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## Uncertainty & Errors in Measurement

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**Keywords**• Uncertainty • Precision • Accuracy • Systematic errors • Random errors • Repeatable • Reproducible • Outliers**Measurements = Errors**• Measurements 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**• ____ significant figures in 62cm3 • ____ significant figures in 100.00 g. The 0s are significant in (2) What is the uncertainty range?**Random (Precision) Errors**• An 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 by**• The 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 Errors**• By 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 • What is the mean temperature? The temperature is reported as as it has a range of Read example in the notes.**Systematic Errors**• An 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 ____________. • 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 Errors**• Control the variables in your lab. • Design a “perfect” procedure ( not ever realistic)**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**systematic error accuracy**random error precision**Putting it together**Example The accurate pH for pure water is 7.00 at 250C. Scenario I You consistently obtain a pH reading of 6.45 +/- 0.05 Accuracy: Precision:**Scenario II**You consistently obtain a pH reading of 8 +/-2 Accuracy: Precision:**Calculations involving addition & subtraction**When 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 & division**When 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)**Example**When 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 ____________**Multiple math operations**Example:**Uncertainties in calculated results**These uncertainties may be estimated by • from the smallest division from a scale • from the last significant figure in a digital measurement • from data provided by the manufacturer**Absolute & Percentage Uncertainty**Consider measuring 25.0cm3 with a pipette that measures to +/- 0.1 cm3. We write Absolute Uncertainty Percentage Uncertainty The uncertainties are themselves approximate and are generally not reported to more than 1 significant fgure.**When adding or subtractingmeasurement , add the absolute**uncertainties Example Initial temperature = 34.500C Final temperature = 45.210C Change in temperature, ΔH**When multiplying or dividing measurement,add the percentage**uncertainties Example Given that mass = 9.24 g and volume = 14.1 cm3 What is the density?**Example**Calculate the following: (a) (b)**Example:**When using a burette , you subtract the initial volume from the final volume. The volume delivered is Final vol = 38.46 Initial vol = 12.15 What is total volume delivered?**Example**The concentration of a solution of hydrochloric acid = moldm-3and the volume = cm3 . Calculate the number of moles and give the absolute uncertainty.**When multiplying or dividing by a pure number,multiply or**divide the uncertainty by that number Example**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**Averaging :**repeated measurements can lead to an average value for a calculated quantity. Example Average ΔH =[+100kJmol-1(10%)+110kJmol-1(10%)+ 108kJmol-1(10%)] 3 = 106kJmol-1(10%)]**Factory made thermometers**Assume 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.**Graphical Technique**• y-axis : values of dependent variable • x-axis : values of independent variables**Plotting Graphs**• Give 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.**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.