1. Topic 11: Measurement and data collection SECTION 11.1 – UNCERTAINTIES AND ERRORS IN MEASUREMENT AND RESULTS

2. Qualitative vs. Quantitative Qualitative data: all non-numerical information obtained from observations not from measurement Quantitative data: all data obtained from measurements ◦Always associated with random errors/uncertainties determined by the apparatus and by human limitations, such as reaction times.

3. Uncertainty in measurement Data can be divided into two groups: exact numbers and inexact numbers. Any experiment involving measurements will always have some uncertainty associated with the measured data.

4. Precision and accuracy Precision – the closeness of agreement between independent test results. The smaller the random part of the experimental errors which affect the results, the more precise the procedure. Accuracy – the closeness of the agreement between the result of a measurement and a true value of the measurand (the particular quantity to be measured).

5. measured value

6. Significant Figures (SF) The number of digits that reflect the precision of a given measurement. ◦The greater the number of SFs, the greater the certainty of the measured value Scientific notation helps to determine the number of SFs. ◦Ex: 6.02 x 10 23 atoms

7. Rules for counting SFs What is counted: 1.Every non-zero digit. 2.Every zero between two non- zero digits. 3.Every zero after a decimal point that follows a non-zero digit ( even if the non-zero digit is before the decimal point). What is not counted: 1.Zeroes before a decimal for numbers less than 1. 2.Every digit between the decimal point and a non-zero digit for numbers less than 1. 3.Every zero that follows a non- zero digit in a number expressed with no decimal.

8. PRACTICE: Rewrite the following measurements in scientific notation and determine the number of significant figures. 1.135.680 g 2.0.00620 dm 3 3.6.00 kg 4.2.0600 m 3 5.0.2 mg 6.300 kg 1.35680 x 10 2 g 6.20 x 10 -3 dm 3 6.00 kg 2.0600 m 3 2 x 10 -1 mg 3 x 10 2 kg Six Three Five One

9. Reporting SFs in calculations For addition/subtraction: The result should be expressed based on the measurement with smallest number of decimal places. Ex: (6.702 g) + (5.2 g) = (11.9 g) For multiplication/division: The result should be expressed based on the measurement with the smallest number of significant figures. Ex: (5.4376 g)/(2.31 cm 3 )=(12.6 g/cm 3 ) 3 decimal places 1 decimal place 5 SFs3 SFs

10. PRACTICE: The mass of a sample bottle and a piece of aluminum metal is 35.4200 g. The mass of the empty sample bottle is 28.9200 g. If the aluminum displaces 2.41 cm 3 of water, calculate the density of aluminum, in g cm -3.

11. Special case: logarithms Log of a number is made of two parts: characteristic (integer part) and mantissa (decimal part). Ex: log 10 (2.7) = 0.43 = 4.3 x 10 -1 Ex: ln(6.28) = 1.837 two SFs mantissa (decimal part) two SFs three SFs mantissa (decimal part) three SFs

12. Experimental errors Systematic errors: flaws in the actual experimental design or with the instrument used. Instrumentation errors (ex: leaky gas syringes, faulty calibration of instruments) Experimental methodology errors (ex: poorly insulated calorimeter, measuring from the top of a meniscus instead of the bottom, occurrence of a side reaction) Personal errors (ex: the exact color of a solution at its end point, reading a graduated cylinder incorrectly)

13. Experimental errors Random errors occur because of uncontrolled variables in an experiment and hence cannot be eliminated. Examples: Estimating a quantity which lies between marked graduations of an instrument Not being able to read an instrument due to fluctuations in reading (temperature variations, airflow, changes in pressure) Reaction time

14. X X X X X X – systematic X – random X – perfect X X X X X X X X X X independent dependent

15. Percentage error

16. Examiner’s hints 1. You should compare your data to the literature value where appropriate. 2. When evaluating investigations (or experiment design/results), distinguish between systematic and random errors. 3. When evaluating procedures, you should discuss the precision and accuracy of the measurements. You should specifically look at the procedure and use of equipment.

17. Absolute and relative uncertainty Every single measurement has a degree of uncertainty. That uncertainty needs to be recorded with all measurements. Analog instruments: ±½ the smallest division ◦Ex: 10 mL graduated cylinder: ±0.5 mL Digital instruments: ± the smallest scale division ◦Ex: top-pan balance: ±0.01 g

18. Absolute and relative uncertainty Experimental results should be reported in the following form: Experimental result = (A ± ΔA) unit Where A represents the measured result and ΔA is the absolute uncertainty. Ex: experimental result = (3.56 ± 0.1) g

19. Absolute and relative uncertainty

20. Examiner’s hint The calculated uncertainty is generally quoted to not more than one SF if it is greater or equal to 2% of the answer and to not more than 2 SFs if it is less than 2%. Intermediate values in calculations should not be rounded off to avoid unnecessary imprecision.

22. PRACTICE: A thermometer has an uncertainty of ±0.1 o C. The boiling point of an unknown substance was measured to be 36.1 o C. Calculate the percentage relative uncertainty.

23. Propagation of uncertainty The process of determining how different uncertainties combine to give the resultant uncertainty. Rule 1: when adding or subtracting measurements, the absolute uncertainties are added. Rule 2: when multiplying or dividing measurements, the percentage relative uncertainties are added.

24. EXAMPLE: During a titration the following titres were recorded for a 0.10 mol dm -3 solution of hydrochloric acid from a burette: initial titre = (5.00±0.02) cm 3 final titre = (21.35±0.02) cm 3 Calculate the volume delivered, in cm 3, and the uncertainty of this volume. Volume delivered = 21.35 – 5.00 = 16.35 cm 3 For uncertainty, use rule 1 since subtraction is used: (±0.02 cm 3 )+(±0.02 cm 3 ) = (±0.04 cm 3 ) So volume would be reported as (16.35±0.04) cm 3.

27. PRACTICE: During a thermodynamic experiment, a change in temperature was measured using a thermometer with an absolute uncertainty of ±0.05 o C. Temperature (initial): 24.32 o C Temperature (final): 78.49 o C Calculate the change in temperature and the uncertainty for this measurement.

28. PRACTICE: The mass of a sample bottle and piece of titanium metal is (33.29±0.005) g. The mass of the empty sample bottle is (26.35±0.005) g. If the metal displaced 1.5±0.05 cm 3 of water, what is the density of the metal. What is the percent uncertainty?

29. Special case: averages For a calculated average, the propagated uncertainty for the component numbers can be used in the final answer. Ex: ΔH mean = [+100 kJ mol -1 (±10%) + 110 kJ mole -1 (±10%) + 108 kJ mol -1 (±10%) / 3 ΔH mean = +106 kJ mol -1 (±10%)

30. Graphical Analysis Independent VariableDependent Variable CauseEffect Plotted on horizontal axis (X)Plotted on vertical axis (Y)

31. Correlation Correlation – a measure of the extent to which the two variables change with one another. Positive correlation – the two variables increase or decrease in parallel to one another. Negative correlation – one variable increases while the second variable decreases or vice versa.

33. Checklist for graphing  Give the graph a title  Label axes with both quantities and units  Use space effectively  Use sensible linear scales – no uneven jumps  Plot points correctly  Line of best fit drawn smoothly/clearly – it should show overall trend  Identify points which do not agree with trend  Think about inclusion of origin – may not always be necessary

34. ‘Best fit’ line This is a line that attempted to demonstrate the correlation between the IV and DV. Keep in mind that this line may not include all the data points. Equation of line: y = mx + c ◦Where m is the gradient (slope) and c is the y-intercept Extrapolation – extension of line beyond range of measurements Interpolation – process of assuming that the trend line applies between two points.

35. X X X X X independent dependent

36. Video: Richard Thornley – Drawing Best Fit Lines

37. Graph the following data. Include an extrapolated best-fit line to show absolute zero. Find the slope and overall line equation of your best-fit line (y=mx+c). Independent VariableDependent Variable Temp ( o C)Volume (cm 3 ) 10.050 50.057 100.066