Measurement Lesson Notes PDF

MEASUREMENT 1  No human endeavor can be called science if it can not be demonstrated mathematically. Leonardo da Vinci (1452-1519) 2  Learning Objectives Use the SI system. Use and report measureme...

MEASUREMENT 1  No human endeavor can be called science if it can not be demonstrated mathematically. Leonardo da Vinci (1452-1519) 2  Learning Objectives Use the SI system. Use and report measurements carefully. Consider the reliability of a measurement in decisions based on measurements. Clearly distinguish between precision and accuracy exact numbers and measurements systematic error and random error 3  Learning Objectives Count the number of significant figures in a recorded measurement. Record measurements to the correct number of digits. Estimate the number of significant digits in a calculated result. Estimate the precision of a measurement by computing a standard deviation. 4  TOPIC OUTLINE Units of measurement Measured numbers and significant number Prefixes Equalities and Conversion Problem Solving Using Conversion Factor Temperature Density 5 Units of Measurement 6  Measurement You are making measurement when you Check your weight Read your watch Take your temperature 7  Measurement in Chemistry Do experiments Measure quantities Use numbers to report measurements 8  SYSTEMS IN MEASUREMENT METRIC (SI) SYSTEM A decimal system based on 10 Used in most part of the world Used by scientists and hospital ENGLISH SYSTEM 9  The SI SYSTEM Le Systéme Internationale (SI) is a set of units and notations that are standard in science. 10  Seven important SI base units (there are others) QUANTITY SI BASE UNITS Length Meter, m Mass Kilogram, kg Time Second, s Temperature Kelvin, K Amount of Matter Mole, mol Luminous Intensity Candela, cd Electric Current Ampere, A 11  Some SI derived units Quantity Dimensions SI Units Area length × length m2 Velocity length/time m/s Density mass/volume kg/m3 Frequency cycles/time s-1 or Hertz (Hz) Acceleration velocity/time m/s2 Force mass × kg m/s 2 or Newton (N) acceleration Work/Energy force × distance kg m2/s2 or Joule (J) 12 Measured Numbers and Significant Figures 13  Measured Numbers and Significant Figures Measured and Exact Numbers Significant Figures in Measurement Precision and Accuracy Rules in Determining Significant Figures Calculations in Significant Figures Scientific Notation Calculations Involving Scientific Notation Engr. Yvonne 14  Measured Numbers When you use a measuring tool is used to determine a quantity such as your height or weight, the numbers you obtain are called measured numbers. 15  Reading Meter Stick. l2.... I.... I3....I.... I4.. cm First digit (known) =2 2.?? cm Second digit (known) = 0.7 2.7? cm Third digit (estimated) between 0.05 - 0.07 Length reported = 2.75 cm or 2.76 cm or 2.77 cm 16  Known + Estimated Digits Known digits 2 and 7 are 100% certain The third digit 6 is estimated (uncertain) In the reported length, all three digits (2.76 cm) are significant including the estimated one 17  Zero as a Measured Number.l....I....I....I....I.. 3 4 5 cm What is the length of the line? First digit 4.?? cm Second digit 4.5? cm Last (estimated) digit is 4.50 cm (not to the left or right of.5) 18  Exact Numbers Obtained when you count objects 2 soccer balls 1 watch 4 pizzas Obtained from a defined relationship 1 foot = 12 inches 1 meter = 100 cm Not obtained with measuring tools 19  Learning Check A. Exact numbers are obtained by 1. measuring 2. counting 3. definition B. Measured numbers are obtained by 1. measuring 2. counting 3. definition 20  Solution A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. Using a measuring tool 21  Learning Check Classify each of the following as an exact (1) or a measured (2) number. A.___Gold melts at 1064°C B.___1 yard = 3 feet C.___A red blood cell with diameter 6 x 10-4 cm D.___There were 6 hats on the shelf E.___A can of soda contains 355 mL of soda 22  Solution Classify each of the following as an exact (1) or a measured(2) number. Give reason. A. 2 Requires a thermometer(measuring tool) B. 1 From a definition in U.S. system C. 2 Need measuring tool to determine D. 1 Counted the hats E. 2 Measured 23 Significant Figures in Measurement 24 Significant Figures in Measurement The numbers reported in a measurement are limited by the measuring tool Significant figures in a measurement include the known digits plus one estimated digit 25  Measurement and Significant Figures When measurement is recorded, all the known with certainty are given along with the last number which is estimated. All the digits are significant because removing any of the digits changes the measurement’s uncertainty 26  Uncertainty of Data All measurement contain some uncertainty We make errors Tools have limits 27  Uncertainty of Data Uncertainty is measured using Accuracy: How close to the true value Precision: How close to each other 28  Precision VS Accuracy good precision & good accuracy but good accuracy poor precision poor accuracy but good precision poor precision & poor accuracy 29  Precision and Accuracy Precision Accuracy Reproducibility Correctness Check by repeating Check by using different measurement method Poor precision results from poor accuracy results from poor technique procedural or equipment flaws poor precision is associated poor accuracy is associated with 'random errors' - error with 'systematic errors' - error has random sign and has a reproducible sign and varying magnitude. Small magnitude. errors more likely than large errors. 30 Methods used to express accuracy and precision You cant report numbers better than the method used to measure them Example 67.2 units = three significant figures Certain Uncertain digits digits 31 Rules in Determining Significant Figures 32 Counting Significant Figures Number of Significant Figures 38.15 cm 4 5.6 ft 2 65.6 lb ___ 122.55 m ___ All non-zero digits in a measured number are significant. 33  Leading Zeros Number of Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb ____ 0.000262 mL ____ Leading zeros in decimal numbers are not significant. 34  Sandwiched Zeros Number of Significant Figures 50.8 mm 3 2001 min 4 0.702 lb ____ 0.00405 m ____ Zeros between nonzero numbers are significant. 35  Trailing Zeros Number of Significant Figures 25,000 in. 2 200 yr 1 48,600 gal 3 25,005,000 g ____ Trailing zeros in numbers without decimals are not significant if they are serving as place holders. 36  Trailing Zeros Number of Significant Figures 4830 km 3 60 g 1 4830. L 4 60. K ____ If such zeros are known to have been measured, however, they are significant and should be specified as such by inserting a decimal point to the right of the zero 37 Some Other Rules of Zero Number of Significant Figures 8.0 dm 2 16.40 g 4 35.000 L 5 1.60 sec ____ All zeros to the right of a decimal point and to the right of a nonzero digit is significant 38  Learning Check A. Which answers contain 3 significant figures? 1) 0.4760 2) 0.00476 3) 4760 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 103 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x 105 39  Solution A. Which answers contain 3 significant figures? 2) 0.00476 3) 4760 B. All the zeros are significant in 2) 25.300 3) 2.050 x 103 C. 534,675 rounded to 3 significant figures is 2) 535,000 3) 5.35 x 105 40 Significant Figures In Calculations 41 Significant Numbers in Calculations A calculated answer cannot be more precise than the measuring tool. A calculated answer must match the least precise measurement. Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing 42  Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. 25.2 one decimal place + 1.34 two decimal places 26.54 answer 26.5 one decimal place 43  Learning Check In each calculation, round the answer to the correct number of significant figures. A. 235.05 + 19.6 + 2.1 = 1) 256.75 2) 256.8 3) 257 B. 58.925 - 18.2 = 1) 40.725 2) 40.73 3) 40.7 44  Solution A. 235.05 + 19.6 + 2.1 = 2) 256.8 B. 58.925 - 18.2 = 3) 40.7 45 Multiplying and Dividing Round (or add zeros) to the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures. 46  Learning Check A. 2.19 X 4.2 = 1) 9 2) 9.2 3) 9.198 B. 4.311 ÷ 0.07 = 1) 61.58 2) 62 3) 60 C. 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.3 2) 11 3) 0.041 47  Solution A. 2.19 X 4.2 = 2) 9.2 B. 4.311 ÷ 0.07 = 3) 60 C. 2.54 X 0.0028 = 2) 11 0.0105 X 0.060 Continuous calculator operation = 2.54 x 0.0028 ÷ 0.0105 ÷ 0.060 48 Scientific Notation 49  Scientific Notation can be used to clearly express significant figures. A properly written number in scientific notation always has the proper number of significant figures 0.00321 = 3.21 x 10-3 (three significant figures) 50 51 52 A typical number in notation Z x 10m Where: Z = coefficient x = multiplication sign 10 = base m = exponent 53  Calculations Involving Scientific Notation Addition and Subtraction Multiplication Division 54  Calculations Involving Scientific Notation Addition and Subtraction of Exponential Numbers Two numbers in scientific notation can only be added or subtracted if both expressions have the same exponent. The coefficient are then added or subtracted while the exponent remains the same. 55  Calculations Involving Scientific Notation Addition and Subtraction of Exponential Numbers 7.2 x 107 7.2 x 107 + 2.1 x 108 + 21.0 x 107 28.2 x 10 7 56  Calculations Involving Scientific Notation Multiplication of Exponential Numbers In multiplying two numbers in exponential form, the coefficients themselves are multiplied and the exponents are added. 57  Calculations Involving Scientific Notation Multiplication of Exponential Number (3.4 x 103) x (2.2 x 105) = 7.48 x 108 or rounded to 7.5 x 108 58  Calculations Involving Scientific Notation Division of Exponential Numbers To divide exponential numbers, divide the coefficients and subtract the exponents. 59 Calculations Involving Scientific Notation Division of Exponential Numbers 60  Calculations Involving Scientific Notation Division of Exponential Numbers The negative exponent means, the coefficient is divided by the number of 10’s The exponential part of a number can be moved from the numerator or denominator and vice versa by simply changing the sign of the exponent. 61 Calculations Involving Scientific Notation Division of Exponential Numbers Example 62 Prefixes 63  PREFIXES A multiple of a unit in the International System is formed by adding a prefix to the name of that unit. The prefixes change the magnitude of the unit by orders of ten from 10 to 10. 24 -24 64  PREFIXES Prefix Symbol Exponential Notation yotta Y 1024 = 1,000,000,000,000,000,000,000,000 zetta Z 1021 = 1,000,000,000,000,000,000,000 exa E 1018 = 1,000,000,000,000,000,000 peta P 1015 = 1,000,000,000,000,000 tera T 1012 = 1,000,000,000,000 giga G 109 = 1,000,000,000 mega M 106 = 1,000,000 kilo K 103 = 1,000 hecto h 102 = 100 deca da 10 = 10 65  PREFIXES Prefix Symbol Exponential Notation deci d 10 -1 = 0.1 centi c 10 -2 = 0.01 milli m 10 -3 = 0.001 micro µ 10 -6 = 0.000,001 nano n 10 -9 = 0.000,000,001 pico p 10 -12 = 0.000,000,000,001 femto f 10 -15 = 0.000,000,000,000,001 atto a 10 -18 = 0.000,000,000,000,000,001 zepto z 10 -21 = 0.000,000,000,000,000,000,001 yocto y 10 -24 = 0.000,000,000,000,000,000,000,001 66 Equalities and Conversion Factor 67  Equalities State the same measurement in two different units length 10.0 in. 25.4 cm 68  Some Metric Equalities Length 1m = 100 cm Mass 1 kg = 1000 g Volume 1L = 1000 mL 69  Some American Equalities 1 ft = 12 inches 1 lb = 16 oz 1 quart = 2 pints 1 quart = 4 cups The quantities in each pair give the same measured amount in two different units. 70 Some Metric-American Equalities 1 in. = 2.54 cm 1 qt = 946 mL 1L = 1.06 qt 1 lb = 454 g 1 kg = 2.20 lb Remember these for exams. 71  Equalities given in a Problem Example 1 At the store, the price of one pound of red peppers is $2.39. Equality: 1 lb red peppers = $2.39 Example 2 At the gas station, one gallon of gas is $1.34. Equality: 1 gallon of gas = $1.34 72  Conversion Factors Fractions in which the numerator and denominator are quantities expressed in an equality between those units Example: 1 in. = 2.54 cm Factors: 1 in. and 2.54 cm 2.54 cm 1 in. 73  Learning Check A. 1000 m = 1 ___ 1) mm 2) km 3) dm B. 0.001 g = 1 ___ 1) mg 2) kg 3) dg C. 0.1 L = 1 ___ 1) mL 2) cL 3) dL D. 0.01 m = 1 ___ 1) mm 2) cm 3) dm 74  Solution A. 1000 m = 1 ___ 2) km B. 0.001 g = 1 ___ 1) mg C. 0.1 L = 1 ___ 3) dL D. 0.01 m = 1 ___ 2) cm 75  Learning Check Give the value of the following units: A. 1 kg = ____ g 1) 10 g 2) 100 g 3) 1000 g B. 1 mm = ____ m 1) 0.001 m 2) 0.01 m 3) 0.1 m 76  Solution A. 1 kg = ____ g 3) 1000 g B. 1 mm = ____ m 1) 0.001 m 77  Learning Check Write conversion factors that relate each of the following pairs of units: A. Liters and mL B. Hours and minutes D. Meters and kilometers 78  Solution A. Liter and mL 1 L = 1000 mL 1L and 1000 mL 1000 mL 1L B. hours and minutes 1 hr = 60 min 1 hr and 60 min 60 min 1 hr C. meters and kilometers 1 km = 1000 m 1 km and 1000 m 1000 m 1 km 79 Problem Solving Using Conversion Factors 80  Initial and Final Units 1. A person has a height of 2.0 meters. What is that height in inches? Initial unit = m Final unit = _______ 2) Blood has a density of 0.05 g/mL. If a person lost 0.30 pints of blood at 18°C, how many ounces of blood would that be? Initial = pints Final unit = _______ 81 How many minutes are in 2.5 hours? Initial unit 2.5 hr Conversion Final factor unit 2.5 hr x 60 min = 150 min 1 hr cancel Answer (2 SF) 82  Learning Check A rattlesnake is 2.44 m long. How long is the snake in cm? 1) 2440 cm 2) 244 cm 3) 24.4 cm 83  Solution A rattlesnake is 2.44 m long. How long is the snake in cm? 2) 244 cm 2.44 m x 100 cm = 244 cm 1m 84  Learning Check How many seconds are in 1.4 days? Unit plan: days hr min seconds 1.4 days x 24 hr x ?? 1 day 85  Solution Unit plan: days hr min seconds 2 SF Exact 1.4 day x 24 hr x 60 min x 60 sec 1 day 1 hr 1 min = 1.2 x 105 sec 86  Unit Check What is wrong with the following setup? 1.4 day x 1 day x 60 min x 60 sec 24 hr 1 hr 1 min 87  Unit Check 1.4 day x 1 day x 60 min x 60 sec 24 hr 1 hr 1 min Units = day2 sec/hr2 Not the final unit needed 88  Learning Check An adult human has 4650 mL of blood. How many gallons of blood is that? Unit plan: mL qt gallon Equalities: 1 quart = 946 mL 1 gallon = 4 quarts Your Setup: 89  Solution Unit plan: mL qt gallon Setup: 4650 mL x 1 qt x 1 gal = 1.23 gal 946 mL 4 qt 3 SF 3 SF exact 3 SF 90  Steps to Problem Solving Read problem Identify data Write down a unit plan from the initial unit to the desired unit Select conversion factors Change initial unit to desired unit Cancel units and check Do math on calculator Give an answer using significant figures 91  Learning Check If the ski pole is 3.0 feet in length, how long is the ski pole in mm? 92  Solution 3.0 ft x 12 in x 2.54 cm x 10 mm = 910 mm 1 ft 1 in. 1 cm 93  Learning Check If your pace on a treadmill is 65 meters per minute, how many seconds will it take for you to walk a distance of 8450 feet? 94  Solution Initial 8450 ft x 12 in. x 2.54 cm x 1 m 1 ft 1 in. 100 cm x 1 min x 60 sec = 2400 sec 65 m 1 min final (2 SF) 95 Measuring Temperature 96  Temperature Particles are always moving. When you heat water, the water molecules move faster. When molecules move faster, the substance gets hotter. When a substance gets hotter, its temperature goes up. 97  Temperature Measures the hotness or coldness of an object Determined by using a thermometer that contains a liquid that expands with heat and contracts with cooling. 98  Temperature Scales Fahrenheit Celsius Kelvin Water boils _____°F _____°C ______K Water freezes _____°F _____°C ______K 99  Temperature Scales Fahrenheit Celsius Kelvin Water boils 212°F 100°C 373 K Water freezes 32°F 0°C 273 K 100 Units of Temperature between Boiling and Freezing Fahrenheit Celsius Kelvin Water boils 212°F 100°C 373 K 180° F 100°C 100K Water freezes 32°F 0°C 273 K 101  Fahrenheit Formula 180°F = 9°F = 1.8°F 100°C 5°C 1°C Zero point: 0°C = 32°F °F = 9/5 T°C + 32 or °F = 1.8 T°C + 32 102  Celsius Formula Rearrange to find T°C °F = 1.8 T°C + 32 °F - 32 = 1.8T°C ( +32 - 32) °F - 32 = 1.8 T°C 1.8 1.8 °F - 32 = T°C 1.8 103  Kelvin Scale On the Kelvin Scale 1K = 1°C 0 K is the lowest temperature 0K = - 273.15°C K °C K = °C + 273.15 104 Kelvin and Rankine Formula Kelvin K = T°C + 273.15 Rankine R = T°F + 460 105 Temperature Conversions A person with hypothermia has a body temperature of 29.1°C. What is the body temperature in °F? °F = 1.8 (29.1°C) + 32 exact tenth's exact = 52.4 + 32 = 84.4°F tenth’s 106  Learning Check The normal temperature of a chickadee is 105.8°F. What is that temperature in °C? 1) 73.8 °C 2) 58.8 °C 3) 41.0 °C 107  Solution 3) 41.0 °C Solution: °C = (°F - 32) 1.8 = (105.8 - 32) 1.8 = 73.8°F 1.8° = 41.0°C 108  Learning Check Pizza is baked at 455°F. What is that in °C? 1) 437 °C 2) 235°C 3) 221°C 109  Solution Pizza is baked at 455°F. What is that in °C? 2) 235°C (455 - 32) = 235°C 1.8 110  Learning Check On a cold winter day, the temperature falls to -15°C. What is that temperature in R? 1) 479 R 2) 519 R 3) 465 R 111  Solution 3) 465 R Solution: °C → °F °F = 1.8(-15°C) + 32 = -27 + 32 = 5°F °F → R R = 5°F + 460 R = 465 R 112  Learning Check What is normal body temperature of 37°C in Kelvin? 1) 236 K 2) 310 K 3) 342 K 113  Solution What is normal body temperature of 37°C in Kelvin? 2) 310 K K = °C + 273 = 37 °C + 273 = 310. K 114 Density 115  Density Density compares the mass of an object to its volume D = mass = g or g volume mL cm3 Note: 1 mL = 1 cm3 116  Learning Check Osmium is a very dense metal. What is its density in g/cm3 if 50.00 g of the metal occupies a volume of 2.22cm3? 1) 2.25 g/cm3 2) 22.5 g/cm3 3) 111 g/cm3 117  Solution 2) Placing the mass and volume of the osmium metal into the density setup, we obtain D = mass = 50.00 g = volume 2.22 cm3 = 22.522522 g/cm3 = 22.5 g/cm3 118  Volume Displacement A solid displaces a matching volume of water when the solid is placed in water. 33 mL 25 mL 119  Learning Check What is the density (g/cm3) of 48 g of a metal if the metal raises the level of water in a graduated cylinder from 25 mL to 33 mL? 1) 0.2 g/ cm3 2) 6 g/m3 3) 252 g/cm3 33 mL 25 mL 120  Solution 2) 6 g/cm3 Volume (mL) of water displaced = 33 mL - 25 mL = 8 mL Volume of metal (cm3) = 8 mL x 1 cm3 = 8 cm3 1 mL Density of metal = mass = 48 g = 6 g/cm3 volume 8 cm3 121  Learning Check Which diagram represents the liquid layers in the cylinder? (K) Karo syrup (1.4 g/mL), (V) vegetable oil (0.91 g/ mL), (W) water (1.0 g/mL) 1) 2) 3) V W K W K V K V W 122  Solution (K) Karo syrup (1.4 g/mL), (W) water (1.0 g/mL), (V) vegetable oil (0.91 g/ mL) 1) V W K 123 Density as Conversion Factors A substance has a density of 3.8 g/mL. Density = 3.8 g/mL Equality 3.8 g = 1 mL Conversion factors. 3.8 g and 1 mL 1 mL 3.8 g 124  Density Connections Mass Volume kg L g mL (cm3) mg 125  Learning Check The density of octane, a component of gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane? 1) 0.614 kg 2) 614 kg 3) 1.25 kg 126  Solution 1) 0.614 kg Unit plan: mL → g → kg Equalities: 1 mL = 0.702 g and 1 kg = 1000 g Setup: 875 mL x 0.702 g x 1 kg = 0.614 kg 1 mL 1000 g density metric factor factor 127  Learning Check If blood has a density of 1.05 g/mL, how many liters of blood are donated if 575 g of blood are given? 1) 0.548 L 2) 1.25 L 3) 1.83 L 128  Solution 1) Unit Plan: g mL L 575 g x 1 mL x 1L = 0.548 L 1.05 g 1000 mL 129  Learning Check A group of students collected 125 empty aluminum cans to take to the recycling center. If 21 cans make 1.0 pound of aluminum, how many liters of aluminum (D=2.70 g/cm3) are obtained from the cans? 1) 1.0 L 2) 2.0 L 3) 4.0 L 130  Solution 1) 1.0 L 125 cans x 1.0 lb x 454 g x 1 cm3 21 cans 1 lb 2.70 g x 1 mL x 1 L = 1.0 L 1 cm3 1000 mL 131  Learning Check You have 3 metal samples. Which one will displace the greatest volume of water? 1 2 3 25 g Al 45 g of gold 75 g of Lead 2.70 g/mL 19.3 g/mL 11.3 g/mL Discuss your choice with another student. 132  Solution 1)25 g Al x 1 mL = 9.2 mL 2.70 g 25 g Al 2.70 g/mL 133 THANKS FOR LISTENING 134