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Uncertainty & Errors in Measurement. Waterfall by M.C. Escher. Keywords. Uncertainty Precision Accuracy Systematic errors Random errors Repeatable Reproducible Outliers. Measurements = Errors. Measurements are done directly by humans or with the help of - PowerPoint PPT Presentation
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Measurements = ErrorsMeasurements are done directly by humans
or with the help of
Humans are behind the development of instruments, thus there will always be
associated with all instrumentation, no matter
how precise that instrument is.
Uncertainty
When a physical quantity is taken, the uncertainty should be stated.
Example If the balance is accurate to +/-
0.001g, the measurement is 45.310g
If the balance is accurate to +/- 0.01g, the
measurement is 45.31g
Exercise
A reward is given for a missing diamond, which has a reported mass of 9.92 +/- 0.05g. You find a diamond and measure its mass as 10.1 +/- 0.2g. Could this be the missing diamond?
Significant Figures
(1)____ significant figures in 62cm3
(2)____ significant figures in 100.00 g.
Measurements
Sig. Fig. Measurements Sig. Fig.
1000s Unspecified 0.45 mol dm-3 2
1 x 103s 1 4.5 x 10-1s mol dm-3 2
1.0 x 103s 2 4.50 x 10-1s mol dm-3 3
1.00 x 103s 3 4.500 x 10-1s mol dm-3
4
1.000 x 103s 4 4.5000 x 10-1s mol dm-3
5
The 0s are significant in (2) What is the uncertainty range?
Random (Precision) ErrorsAn error that can based on
individual interpretation.Often, the error is the result of mistakes or errors.Random error is not ______ and can
fluctuate up or down. The smaller your random error is, the greater your ___________ is.
Random Errors are caused byThe readability of the measuring
instrument.The effects of changes in the
surroundings such as temperature variations and air currents.
Insufficient data.The observer misinterpreting the
reading.
Minimizing Random ErrorsBy repeating measurements. If the same person duplicates the
experiment with the same results, the results are repeatable.
If several persons duplicate the results, they are reproducible.
10 readings of room temperature
19.9 , 20.2 , 20.0, 20.0, 20.1, 19.9, 20.3, 19.9, 20.2, 22.3
(a) What is the mean temperature?
The temperature is reported as
as it has a range of
Read example in the notes.
Systematic ErrorsAn error that has a fixed margin,
thus producing a result that differs from the true value by a fixed amount.
These errors occur as a result of poor experimental design or procedure.
They cannot be reduced by repeating the experiment.
10 readings of room temperature
20.0 , 20.3 , 20.1, 20.1, 20.2, 20.0, 20.4, 20.0, 20.3
All the values are ____________.(a) What is the mean temperature?
The temperature is reported as
19.9 , 20.2 , 20.0, 20.0, 20.1, 19.9, 20.3, 19.9, 20.2
Examples of Systematic Errors
Measuring the volume of water from the top of the meniscus rather than the bottom will lead to volumes which are too ________.
Heat losses in an exothermic reaction will lead to ______ temperature changes.
Overshooting the volume of a liquid delivered in a titration will lead to volumes which are too ______ .
Minimizing Systematic ErrorsControl the variables in your lab.Design a “perfect” procedure
( not ever realistic)
Errors
Systematic errors
Apparatus
Way in which readings are taken
Random errors
Equal chance of reading being high or
low from 1 measurement to the
next
How trustworthy is your reading?
Accuracy•How close to the accepted(true) value your reading is.
Precision•The reproducibility of your reading•Reproducibility does not guarantee accuracy. It could simply mean you have a very determinate systematic error.
If all the temperature reading is 200C but the true reading is 190C .
This gives us a precise but inaccurate reading.
If you have consistently obtained a reading of 200C in five trials. This could mean that your thermometer has a large systematic error.
systematic error accuracy
random error precision
Putting it together
Example The accurate pH for pure water is 7.00
at 250C.
Scenario IYou consistently obtain a pH
reading of 6.45 +/- 0.05Accuracy:
Precision:
Calculations involving addition & subtractionWhen adding and subtracting, the final
result should be reported to the same number of decimal places as the least no. of decimal places.
Example:(a) 35.52 + 10.3 (b) 3.56 – 0.021
Calculations involving multiplication & divisionWhen adding and subtracting, the final
result should be reported to the same number of significant figures as the least no. of significant figures .
Example:(a) 6.26 x 5.8 (b) 5.27
12
ExampleWhen the temperature of 0.125kg of
water is increased by 7.20C. Find the heat required.
Heat required = mass of water x specific heat capacity x
temperature rise= 0.125 kg x 4.18 kJ kg-1 0C-1 x 7.20C=
Since the temperature recorded only has 2 sig fig, the answer should be written as ____________
Uncertainties in calculated resultsThese uncertainties may be
estimated byfrom the smallest division from a
scalefrom the last significant figure in
a digital measurementfrom data provided by the
manufacturer
Absolute & Percentage Uncertainty
Consider measuring 25.0cm3 with a pipette that measures to +/- 0.1 cm3.We write
325.0 0.1cmAbsolute Uncertainty
0.1100% 0.4%
25.0
Percentage Uncertainty
The uncertainties are themselves approximate and are generally not reported to more than 1 significant fgure.
Percentage Uncertainty & Percentage Error
absolute uncertaintyPercentage uncertainty = 100%
measured value
accepted value-experimental valuePercentage error = 100%
accepted value
When adding or subtracting measurement , add the absolute uncertainties
ExampleInitial temperature = 34.500CFinal temperature = 45.210CChange in temperature, ΔH
00.05 C
00.05 C
When multiplying or dividing measurement, add the percentage uncertainties
ExampleGiven that mass = 9.24 g and volume = 14.1 cm3
What is the density?
0.005g
30.05cm
Example:When using a burette , you
subtract the initial volume from the final volume. The volume delivered is
Final vol = 38.46Initial vol = 12.15What is total volume delivered?
30.02cm
ExampleThe concentration of a solution of
hydrochloric acid = moldm-
3 and the volume = cm3 . Calculate the number of moles and give
the absolute uncertainty.
1.00 0.0510.0 0.1
When multiplying or dividing by a pure number, multiply or divide the uncertainty by that number
Example
4.95±0.05 ×10
Powers : When raising to the nth power,
multiply the percentage uncertainty by n.
When extracting the nth root, divide the percentage uncertainty by n.
Example 34.3±0.5cm
Averaging : repeated measurements can lead to an
average value for a calculated quantity.ExampleAverage ΔH=[+100kJmol-1( 10%)+110kJmol-1 ( 10%)+ 108kJmol-1 ( 10%)] 3= 106kJmol-1 ( 10%)]
Factory made thermometersAssume that the liquid in the
thermometer is calibrated by taking the melting point at 00C and boiling point at 1000C (1.01kPa).
If the factory made a mistake, the reading will be biased.
Instruments have measuring scale identified and also the tolerance.
Manufacturers claim that the thermometer reads from -100C to 1100C with uncertainty +/- 0.20C.
Upon trust, we can reasonably state the room temperature is
20.10C +/- 0.20C.
Plotting GraphsGive the graph a title.Label the axes with both quantities and
units.Use sensible linear scales – no uneven
jumps.Plot all the points correctly.A line of best fit should be drawn clearly.
It does not have to pass all the points but should show the general trend.
Identify the points which do not agree with the general trend.
Line of Best Equation
Temperature (0 C) Volume of Gas (cm3)
20.0 60.0
30.0 63.0
40.0 64.0
50.0 67.0
60.0 68.0
70.0 72.0
10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.054.0
56.0
58.0
60.0
62.0
64.0
66.0
68.0
70.0
72.0
74.0
Change in volume of a fixed gas heated at a constant pressure
temperature (0C)
Volu
me (
cm
3)
Graphs can be useful to us in predicting values.
Interpolation – determining an unknown value within the limits of the values already measured.
Extrapolation – requires extending the graph to determine an unknown value that lies outside the range of the values measured.
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