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3.1 3.1 Measurements and Their Uncertainty 1 Connecting to Your World On January 4, 2004, the Mars Exploration Rover Spirit landed on Mars. Equipped with ﬁve scientiﬁc instruments and a rock abrasion tool (shown at left), Spirit was sent to examine the Martian surface around Gusev Crater, a wide basin that may have once held a lake. Each day of its mission, Spirit recorded measurements for analysis. This data helped scientists learn about the geology and climate on Mars. All measurements have some uncertainty. In the chemistry laboratory, you must strive for accuracy and precision in your measurements. Guide for Reading Key Concepts • How do measurements relate to science? • How do you evaluate accuracy and precision? • Why must measurements be reported to the correct number of signiﬁcant ﬁgures? • How does the precision of a calculated answer compare to the precision of the measurements used to obtain it? Vocabulary Using and Expressing Measurements Your height (67 inches), your weight (134 pounds), and the speed you drive on the highway (65 miles/hour) are some familiar examples of measurements. A measurement is a quantity that has both a number and a unit. Everyone makes and uses measurements. For instance, you decide how to dress in the morning based on the temperature outside. If you were baking cookies, you would measure the volumes of the ingredients as indicated in the recipe. Such everyday situations are similar to those faced by scientists. Measurements are fundamental to the experimental sciences. For that reason, it is important to be able to make measurements and to decide whether a measurement is correct. The units typically used in the sciences are those of the International System of Measurements (SI). In chemistry, you will often encounter very large or very small numbers. A single gram of hydrogen, for example, contains approximately 602,000,000,000,000,000,000,000 hydrogen atoms. The mass of an atom of gold is 0.000 000 000 000 000 000 000 327 gram. Writing and using such large and small numbers is very cumbersome. You can work more easily with these numbers by writing them in scientific, or exponential, notation. In scientiﬁc notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power. For example, the number 602,000,000,000,000,000,000,000 written in scientific notation is 6.02 1023. The coefficient in this number is 6.02. In scientific notation, the coefficient is always a number equal to or greater than one and less than ten. The power of 10, or exponent, in this example is 23. Figure 3.1 illustrates how to express the number of stars in a galaxy by using scientific notation. For more practice on writing numbers in scientific notation, refer to page R56 of Appendix C. measurement scientiﬁc notation accuracy precision accepted value experimental value error percent error signiﬁcant ﬁgures Reading Strategy Building Vocabulary As you read, write a deﬁnition of each key term in your own words. FOCUS Objectives 3.1.1 Convert measurements to scientific notation. 3.1.2 Distinguish among accuracy, precision, and error of a measurement. 3.1.3 Determine the number of significant figures in a measurement and in a calculated answer. Guide for Reading Build Vocabulary Paraphrase Have students write definitions of the words accurate and precise in their own words. As they read the text, have students compare the definitions with those of accuracy and precision given in the text. Reading Strategy 200,000,000,000. 2 1011 Decimal moves 11 places to the left. Exponent is 11 Figure 3.1 Expressing very large numbers, such as the estimated number of stars in a galaxy, is easier if scientific notation is used. Section 3.1 Measurements and Their Uncertainty 63 L2 L2 Use Prior Knowledge Ask, What everyday activities involve measuring? (Examples include buying consumer products, doing sports activities, and cooking.) Ask students to recall which units of measure are related to each of the examples they give. Have some students estimate their height in inches. Have other students measure the same students with a yardstick or tape measure. Compare the estimates with measured values. Ask, What do you think your height is in centimeters? (Acceptable answers range from 130 to 200 centimeters.) 2 INSTRUCT Section Resources Print • Guided Reading and Study Workbook, Section 3.1 • Core Teaching Resources, Section 3.1 Review • Transparencies, T20–T26 Technology • Interactive Textbook with ChemASAP, Animation 2, Problem-Solving 3.2, 3.3, 3.6, 3.8, Assessment 3.1 Have students study the photograph and read the text that opens the section. Ask, How do you think scientists ensure measurements are accurate and precise? (Acceptable answers include that scientists make multiple measurements by using the most precise equipment available. They use samples with known values to check the reliability of the equipment.) Scientific Measurement 63 Section 3.1 (continued) Using and Expressing Measurements L1 Use Visuals Figure 3.1 Have students study the photograph and read the text that opens the section. Ask, How is the exponent of a number expressed in scientific notation related to the number of places the decimal point is moved to the left in a number larger than 1? (They are equal.) Figure 3.2 The distribution of darts illustrates the difference between accuracy and precision. a Good accuracy and good precision: The darts are close to the bull’s-eye and to one another. b Poor accuracy and good precision: The darts are far from the bull’s-eye but close to one another. c Poor accuracy and poor precision: The darts are far from the bull’s-eye and from one another. Poor accuracy Good precision Poor accuracy Poor precision Accuracy, Precision, and Error Your success in the chemistry lab and in many of your daily activities depends on your ability to make reliable measurements. Ideally, measurements should be both correct and reproducible. Accuracy and Precision Correctness and reproducibility relate to the concepts of accuracy and precision, two words that mean the same thing to many people. In chemistry, however, their meanings are quite different. Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured. Precision is a measure of how close a series of measurements are to one another. To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements. Darts on a dartboard illustrate accuracy and precision in measurement. Let the bull’s-eye of the dartboard represent the true, or correct, value of what you are measuring. The closeness of a dart to the bull’s-eye corresponds to the degree of accuracy. The closer it comes to the bull’seye, the more accurately the dart was thrown. The closeness of several darts to one another corresponds to the degree of precision. The closer together the darts are, the greater the precision and the reproducibility. Look at Figure 3.2 as you consider the following outcomes. Activity L2 Purpose To illustrate the concepts of precision and accuracy Materials a small object (such as a lead fishing weight), triple-beam balance Procedure Place the object and a triple-beam balance in a designated area. Set a deadline by which each student will have measured the mass of the object. After everyone has had an opportunity, have students compile a summary of all the measurements. Illustrate precision by having the students find the average and compare their measurement to it. Expected Outcome Measured values should be similar, but not necessarily identical for all students. Precision and Accuracy a. All of the darts land close to the bull’s-eye and to one another. Closeness to the bull’s-eye means that the degree of accuracy is great. Each dart in the bull’s-eye corresponds to an accurate measurement of a value. Closeness of the darts to one another indicates high precision. b. All of the darts land close to one another but far from the bull’s-eye. The precision is high because of the closeness of grouping and thus the high level of reproducibility. The results are inaccurate, however, because of the distance of the darts from the bull’s-eye. c. The darts land far from one another and from the bull’s-eye. The results are both inaccurate and imprecise. FYI Other analogies that may be useful in explaining precision vs. accuracy: • casting a fishing line • pitching horseshoes • a precision marching band c b Good accuracy Good precision Accuracy, Precision, and Error CLASS a Checkpoint How does accuracy differ from precision? 64 Chapter 3 Facts and Figures Striving for Scientific Accuracy The French chemist, Antoine Lavoisier, worked hard to establish the importance of accurate measurement in scientific inquiry. Lavoisier devised an experiment to test the Greek scientists’ idea that when water was heated, it could turn into earth. For 100 days, Lavoisier boiled water in a glass flask constructed to allow steam to condense without 64 Chapter 3 escaping. He weighed the water and the flask separately before and after boiling. He found that the mass of the water had not changed. The flask, however, lost a small mass equal to the sediment he found in the bottom of it. Lavosier proved that the sediment was not earth, but part of the flask etched away by the boiling water. Determining Error Note that an individual measurement may be accurate or inaccurate. Suppose you use a thermometer to measure the boiling point of pure water at standard pressure. The thermometer reads 99.1°C. You probably know that the true or accepted value of the boiling point of pure water under these conditions is actually 100.0°C. There is a difference between the accepted value, which is the correct value based on reliable references, and the experimental value, the value measured in the lab. The difference between the accepted value and the experimental value is called the error. Word Origins Percent comes from the Latin words per, meaning “by” or “through,” and centum, meaning “100.” What do you think the phrase per annum means? Error experimental value accepted value Error can be positive or negative depending on whether the experimental value is greater than or less than the accepted value. For the boiling-point measurement, the error is 99.1°C 100.0°C, or 0.9° C. The magnitude of the error shows the amount by which the experimental value differs from the accepted value. Often, it is useful to calculate the relative error, or percent error. The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%. Percent error 0error 0 100% accepted value Word Origins L2 The phrase per annum means “by the year.” Use Visuals L1 Figure 3.2 Have students inspect Figure 3.2. Ask, If one of the darts in Figure 3.2c were closer to the bull’s-eye, what would happen to the accuracy? (The accuracy would increase.) What would happen to the precision? (The precision would increase.) What is the operational definition of error implied by this figure? (The error is the distance between the dart and the bull’s-eye.) L2 Discuss Review the concept of absolute value. Ask, What is the meaning of a positive error? (Measured value is greater than accepted value.) What is the meaning of a negative error? (Measured value is less than accepted value.) Explain that the absolute value of the error is a positive value that describes the difference between the measured value and the accepted value, but not which is greater. Using the absolute value of the error means that the percent error will always be a positive value. For the boiling-point measurement, the percent error is calculated as follows. 099.1C - 100.0C 0 100% 100.0C 0.9C 100% 100.0C Percent error 0.009 100% 0.9% Just because a measuring device works doesn’t necessarily mean that it is accurate. As Figure 3.3 shows, a weighing scale that does not read zero when nothing is on it is bound to yield error. In order to weigh yourself accurately, you must first make sure that the scale is zeroed. Figure 3.3 The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate. There is a difference between the person’s correct weight and the measured value. Calculating What is the percent error of a measured value of 114 lb if the person’s actual weight is 107 lb? Section 3.1 Measurements and Their Uncertainty 65 Answers to... Figure 3.3 7% Checkpoint Accuracy is determined by comparing a measured value to the correct value. Precision is determined by comparing the values of two or more repeated measurements. Scientific Measurement 65 Section 3.1 (continued) Signiﬁcant Signiﬁcant FiguresFigures in Measurements in Measurements Supermarkets Supermarkets often provide often scales provide like the scales onelike in Figure the one3.4. in Figure Customers 3.4. Customers use use these scales these to measure scales tothe measure weightthe of produce weight ofthat produce is priced thatper is priced pound.per If pound. If you use a scale you use thataisscale calibrated that is in calibrated 0.1-lb intervals, in 0.1-lbyou intervals, can easily you read can easily the read the scale to thescale nearest to the tenth nearest of a pound. tenth ofWith a pound. such With a scale, such however, a scale,you however, can you can also estimate alsothe estimate weight the to the weight nearest to the hundredth nearest hundredth of a poundof byanoting poundthe by noting the position ofposition the pointer of the between pointercalibration between calibration marks. marks. Suppose you Suppose estimate youa estimate weight that a weight lies between that lies2.4 between lb and 2.5 2.4 lb lb to and be2.5 lb to be 2.46 lb. The2.46 number lb. The innumber this estimated in this estimated measurement measurement has three digits. has three Thedigits. The first two digits first in two the digits measurement in the measurement (2 and 4) are (2 known and 4) are with known certainty. withBut certainty. But the rightmost the digit rightmost (6) has digit been (6)estimated has been and estimated involves and some involves uncertainty. some uncertainty. These threeThese reported threedigits reported all convey digits useful all convey information, useful information, however, and however, are and are called significant called significant figures. The figures. signiﬁcant The signiﬁcant ﬁgures in aﬁgures measurement in a measurement include include all of the digits all ofthat the are digits known, that are plus known, a last digit plus athat lastisdigit estimated. that is estimated. MeaMeasurements surements must always must be reported always betoreported the correct to the number correct of number signiﬁcant of signiﬁcant ﬁgﬁgures because ures calculated because answers calculated often answers depend often ondepend the number on the ofnumber signiﬁcant of signiﬁcant ﬁgures in the ﬁgures values inused the values in theused calculation. in the calculation. Instruments Instruments differ in the differ number in theofnumber significant of significant figures that figures can be that can be obtained from obtained their from use and their thus usein and thethus precision in the of precision measurements. of measurements. The The three meterthree sticks meter in Figure sticks3.5 in can Figure be used 3.5 can to be make used successively to make successively more pre- more precise measurements cise measurements of the board. of the board. Significant Figures in Measurements L2 Discuss Point out that the concept of significant figures applies only to measured quantities. If students ask why an estimated digit is considered significant, tell them a significant figure is one that is known to be reasonably reliable. A careful estimate fits this definition. FYI When calibration marks on an instrument are spaced very close together (e.g., on certain thermometers and graduated cylinders), it is sometimes more practical to estimate a measurement to the nearest half of the smallest calibrated increment, rather than to the nearest tenth. CLASS Rules forRules determining for determining whether whether a digit inaadigit measured in a measured value is value signiﬁcant: is signiﬁcant: 1 Every nonzero 1 Every digit nonzero in a reported digit in measurement a reported measurement is assumed is toassumed be to be signiﬁcant. The signiﬁcant. measurements The measurements 24.7 meters,24.7 0.743meters, meter,0.743 and 714 meter, meters and 714 meters each express each a measure expressof a measure length toofthree length signiﬁcant to threeﬁgures. signiﬁcant ﬁgures. Activity Olympic Times L2 Purpose To illustrate how similar measurements fom different eras may vary in precision Materials Almanacs or Internet access Procedure Have students look up the winning times for the men’s and women’s 100-meter dashes at the 1948 and 2000 Olympic Games. Then have them answer the following question. Why do the more recently recorded race times contain more digits to the right of the decimal? (Because the technology used for timekeeping improved to allow for more precise measurements.) Expected Outcome Students should find that the race times from 1948 were recorded to the nearest tenth of a second. The race times from 2000 were recorded to the nearest hundredth of a second. 66 Chapter 3 Figure 3.4 Figure The precision 3.4 The ofprecision a of a weighing scale weighing depends scale ondepends on how finely ithow is calibrated. finely it is calibrated. 2 Zeros appearing 2 Zerosbetween appearing nonzero between digits nonzero are signiﬁcant. digits areThe signiﬁcant. measureThe measurements 7003ments meters, 7003 40.79 meters, meters, 40.79 andmeters, 1.503 meters and 1.503 eachmeters have four each have four signiﬁcant ﬁgures. signiﬁcant ﬁgures. Animation Animation 2 See 2 See how the accuracy how the of aaccuracy of a calculated result calculated depends result depends on the sensitivity on theofsensitivity the of the measuring instruments. measuring instruments. with ChemASAP with ChemASAP 3 Leftmost3zeros Leftmost appearing zerosinappearing front of nonzero in front digits of nonzero are not digits signiﬁcant. are not signiﬁcant. They act as placeholders. They act as placeholders. The measurements The measurements 0.0071 meter, 0.0071 0.42 meter, meter, 0.42 meter, and 0.000 099 andmeter 0.000each 099 meter have only eachtwo have signiﬁcant only twoﬁgures. signiﬁcant The ﬁgures. zeros The zeros to the left are to not the left signiﬁcant. are not By signiﬁcant. writing the By writing measurements the measurements in scien- in scientiﬁc notation, tiﬁc you notation, can eliminate you cansuch eliminate placeholding such placeholding zeros: in thiszeros: case,in this case, 1 5 4.23meter, 101 meter, 4.2 10 and 9.9 meter, 10 and meter. 9.9 105 meter. 7.1 103 meter, 7.1 10 4 Zeros at the 4 Zeros end of atathe number end ofand a number to the right and to of the a decimal right ofpoint a decimal are point are always signiﬁcant. alwaysThe signiﬁcant. measurements The measurements 43.00 meters, 43.00 1.010 meters, meters, 1.010 andmeters, and 9.000 meters 9.000 eachmeters have four eachsigniﬁcant have fourﬁgures. signiﬁcant ﬁgures. 66 Chapter 663 Chapter 3 a Measured length = 0.6 m 1m b Measured length = 0.61 m 10 c 20 30 40 50 60 70 80 90 1m 50 60 70 80 90 1m Figure 3.5 Three differently calibrated meter sticks are used to measure the length of a board. a A meter stick calibrated in a 1-m interval. b A meter stick calibrated in 0.1-m intervals. c A meter stick calibrated in 0.01-m intervals. Measuring How many significant figures are reported in each measurement? Measured length = 0.607 m 10 20 30 40 5 Zeros at the rightmost end of a measurement that lie to the left of an understood decimal point are not signiﬁcant if they serve as placeholders to show the magnitude of the number. The zeros in the measurements 300 meters, 7000 meters, and 27,210 meters are not signiﬁcant. The numbers of signiﬁcant ﬁgures in these values are one, one, and four, respectively. If such zeros were known measured values, however, then they would be signiﬁcant. For example, if all of the zeros in the measurement 300 meters were signiﬁcant, writing the value in scientiﬁc notation as 3.00 102 meters makes it clear that these zeros are signiﬁcant. Use Visuals L1 Figure 3.5 As students inspect Figure 3.5, model the use of meter stick A by pointing out that one can be certain that the length of the board is between 0 and 1 m, and one can say that the actual length is closer to 1 m. Thus, one can estimate the length as 0.6 m. Similarly, using meter stick B, one can say with certainty that the length is between 60 and 70 cm. Because the length is very close to 60 cm, one should estimate the length as 61 cm or 0.61 m. Have students study meter stick C and use similar reasoning to describe the measurement and estimation process. Ask, If meterstick C were divided into 0.001-m intervals, as are most meter sticks, what would be the estimated length of the board in meters? (Acceptable answers range from 0.6065 to 0.6074 m) In millimeters? (606.5 to 607.4 mm) L2 Discuss Be sure to review the role of zeros in determining the number of significant figures. When adding or subtracting numbers expressed in scientific notation, remind students that the numbers must all have the same exponent. 6 There are two situations in which numbers have an unlimited number of signiﬁcant ﬁgures. The ﬁrst involves counting. If you count 23 people in your classroom, then there are exactly 23 people, and this value has an unlimited number of signiﬁcant ﬁgures. The second situation involves exactly deﬁned quantities such as those found within a system of measurement. When, for example, you write 60 min 1 hr, or 100 cm 1 m, each of these numbers has an unlimited number of signiﬁcant ﬁgures. As you shall soon see, exact quantities do not affect the process of rounding an answer to the correct number of signiﬁcant ﬁgures. Section 3.1 Measurements and Their Uncertainty 67 Differentiated Instruction L1 Have students write in their own words the rules for determining the number of significant digits. Help them if necessary. Have them measure several lengths, masses, and volumes; then have them use their rules to determine the correct number of significant figures for each measurement. Less Proficient Readers Answers to... Figure 3.5 The measurements are: a. 0.6 m (1 significant figure), b. 0.61 m (2 significant figures), c. 0.607 m (3 significant figures) Scientific Measurement 67 Section 3.1 (continued) CLASS CONCEPTUAL PROBLEM 3.1 Counting Signiﬁcant Figures in Measurements Activity L2 Purpose To provide practice in applying the rules governing the significance of zeros in measurements Materials textbook and scientific literature, index cards Procedure Have students search their textbooks and other sources for length, mass, volume, or temperature measurements that contain zeros. Have them include some examples written in scientific notation. Ask them to write each measurement on the front of an index card; on the back of each card, have them write (1) all the rules governing the significance of zeros that apply to the measurement, and (2) the number of significant figures in the measurement. Have pairs of students exchange index cards and agree on the appropriateness of the rules and the answers. Expected Outcome Students should be able to apply correctly rules 2–5 listed on pages 66 and 67. Significant Zeros How many signiﬁcant ﬁgures are in each measurement? a. 123 m b. 40,506 mm c. 9.8000 104 m d. 22 meter sticks e. 0.070 80 m f. 98,000 m Analyze Identify the relevant concepts. Solve Apply the concepts to this problem. The location of each zero in the measurement and the location of the decimal point determine which of the rules apply for determining signiﬁcant ﬁgures. All nonzero digits are signiﬁcant (rule 1). Use rules 2 through 6 to determine if the zeros are signiﬁcant. a. three (rule 1) b. ﬁve (rule 2) c. ﬁve (rule 4) d. unlimited (rule 6) e. four (rules 2, 3, 4) f. two (rule 5) Practice Problems 1.Practice Count theProblems signiﬁcant ﬁgures in each length. a. 0.057 30 meters c. 0.000 73 meters 2. How many signiﬁcant ﬁgures are in each measurement? a. 143 grams c. 8.750 102 gram with ChemASAP Suppose you use a calculator to ﬁnd the area of a ﬂoor that measures 7.7 meters by 5.4 meters. The calculator would give an answer of 41.58 square meters. The calculated area is expressed to four signiﬁcant ﬁgures. However, each of the measurements used in the calculation is expressed to only two signiﬁcant ﬁgures. So the answer must also be In general, a calculated reported to two signiﬁcant ﬁgures (42 m2). answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated. Rounding To round a number, you must ﬁrst decide how many signiﬁ- Answers cant ﬁgures the answer should have. This decision depends on the given measurements and on the mathematical process used to arrive at the answer. Once you know the number of signiﬁcant ﬁgures your answer should have, round to that many digits, counting from the left. If the digit immediately to the right of the last signiﬁcant digit is less than 5, it is simply dropped and the value of the last signiﬁcant digit stays the same. If the digit in question is 5 or greater, the value of the digit in the last significant place is increased by 1. 1. a. 4 b. 4 c. 2 d. 5 2. a. 3 b. 2 c. 4 d. 4 L2 Chapter 3 Assessment problem 58 is related to Sample Problem 3.1. FYI Due to rounding, there will often be discrepancies between actual values and calculated values derived from measurements. Some students may conclude (correctly) that following the rules for significant figures introduces error in calculated values. Such rounding errors are generally small, but should nonetheless be acknowledged when performing calculations. 68 Chapter 3 b. 0.074 meter d. 1.072 meter Problem-Solving 3.2 Solve Problem 2 with the help of an interactive guided tutorial. Signiﬁcant Figures in Calculations CONCEPTUAL PROBLEM 3.1 Practice Problems Plus b. 8765 meters d. 40.007 meters Checkpoint Why must a calculated answer generally be rounded? 68 Chapter 3 Significant Figures in Calculations Sample Problem 3.1 SAMPLE PROBLEM 3.1 Answers Rounding Measurements 3. a. 8.71 × 101 m b. 4.36 × 108 m c. 1.55 × 10–2 m d. 9.01 × 103 m e. 1.78 × 10–3 m f. 6.30 × 102 m 4. a. 9 × 101 m b. 4 × 108 m c. 2 × 10–2 m d. 9 × 103 m e. 2 × 10–3 m f. 6 × 102 m Round off each measurement to the number of signiﬁcant ﬁgures shown in parentheses. Write the answers in scientiﬁc notation. a. 314.721 meters (four) b. 0.001 775 meter (two) c. 8792 meters (two) Analyze Identify the relevant concepts. Round off each measurement to the number of signiﬁcant ﬁgures indicated. Then apply the rules for expressing numbers in scientiﬁc notation. Solve Apply the concepts to this problem. Count from the left and apply the rule to the digit immediately to the right of the digit to which you are rounding. The arrow points to the digit immediately following the last signiﬁcant digit. a. 314.721 meters c 2 is less than 5, so you do not round up. 314.7 meters 3.147 102 meters b. 0.001 775 meter c 7 is greater than 5, so round up. 0.0018 meter 1.8 103 meter c. 8792 meters c 9 is greater than 5, so round up 8800 meters 8.8 103 meters Practice Problems Plus Math Handbook For help with scientiﬁc notation, go to page R56. Practice Problems Evaluate Do the results make sense? The rules for rounding and for writing numbers in scientiﬁc notation have been correctly applied. 3. Round each measurement to three signiﬁcant ﬁgures. Write your answers in scientiﬁc notation. a. 87.073 meters b. 4.3621 108 meters c. 0.01552 meter d. 9009 meters e. 1.7777 103 meter f. 629.55 meters 4. Round each measurement in Practice Problem 3 to one signiﬁcant ﬁgure. Write each of your answers in scientiﬁc notation. Round each measurement to two significant figures. Write your answers in scientific notation. a. 94.592 grams (9.5 × 101g) b. 2.4232 × 103 grams (2.4 × 103g) c. 0.007 438 grams (7.4 × 10–3g) d. 54 752 grams (5.5 × 104g) e. 6.0289 × 10–3 grams (6.0 × 10–3 g) f. 405.11 grams (4.1 × 102g) Math Practice Problems Problem-Solving 3.3 Solve Problem 3 with the help of an interactive guided tutorial. L2 Handbook For a math refresher and practice, direct students to scientific notation, page R56. with ChemASAP Section 3.1 Measurements and Their Uncertainty 69 Answers to... Checkpoint A calculated answer must be rounded in order to make it consistent with the measurements from which it was calculated. The calculated answer cannot be more precise than the least precise measurement used in the calculation. Scientific Measurement 69 Section 3.1 (continued) Addition and Subtraction The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. Work through Sample Problem 3.2 below which provides an example of rounding in an addition calculation. L2 Discuss The rules for rounding calculated numbers can be compared with the old adage,“A chain is only as strong as its weakest link.” Explain that an answer cannot be more precise than the least precise value used to calculate the answer. Ask, In addition and subtraction, what is the least precise value? (The measurement with the fewest digits to the right of the decimal point.) In multiplication and division, what is the least precise value? (The measurement with the fewest significant figures.) If students wonder why addition and subtraction rules differ from multiplication and division rules, point out that in addition and subtraction of measurements, the measurements are of the same property, such as length or volume. However, in the multiplication and division of measurements, new quantities or properties are being described, such as speed (length ÷ time), area (length × length), and density (mass ÷ volume). SAMPLE PROBLEM 3.2 Signiﬁcant Figures in Addition Calculate the sum of the three measurements. Give the answer to the correct number of signiﬁcant ﬁgures. 12.52 meters ⫹ 349.0 meters ⫹ 8.24 meters Analyze Identify the relevant concepts. Calculate the sum and then analyze each measurement to determine the number of decimal places required in the answer. Solve Apply the concepts to this problem. Align the decimal points and add the numbers. Round the answer to match the measurement with the least number of decimal places. Math Handbook For help with signiﬁcant ﬁgures, go to page R59. 369.76 meters The second measurement (349.0 meters) has the least number of digits (one) to the right of the decimal point. Thus the answer must be rounded to one digit after the decimal point. The answer is rounded to 369.8 meters, or 3.698 ⫻ 102 meters. Practice Problems Answers 5. Perform each operation. 5. a. 79.2 m b. 7.33 m c. 11.53 m d. 17.3 m 6. 23.8 g Practice Problems Plus L2 Problem-Solving 3.6 Solve Problem 6 with the help of an interactive guided tutorial. with ChemASAP Find the total mass of four stones with the following masses: 10.32 grams, 11.81 grams, 124.678 grams, and 0.9129 gram. (147.72 g) Express your answers to the correct number of signiﬁcant ﬁgures. a. 61.2 meters ⫹ 9.35 meters ⫹ 8.6 meters b. 9.44 meters ⫺ 2.11 meters c. 1.36 meters ⫹ 10.17 meters d. 34.61 meters ⫺ 17.3 meters 6. Find the total mass of three diamonds that have masses of 14.2 grams, 8.73 grams, and 0.912 gram. Handbook For a math refresher and review, direct students to scientific notation, page R56. 70 Chapter 3 Sample Problem 3.3 Math Answers Practice Problems Plus Handbook For a math refresher and practice, direct students to using a calculator, page R62. 7. a. 1.8 × 101 m2 b. 6.75 × 102 m c. 5.87 × 10–1 min 8. 1.3 × 103 m3 L2 Calculate the volume of a house that has dimensions of 12.52 meters by 36.86 meters by 2.46 meters. (1.14 × 103 m3) 70 Chapter 3 meters meters meters Evaluate Does the result make sense? The mathematical operation has been correctly carried out and the resulting answer is reported to the correct number of decimal places. Sample Problem 3.2 Math 12.52 349.0 + 8.24 Multiplication and Division In calculations involving multiplication Quick LAB and division, you need to round the answer to the same number of signiﬁcant ﬁgures as the measurement with the least number of signiﬁcant ﬁgures. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements. The position of the decimal point is important only in rounding the answers of addition or subtraction problems. L2 Objectives After completing this activity, students will be able to • measure length with accuracy and precision. • apply rules for rounding answers calculated from measurements. • determine experimental error and express it as percent error. Accuracy and Precision Checkpoint How many signiﬁcant ﬁgures must you round an answer to when performing multiplication or division? SAMPLE PROBLEM 3.3 Signiﬁcant Figures in Multiplication and Division Perform the following operations. Give the answers to the correct number of signiﬁcant ﬁgures. a. 7.55 meters ⫻ 0.34 meter b. 2.10 meters ⫻ 0.70 meter c. 2.4526 meters ⫼ 8.4 Analyze Identify the relevant concepts. Students may contend that making one measurement of some property, such as length, is satisfactory. Ask, What possible errors may occur when making only one length measurement? (Acceptable answers include misreading the ruler or not holding the ruler parallel to the length of the object.) Perform the required math operation and then analyze each of the original numbers to determine the correct number of signiﬁcant ﬁgures required in the answer. Solve Apply the concepts to this problem. Round the answers to match the measurement with the least number of signiﬁcant ﬁgures. a. 7.55 meters ⫻ 0.34 meter ⫽ 2.567 (meter)2 ⫽ 2.6 meters2 (0.34 meter has two signiﬁcant ﬁgures) b. 2.10 meters ⫻ 0.70 meter ⫽ 1.47 (meter)2 ⫽ 1.5 meters2 (0.70 meter has two signiﬁcant ﬁgures) c. 2.4526 meters ⫼ 8.4 ⫽ 0.291 976 meter ⫽ 0.29 meter (8.4 has two signiﬁcant ﬁgures) Math Handbook For help with using a calculator, go to page R62. Skills Focus Measuring, calculating Prep Time 5 minutes Materials 3 inch × 5 inch index cards, metric rulers Class Time 15 minutes Evaluate Do the results make sense? The mathematical operations have been performed correctly, and the resulting answers are reported to the correct number of places. Teaching Tips Practice Problems 7. Solve each problem. Give your answers to the correct number of signiﬁcant ﬁgures and in scientiﬁc notation. a. 8.3 meters ⫻ 2.22 meters b. 8432 meters ⫼ 12.5 c. 35.2 seconds ⫻ 1 minute 60 seconds 8. Calculate the volume of a warehouse that has inside dimensions of 22.4 meters by 11.3 meters by 5.2 meters. (Volume ⫽ l ⫻ w ⫻ h) Problem-Solving 3.8 Solve Problem 8 with the help of an interactive guided tutorial. with ChemASAP Emphasize that students should use an interior, marked line, such as 10.0 cm, as the initial point, instead of the end of the ruler, which may be damaged. Expected Outcome Measured values should be similar, but not necessarily identical for all students. Analyze and Conclude 71 For Enrichment L3 Have students devise methods of calculating the volume of one card. Point out that measuring the thickness of one card with a ruler would be very inaccurate. Ask, How might the measurement of the thickness of the card be improved? (Use a more precise instrument, such as a micrometer, or measure the thickness of a number of cards and divide by the number of cards.) Have students determine the thickness of one card and calculate its volume. Using the class average of the calculated volumes, have each student determine the percent error using the average as the accepted value. 1. Four for length; three for width 2. See Expected Outcome. 3. Significant digits for rounded-off answers are area, 3, and perimeter, 4. Some students may not round to the proper number of digits. 4. Errors of ±0.03 cm are acceptable. Such errors yield percent errors of 0.2% for length and 0.4% for width. Answers to... Checkpoint The same number of significant figures as the measurement with the least number of significant figures. Scientific Measurement 71 Quick LAB Section 3.1 (continued) 3 Accuracy and Precision ASSESS Evaluate Understanding L2 Write the following sets of measurements on the board. (1) 78°C, 76°C, 75°C (2) 77°C, 78°C, 78°C (3) 80°C, 81°C, 82°C Ask, If these sets of measurements were made of the boiling point of a liquid under similar conditions, explain which set is the most precise? (Set 2 is the most precise because the three measurements are closest together.) What would have to be known to determine which set is the most accurate? (the accepted value of the liquid’s boiling point) Reteach Purpose Procedure To measure the dimensions of an object as accurately and precisely as possible and to apply rules for rounding answers calculated from the measurements. 1. Use a metric ruler to measure in centimeters the length and width of an index card as accurately and precisely as you can. The hundredths place in your measurement should be estimated. Materials 3 inch 5 inch index • card • metric ruler 2. Calculate the perimeter [2 (length width)] and the area (length width) of the index card. Write both your unrounded answers and your correctly rounded answers on the chalkboard. Analyze and Conclude 1. How many signiﬁcant ﬁgures are in your measurements of length and of width? 2. How do your measurements compare with those of your classmates? 3. How many signiﬁcant ﬁgures are in your calculated value for the area? In your calculated value for the perimeter? Do your rounded answers have as many signiﬁcant ﬁgures as your classmates’ measurements? 4. Assume that the correct (accurate) length and width of the card are 12.70 cm and 7.62 cm, respectively. Calculate the percent error for each of your two measurements. L1 Use Figure 3.5 to reteach the method of correctly recording the number of significant figures in a measurement. Then have students convert each measurement into scientific notation. (6 × 10–1 m, 6.1 × 10–1 m, 6.07 × 10–1 m) Acceptable answers will include the following information: Accuracy compares a measured value to an accepted value of the measurement, precision compares a measured value to a set of measurements made under similar conditions, and error is the difference between the measured and accepzzted values. 3.1 Section Assessment 9. Key Concept How do measurements relate to experimental science? 10. Key Concept How are accuracy and precision evaluated? 11. Key Concept Why must a given measurement always be reported to the correct number of signiﬁcant ﬁgures? 12. Key Concept How does the precision of a calculated answer compare to the precision of the measurements used to obtain it? 13. A technician experimentally determined the boiling point of octane to be 124.1°C. The actual boiling point of octane is 125.7°C. Calculate the error and the percent error. 15. Solve the following and express each answer in scientiﬁc notation and to the correct number of signiﬁcant ﬁgures. a. (5.3 104) (1.3 104) b. (7.2 104) (1.8 103) c. 104 103 106 d. (9.12 101) (4.7 102) e. (5.4 104) (3.5 109) Explanatory Paragraph Explain the differences between the accuracy, precision, and error of a measurement. 14. Determine the number of signiﬁcant ﬁgures in each of the following. a. 11 soccer players b. 0.070 020 meter c. 10,800 meters d. 5.00 cubic meters Assessment 3.1 Test yourself on the concepts in Section 3.1. with ChemASAP 72 Chapter 3 If your class subscribes to the Interactive Textbook, use it to review key concepts in Section 3.1. with ChemASAP 72 Chapter 3 Section 3.1 Assessment 9. Making correct measurements is fundamental to the experimental sciences. 10. Accuracy is the measured value compared to the correct values. Precision is comparing more than one measurement. 11. The significant figures in a calculated answer often depend on the number of significant figures of the measurements used in the calculation. 12. A calculated answer cannot be more precise than the least precise measurement used in the calculation. 13. error = –1.6°C; percent error = 1.3% 14. a. unlimited b. 5 c. 3 d. 3 15. a. 6.6 × 104 b. 4.0 × 10–7 c. 107 d. 8.65 × 10–1 e. 1.9 × 1014