Montag, 26. September 2011

Error, Uncertainty and Significant figures

Error and uncertainty in practical work



The error is the difference between the result obtained and the generally accepted 'correct' result found in the data book or other literature. If the 'correct' result is available it should be recorded and the percentage error calculated and commented upon in your conclusion. Without the 'correct ' value no useful comment on the error can be made.


The percentage error is equal to:

the difference between the value obtained and the literature value    x 100
                              the literature value

Uncertainty occurs due to the limitations of the apparatus itself and the taking of readings from scientific apparatus. For example during a titration there are generally four separate pieces of apparatus, each of which contributes to the uncertainty.

When making a single measurement with a piece of apparatus then the absolute uncertainty and the percentage uncertainty can both be stated relatively easily. For example consider measuring 25.0 cm3 with a 25 cm3 pipette which measures to + 0.1 cm3. The absolute uncertainty is 0.1 cm3 and the percentage uncertainty is equal to:

                        0.1  x 100   =  0.4%
                       25.0



If two volumes or two masses are simply added or subtracted then the absolute uncertainties are added. For example suppose two volumes of 25.0 cm3 + 0.1 cm3 are added. In one extreme case the first volume could be 24.9 cm3 and the second volume 24.9 cm3 which would give a total volume of 48.8 cm3. Alternatively the first volume might have been 25.1 cm3 which when added to a second volume of 25.1 cm3 gives a total volume of 50.2 cm3. The final answer therefore can be quoted between 48.8 cm3 and 50.2 cm3, that is, 50.0 cm3 + 0.2 cm3.


 
When using multiplication, division or powers then percentage uncertainties should be used during the calculation and then converted back into an absolute uncertainty when the final result is presented. For example, during a titration there are generally four separate pieces of apparatus, each of which contributes to the uncertainty.

e.g. when using a balance that weighs to + 0.001 g the uncertainty in weighing 2.500 g will equal

                               0.001 x 100 = 0.04%
                               2.500

Similarly a pipette measures 25.00 cm3 + 0.04 cm3.

The uncertainty due to the pipette is thus 0.04 x 100 = 0.16%
                                                                  25.00

Assuming the uncertainty due to the burette and the volumetric flask is 0.50% and 0.10% respectively the overall uncertainty is obtained by summing all the individual uncertainties:

Overall uncertainty = 0.04 + 0.16 + 0.50 + 0.10 = 0.80% ~ 1.0%

Hence if the answer is 1.87 mol dm-3 the uncertainty is 1.0% or 0.0187 mol dm-3

The answer should be given as 1.87 + 0.02 mol dm-3.


 
If the generally accepted ‘correct’ value (obtained from the data book or other literature) is known then the total error in the result is the difference between the literature value and the experimental value divided by the literature value expressed as a percentage. For example, if the ‘correct’ concentration for the concentration determined above is 1.90 mol dm-3 then:


the total error = (1.90 – 1.87)  x 100  = 1.6%.
                             1.9

Significant figures

Whenever a measurement of a physical quantity is taken there will be uncertainty in the reading. The measurement quoted should include the first figure that is uncertain. This should include zero if necessary. Thus a reading of 25.30oC indicates that the temperature was taken with a thermometer that is accurate to + or - 0.01oC. If a thermometer accurate to only + or - 0.1oC was used the temperature should be recorded as 25.3oC.



Zero can cause problems when determining the number of significant figures. Essentially zero only becomes significant when it comes after a non-zero digit (1,2,3,4,5,6,7,8,9).



000123.4        0.0001234                                1.0234         1.2340

              zero not a significant figure               zero is a significant figure

              values quoted to 4 sig. figs.              values quoted to 5 sig. figs.



Zeros after a non-zero digit but before the decimal point may or may not be significant depending on how the measurement was made. For example 123 000 might mean exactly one hundred and twenty three thousand or one hundred and twenty three thousand to the nearest thousand. This problem can be neatly overcome by using scientific notation.

            1.23000 x 106  quoted to six significant figures

            1.23 x 106        quoted to three significant figures.



Calculations.

1. When adding or subtracting it is the number of decimal places that is important.

e.g.      7.10 g          +          3.10 g          =         10.20 g

          3 sig. figs.                3 sig. figs.                4 sig. figs.

This answer can be quoted to four significant figures since the balance used in both cases was accurate to + or - .01g.



2.  When multiplying or dividing it is the number of significant figures that is important. The number with the least number of significant figures used in the calculation determines how many significant figures should be used when quoting the answer.

e.g. When the temperature of 0.125 kg of water is increased by 7.2oC the heat required =

       0.125 kg  x  7.2oC  x 4.18 kJ kg-1 oC-1 = 3.762 kJ.

Since the temperature was only recorded to two significant figures the answer should strictly be given as 3.8 kJ.

In practice the IB does not tend to penalise in exams if the number of significant figures in an answer differs by one from the correct number (unless the question specifically asks for them) but will penalise if they are grossly wrong.





 Source: Old file, however I think it's from the Geoff Nuess book, Chemistry Companion

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