Scientific Measurement 1 Measurement in Chemistry In chemistry
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Scientific Measurement 1
Measurement in Chemistry In chemistry we § Measure quantities. § Do experiments. § Calculate results. § Use numbers to report measurements. § Compare results to standards. 2
Stating a Measurement In every measurement, a number is followed by a unit. Observe the following examples of measurements: Number and Unit 35 m 0. 25 L 225 lb 3. 4 hr 3
Qualitative vs. Quantitative n Qualitative – descriptive, non-numerical form n Quantitative – definite form, usually with numbers and units n Fever example 4
Scientific Notation Scientific notation § Is used to write very large or very small numbers. § For the width of a human hair (0. 000 008 m) is written 8 x 10 -6 m § For a large number such as 4 500 000 s is written 4. 5 x 106 s 5
Writing Numbers in Scientific Notation § A number in scientific notation contains a coefficient and a power of 10. coefficient 1. 5 power of ten x 102 coefficient 7. 35 power of ten x 10 -4 § To write a number in scientific notation, the decimal point is placed after the first digit. § The spaces moved are shown as a power of ten. 52 000. = 5. 2 x 104 4 spaces left 0. 00378 = 3. 78 x 10 -3 3 spaces right 6
Some Powers of Ten TABLE 1. 2 7
Comparing Numbers in Standard and Scientific Notation Here are some numbers written in standard format and in scientific notation. Number in Standard Format Scientific Notation Diameter of the Earth 12 800 000 m 1. 28 x 107 m Mass of a human 68 kg 6. 8 x 101 kg Length of a virus 0. 000 03 cm 3 x 10 -5 cm 8
Learning Check Select the correct scientific notation for each. A. 0. 000 008 1) 8 x 106 2) 8 x 10 -6 3) 0. 8 x 10 -5 B. 72 000 1) 7. 2 x 104 2) 72 x 103 3) 7. 2 x 10 -4 9
Solution Select the correct scientific notation for each. A. 0. 000 008 2) 8 x 10 -6 B. 72 000 1) 7. 2 x 104 10
Learning Check Write each as a standard number. A. 2. 0 x 10 -2 1) 200 2) 0. 0020 3) 0. 020 B. 1. 8 x 105 1) 180 000 3) 18 000 2) 0. 000 018 11
Solution Write each as a standard number. A. 2. 0 x 10 -2 3) 0. 020 B. 1. 8 x 105 1) 180 000 12
Accuracy, Precision, and Error n Accuracy is how close a measurement is to its true value. If weigh yourself and know you weigh 170 lbs, and scale says 20 lbs, not accurate. n Precision is how close a series of measurements are to each other. 13
Accuracy and Precision 14
Significant Figures in Measured Numbers Significant figures § Obtained from a measurement include all of the known digits plus the estimated digit. § Reported in a measurement depend on the measuring tool. 15
n Error = experimental value – accepted value (can be negative or positive) n % error = error / accepted value x 100 n Thermometer examples 16
Significant Figures TABLE 1. 4 17
Counting Significant Figures All non-zero numbers in a measured number are significant. Measurement 38. 15 cm 5. 6 ft 65. 6 lb 122. 55 m Number of Significant Figures 4 2 3 5 18
Sandwiched Zeros Sandwiched zeros § Occur between nonzero numbers. § Are significant. Measurement 50. 8 mm 2001 min 0. 0702 lb 0. 40505 m Number of Significant Figures 3 4 3 5 19
Trailing Zeros Trailing zeros § Follow non-zero numbers in numbers without decimal points. § Are usually place holders. § Are not significant. Measurement 25 000 cm 200 kg 48 600 m. L 25 000 g Number of Significant Figures 2 1 3 5 20
Leading Zeros Leading zeros § Precede non-zero digits in a decimal number. § Are not significant. Measurement 0. 008 mm 0. 0156 oz 0. 0042 lb 0. 000262 m. L Number of Significant Figures 1 3 21
Significant Figures in Scientific Notation In scientific notation § All digits including zeros in the coefficient are significant. Scientific Notation 8 x 104 m 8. 00 x 104 m Number of Significant Figures 1 2 3 22
Learning Check State the number of significant figures in each of the following measurements: A. 0. 030 m B. 4. 050 L C. 0. 0008 g D. 2. 80 m 23
Solution State the number of significant figures in each of the following measurements: A. 0. 030 m 2 B. 4. 050 L 4 C. 0. 0008 g 1 D. 2. 80 m 3 24
Learning Check A. Which answer(s) contains 3 significant figures? 1) 0. 4760 2) 0. 00476 3) 4. 76 x 103 B. All the zeros are significant in 1) 0. 00307 2) 25. 300 3) 2. 050 x 103 C. The number of significant figures in 5. 80 x 102 is 1) one 3) two 3) three 25
Solution A. Which answer(s) contains 3 significant figures? 2) 0. 00476 3) 4. 76 x 103 B. All the zeros are significant in 2) 25. 300 3) 2. 050 x 103 C. The number of significant figures in 5. 80 x 102 is 3) three 26
Learning Check In which set(s) do both numbers contain the same number of significant figures? 1) 22. 0 and 22. 00 2) 400. 0 and 4. 00 x 102 3) 0. 000015 and 150 000 27
Solution In which set(s) do both numbers contain the same number of significant figures? 3) 0. 000015 and 150 000 Both numbers contain two (2) significant figures. 28
Calculations with Measured Numbers In calculations with measured numbers, significant figures or decimal places are counted to determine the number of figures in the final answer. 29
Multiplication and Division When multiplying or dividing use § The same number of significant figures as the measurement with the fewest significant figures. § Rounding rules to obtain the correct number of significant figures. Example: 110. 5 4 SF x 0. 048 = 5. 304 2 SF calculator = 5. 3 (rounded) 2 SF 30
Learning Check Give an answer for the following with the correct number of significant figures: A. 2. 19 x 4. 2 1) 9 = 2) 9. 2 3) 9. 198 B. 4. 311 ÷ 0. 07 = 1) 61. 59 2) 62 3) 60 C. 2. 54 x 0. 0028 = 0. 0105 x 0. 060 1) 11. 3 2) 11 3) 0. 041 31
Solution A. 2. 19 x 4. 2 B. 4. 311 ÷ 0. 07 C. 2. 54 x 0. 0028 0. 0105 x 0. 060 = 2) 9. 2 = 3) 60 = 2) 11 On a calculator, enter each number followed by the operation key. 2. 54 x 0. 0028 0. 0105 0. 060 = 11. 28888889 = 11 (rounded) 32
Addition and Subtraction When adding or subtracting use § The same number of decimal places as the measurement with the fewest decimal places. § Rounding rules to adjust the number of digits in the answer. 25. 2 + 1. 34 26. 5 one decimal place two decimal places calculated answer with one decimal place 33
Learning Check For each calculation, round the answer to give the correct number of significant figures. A. 235. 05 + 19. 6 + 2 = 1) 257 2) 256. 7 B. 58. 925 - 18. 2 = 1) 40. 725 2) 40. 73 3) 256. 65 3) 40. 7 34
Solution A. 235. 05 +19. 6 + 2 256. 65 rounds to 257 B. 58. 925 -18. 2 40. 725 round to 40. 7 Answer (1) Answer (3) 35
The Metric System (SI) The metric system or SI (international system) is § A decimal system based on 10. § Used in most of the world. § Used everywhere by scientists. 36
Units in the Metric System In the metric and SI systems, one unit is used for each type of measurement: Measurement Length Volume Mass Time Temperature Metric meter (m) liter (L) gram (g) second (s) Celsius ( C) SI meter (m) cubic meter (m 3) kilogram (kg) second (s) Kelvin (K) 37
Learning Check For each of the following, indicate whether the unit describes 1) length 2) mass or 3) volume. ____ A. A bag of tomatoes is 4. 6 kg. ____ B. A person is 2. 0 m tall. ____ C. A medication contains 0. 50 g aspirin. ____ D. A bottle contains 1. 5 L of water. 38
Solution For each of the following, indicate whether the unit describes 1) length 2) mass or 3) volume. 2 A. A bag of tomatoes is 4. 6 kg. 1 B. A person is 2. 0 m tall. 2 C. A medication contains 0. 50 g aspirin. 3 D. A bottle contains 1. 5 L of water. 39
Learning Check Identify the measurement that has a SI unit. A. John’s height is 1) 1. 5 yd 2) 6 ft 3) 2. 1 m B. The race was won in 1) 19. 6 s 2) 14. 2 min 3) 3. 5 hr C. The mass of a lemon is 1) 12 oz 2) 0. 145 kg 3) 0. 6 lb D. The temperature is 1) 85 C 2) 255 K 3) 45 F 40
Solution A. John’s height is 3) 2. 1 m B. The race was won in 1) 19. 6 s C. The mass of a lemon is 2) 0. 145 kg D. The temperature is 2) 255 K 41
Prefixes A prefix § In front of a unit increases or decreases the size of that unit. § Make units larger or smaller than the initial unit by one or more factors of 10. § Indicates a numerical value. prefix 1 kilometer = value 1000 meters 1 kilogram = 1000 grams 42
Metric and SI Prefixes TABLE 1. 6 Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cmings 43
Learning Check Indicate the unit that matches the description: 1. A mass that is 1000 times greater than 1 gram. 1) kilogram 2) milligram 3) megagram 2. A length that is 1/100 of 1 meter. 1) decimeter 2) centimeter 3) millimeter 3. A unit of time that is 1/1000 of a second. 1) nanosecond 2) microsecond 3) millisecond 44
Solution Indicate the unit that matches the description: 1. A mass that is 1000 times greater than 1 gram. 1) kilogram 2. A length that is 1/100 of 1 meter. 2) centimeter 3. A unit of time that is 1/1000 of a second. 3) millisecond 45
Learning Check Select the unit you would use to measure A. Your height 1) millimeters 2) meters 3) kilometers B. Your mass 1) milligrams 2) grams 3) kilograms C. The distance between two cities 1) millimeters 2) meters 3) kilometers D. The width of an artery 1) millimeters 3) kilometers 2) meters 46
Solution A. Your height 2) meters B. Your mass 3) kilograms C. The distance between two cities 3) kilometers D. The width of an artery 1) millimeters 47
Metric Equalities An equality § States the same measurement in two different units. § Can be written using the relationships between two metric units. Example: 1 meter is the same as 100 cm and 1000 mm. 1 m = 100 cm 1 m = 1000 mm 48
Measuring Length Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings 49
Measuring Volume Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings 50
Measuring Mass § Several equalities can be written for mass in the metric (SI) system 1 kg = 1 mg = 1000 g 1000 mg 0. 001 g 1000 µg Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cmings 51
Learning Check Indicate the unit that completes each of the following equalities: A. 1000 m = 1) 1 mm 2) 1 km 3) 1 dm B. 0. 001 g = 1) 1 mg 2) 1 kg 3) 1 dg C. 0. 1 s 1) 1 ms 2) 1 cs 3) 1 ds 1) 1 mm 2) 1 cm 3) 1 dm = D. 0. 01 m = 52
Solution Indicate the unit that completes each of the following equalities: A. 2) 1000 m = 1 km B. 1) 0. 001 g = 1 mg C. 3) 0. 1 s = 1 ds D. 2) 0. 01 m = 1 cm 53
Learning Check Complete each of the following equalities: A. 1 kg = 1) 10 g 2) 100 g 3) 1000 g B. 1 mm = 1) 0. 001 m 2) 0. 01 m 3) 0. 1 m 54
Solution Complete each of the following equalities: A. 1 kg = 1000 g B. 1 mm = 0. 001 m (3) (1) 55
Writing Conversion Factors 56
Equalities § Use two different units to describe the same measured amount. § Are written for relationships between units of the metric system, U. S. units, or between metric and U. S. units. For example, 1 m = 1000 mm 1 lb = 16 oz 2. 20 lb = 1 kg 57
Some Common Equalities TABLE 1. 9 58
Equalities on Food Labels The contents of packaged foods § In the U. S. are listed as both metric and U. S. units. § Indicate the same amount of a substance in two different units. 59
Conversion Factors A conversion factor § Is a fraction obtained from an equality. Equality: 1 in. = 2. 54 cm § Is written as a ratio with a numerator and denominator. § Can be inverted to give two conversion factors for every equality. 1 in. and 2. 54 cm 1 in. 60
Learning Check Write conversion factors for each pair of units: A. liters and m. L B. hours and minutes C. meters and kilometers 61
Solution Write conversion factors for each pair of units: A. liters and m. L Equality: 1 L = 1000 m. L 1 L and 1000 m. L 1 L B. hours and minutes Equality: 1 hr = 60 min 1 hr and 60 min 1 hr C. meters and kilometers Equality: 1 km and 1000 m 1 km = 1000 m 62
Conversion Factors in a Problem A conversion factor § May be obtained from information in a word problem. § Is written for that problem only. Example 1: The price of one pound (1 lb) of red peppers is $2. 39. 1 lb red peppers and $2. 39 1 lb red peppers Example 2: The cost of one gallon (1 gal) of gas is $2. 94. 1 gallon of gas and $2. 94 1 gallon of gas 63
Percent as a Conversion Factor A percent factor § Gives the ratio of the parts to the whole. % = Parts x 100 Whole § Use the same units for the parts and whole. § Uses the value 100 and a unit for the whole. § Can be written as two factors. Example: A food contains 30% (by mass) fat. 30 g fat and 100 g food 30 g fat 64
Learning Check Write the equality and conversion factors for each of the following: A. meters and centimeters B. jewelry that contains 18% gold C. one gallon of gas is $ 2. 95 65
Solution A meters and centimeters 1 m and 100 cm 1 m B. jewelry that contains 18% gold 18 g gold and 100 g jewelry 18 g gold C. one gallon of gas is $2. 95 1 gal $2. 95 and $2. 95 1 gal 66
Density 67
Problem Solving 68
Initial and Final Units To solve a problem § Identify the initial unit. § Identify the final unit. Problem: A person has a height of 2. 0 meters. What is that height in inches? The initial unit is the given unit of height. initial unit = meters (m) The final unit is the unit for the answer. final unit = inches (in. ) 69
Learning Check An injured person loses 0. 30 pints of blood. How many milliliters of blood would that be? Identify the initial and final units given in this problem. Initial unit = _______ Final unit = _______ 70
Solution An injured person loses 0. 30 pints of blood. How many milliliters of blood would that be? Identify the initial and final units given in this problem. Initial unit = pints Final unit = milliliters 71
Problem Setup § Write the initial and final units. § Write a unit plan to convert the initial unit to the final unit. § Write equalities and conversion factors. § Use conversion factors to cancel the initial unit and provide the final unit. Unit 1 x Unit 2 = Unit 2 Unit 1 Initial x Conversion = Final unit factor unit 72
Guide to Problem Solving The steps in the Guide to Problem Solving are useful in setting up a problem with conversion factors. 73
Setting up a Problem How many minutes are 2. 5 hours? Initial unit = 2. 5 hr Final unit = ? min Plan = hr min Setup problem to cancel hours (hr). Initial Conversion Final unit factor unit 2. 5 hr x 60 min = 150 min (2 SF) 1 hr 74
Learning Check A rattlesnake is 2. 44 m long. How many centimeters long is the snake? 1) 2440 cm 2) 244 cm 3) 24. 4 cm 75
Solution A rattlesnake is 2. 44 m long. How many centimeters long is the snake? 2) 244 cm 2. 44 m x 100 cm 1 m = 244 cm 76
Using Two or More Factors § Often, two or more conversion factors are required to obtain the unit needed for the answer. Unit 1 Unit 2 Unit 3 § Additional conversion factors are placed in the setup to cancel each preceding unit Initial unit x factor 1 x factor 2 = Final unit Unit 1 x Unit 2 x Unit 3 = Unit 3 Unit 1 Unit 2 77
Example: Problem Solving How many minutes are in 1. 4 days? Initial unit: 1. 4 days Factor 1 Factor 2 Plan: days hr min Set up problem: 1. 4 days x 24 hr x 60 min = 2. 0 x 103 min 1 day 1 hr 2 SF Exact = 2 SF 78
Check the Unit Cancellation § Be sure to check your unit cancellation in the setup. § The units in the conversion factors must cancel to give the correct unit for the answer. What is wrong with the following setup? 1. 4 day x 1 hr 24 hr 60 min Units = day 2/min is not the unit needed Units don’t cancel properly. 79
Guide to Problem Solving What is 165 lb in kg? STEP 1 Initial 165 lb Final: kg STEP 2 Plan lb kg STEP 3 Equalities/Factors 1 kg = 2. 20 lb and 1 kg 2. 20 lb STEP 4 Set Up Problem 165 lb x 1 kg = 2. 20 lb 74. 8 kg 80
Learning Check A bucket contains 4. 65 L of water. How many gallons of water is that? Unit plan: L Equalities: 1. 06 qt = 1 L 1 gal = 4 qt qt gallon 81
Solution Initial : 4. 65 L Plan: Final: gallons L qt gallon Equalities: 1. 06 qt = 1 L; 1 gal = 4 qt Set Up Problem: 4. 65 L x x 1. 06 qt 1 L 3 SF x 1 gal 4 qt exact = 1. 23 gal 3 SF 82
Learning Check If a ski pole is 3. 0 feet in length, how long is the ski pole in mm? (2. 54 cm = 1 inch) 83
Solution 3. 0 ft x 12 in x 2. 54 cm x 10 mm = 1 ft 1 in. 1 cm Calculator answer: 914. 4 mm Final answer: 910 mm (2 SF rounded) Check factor setup: Check final unit: Units cancel properly mm 84
Learning Check If your pace on a treadmill is 65 meters per minute, how many minutes will it take for you to walk a distance of 7500 feet? (1 inch = 2. 54 cm) 85
Solution Initial: 7500 ft Plan: ft 65 m/min in. Equalities: 1 ft = 12 in. Final: min cm m 1 in. = 2. 54 cm min 1 m = 100 cm 1 min = 65 m (walking pace) Set Up Problem 7500 ft x 12 in. x 1 ft 2. 54 cm 1 in. x 1 m x 1 min 100 cm 65 m = 35 min final answer (2 SF) 86
Percent Factor in a Problem If the thickness of the skin fold at the waist indicates an 11% body fat, how much fat is in a person with a mass of 86 kg? percent factor 86 kg x 11 kg fat 100 kg = 9. 5 kg fat 87
Learning Check How many lb of sugar are in 120 g of candy if the candy is 25% (by mass) sugar? 88
Solution How many lb of sugar are in 120 g of candy if the candy is 25%(by mass) sugar? % factor 120 g candy x 1 lb candy x 25 lb sugar 454 g candy 100 lb candy = 0. 066 lb sugar 89
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