Chapter 11

1. Chapter 11 Measurement & Data Processing IB Chemistry Tam Fall 2010

2. Standards • 11.1.1 • Describe and give examples of random uncertainties and systematic errors. • 11.1.2 • Distinguish between precision and accuracy. • 11.1.3 • Describe how the effects of random uncertainties may be reduced. • 11.1.4 • State random uncertainty as an uncertainty range (±). • 11.1.5 • State the results of calculations to the appropriate number of significant figures. • 11.2.1 • State uncertainties as absolute and percentage uncertainties. • 11.2.2 • Determine the uncertainties in results. • 11.3.1 • Sketch graphs to represent dependences and interpret graph behaviour. • 11.3.2 • Construct graphs from experimental data. • 11.3.3 • Draw best-fit lines through data points on a graph. • 11.3.4 • Determine the values of physical quantities from graphs.

3. Uncertainty in Measurement • Different levels of uncertainty depending upon instrument used. • Uncertainty range applies to any experimental value. • Some apparatus state degree of uncertainty.

4. Analogue instruments uncertainty is ± half the smallest division. • Ex. Smallest division of graduated cylinder is 4 cm3 therefore uncertainty is ±2 cm3 • Uncertainty of a digital scale is ± the smallest scale division. • Ex. 100.00 g, last digit uncertain, therefore, degree of uncertainty is ±0.01 g.

5. Significant Figures • Digits in measurement up to and including the first uncertain digit are significant. • Follow the Atlantic –Pacific Rule to help determine number of sig. figs. • Put values into scientific notation to help avoid confusion.

6. Experimental Errors • Difference between recorded value and generally accepted or literature value. • Errors can be random or systematic.

7. Random Errors • Ex. Experimenter approximates reading, an equal probability of being too high or too low. • Caused by: readability of measuring instrument, effects of changes in the surroundings such as temperature variations & air currents, insufficient data, observer misinterpreting the reading.

8. Random errors • Can be reduced through repeated measurements. • If the same person duplicates the experiment with the same result the results are repeatable. • If several experimenters duplicate the results they are reproducible.

9. Systematic Errors • Occur as a result of poor experimental design or procedure. • Can’t be reduced by repeating experiments. • Can be reduced by careful experimental design. • Ex. Measuring volume of water for the top of the meniscus rather than the bottom; overshooting the volume of a liquid delivered in a titration will lead to volumes that are too high; heat losses in an exothermic reaction will lead to smaller temperatures changes; balance not properly zeroed resulting in incorrect measurements.

10. Accuracy & Precision • The smaller the systematic error, the greater will be the accuracy. • The smaller the random uncertainties, the greater will be the precision. • Precise measurements have small random errors & are reproducible in repeated trials. • Accurate measurements have small systematic errors and give a result close to the accepted value.

11. Sig. Figs in Calculations • Multiplication & Division • Ex. What is the density of sodium chloride with a mass of 5.00 ±0.01 g and a volume of 2.3 ±0.1 cm3 ? (D = m/v) • Give range of values for density from ranges of mass & volume.

12. Max value is obtained when max mass is combined with minimum volume. (2.277 gcm-3) • Minimum value is obtained by combining min. mass with max volume. (2.079 gcm-3 ) • Using correct sig. figs (2nd sig. fig is uncertain so 2.2 gcm-3 ) • Precision of density is limited by the volume measurement as it is least precise.

13. Addition & Subtraction • Number of decimal places determine the precision of the calculated value. • Ex. Suppose we need the total mass of two pieces of zinc of mass 1.21 g and 0.56 g. The total mass = 1.77 g can be given to two decimal places as the balance was precise to ±0.01 in both cases. • Ex. Calculate temp. increase from 25.2 to 35.2 oC. (10.0 oC)

14. Percentage Uncertainties & Errors • An uncertainty of 1s is more significant for time measurements of 10s than for 100s. • Helpful to express uncertainty using absolute, fractional, or percentage values. • Fractional uncertainty = absolute uncertainty • measured value

15. Percentage uncertainty = • absolute uncertainty x 100% • measured value • Percent error is a measure of how close the experimental value is to the literature or accepted value. • % Error = accpt. Value – expt. Value x 100% • accpt. value

16. Propagation of Uncertainties • Addition & Subtraction • The uncertainty is the sum of the absolute uncertainties. • Ex. Consider two burette readings: • Initial reading/ ±0.05 cm3 = 15.05 • Final reading/ ±0.05 cm3 = 37.20 • What value should be reported for the volume delivered?

17. Initial range: 15.00-15.10 • Final range: 37.15-37.25 • Max volume is formed by combining the max. final volume with min. initial reading. • Vol max = 37.25 -15.00 = 22.25 cm3 • Min. volume is formed by combining the min. final vol. with the max initial reading. • Vol min = 37.15 – 15.10 = 22.05 cm3 • Vol = 22.15 ±0.1 cm3

18. Addition & Subtraction • The volume depends on two measurements and the uncertainty is the sum of the two absolute uncertainties.

19. Multiplication & Division • Total percentage uncertainty is the sum of the individual percentage uncertainties, when multiplying or dividing measurements. • The absolute uncertainty can then be calculated from the percentage uncertainty. • Ex. Density uncertainty in calculated value is equal to the sum of the uncertainties in the mass and volume values.

20. Mass= 24.0 g ±0.5 • % Uncertainty = (0.5/24.0)x100% = 2% • Volume = 2.0 cm ±0.1 • % Uncertainty = (0.1/2.0)x100% = 5% • Density = 24.0/2.0 = 12.00 g cm-3 • Max Density = 24.5/1.9 = 12.89 g cm-3 • Min Density = 23.5/2.1 = 11.19 g cm-3 • Absolute uncertainty = 12.89 – 12.00 = ±0.89 • % Uncertainty = (0.89/12.00) x 100% = 7.4% • The uncertainty in the calculated value of the density is 7% from the sum of the uncertainties in mass and volume values. This approx. result provides a simple treatment of propagating uncertainties when multiplying & dividing measurements.

21. Graphical Techniques • Shows relationship between the independent (x-axis) and dependent variable (y-axis). • Be sure to include in your graph: a title, label axes with both quantities and units, use the space as effectively as possible, use sensible linear scales (no uneven jumps), plot all the points correctly, best fit line drawn smoothly & clearly to show overall trend, identify outliers, think carefully about the inclusion of the origin.

22. Best fit Straight line • Passes as near to as many of the points as possible. • Does not necessarily pass through any of the points plotted. • Two important properties of a straight line that are useful: the gradient & intercept.

23. Finding the gradient & intercept • Equation for straight line is y = mx + c • X is independent variable, y is dependent variable, m is the gradient, and c is intercept on vertical axis. • The gradient of a straight line is the increase in the dependent variable divided by the increase in the independent variable. • m = ∆y/ ∆x • The gradient units are the units of the vertical axis divided by the units of the horizontal axis.

24. Extrapolation • Sometimes a line has to be extended beyond the range of measurements of the graph. • Ex. Absolute zero can be found by extrapolating the volume/temperature graph for an ideal gas.

25. Interpolation • The process of assuming that the trend line applies between two points. • The gradient of a curve at any point is the gradient of the tangent to the curve at that point.

26. Errors & Graphs • Systematic errors & random measurements can often be recognized from a graph. • A graph combines the results of many measurements and so minimizes the effects of random uncertainties in the measurements.

27. Choosing what to plot • The best way to analyze measurements is to find a way of plotting the data to produce a straight line. Try to rearrange the equations to give a straight line graph. • Ex. PV = nRT becomes a straight line graph when equation is rearranged to: P = nRT(1/V)

28. Using Spreadsheets to Plot Graphs • Many software packages allow graphs to be plotted and analyzed, the equation of the best fit line given and other properties calculated. • Care should be taken • The set of data points can either be joined by a best fit straight line which does not pass through any point except the origin or a polynomial which gives a perfect fit as indicated by the R value of 1. • The polynomial equation is unlikely to be physically significant as any series of random point can fit a polynomial of sufficient length, just as any two points define a straight line.