1.43 mL with a 10.0 mL graduated cylinder? Why or why not? What is the difference between a 50 mL beaker and a 50.00 mL volumetric flask? How is this difference reflected in the way their volumes are written (i.e. 50 vs. 50.00 mL?)
PART A) “Reading between the lines: recording measured numbers”: There are 4 containers in front of you:
- 25.0 mL graduated cylinder
- 50 mL beaker
- 100.0 mL graduated cylinder
- 125 mL Erlenmyer flask
- In groups of two, determine how much liquid is in each container. Record your answers.
- Now flip over the card in front of each station and choose from the four options which you think is the best way of reporting the amount of liquid in the container. Be prepared to explain to the class why you chose the option you did and why the other options weren’t correct.
Questions for discussion:
- Which of the four pieces of glassware was the least precise? Which was the most precise?
- Suppose you wanted to design a container that was more precise than any of the ones you just used. What features might it have?
- How does estimation play a role in making measurements?
- All four containers had the same amount of liquid. Why must we report them differently?
PART C) Propagation of uncertainty: addition and subtraction
Does 1+ 1 = 2? Maybe, but perhaps not in a given measurement!
Split up into four groups, and do the experiment corresponding to your group:
Group 1:
- Take four 10.0 mL graduated cylinders and one 100.0 mL graduated cylinder.
- Measure out “exactly” (get as close to the line as possible) 5.0 mL of water in each of the four graduated cylinders and pour into the 100.0 mL graduated cylinder. Put parafilm over the graduated cylinders when not pouring to prevent evaporation. Make sure to transfer every drop into the 100.0 mL cylinder.
- Repeat this process again, so now you should expect to have 40.0 mL of water in the 100.0 mL cylinder.
- Do this a third time, so you should expect to have 60.0 mL of water in the 100.0 mL cylinder.
- Record the amount of water you actually measure in the final graduated cylinder. Is it right on the 60.0 mL line? Be prepared to show your data to the rest of the class.
Group 2:
- Take five 100.0 mL graduated cylinders.
- Measure out “exactly” (get as close to the line as possible) 5.0 mL of water in four graduated cylinders and pour into the fifth 100.0 mL graduated cylinder. Put parafilm over the graduated cylinders when not pouring to prevent evaporation. Make sure to transfer every drop when transferring to the fifth 100.0 mL cylinder.
- Repeat this process again, so now you should expect to have 40.0 mL of water in the 100.0 mL cylinder.
- Do this a third time, so you should expect to have 60.0 mL of water in the 100.0 mL cylinder.
- Record the amount of water you actually measure in the final graduated cylinder. Is it right on the 60.0 mL line? Be prepared to show your data to the rest of the class.
Group 3:
- Each group member should obtain a 10.0 mL graduated cylinder and measure 2.0 mL of water.
- Obtain a weigh boat (1 per group) and weigh it on the top loader balance, record weight.
- Tare the balance to re-zero the weight and add one group member’s 2.0 ml of water to the weigh boat, record the weight.
- Tare the balance again without removing the weigh boat or the water in it and have the next person add their 2.0 ml of water and record the weight.
- Repeat Step 4 until all the people in your group have recorded the weight of their individual 2.0 mL of water.
- Remove the weigh boat from the balance and re-zero
- Replace the weigh boat with the combined solutions and record the final weight.
- Does the final weight exactly equal the sum of the individual masses of the four solutions? If not, by how much is it off? Be prepared to share your data with the rest of the class.
Group 4)
- Each group member should obtain a 10.0 mL graduated cylinder and measure 2.0 mL of water.
- Obtain a weigh boat (1 per group) and weigh it on the analytical balance, record weight. ( Be sure to close the door to the balance before recording weight)
- Keeping the doors closed, tare the balance to re-zero the weight and then open a door to add one group member’s 2.0 ml of water to the weigh boat, record the weight (again with doors closed).
- Tare the balance again without removing the weigh boat or the solution in it and have the next person add their 2.0 ml of water and record the weight.
- Repeat Step 4 until all the people in your group have recorded the weight of their solution.
- Remove the weigh boat from the balance and re-zero
- Replace the weigh boat with the combined solutions and record the final weight.
- Does the final weight exactly equal the sum of the individual masses of the four solutions? If not, by how much is it off? Be prepared to share your data with the rest of the class.
Questions for group discussion after making a chart of the data:
a) Why wasn’t the measurement of the sum exactly equal to the sum of the measurements in each case? If it was, why? b) The top loading balance give masses good to +/-0.01 g. Suppose you make 4 individual measurements and you want to know what the mass of the sum is, so you add the four numbers together. What is the maximum amount by which this sum could be off by? How did you arrive at this number? c) Can you come up with a general rule for determining the uncertainty of a value when it’s obtained by adding or subtracting measured values? Go back and use this rule to determine the uncertainties in
PART D) Propagation of uncertainty: multiplication and division.
You are looking after a patient that needs to get a precise concentration of a particular drug. To make this solution, you need to dissolve 0.1000 g of substance A in 20.00 mL of water. This solution can be made in many ways, some of which will lead to a more precise concentration of final drug than others. We will explore 10 different ways of making this solution.
Directions: in partners, dissolve 0.1000 g of “drug A” in 20.00 mL of water using one of the eight methods below (each group should sign up for a method). Record the correct level of precision for the mass and volume (for example, if you were using an analytical balance, you should measure out and write down “0.10000g”, but if you were using a top-loading balance, you’d measure out and write down “0.10g.”You will need to stir for a few minutes with a clean glass stir-rod for all of the compound to dissolve.
Method 1: 25.0 mL graduated cylinder, top-loading balance Method 2: 25.0 mL graduated cylinder, analytical balance Method 3: 100.0 mL graduated cylinder, top-loading balance Method 4: 100.0 mL graduated cylinder, analytical balance Method 5: 50 mL beaker, top-loading balance Method 6: 50 mL beaker, analytical balance Method 7: 100 mL beaker, top-loading balance Method 8: 100 mL beaker, analytical balance
Method 9: (done by instructor): volumetric pipette (20.00 ml), top-loading balance Method 10: (done by instructor): volumetric pipette, analytical balance
After making your solution, fill a test tube 2/3 full with it and bring it to your instructor who will quantitatively measure the concentration using something called a Spec 20 (you will use this next week yourself!).
As a class, make a data table of your measured concentrations on the board.
Questions for discussion: a) Assume drug A has a molar mass of 588.24 g/mol. Calculate the concentration of drug solution you wanted to make, assuming 20.00 mL of total solution. b) Which method came closest to the desired concentration? c) Which method took the longest? d) If the patient can actually tolerate a wide range of concentrations, which method might a company adopt? e) Relate this statement to uncertainty in this experiment: “You’re only as strong as the weakest link”
How could you assign uncertainty to the concentration you calculated in question A? This value required multiplication/division of two measured quantities (assume there is no uncertainty in the molar mass). Can you come up with a rule to assign uncertainty to a value obtained by multiplying or dividing two numbers? Discuss your rule with your lab instructor, then go back and calculate the uncertainties in each of the concentrations reported in the data table containing the class concentration data. Do the actual concentrations fall within the uncertainties?
You will be using the rule you devised as a class to propagate uncertainty when multiplying and dividing measured values in Chem 105.
Optional Material: Preview of Chem 205: Similarly to the last part, your lab instructor will show you a formula – based on the same ideas you just explored – that is used to determine the “probable” range of a value acquired by multiplying or dividing two measured quantities.
Just like you did for standard deviation, the formula for propagation of error in addition and subtraction uses the square root of the sum of the variance in the data, only this time instead of using the absolute errors you use the relative errors. e.g. S = A * B * C
2 2 2 ⎟ ⎠
C
e B
e A
e S
e (^) S A B C 2 2 2
C
e B
e A
e e (^) S S A B C
i .e. the RU of the product or quotient is the square root of the sum of the squares of the RU’s of each measurement.
Question for discussion: Why are absolute uncertainties used to propagate error when adding and subtracting, and why are relative uncertainties used to propagate error when multiplying and dividing? (Think about units as well as mathematical functions.)
