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**Chapter 1 Chemistry and You**

‘SI Units of Measure and Uncertainties'

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**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)

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**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 m2 s-2 Watt (W) = kg m2 s-3 Coulomb (C) = A s

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**Estimation with SI Units**

Fundamental Units Mass: 1 kg – 2.2lbs / 1 L of H2O / 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

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**Scientific Notation and Prefixes**

SI prefixes Table 1 Gm = 1,000,000,000 m = 1,000,000 km 1 GM = 1 x 109 m = 1 x 106 km s = 1 ?s = ? ms

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**Uncertainties & Errors**

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

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**Uncertainties & Errors (cont.)**

An experiment is accurate if…… it has a small systematic error An experiment is precise if…… it has a small random error Systematic error x Perfect Random errors

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**Uncertainties & Errors (cont.)**

Accuracy and Precision: Precise but not accurate Accurate but not precise Precise and accurate! Precision– uniformity Accuracy- conformity to a standard

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**Determining the Range of Uncertainty**

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

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**Range of Uncertainty (cont.)**

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

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**Range of Uncertainty (cont.)**

4. From repeated measurements (an average) 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 Find the deviations between the average value and the largest and smallest values. Largest: = 0.11 s Smallest: = 0.09 s The average is the best value and the largest deviation is taken as the uncertainty range: 1.98 ± 0.11 s

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**Mathematical Representation of Uncertainty**

For calculations, compare the calculated value without uncertainties (the best value) with the max and min values with uncertainties in the calculation. Example 1: Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3 Best value m v Density = = 15 g 5.0 cm3 = 3.0 g cm-3

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**Mathematical Representation of Uncertainty**

Example 1 (cont.): Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3 Maximum value: m v Density = = 16 g 4.7 cm3 = 3.40 g cm-3 Minimum value: m v Density = = 14 g 5.3 cm3 = 2.64 g cm-3

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**Mathematical Representation of Uncertainty (cont.)**

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 The uncertainty in the previous problem could have been written as a percentage Dy y = 0.4 3 X 100% = 13% In this case, the density is 3.0 g cm-3 ± 13%

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**Mathematical Representation of Uncertainty (cont.)**

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

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**Mathematical Representation of Uncertainty: Shortcuts!**

Addition and Subtraction: When 2 or more quantities are added or subtracted, the overall uncertainty is equal to the sum of the individual uncertainties. Uncertainty of 2nd quantity Uncertainty of 1st quantity Total uncertainty Dy = Da + Db

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**Mathematical Representation of Uncertainty: Shortcuts! (cont.)**

Example for Addition and Subtraction: 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 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 cm = 0.2 cm Thickness with uncertainty: 0.4 ± 0.2 cm OR 0.4 cm ± 50%

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**Mathematical Representation of Uncertainty: Shortcuts! (cont.)**

Multiplication and Division: The overall uncertainty is approximately equal to the sum of the percentage (or fractional) uncertainties of each quantity. Dy = Da + Db + Dc y a b c Denominators represent best values Total percentage/ fractional uncertainty Fractional Uncertainties of each quantity

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**Mathematical Representation of Uncertainty: Shortcuts! (cont.)**

Example for Multiplication and Division: Using the density example from before (where the mass was 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3) Dy = Da + Db y a b = = = .13 ( this means 13%) 13% of 3 g cm-3 is 0.4 g cm-3 The result of this calculation with uncertainty is: 3.0 ± 0.4 g cm-3 or 3.0 g cm-3 ± 13%

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**Mathematical Representation of Uncertainty: Shortcuts! (cont.)**

For exponential calculations (x2, x3): Just multiply the exponent by the percentage (or fractional) uncertainty of the number. Example: Cube- each side is 6.0 ± 0.1 cm Volume = (6 cm)3 = 216 cm3 Percent uncertainty 0.1 6 x 100 % = = 1.7% Uncertainty in value = 3 (1.7%) = ± 5.1% (or 11 cm3) Therefore the volume is 216 ± 11 cm3

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Problems: 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

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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

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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. +_0.02% c. +_3% +_1 % d. +_5%

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