Chapter 1 Chemistry and You

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

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)

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

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

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

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

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

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

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

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

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

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

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

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%

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%

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

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%

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

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%

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

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

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

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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