Measurement Chapter 1.6-7

Unit 1 Essential Skills Measurement Chapter 1.6-7 The Unit 1 Test will cover material from the following Chapters and Sections: 1.all 2.5-8 3.all 2 Two types of Data: When we make observations of matter, we are collecting Data. Qualitative - Attributes Properties Purity (Observations) Quantitative - Amount (Measurements) Measurement A measuring tool is used to compare some dimension of an object to a standard. An electronic balance is the measuring tool used to determine the mass of a nickel. 4 Stating a Measurement In every measurement, a number is followed by a unit. Observe the following examples of measurements: Number Unit 35 m 0.25 L 225 lb 3.4 h The Metric System (SI) The metric system and SI (international system) are decimal systems based on 10 used everywhere by scientists used in most of the world 5 6 1

Units in the Metric System for Measured Quantities In the metric and SI systems, one unit is used for each type of measurement: Measurement Metric SI Length meter (m) meter (m) Volume liter (L) cubic meter (m 3 ) Mass gram (g) kilogram (kg) Time second (s) second (s) Temperature degree Celsius ( C) Kelvin (K) Length Measurement Length is measured using a meterstick has the unit meter (m) in both the metric and SI systems 1 in. = 2.54 cm 7 8 Volume Measurement Volume is the space occupied by a substance has the unit liter (L) in the metric system 1 L = 1.057 qt has the unit m 3 (cubic meter) in the SI system is measured using a graduated cylinder Mass Measurement The mass of an object is the quantity of material it contains is measured on a balance has the unit gram(g) in the metric system has the unit kilogram(kg) in the SI system 9 10 Time Measurement Time measurement uses the unit second(s) in both the metric and SI systems is based on an atomic clock that uses a frequency emitted by cesium atoms Temperature Measurement The temperature indicates how hot or cold it is is measured on the Celsius ( C) scale in the metric system is measured on the Kelvin(K) scale in the SI system on this thermometer is 18 ºC or 64 ºF 11 12 2

Uncertainty in Measurements Different measuring devices have different uses and different degrees of accuracy. Therefore, It is important to make accurate measurements Measured Numbers A measuring tool is used to determine a quantity such as the length or the mass of an object provides numbers for a measurement called measured numbers 14 Reading a Meterstick. l 2.... l.... l 3.... l.... l 4.. cm The markings on the meterstick at the end of the orange line are read as The first digit 2 plus the second digit 2.7 The last digit is obtained by estimating. The end of the line might be estimated between 2.7 2.8 as half-way (0.05) or a little more (0.06), which gives a reported length of 2.75 cm or 2.76 cm. Known + Estimated Digits In the length reported as 2.76 cm, the digits 2 and 7 are certain (known) the final digit 6 was estimated (uncertain) all three digits (2.76) are significant including the estimated digit 15 16 Zero as a Measured Number. l 3.... l.... l 4.... l.... l 5.. cm For this measurement, the first and second known digits are 4.5. Because the line ends on a mark, the estimated digit in the hundredths place is 0. This measurement is reported as 4.50 cm. Significant Figures in Measured Numbers Significant figures obtained from a measurement include all of the known digits plus the estimated digit. The number of significant figures reported in a measurement depends on the measuring tool. 17 18 3

Let s Make Some Accurate Measurements Measure the Following: Width of your book with a ruler. Temperature of the room. Volume of water in a graduated cylinder. Time to start and stop a stopwatch. Significant Figures The term significant figures refers to digits in a number that were measured. When rounding calculated numbers, we pay attention to significant figures so we do not overstate the accuracy of our answers. Counting Significant Figures All nonzero numbers in a measured number are significant. Measurement Number of Significant Figures 38.15 cm 4 5.6 ft 2 65.6 lb 3 122.55 m 5 Sandwiched Zeros Sandwiched zeros occur between nonzero numbers are significant Measurement Number of Significant Figures 50.8 mm 3 2001 min 4 0.0702 lb 3 0.40505 m 5 21 22 Trailing Zeros Trailing zeros follow nonzero numbers in numbers without decimal points are placeholders are not significant Measurement 25 000 cm 2 200 kg 1 48 600 ml 3 25 005 000 g 5 Number of Significant Figures Leading Zeros Leading zeros precede nonzero digits in a decimal number are placeholders are not significant Measurement Number of Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb 2 0.000262 ml 3 23 24 4

Significant Figures in Scientific Notation In scientific notation all digits including zeros in the coefficient are significant Scientific Notation Number of Significant Figures 8 x 10 4 m 1 8.0 x 10 4 m 2 8.00 x 10 4 m 3 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 25 26 A. Which answer(s) contain three significant figures? 1) 0.4760 2) 0.00476 3) 4.76 x 10 3 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 10 3 C. The number of significant figures in 5.80 x 10 2 is 1) one 3) two 3) three 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 10 2 3) 0.000015 and 150 000 27 28 Exact Numbers An exact number is obtained when objects are counted Example: counting objects 2 baseballs 4 pizzas from numbers in a defined relationship Example: defined relationships 1 foot = 12 inches 1 meter = 100 cm Classify each of the following as exact (E) or measured numbers (M). Explain your answer. A. Gold melts at 1064 C. B. 1 yd = 3 ft C. The diameter of a red blood cell is 6 x 10 4 cm. D. There are 6 hats on the shelf. E. A can of soda contains 355 ml of soda. 29 30 5

Rounding Off Answers In calculations answers must have the same number of significant figures as the measured numbers calculated answers are usually rounded off rounding rules are used to obtain the correct number of significant figures Rounding Off Calculated Answers When the first digit dropped is 4 or less, the retained numbers remain the same. to round off 45.832 to 3 significant figures drop the digits 32 = 45.8 When the first digit dropped is 5 or greater, the last retained digit is increased by 1. to round off 2.4884 to 2 significant figures drop the digits 884 = 2.5 (increase by 0.1) Odd/Even Rule 31 32 Adding Significant Zeros Sometimes a calculated answer requires more significant digits. Then one or more zeros are added. Calculated Zeros Added to Answer Give 3 Significant Figures 4 4.00 1.5 1.50 0.2 0.200 12 12.0 Adjust the following calculated answers to give answers with three significant figures: A. 824.75 cm B. 0.112486 g C. 8.2 L 33 34 Improving Accuracy The accuracy in a measurement may be increased by using a more precise instrument. Using the first thermometer, the temperature is 24.3 ºC (3 significant digits). Using the more precise (second) thermometer, the temperature is 24.32 ºC (4 significant digits) Calculations with Measured Numbers Remember, each measured quantity must have an uncertain digit. The uncertain digit relays the accuracy of a measurement. That accuracy is maintained when converting between units by ensuring you have as many digits after the conversion as you had before. How then does the accuracy of a set of measurements become relayed in a calculated, or derived, quantity? A derived quantity can not be more accurate than the least accurate measurement! 6

Calculations with Measured Numbers So, how is accuracy relayed to calculated values from measured values? The answer: By use of Significant Figures. Observe: 21.32 cm x 1.3 cm If the answer may only contain 1 uncertain digit, what is the accuracy of the measurement? Multiplication and Division When multiplying or dividing, use the same number of significant figures (SFs) as in the measurement with the fewest significant figures rounding rules to obtain the correct number of significant figures Example 110.5 x 0.048 = 5.304 = 5.3 (rounded) 4 SFs 2 SFs calculator 2 SFs 38 Give an answer for each 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 Calculations with Measured Numbers O.K. That works for multiplication but what about addition: 1.3 cm + 21.32 cm + 1.3 cm + 21.3 cm =? C. 2.54 x 0.0028 = 0.0105 x 0.060 1) 11.3 2) 11 3) 0.041 39 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 one decimal place + 1.34 two decimal places 26.54 calculated answer 26.5 final answer (one decimal place) For each calculation, round the answer to give the correct number of digits. A. 235.05 + 19.6 + 2 = 1) 257 2) 256.7 3) 256.65 B. 58.925-18.2 = 1) 40.725 2) 40.73 3) 40.7 41 42 7

Prefixes Metric and SI Prefixes A prefix in front of a unit increases or decreases the size of that unit makes units larger or smaller that the initial unit by one or more factors of 10 indicates a numerical value prefix = value 1 kilometer = 1000 meters 1 kilogram = 1000 grams 43 44 Education, Inc. Copyright 2011 Pearson 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 length as 100 cm and 1000 mm. 1 m = 100 cm 1 m = 1000 mm 100 cm = 1000 mm Equalities 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 Examples: 1 m = 1000 mm 1 lb = 16 oz 2.205 lb = 1 kg 45 46 Exact and Measured Numbers in Equalities Some Common Equalities Equalities between units of the same system are definitions and are exact numbers different systems (metric and U.S.) use measured numbers and count as significant figures 47 48 8

Conversion Factors A conversion factor is a fraction obtained from an equality Equality: 1 in. = 2.54 cm written as a ratio with a numerator and denominator is inverted to give two conversion factors for every equality 1 in. and 2.54 cm 2.54 cm 1 in. Factors with Powers A conversion factor can be squared or cubed on both sides of the equality. Equality 1 in. = 2.54 cm 1 in. and 2.54 cm 2.54 cm 1 in. Equality squared (1 in.) 2 = (2.54 cm) 2 (1 in.) 2 and (2.54 cm) 2 (2.54 cm) 2 (1 in.) 2 Equality cubed (1 in.) 3 = (2.54 cm) 3 (1 in.) 3 and (2.54 cm) 3 (2.54 cm) 3 (1 in.) 3 49 50 Conversion Factors in a Problem An equality and conversion factors may be obtained from information in a word problem are for that problem only Example: The price of one pound (1 lb) of red peppers is $2.39. Equality: 1 lb peppers = $2.49 Conversion factors: 1 lb red peppers and $2.39 $2.39 1 lb red peppers Given and Needed Units To solve a problem identify the given unit identify the needed unit Problem: A person has a height of 2.0 m. What is that height in inches? The given unit is the initial unit of height. given unit = meters (m) The needed unit is the unit for the answer. needed unit = inches (in.) 51 52 Dimensional Analysis Simple but important method used to solve chemistry and physics problems based on units. Uses the form of the conversion factor that puts the sought-for unit in the numerator: Given unit desired unit given unit desired unit Guide to Problem Solving The steps in the Guides to Problem Solving Using Conversion Factors are useful in setting up a problem with conversion factors. Conversion factor 54 9

Setting up a Problem How many minutes are 2.5 hours? STEP 1 Given 2.5 h Need min STEP 2 Plan hours minutes STEP 3 Equalities 1 h = 60 min STEP 4 Set up problem 2.5 h x 60 min = 150 min 1 h (2 SFs) given conversion needed unit factor unit A rattlesnake is 2.44 m long. How long is the snake in centimeters? 1) 2440 cm 2) 244 cm 3) 24.4 cm 55 56 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. Given unit x factor 1 x factor 2 = needed unit Unit 1 x Unit 2 x Unit 3 = Unit 3 Unit 1 Unit 2 Example: Problem Solving How many minutes are in 1.4 days? STEP 1 Given 1.4 days Need minutes STEP 2 Plan days hours minutes STEP 3 Equalties: 1 day = 24 h 1 h = 60 min STEP 4 Set up problem: 1.4 days x 24 h x 60 min = 2.0 x 10 3 min 1 day 1 h 2 SFs Exact Exact = 2 SFs 57 58 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. Example: What is wrong with the following setup? 1.4 day x 1 day x 1 h 24 h 60 min Units = day 2 /min, which is not the unit needed. Units do not cancel properly; the setup is wrong. Using the GPS What is 165 lb in kilograms? STEP 1 Given 165 lb Need kg STEP 2 Plan lb kg STEP 3 Equalities/conversion factors 1 kg = 2.20 lb 2.20 lb and 1 kg STEP 4 Set up problem 1 kg 2.20 lb 165 lb x 1 kg = 74.8 kg (3 SFs) 2.20 lb 59 60 10

A bucket contains 4.65 L water. How many gallons of water is that? If a ski pole is 3.0 feet in length, how long is the ski pole in mm? 61 62 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? Units in the Metric System for Derived Quantities In the metric and SI systems, one unit is used for each type of measurement: Measurement Metric SI Volume liter (L) cubic meter (m 3 ) Weight Newtons (N) Newtons (N) Energy calorie (cal) Kilojoule (KJ) Density (g/ml) (Kg/L) 63 64 The Calorie The Joule Commonly, energies are described in the units of calories. A calorie is the amount of heat needed to raise the temperature of 1 gram H 2 0, 1 degree centigrade 1 cal = 4.184 J nutritional calorie (Cal) 1 Cal = 1 Kcal the kinetic energy required to move a 2Kg mass a distance of one meter in one sec. From the equation of KE: J 2 kg m 2 s KE = ½ mv 2 James Joule 1818-1889 11

Density Density compares the mass of an object to its volume is the mass of a substance divided by its volume Density expression: D = mass = g or g = g/cm 3 volume ml cm 3 Note: 1 ml = 1 cm 3 Sink or Float Objects that sink in water are more dense than water. Ice floats in water because the density of ice is less than the density of water. Aluminum sinks because its density is greater than the density of water. 67 68 Density Using Volume Displacement The density of the object is calculated from its mass and volume. mass = 68.60 g = 7.2 g/cm 3 volume 9.5 cm 3 Volume by Displacement A solid completely submerged in water displaces its own volume of water. The volume of the solid is calculated from the volume difference. 45.0 ml - 35.5 ml = 9.5 ml = 9.5 cm 3 The density of a solid is determined from its mass and the volume it displaces when submerged in water. 69 70 Density as a Conversion Factor What is the density (g/cm 3 ) of 48.0 g of a metal if the level of water in a graduated cylinder rises from 25.0 ml to 33.0 ml after the metal is added? 1) 0.17 g/cm 3 2) 6.0 g/cm 3 3) 380 g/cm 3 For a density of 3.8 g/ml, an equality is written as 3.8 g = 1 ml and two conversion factors are written as 25.0 ml 33.0 ml object Conversion 3.8 g and 1 ml factors 1 ml 3.8 g 71 72 12

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 If olive oil has a density of 0.92 g/ml, how many liters of olive oil are in 285 g of olive oil? 1) 0.26 L 2) 0.31 L 3) 310 L 73 74 Accuracy vs. Precision Accuracy vs. Precision The goal of all measurements is to be both accurate and precise! Accuracy refers to the proximity of a measurement to the true value of a quantity. Precision refers to the proximity of several measurements to each other. Absolute error (AE) -The difference between an experimental value and accepted value A.E. = exp known % Absoulte Error (%AE) exp known %AE 100 known % Accuracy - Percentage your value differs from 100. % Accuracy = 100 - %AE Accuracy vs. Precision What is the difference between the percent accuracy of a measurement and the percent error of a measurement? Accuracy vs. Precision Relative error (RE) - The difference between an experimental value and the average for a set of experimental values R.E. = exp avg % Relative Error (%RE) exp avg %RE 100 avg % Precision - The closeness of a value to a set of values in terms of percentage. % Precision = 100 - %RE 13