Topic 11 Measurement and data processing IB CHEMISTRY Topic 11 Measurement and data processing

11.1 Uncertainties and errors in measurement and results OBJECTIVES Qualitative data includes all non-numerical information obtained from observations not from measurement. Quantitative data are obtained from measurements, and are always associated with random errors/uncertainties, determined by the apparatus, and by human limitations such as reaction times. Propagation of random errors in data processing shows the impact of the uncertainties on the final result. Experimental design and procedure usually lead to systematic errors in measurement, which cause a deviation in a particular direction. Repeat trials and measurements will reduce random errors but not systematic errors. Distinction between random errors and systematic errors. Record uncertainties in all measurements as a range (+) to an appropriate precision. Discussion of ways to reduce uncertainties in an experiment. Propagation of uncertainties in processed data, including the use of percentage uncertainties. Discussion of systematic errors in all experimental work, their impact on the results and how they can be reduced. Estimation of whether a particular source of error is likely to have a major or minor effect on the final result. Calculation of percentage error when the experimental result can be compared with a theoretical or accepted result. Distinction between accuracy and precision in evaluating results.

Errors and uncertainties

Key terms Random error – above or below true value, usually due to limitations of equipment Systematic error – in one direction, usually due to instrument or method error Precision – a measure of the certainty (±) Accuracy – how close the value is to the accepted value

Some causes of systematic error Physical errors in the measuring device Thermometer was dropped and has small air bubbles in it, leaking gas syringe. Improper or sloppy use of measuring device You measured the values in Fahrenheit instead of Celsius, you didn’t select the right size/range for/of the instrument, the instrument wasn’t calibrated or cleaned, parallax error, chemical splashes (can be random error too). Ambient conditions The temperature, pressure, or air currents changed during the experiment. Evaporation.

How to reduce random uncertainty Repetition of at least 3! Range of 5 (to determine a relationship)

Determining uncertainty UNLESS THE INSTRUMENT TELLS YOU, UNCERTAINTY IS MEASURED IN ONE OF TWO WAYS: 1. For glassware and similar instruments, the uncertainty is half the smallest increment of the instrument. 2. For digital instruments, the uncertainty is the smallest digit.

The volume is between 17cm3 and 18cm3. The uncertainty is half the smallest digit = 1/2 = 0.5 So the answer is 17.5 ± 0.5cm3. 9

We can see the markings between 1.6-1.7cm The uncertainty is half the smallest digit 0.1/2 = 0.05 We record 1.65 +/- 0.05cm as our measurement

Example 1 What is the length of the wooden stick? 1) 4.50 cm ± 0.05cm

Example 2 What is the mass on the scale? 1) 0.025 ± 0.0005g

Uncertainty – halfway method If there is one factor with significant uncertainty that will override all other uncertainties use the halfway method. Take the lowest value away from the highest value, divide it by two, and that is your uncertainty For equipment like a stopwatch, use the halfway method (estimates on reaction times are 0.1s)

Problem 1: What is the uncertainty for this data. 8. 00, 6. 00, 10 Problem 1: What is the uncertainty for this data? 8.00, 6.00, 10.00, 12.00 (± 0.01 units) Average = 9 Error = 12-6/2 = 3 Therefore answer is 9 ± 3.

Rules for significant zeros RULE 1. Zeros in the middle of a number are significant. Eg. 94.072 or 94072 RULE 2. Zeros at the beginning of a number are not significant. Eg. 0.0834 RULE 3. Zeros at the end of a number and after the decimal point are significant. Eg. 138.200 RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. Eg. 138200

Practice: Significant zeros All digits count Leading 0’s don’t Trailing 0’s do 0’s count in decimal form 0’s don’t count w/o decimal 0’s between digits count as well as trailing in decimal form 45.8736 .000239 .00023900 48000. 48000 3.982106 1.00040 6 3 5 2 4 16

Rules for significant figures (multiplication and division) RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.

Rules for significant figures (addition and subtraction) RULE 2. Your answer should only have the same number of unit placings as the most imprecise number. Eg. 0.011 + 0.01 = 0.021  0.02 Eg. 90 000 + 900 = 90 900  90 000 18

Practice: Multiplication and division of significant figures 32.27  1.54 = 49.6958 3.68  .07925 = 46.4353312 1.750  .0342000 = 0.05985 3.2650106  4.858 = 1.586137  107 6.0221023  1.66110-24 = 1.000000 49.7 46.4 .05985 1.586 107 1.000 19

Practice: Addition/Subtraction of significant figures 25.5 32.72 320 +34.270 ‑ 0.0049 + 12.5 59.770 32.7151 332.5 59.8 32.72 330

Practice: Addition and Subtraction of significant figures Look for the last important unit placing Most imprecise number .71 82000 .1 .56 + .153 = .713 82000 + 5.32 = 82005.32 10.0 - 9.8742 = .12580 10 – 9.8742 = .12580 21

Key terms Absolute uncertainty – the ± value in a reading Percentage uncertainty – the ± value in a reading divided by the reading Absolute error – measured value less accepted value Percentage error – measured value less accepted value all divided by accepted value

Propagation of uncertainty For addition and subtraction you add the uncertainties together For division and multiplication you add the percentage uncertainties together The final uncertainty is a single significant figure, and the final answer is rounded off to a similar number of places as the final uncertainty

Mass of beaker plus water Example: Table 1: Raw data to determine density of water Trial Mass of beaker (± 0.01g) Mass of beaker plus water (± 0.01g) Volume (± 0.5mL) 1 20.02 30.05 10.0 2 20.00 30.04 3 20.01 30.03 10.5 Average 10.2

Mass of water = Mass of beaker plus water – Mass of beaker = (30.04 ± 0.1) - (20.01 ± 0.01) = 30.04 -20.01 ± 0.1 + 0.01 = 10.03 ± 0.11g

Density of water= Mass of water Volume of water = 10.03 ±0.11 10.2 ±0.5 = 10.03 10.2 ± 0.11 10.03 + 0.5 10.2 x 100% = 10.03 10.2 ± 0.01097+0.04901 x 100% = 10.03 10.2 ± 0.059887 x 100% = 0.9833 ± 5.9887% = 0.9833 ± 0.059887 x 0.9833 = 0.9833 ± 0.058987 = 0.98 ± 0.06g/cm3 Final answer: Density of water is 0.98 ± 0.06g/cm3

Percentage error Used to determine how accurate you have been in the final value arrived at in your experiment. (yes it is similar to percentage uncertainty) Takes your answer and compares it to the ‘true’ value.

Percentage error calculation In an experiment you found the density of a solution to be 1.017g/cm3. The true value is 1.011g/cm3. What is the percentage error? % error = (accepted – measured)/accepted x 100 = (1.011-1.017)/1.011 x 100 = 0.005935 x 100 = 0.5935% (absolute values) = 0.6% (to one significant figure)

Measuring with glassware

Using glassware – graduated or volumetric pipette

Using glassware – burette

Using glassware – volumetric flask

11.2 Graphical techniques OBJECTIVES Graphical techniques are an effective means of communicating the effect of an independent variable on a dependent variable, and can lead to determination of physical quantities. Sketched graphs have labelled but unscaled axes, and are used to show qualitative trends, such as variables that are proportional or inversely proportional. Drawn graphs have labelled and scaled axes, and are used in quantitative measurements. Drawing graphs of experimental results including the correct choice of axes and scale. Interpretation of graphs in terms of the relationships of dependent and independent variables. Production and interpretation of best-fit lines or curves through data points, including an assessment of when it can and cannot be considered as a linear function. Calculation of quantities from graphs by measuring slope (gradient) and intercept, including appropriate units.

Types of graphs Directly proportional

Inversely proportional

Manual calculation of slope

IB graph expectations Missing line of best fit and error bars

IB graph expectations