data practice workbook

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IB ChemistryData

PracticeWorkbook


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Uncertainties in Calculated ResultsGoals: 1.S tate uncertainties as absolute and percentage uncertainties .

2. De termine the overall uncertainties in calculatedresults .The uncertainties in individual measurements can be used to determine the uncertainty in acalculated result. One way to estimate this is to apply the following simple rules:

Rule #1 Add absolute unce rtaintie swhen adding or subtracting numbers.Rule #2 Add percent unce rtaintieswhen multiplying or dividing numbers.

The absolute uncertainty is the recorded uncertainty in the measurement due to the limitationsof the instrument or the user, for example, 0.02 for a quantity that has the value 12.70 0.02.The Qercentuncertainty is the absolute uncertainty expressed as a percentage of the value. Forexample, the percent uncertainty of 123 0.5 is (0.5/123}100 = 0.4%.

Suppose you want to calculate the value and overall uncertainty of X , ignoring units for thesake of simplicity, where

x =A (B-C)Given the values:

A == 123 O .S ; B :::12.70 0.02; c = 4.30 O .O Sx = (123) (12.70 - 4.30) = 123(8.40)X = 1030 ( note correct use of sig figs!)

Now, le t's de termine the overall uncertainty in the calculated result X =1030Absolute uncertainty in (B - C) = 0.02 +0.05 = = 0.07 (add absolute uncertainties)Percent uncertainty in A == (0.5/123)100 = 0.4%Percent uncertainty in (B - C) ::: (0.07/8.40}100 = = 0.8%So....using Rule #2; OVERALL Percent Uncertainty in X = 0.4% + 0.8% = 1.2%-----------------------.----.

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Note : A common protocol is that the OVERALL pe rce nt uncertainty should be c ited to no morethan one significant figure if it is greater tha n or equa l to 2% and to no more than two significa ntfigure s if it is less tha n 2%.

1.2% of 1030 = 12 which rounds to 10So, the answer X ca n be e xpressed as .

x = 1030 1.2% or x = 1030 10This propaga ted overa ll unce rta inty ca n be visua lize d as

~ : J ~ - \ . - - - - - ) ~ ~ ~Y Y-1.2% = -10 +1.2% = +10

True Value = 1057If the actua l value is 1057, then% error = 1030-1057 )( 100 =

1057- 30 )( 100 = -3%1057

You try it!Cons ide r a ca lorime try experiment tha t gave the following results . Calculate the ene rgy givenoff by a particular chemical reaction in kl/rnol. (The "true" va lue is 40.5 kl/mol)

Absolute Perce ntuncertainty unce rta inty

Initia l tempe rature 21 .60.1C L\T= L\T =Final tempera ture 24 .2 0.1 cMass of wa te r heated 200.0 g 0.5 gAmount of limiting 0.0500 mol 0 .0005 molreagent chemicalGiven; Specific Heat ca pacity of water = 4.184 0.005 ligO C


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Simple Propagation of Error

Note: Unit are ignored on this worksheet to focus on the process of determining OVERALLuncertaintyin a result.1. Solve for W;

K= 0.0162 0.0001L = 0.0371 0.0001

2. Solve for V;

M = 16.2 0.2N = 11.80.2P = 2555 5

W = K+L

V = M NP


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3. Solve for Y

D= 17.1 0.1E=10.10.1F = 15.0 0.5

4. Solve for Z;

G = 5.70 0.05H =2.60 0.05I = 6.22 0.02J = 3.14 0.02

Y = {D +E }F

Z = (G) (H)(I - J )


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Practicing P ropagating ErrorThe following data has been collected in the lab. A group of students used a dry 100 mlgraduated cylinder to record the mass and the volume of various samples of pure ethanol. Thegoal of this "dry" lab activity is to determine the density of the ethanol and propagate error dueto the uncertainties in the instruments through the calculations.The table below lists sample data collected by a pair of chemistry students using standard labequipment. The graduated cylinder was marked in 1 mL intervals and the electronic balancehad a precision of 0.01 g. Thus, all volumes were recorded to the nearest 0.1 mL and all massesare recorded to 0.01 g.The students found the mass and volume of 5 different samples of ethanol using a range ofdifferent volumes.Data Tab le 1 :The Ma s s and Volume Measureme nts of Samplesof Pure Ethanol a t RoomTemperature._--Measurement Trial 1 Trial 2 Trial 3 Trial 4 TrialS Trial 6Mass of grad. cylinderonly/g (O.Olg) 43.43 43.43 43.43 43.43 43.43Mass of grad. cylinderand ethanol / g 47.78 55.22 65.96 73.27 124.29(O.Olg)Volume of ethanol / mL(O.l mL) 5.0 15.0 27.0 37.3 100.0. _ . .Qualitative Data; The ethanol was a clear, colorless, transparent liquid. The odor was that of alcohol,pungent and strong.

Re sultsTable 1: Ca lculatedResults for the Mass and Density of Pure Ethano lMeasurement Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6Mass of ethanol /g(+ g)Density of the ethanol /g/mL( -.... g/ml)

Note to s tudents ; as in all recording of data, calculations, and res ults , pay particular attention tothe proper use of significant figure s. S how s ample work in space provided.

1. Calculate the mass of ethanol used in each trial. Fill in the masses in Results Table 1.


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2. Consider the propagation of error in each mass calculation. Did you add, subtract, multiply,or divide to determine each mass? What rule do you follow to determine the uncertainty ineach mass? Record this uncertainty for the mass of ethanol ( g) in Results Table 1.

3. Calculate the density of ethanol in each trial. Fill in the densities in Results Table 1.

4. Consider the propagation of error for each density calculation. Did you add, subtract,multiply, or divide to determine each density? What rule do you follow to determine theuncertainty in each density? We will come back later to consider what to write in the( g/mL).

5. Compare the precision of the mass and the volume readings of Trials 1-5. Identify which ofthe five trials is the least precise and which is the most precise estimate for the density ofethanol.

6. Now this is important and requires insightful thinking you can do it.. ; Look at how thedata was collected. What aspects of collecting the data for volume and/or mass caused theprecision in the calculated densities to vary?

7. At 20C, the density of ethanol is 0.789 g/rnl. Which of the five trials is the most accurate?Defend your choice.

8. Determine the average density of the five trials and calculate the % error.


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9 Which trial is limiting your significant figures in the average? What is the fundamental issuewith this trial that reduces the number of sig figs?

10.Assume a 6th trial is done and the 1s t trial is omitted in the calculations. Add the followingdata to Data Table 1and continue the calculations for Trial 6 in Result Table 1;

mass of cylinder and ethanol = 101.21g volume of ethanol =72.5 mL the same graduated cylinder and balance were used

What is the new average density of the ethanol using Trials 2-6?

11.Now we will propagate the error to determine the OVERALL percent uncertainty and theOVERALL absolute uncertainty in the new average density.

Density =MassVolume

First consider the mass in the numerator of the density equation. This mass is the resultof subtraction, so the propagated absolute uncertainty is already recorded in ResultsTable 1as; [f;{ass of ethanol/g ( g)j

12. Each of the masses of ethanol in Trials 2-6 would have the same absolute uncertainty and adifferent percent uncertainty. Calculate only the value for the mass measurement thatwould have the largest percent uncertainty and write it below;

13.Consider the volume measurements in the denominator of the density equation. Thisvolume is the result of a direct reading, so the uncertainty as reported in Data Table 1is;Rolume of ethanol / mL (O.l mL)1Each of the volumes of ethanol in Trials 2-6 would have the same absolute uncertainty anda different percent uncertainty. Calculate only the value for the volume measurement thatwould have the largest percent uncertainty and show that calculation below;


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14. Use your answers from #12 and #13 to determine the largest OVERALL percent uncertaintyin the average density of ethanol.

Density =MassVolume

15. Organize the results of your calculations results into a visual representing the averagedensity and the range considering uncertainties.

Overall Percent Overall AbsoluteAverage Uncertainty UncertaintyDensity =

Average Density

' - - - - - I 1 1 -


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9 Classify each of the following laboratory errors1 ---- Description of Experimental Error in a lab Type of er ro r (b lunder,I~

systematic or randomi a. Youmass the same aluminum bar 4 times and get massesof10.34, 10.33g, 10.33 g, and 10.34g. You wonder why the mass isnot consistent.~-----b. The thermometer you are using does not record a temperature of

! 100.0C O.5C in boiling water. Instead your thermometerdisplays a temperature of 97.SoCin boiling water. Now you usethe same thermometer to read the room temperature as 21.0C. .. , --_ .,._ ' _ . . ,-,_ . .- --------c . Tim carefully masses out 10.5634 grams of a salt to dissolve in

Iwater and make a particular concentration of a solution.However, he misses the opening of the bottle as he pours the saltand ends up spilling some of the salt on the lab counter.i-i d. Vicki is reading a graduated cylinder of water by looking down atI e . the cylinder which is placed on the lab counter at waist height.The temperature of water in a can being heated with a Bunsenburner is measured with the thermometer touching the insidebottom of the can instead of being suspended in the water.

f. 5 different students measure the length of a strip of magnesiumwith a centimeter ruler and the lengths range from 3S.10cmto3S.20cm O.OScm.The average length is 3S.1Scm. - - --_._ .. .-g. Bill uses a ruler to measure the length of a block of metal butnotices that the end of the ruler isworn down. Severalmillimeters are missing fr?m t~e end of the ruler.

h. Suemisreads a 6 for a 9 on the balance with a digital readout.i. The thermostat to regulate your furnace is set to turn on at 65F.

However, the thermostat is located in the same room as thewood burning fireplace.

j. Beckyuses a beaker to measure 25 ml of water for a labexperiment in which a precise quantity of water is required.

True or False10. Increasing the number of measurements will improve random error.11. Personal careless errors are sometimes called bias.12. Great care in lab techniques will produce perfect measurements.13. Data considered to be an outlier might be discarded if justified with statistical work.14. Percent error measures accuracy of your results.15. Random errors always skew data in one direction.16. Systematic error can be identified and corrected.17. Random errors can be identified and corrected.18. Random errors scatter the data around a central value.19. Assessment of errors is addressed in more than one section of your IB lab report.20. Error analysis is useful for becoming critical, creative thinkers and problem solvers.

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