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Chapter 1 Chemistry and You ‘SI Units of Measure and Uncertainties'

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1 Chapter 1 Chemistry and You ‘SI Units of Measure and Uncertainties'

2 SI Units and Uncertainties  SI Unit (Le Système International d’Unités)  Fundamental units meter (m) kilogram (kg) second (s) ampere (A) Kelvin (K) mole (mol) candela (cd)

3 SI Units and Uncertainties  Derived Units Any unit made of 2 or more fundamental units  m s -1  m s -2  Newton (N) = kg m s -2  Joule (J) = kg m 2 s -2  Watt (W) = kg m 2 s -3  Coulomb (C) = A s

4 Estimation with SI Units  Fundamental Units Mass: 1 kg – 2.2lbs / 1 L of H 2 O / An avg. person is 50 kg Length: 1 m - Distance between one’s hands with outstretched arms Time: 1 s - Duration of resting heartbeat  Derived Units Force: 1 N- weight of an apple Energy: 1 J- Work lifting an apple off of the ground

5 Scientific Notation and Prefixes  SI prefixes Table 1 Gm = 1,000,000,000 m = 1,000,000 km 1 GM = 1 x 10 9 m = 1 x 10 6 km 0.0000000001 s = 1 ?s = ? ms

6 Uncertainties & Errors A. Random Errors 1.Readability of an instrument 2.A less than perfect observer 3.Effects of a change in the surroundings Can be reduced by repeated readings B. Systematic Errors 1.A wrongly calibrated instrument 2.An observer is less than perfect for every measurement in the same way Cannot be reduced by repeated readings

7 Uncertainties & Errors (cont.) An experiment is accurate if…… it has a small systematic error it has a small random error x x x x Systematic error Random errors Perfect An experiment is precise if……

8 Uncertainties & Errors (cont.) Accuracy and Precision: Precise but not accurate Accurate but not precise Precise and accurate! Precision– uniformity Accuracy- conformity to a standard

9 Determining the Range of Uncertainty 1) Analogue scales (rulers,thermometers meters with needles) ± half of the smallest division 2) Digital scales ± the smallest division on the readout If the digital scale reads 5.052g, then the uncertainty would be ± 0.001g 10 40 30 20 50 Since the smallest division on the cylinder is 10 ml, the reading would be 32 ± 5 ml Absolute Uncertainty- has units of the measurement

10 Range of Uncertainty (cont.) 3. Significant Figures The measurement is 14.742 g, the uncertainty of the measurement is 14.742 ±.001 g The measurement is 50ml, the uncertainty of the measurement is 50 ± 1 ml If you are given a value without an uncertainty, assume its uncertainty is ±1 of the last significant figure Examples:

11 Range of Uncertainty (cont.) 4. From repeated measurements (an average) Find the deviations between the average value and the largest and smallest values. Example: A student times a cart going down a ramp 5 times, and gets these numbers: 2.03 s, 1.89 s, 1.92 s, 2.09 s, 1.96 s Average: 1.98 s The average is the best value and the largest deviation is taken as the uncertainty range: Largest: 2.09 - 1.98 = 0.11 s Smallest: 1.98 - 1.89 = 0.09 s 1.98 ± 0.11 s

12 Mathematical Representation of Uncertainty Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm 3 =  g 5.0 cm 3 = 3.0 g cm -3 For calculations, compare the calculated value without uncertainties (the best value) with the max and min values with uncertainties in the calculation. Example 1: Best value m v Density =

13 Mathematical Representation of Uncertainty Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm 3 =  g 4.7 cm 3 = 3.40 g cm -3 Example 1 (cont.) : Maximum value: m v Density = Minimum value: m v Density = =  g 5.3 cm 3 = 2.64 g cm -3

14 Mathematical Representation of Uncertainty (cont.) The uncertainty in the previous problem could have been written as a percentage In this case, the density is 3.0 g cm -3 ± 13% yy y =  3 X 100% = 13% The uncertainty range of our calculated value is the largest difference from the best value.. In this case, the density is 3.0 ± 0.4 g cm - 3

15 Mathematical Representation of Uncertainty (cont.) Example #2: What is the uncertainty of cos  if = 60 o ± 5 o ? Best value of cos  = cos 60 o = 0.50 Max value of cos  = cos 55 o = 0.57 Min value of cos  = cos 65 o = 0.42 The largest deviation is taken as the uncertainty range: In this case, it is 0.50 ±.08 OR 0.50 ± 16% Deviates 0.07 Deviates 0.08

16 Mathematical Representation of Uncertainty: Shortcuts! When 2 or more quantities are added or subtracted, the overall uncertainty is equal to the sum of the individual uncertainties. Addition and Subtraction: y = a + b Uncertainty of 2 nd quantity Uncertainty of 1 st quantity Total uncertainty

17 Mathematical Representation of Uncertainty: Shortcuts! (cont.) Determine the thickness of a pipe wall if the external radius is 4.0 ± 0.1 cm and the internal radius is 3.6 ± 0.1 cm Example for Addition and Subtraction: Internal radius = 3.6 ± 0.1 cm External radius = 4.0 ± 0.1 cm Thickness of pipe: 4.0 cm – 3.6 cm = 0.4 cm Uncertainty = 0.1 cm + 0.1 cm = 0.2 cm Thickness with uncertainty: 0.4 ± 0.2 cm OR 0.4 cm ± 50%

18 Mathematical Representation of Uncertainty: Shortcuts! (cont.) The overall uncertainty is approximately equal to the sum of the percentage (or fractional) uncertainties of each quantity. Multiplication and Division: y = a + b + c y a b c Denominators represent best values Total percentage/ fractional uncertainty Fractional Uncertainties of each quantity

19 Mathematical Representation of Uncertainty: Shortcuts! (cont.) Using the density example from before (where the mass was 15 g ± 1 g and its volume is 5.0 ± 0.3 cm 3 ) Example for Multiplication and Division: y = a + b y a b = 1 + 0.3 15 5 = 0.07 + 0.06 =.13 ( this means 13%) 13% of 3 g cm -3 is 0.4 g cm -3 3.0 ± 0.4 g cm -3 or 3.0 g cm -3 ± 13% The result of this calculation with uncertainty is:

20 Mathematical Representation of Uncertainty: Shortcuts! (cont.) Just multiply the exponent by the percentage (or fractional) uncertainty of the number. For exponential calculations (x 2, x 3 ): Cube- each side is 6.0 ± 0.1 cm Example: Percent uncertainty = 1.7% 0.1 6 x 100 %= Volume = (6 cm) 3 = 216 cm 3 Uncertainty in value= 3 (1.7%) = ± 5.1% (or 11 cm 3 ) Therefore the volume is 216 ± 11 cm 3

21 Problems: 1. If a cube is measured to be 4.0+_ 0.1 cm in length along each side. Calculate the uncertainty in volume. Answer: 64+_5 Cm

22 Problem ( IB 2010) The length of each side of a sugar cube is measured as 10 mm with an uncertainty of +_2mm. Which of the following is the absolute uncertainty in the volume of the sugar cube? a.+_6 mmc. +_400 mm b. +_8 mmd. +_600 mm

23 Problem: 3. The lengths and width of a rectangular plates are 50+_0.5 mm and 25+_0.5 mm. Calculate the best estimate of the percentage uncertainty in the calculated area. a. +_0.02%c. +_3% b. +_1 %d. +_5%


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