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Measurements and Calculations

Measurements and Calculations. I. Using Scientific Measurements Scientific Method Percent Error – accuracy / precision Significant Figures Scientific Notation Density. Steps in the Scientific Method. 1. Observations - quantitative - qualitative 2. Formulating hypotheses

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Measurements and Calculations

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  1. Measurements and Calculations I. Using Scientific Measurements Scientific Method Percent Error – accuracy / precision Significant Figures Scientific Notation Density

  2. Steps in the Scientific Method 1. Observations - quantitative - qualitative 2. Formulating hypotheses - possible explanation for the observation 3. Performing experiments - gathering new information to decide whether the hypothesis is valid

  3. Outcomes Over the Long-Term Theory (Model) - A set of tested hypotheses that give an overall explanation of some natural phenomenon. Natural Law - The same observation applies to many different systems - Example - Law of Conservation of Mass

  4. Law vs. Theory • A law summarizes what happens • A theory (model) is an attempt to explain why it happens.

  5. Nature of Measurement Measurement - quantitative observation consisting of 2 parts Part 1 - number Part 2 - scale (unit) Examples: 20 grams 6.63 x 10-34 Joule seconds

  6. Uncertainty in Measurement • A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty.

  7. Why Is there Uncertainty? • Measurements are performed with instruments • No instrument can read to an infinite number of decimal places Which of these balances has the greatest uncertainty in measurement?

  8. A. Accuracy vs. Precision • Accuracyrefers to the agreement of a particular value with the truevalue. • Precisionrefers to how closely individual measurements agree with each other. ACCURACY = CORRECTNESS PRECISION = CONSISTENCY

  9. Accuracy vs. Precision Precise but not accurate Precise AND accurate Neither accurate nor precise

  10. Types of Error • Random Error • (Indeterminate Error) - measurement has an equal probability of being high or low. • Systematic Error • (Determinate Error) - Occurs in the same direction each time (high or low), often resulting from poor technique or incorrect calibration.

  11. B. Percent Error • Indicates accuracy of a measurement.

  12. % error = 7.51 % B. Percent Error • Julie determines the density of copper to be 8.25 g/cm3. Find the % error if the actual density of copper is 8.92 g/cm3.

  13. C. Significant Figures • Indicate precision of a measurement. • Recording Sig Figs • Sig figs in a measurement include the known digits plus a final estimated digit 2.33 cm

  14. C. Significant Figures • Counting Sig Figs • Count all numbers EXCEPT: • Leading zeros -- 0.0025 • Trailing zeros without a decimal point -- 2,500

  15. Rules for Counting Significant Figures - Details • Nonzero integersalways count as significant figures. • 3456has4sig figs.

  16. Rules for Counting Significant Figures - Details • Zeros • Leading zeros do not count as • significant figures. • 0.0486 has 3 sig figs.

  17. Rules for Counting Significant Figures - Details • Zeros • Captive zeros always count as significant figures. • 16.07 has 4 sig figs.

  18. Rules for Counting Significant Figures - Details • Zeros • Trailing zerosare significant only if the number contains a decimal point. • 9.300 has 4 sig figs.

  19. Rules for Counting Significant Figures - Details • Exact numbershave an infinite number of significant figures. • 1 inch = 2.54cm, exactly

  20. C. Significant Figures Counting Sig Fig Examples 23.50 23.50 4 sig figs 3 sig figs 402 402 5,280 5,280 3 sig figs 2 sig figs 0.080 0.080

  21. Sig Fig Practice #1 How many significant figures in each of the following? 1.0070 m  5 sig figs 17.10 kg  4 sig figs 100,890 L  5 sig figs 3.29 x 103 s  3 sig figs 0.0054 cm  2 sig figs 3,200,000  2 sig figs

  22. C. Significant Figures • Calculating with Sig Figs • Multiplication and Division:# sig figs in the result equals the number in the least precise measurement used in the calculation. (13.91g/cm3)(23.3cm3) = 324.103g (13.91g/cm3)(23.3cm3) = 324.103g (13.91g/cm3)(23.3cm3) = 324.103g 324g

  23. Sig Fig Practice #2 Calculation Calculator says: Answer 22.68 m2 3.24 m x 7.0 m 23 m2 100.0 g ÷ 23.7 cm3 4.22 g/cm3 4.219409283 g/cm3 0.02 cm x 2.371 cm 0.05 cm2 0.04742 cm2 710 m ÷ 3.0 s 236.6666667 m/s 240 m/s 5870 lb·ft 1818.2 lb x 3.23 ft 5872.786 lb·ft 2.9561 g/mL 2.96 g/mL 1.030 g ÷ 2.87 mL

  24. C. Significant Figures • Calculating with Sig Figs • Addition and Subtraction: The number of decimal places in the result equals the number of decimal places in the least precise measurement. 224 g + 130 g 354 g 224 g + 130 g 354 g 224 g + 130 g 354 g 3.75 mL + 4.1 mL 7.85 mL 3.75 mL + 4.1 mL 7.85 mL 3.75 mL + 4.1 mL 7.85 mL  350 g  7.9 mL

  25. Sig Fig Practice #3 Calculation Calculator says: Answer 10.24 m 3.24 m + 7.0 m 10.2 m 100.0 g - 23.73 g 76.3 g 76.27 g 0.02 cm + 2.371 cm 2.39 cm 2.391 cm 713.1 L - 3.872 L 709.228 L 709.2 L 1821.6 lb 1818.2 lb + 3.37 lb 1821.57 lb 0.160 mL 0.16 mL 2.030 mL - 1.870 mL

  26. C. Significant Figures • Calculating with Sig Figs • Exact Numbers do not limit the # of sig figs in the answer. exact number (not estimated) (15 students)(1.25 g/st.) = 18.75 g (15 students)(1.25 g/st.) = 18.75 g (15 students)(1.25 g/st.) = 18.75 g 18.8g

  27. 5. (15.30 g) ÷ (6.4 mL) C. Significant Figures Practice Problems = 2.390625 g/mL  2.4 g/mL 6. (18.9 g) - (0.86 g) = 18.04 g  18.0 g

  28. D. Scientific Notation • Converting into Sci. Notation: • Move decimal until there’s 1 digit to its left. Places moved = exponent. • Large # > 1  positive exponentSmall # < 1  negative exponent • Only show sig figs.

  29. 7. 2,400,000 g 8. 0.00256 kg 9. 7  10-5 km 10. 6.2  104 mm D. Scientific Notation Practice Problems 2.4  106 g 2.56  10-3 kg 0.00007 km 62,000 mm

  30. Density Density – the amount of matter present in a given volume of a substance Density = mass of an object volume of object

  31. Algebra D M V

  32. Density Units The mass of the object is expressed in grams and the volume is expressed in: mL or cm3 for solids and liquids L for gases Density units: g/mL or g/cm3 – solids and liquids g/L - gases Other: Pounds/foot3 (English system)

  33. Density Density of water is ~ 1g/mL at room temperature. What is the approximate mass of water in a 0.5 L water bottle? What is the volume of 150 grams of water?

  34. Density Objects with a density greater than 1g/mL sink in water. Objects with a density less than 1g/mL float in water.

  35. Density The density of the elements can be found on the periodic table. Examples: The density of compounds must be looked up in reference “books”.

  36. Need More help with Exponents?

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