3.1 measurements and their uncertainty 3 - henry...

Scientific Measurement

63

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

3.1

FOCUS

Objectives

3.1.1 Convert

measurements to sci-entific notation.

3.1.2 Distinguish

among accuracy, precision, and error of a mea-surement.

3.1.3 Determine

the number of sig-nificant figures in a measure-ment and in a calculated answer.

Guide for Reading

Build Vocabulary

Paraphrase

Have students write defi-nitions of the words

accurate

and

pre-cise

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

Use Prior Knowledge

Ask,

What everyday activities involve measur-ing?

(Examples include buying con-sumer 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 centime-ters?

(Acceptable answers range from 130 to 200 centimeters.)

INSTRUCT

Have students study the photograph and read the text that opens the section. Ask,

How do you think scientists ensure measurements are accurate and pre-cise?

(Acceptable answers include that sci-entists make multiple measurements by using the most precise equipment avail-able. They use samples with known values to check the reliability of the equipment.)

1

L2

L2

2

Section Resources

Connecting to Your World

Section 3.1 Measurements and Their Uncertainty 63

3.1 Measurements and Their Uncertainty

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?

Vocabularymeasurement

scientific notation

accuracy

precision

accepted value

experimental value

error

percent error

significant figures

Reading StrategyBuilding Vocabulary As you read, write a definition of each vocabulary term in your own words.

Using and Expressing MeasurementsYour height (67 inches), your weight (134 pounds), and the speed you drive onthe highway (65 miles/hour) are some familiar examples of measurements. Ameasurement is a quantity that has both a number and a unit. Everyonemakes and uses measurements. For instance, you decide how to dress in themorning based on the temperature outside. If you were baking cookies, youwould 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 decidewhether a measurement is correct. The units typically used in the sciencesare those of the International System of Measurements (SI).

In chemistry, you will often encounter very large or very small num-bers. A single gram of hydrogen, for example, contains approximately602,000,000,000,000,000,000,000 hydrogen atoms. The mass of an atom ofgold is 0.000 000 000 000 000 000 000 327 gram. Writing and using suchlarge and small numbers is very cumbersome. You can work more easilywith these numbers by writing them in scientific, or exponential, notation.

In scientific notation, a given number is written as the product oftwo numbers: a coefficient and 10 raised to a power. For example, thenumber 602,000,000,000,000,000,000,000 written in scientific notation is6.02 � 1023. The coefficient in this number is 6.02. In scientific notation, thecoefficient is always a number equal to or greater than one and less thanten. The power of 10, or exponent, in this example is 23. Figure 3.1 illus-trates how to express the number of stars in a galaxy by using scientificnotation. For more practice on writing numbers in scientific notation, referto page R56 of Appendix C.

Figure 3.1 Expressing very large numbers, such as the estimated number of stars in a galaxy, is easier if scientific notation is used.

200,000,000,000. � 2 � 1011

Decimalmoves11 placesto the left.

Exponent is 11

chem_TE_ch03.fm Page 63 Saturday, April 15, 2006 8:51 AM

64 Chapter 3

Section 3.1 (continued)

Using and Expressing MeasurementsUse VisualsUse VisualsFigure 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.)

Accuracy, Precision, and Error

CLASS ActivityActivityCLASS

Precision and AccuracyPurpose To illustrate the concepts of precision and accuracyMaterials a small object (such as a lead fishing weight), triple-beam balanceProcedure Place the object and a triple-beam balance in a designated area. Set a deadline by which each stu-dent 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 stu-dents find the average and compare their measurement to it.Expected Outcome Measured values should be similar, but not necessarily identical for all students.

FYIFYIOther analogies that may be useful in explaining precision vs. accuracy:• casting a fishing line• pitching horseshoes• a precision marching band

L1

L2

Striving for Scientific AccuracyThe 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 con-structed to allow steam to condense without

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 sedi-ment was not earth, but part of the flask etched away by the boiling water.

Facts and Figures

64 Chapter 3

Accuracy, Precision, and ErrorYour success in the chemistry lab and in many of your daily activitiesdepends on your ability to make reliable measurements. Ideally, measure-ments should be both correct and reproducible.

Accuracy and PrecisionAccuracy and Precision Correctness and reproducibility relate to theconcepts of accuracy and precision, two words that mean the same thing tomany people. In chemistry, however, their meanings are quite different.Accuracy is a measure of how close a measurement comes to the actual ortrue value of whatever is measured. Precision is a measure of how close aseries of measurements are to one another. To evaluate the accuracyof a measurement, the measured value must be compared to the correctvalue. To evaluate the precision of a measurement, you must comparethe values of two or more repeated measurements.

Darts on a dartboard illustrate accuracy and precision in measure-ment. 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-eyecorresponds to the degree of accuracy. The closer it comes to the bull’s-eye, the more accurately the dart was thrown. The closeness of severaldarts to one another corresponds to the degree of precision. The closertogether the darts are, the greater the precision and the reproducibility.

Look at Figure 3.2 as you consider the following outcomes.

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.

Checkpoint How does accuracy differ from precision?

Figure 3.2 The distribution of darts illustrates the difference between accuracy and precision.

Good accuracy and good precision: The darts are close to the bull’s-eye and to one another. Poor accuracy and good precision: The darts are far from the bull’s-eye but close to one another. Poor accuracy and poor precision: The darts are far from the bull’s-eye and from one another.

a

b

c

Good accuracyGood precision

Poor accuracyGood precision

Poor accuracyPoor precision

a b c

chem_TE_ch03_FPL.fm Page 64 Monday, August 2, 2004 1:09 PM


65

Word

Origins

The phrase

per annum

means “by the year.”

Use Visuals

Figure 3.2

Have students inspect Fig-ure 3.2. Ask,

If one dart 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 fig-ure?

(The error is the distance between the dart and the bull’s-eye.)

Discuss

Review the concept of absolute value. Ask,

What is the meaning of a

posi-

tive

error?

(Measured value is greater than accepted value.)

What is the meaning of a

negative

error?

(Mea-sured 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.

L2

L1

L2

Answers to...

Figure 3.3

7%

Checkpoint

Accuracy is determined by comparing a mea-sured value to the correct value. Pre-cision is determined by comparing the values of two or more repeated measurements.


Determining Error Note that an individual measurement may be accu-rate or inaccurate. Suppose you use a thermometer to measure the boilingpoint 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 ofpure water under these conditions is actually 100.0°C. There is a differencebetween the accepted value, which is the correct value based on reliablereferences, and the experimental value, the value measured in the lab. Thedifference between the experimental value and the accepted value is calledthe error.

Error � experimental value � accepted value

Error can be positive or negative depending on whether the experimentalvalue 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 exper-imental value differs from the accepted value. Often, it is useful to calculatethe relative error, or percent error. The percent error is the absolute value ofthe error divided by the accepted value, multiplied by 100%.

Using the absolute value of the error means that the percent error willalways be a positive value. For the boiling-point measurement, the percenterror is calculated as follows.

� 0.009 � 100%

� 0.9%

Just because a measuring device works doesn’t necessarily mean that itis accurate. As Figure 3.3 shows, a weighing scale that does not read zerowhen nothing is on it is bound to yield error. In order to weigh yourselfaccurately, you must first make sure that the scale is zeroed.

Percent error �0error 0

accepted value � 100%

Percent error �099.1�C - 100.0�C 0

100.0�C � 100%

� 0.9�C100.0�C � 100%

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?

Word OriginsPercent comes from the Latin words per, meaning “by” or “through,” and centum, meaning “100.” What do you think the phrase per annum means?

chem_TE_ch03.fm Page 65 Wednesday, April 26, 2006 9:12 AM

66

Chapter 3


Significant Figures in Measurements

Discuss

Point out that the concept of signifi-cant figures applies only to measured quantities. If students ask why an esti-mated 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 instru-ment are spaced very close together (e.g., on certain thermometers and graduated cylinders), it is sometimes more practical to estimate a measure-ment to the nearest half of the smallest calibrated increment, rather than to the nearest tenth.

CLASS ActivityCLASS

Olympic Times

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.

L2

L2

scale to the nearest tenth of a pound. With such a scale, however, you canalso estimate the weight to the nearest hundredth of a pound by noting theposition of the pointer between calibration marks.

Suppose you estimate a weight that lies between 2.4 lb and 2.5 lb to be2.46 lb. The number in this estimated measurement has three digits. Thefirst two digits in the measurement (2 and 4) are known with certainty. Butthe rightmost digit (6) has been estimated and involves some uncertainty.These three reported digits all convey useful information, however, and arecalled significant figures. The significant figures in a measurement includeall of the digits that are known, plus a last digit that is estimated. Mea-surements must always be reported to the correct number of significant fig-ures because calculated answers often depend on the number of significantfigures in the values used in the calculation.

Instruments differ in the number of significant figures that can beobtained from their use and thus in the precision of measurements. Thethree meter sticks in Figure 3.5 can be used to make successively more pre-cise measurements of the board.

igure 3.4 The precision of a weighing scale depends on

ow finely it is calibrated.

Rules for determining whether a digit in a measured value is significant:

1 Every nonzero digit in a reported measurement is assumed to be significant. The measurements 24.7 meters, 0.743 meter, and 714 meters each express a measure of length to three significant figures.

2 Zeros appearing between nonzero digits are significant. The measure-ments 7003 meters, 40.79 meters, and 1.503 meters each have four significant figures.

3 Leftmost zeros appearing in front of nonzero digits are not significant. They act as placeholders. The measurements 0.0071 meter, 0.42 meter, and 0.000 099 meter each have only two significant figures. The zeros to the left are not significant. By writing the measurements in scien-tific notation, you can eliminate such placeholding zeros: in this case, 7.1 � 10�3 meter, 4.2 � 10�1 meter, and 9.9 � 10�5 meter.

4 Zeros at the end of a number and to the right of a decimal point are always significant. The measurements 43.00 meters, 1.010 meters, and 9.000 meters each have four significant figures.

withChemASAP

Animation 2 Seehow the precision of a calculated result depends on the sensitivity of the measuring instruments.

scale to the nearest tenth of a pound. With such a scale, however, you canalso estimate the weight to the nearest hundredth of a pound by noting thposition of the pointer between calibration marks.

Suppose you estimate a weight that lies between 2.4 lb and 2.5 lb to b2.46 lb. The number in this estimated measurement has three digits. Thfirst two digits in the measurement (2 and 4) are known with certainty. Buthe rightmost digit (6) has been estimated and involves some uncertaintyThese three reported digits all convey useful information, however, and arcalled significant figures. The significant figures in a measurement includall of the digits that are known, plus a last digit that is estimated. Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significanfigures in the values used in the calculation.

Instruments differ in the number of significant figures that can bobtained from their use and thus in the precision of measurements. Ththree meter sticks in Figure 3.5 can be used to make successively more precise measurements of the board.

igure 3.4 The precision of a weighing scale depends on

ow finely it is calibrated.

Rules for determining whether a digit in a measuredvalue is significant:

1 Every nonzero digit in a reported measurement is assumed to be significant. The measurements 24.7 meters, 0.743 meter, and 714 meterseach express a measure of length to three significant figures.

2 Zeros appearing between nonzero digits are significant. The measurements 7003 meters, 40.79 meters, and 1.503 meters each have four significant figures.

3 Leftmost zeros appearing in front of nonzero digits are not significantThey act as placeholders. The measurements 0.0071 meter, 0.42 meterand 0.000 099 meter each have only two significant figures. The zerosto the left are not significant. By writing the measurements in scien-tific notation, you can eliminate such placeholding zeros: in this case7.1 � 10�3 meter, 4.2 � 10�1 meter, and 9.9 � 10�5 meter.

4 Zeros at the end of a number and to the right of a decimal point arealways significant. The measurements 43.00 meters, 1.010 meters, and9.000 meters each have four significant figures.

withChemASAP


66 Chapter 3

Significant Figures in MeasurementsSupermarkets often provide scales like the one in Figure 3.4. Customers usethese scales to measure the weight of produce that is priced per pound. Ifyou use a scale that is calibrated in 0.1-lb intervals, you can easily read thescale to the nearest tenth of a pound. With such a scale, however, you canalso estimate the weight to the nearest hundredth of a pound by noting theposition of the pointer between calibration marks.

Suppose you estimate a weight that lies between 2.4 lb and 2.5 lb to be2.46 lb. The number in this estimated measurement has three digits. Thefirst two digits in the measurement (2 and 4) are known with certainty. Butthe rightmost digit (6) has been estimated and involves some uncertainty.These three reported digits all convey useful information, however, and arecalled significant figures. The significant figures in a measurement includeall of the digits that are known, plus a last digit that is estimated. Mea-surements must always be reported to the correct number of significant fig-ures because calculated answers often depend on the number of significantfigures in the values used in the calculation.

Instruments differ in the number of significant figures that can beobtained from their use and thus in the precision of measurements. Thethree meter sticks in Figure 3.5 can be used to make successively more pre-cise measurements of the board.

Figure 3.4 The precision of a weighing scale depends on how finely it is calibrated.

Rules for determining whether a digit in a measured value is significant:

1 Every nonzero digit in a reported measurement is assumed to be significant. The measurements 24.7 meters, 0.743 meter, and 714 meters each express a measure of length to three significant figures.

2 Zeros appearing between nonzero digits are significant. The measure-ments 7003 meters, 40.79 meters, and 1.503 meters each have four significant figures.

3 Leftmost zeros appearing in front of nonzero digits are not significant. They act as placeholders. The measurements 0.0071 meter, 0.42 meter, and 0.000 099 meter each have only two significant figures. The zeros to the left are not significant. By writing the measurements in scien-tific notation, you can eliminate such placeholding zeros: in this case, 7.1 � 10�3 meter, 4.2 � 10�1 meter, and 9.9 � 10�5 meter.

4 Zeros at the end of a number and to the right of a decimal point are always significant. The measurements 43.00 meters, 1.010 meters, and 9.000 meters each have four significant figures.

withChemASAP


chem_TE_ch03.fm Page 66 Wednesday, April 13, 2005 1:54 PM


67

Use Visuals

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. Simi-larly, 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 estima-tion process. Ask,

If meter stick 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 millime-ters?

(606.5 to 607.4 mm)

Discuss

Be sure to review the role of zeros in determining the number of significant figures. When adding or subtracting numbers expressed in scientific nota-tion, remind students that the num-bers must all have the same exponent.

L1

L2

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)

Less Proficient Readers

Have students write in their own words the rules for determining the number of signifi-cant 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.

L1

Differentiated Instruction


Figure 3.5 Three differently calibrated meter sticks are used to measure the length of a board. A meter stick calibrated in a 1-m interval.

A meter stick calibrated in 0.1-m intervals. A meter stick calibrated in 0.01-m intervals. Measuring How many signifi-cant figures are reported in each measurement?

a

bc

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 place-holders 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.

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 unlim-ited number of significant figures. As you shall soon see, exact quan-tities do not affect the process of rounding an answer to the correct number of significant figures.

1m10 20 30 40 50 60 70 80 90

1m10 20 30 40 50 60 70 80 90

1m

a

b

c

Measured length = 0.6 m



chem_TE_ch03.fm Page 67 Saturday, April 15, 2006 8:59 AM

68 Chapter 3


CLASS ActivityActivityCLASS

Significant ZerosPurpose To provide practice in apply-ing the rules governing the signifi-cance of zeros in measurements

Materials textbook and scientific liter-ature, 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 fig-ures 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.

CONCEPTUAL PROBLEM 3.1

Answers1. a. 4 b. 4 c. 2 d. 52. a. 3 b. 2 c. 4 d. 4

Practice Problems PlusChapter 3 Assessment problem 58 is related to Sample Problem 3.1.

FYIFYIDue 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 round-ing errors are generally small, but should nonetheless be acknowledged when performing calculations.

L2

L2

68 Chapter 3

withChemASAP

CONCEPTUAL PROBLEM 3.1

Counting Significant Figures in MeasurementsCounting Significant Figures in MeasurementsHow many significant figures are in each measurement?a. 123 m b. 40,506 mmc. 9.8000 � 104 m d. 22 meter stickse. 0.070 80 m f. 98,000 m

Analyze Identify the relevant concepts.

The location of each zero in the measurement and the location of the decimal point deter-mine which of the rules apply for determining significant figures.

Solve Apply the concepts to this problem.

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. Count the significant figures in each length.a. 0.057 30 meters b. 8765 metersc. 0.000 73 meters d. 40.007 meters

2. How many significant figures are in each measurement?a. 143 grams b. 0.074 meterc. 8.750 � 10�2 gram d. 1.072 meter

Problem-Solving 3.2 Solve Problem 2 with the help of an interactive guided tutorial.

Significant Figures in CalculationsSuppose you use a calculator to find the area of a floor that measures7.7 meters by 5.4 meters. The calculator would give an answer of41.58 square meters. The calculated area is expressed to four significant fig-ures. However, each of the measurements used in the calculation isexpressed to only two significant figures. So the answer must also bereported to two significant figures (42 m2). In general, a calculatedanswer cannot be more precise than the least precise measurement fromwhich it was calculated. The calculated value must be rounded to make itconsistent with the measurements from which it was calculated.

RoundingRounding To round a number, you must first decide how many signifi-cant figures the answer should have. This decision depends on the givenmeasurements and on the mathematical process used to arrive at theanswer. Once you know the number of significant figures your answershould have, round to that many digits, counting from the left. If the digitimmediately to the right of the last significant digit is less than 5, it issimply dropped and the value of the last significant digit stays the same. Ifthe digit in question is 5 or greater, the value of the digit in the last signif-icant place is increased by 1.

Checkpoint Why must a calculated answer generally be rounded?

chem_TE_ch03_FPL.fm 8 Monday, August 2, 2004 1:09 PM


69

Significant Figures in Calculations

Sample Problem 3.1

Answers

3. a.

8.71

×

10

1

m

b.

4.36

×

10

8

m

c.

1.55

×

10

–

2

m

d.

9.01

×

10

3

m

e.

1.78

×

10

–

3

m

f.

6.30

×

10

2

m

4. a.

9

×

10

1

m

b.

4

×

10

8

m

c.

2

×

10

–

2

m

d.

9

×

10

3

m

e.

2

×

10

–

3

m

f.

6

×

10

2

m

Practice Problems Plus

Round each measurement to two sig-nificant figures. Write your answers in scientific notation.

a. 94.592 grams

(9.5

×

10

1

g)

b. 2.4232

×

10

3

grams

(2.4

×

10

3

g)

c. 0.007 438 grams

(7.4

×

10

–3

g)

d. 54,752 grams

(5.5

×

10

4

g)

e. 6.0289

×

10

–3

grams

(6.0

×

10

–3

g)

f. 405.11 grams

(4.1

×

10

2

g)

Math Handbook

For a math refresher and practice, direct students to scientific notation, page R56.

L2

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.


Math Handbook

withChemASAP

Practice Problems

Practice Problems

SAMPLE PROBLEM 3.1

Rounding MeasurementsRound off each measurement to the number of significant figuresshown 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 figuresindicated. Then apply the rules for expressing numbers in scientificnotation.

Solve Apply the concepts to this problem.

Count from the left and apply the rule to the digit immediately to theright of the digit to which you are rounding. The arrow points to thedigit immediately following the last significant digit.a.

2 is less than 5, so you do not round up.314.7 meters � 3.147 � 102 meters

b.

7 is greater than 5, so round up.0.0018 meter � 1.8 � 10�3 meter

c.

9 is greater than 5, so round up8800 meters � 8.8 � 103 meters

Evaluate Do the results make sense?

The rules for rounding and for writing numbers in scientific notationhave been correctly applied.

314.721 metersc

0.001 775 meterc

8792 metersc

3. Round each measurement to three significant figures. Write your answers in scientific notation.a. 87.073 metersb. 4.3621 � 108 metersc. 0.01552 meterd. 9009 meterse. 1.7777 � 10�3 meterf. 629.55 meters

4. Round each measurement in Practice Problem 3 to one sig-nificant figure. Write each of your answers in scientific notation.


For help with scientific notation, go to page R56.

chem_TE_ch03.fm Page 69 Wednesday, April 13, 2005 1:29 PM

70 Chapter 3


DiscussDiscussThe rules for rounding calculated num-bers 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 subtrac-tion, what is the least precise value?(The measurement with the fewest digits to the right of the decimal point.) In mul-tiplication and division, what is the least precise value? (The measurement with the fewest significant figures.) If stu-dents wonder why addition and subtrac-tion 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 mea-surements, new quantities or properties are being described, such as speed (length ÷ time), area (length × length), and density (mass ÷ volume).

Sample Problem 3.2

Answers5. a. 79.2 m b. 7.33 m c. 11.53 m

d. 17.3 m6. 23.8 g

Practice Problems PlusFind 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)

MathMath HandbookFor a math refresher and review, direct students to scientific notation, page R56.

L2

L2

Sample Problem 3.3

Answers7. a. 1.8 × 101 m2 b. 6.75 × 102 m

c. 5.87 × 10–1 min8. 1.3 × 103 m3

Practice Problems PlusCalculate the volume of a house that has dimensions of 12.52 meters by 36.86 meters by 2.46 meters. (1.14 × 103 m3)

MathMath HandbookFor a math refresher and practice, direct students to using a calculator, page R62.

L2

70 Chapter 3

Practice Problems

Significant Figures in AdditionCalculate the sum of the three measurements. Give the answer to thecorrect 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 determinethe 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 tomatch the measurement with the least number of decimal places.

The second measurement (349.0 meters) has the least number of digits(one) to the right of the decimal point. Thus the answer must berounded to one digit after the decimal point. The answer is rounded to369.8 meters, or 3.698 � 102 meters.

Evaluate Does the result make sense?The mathematical operation has been correctly carried out and theresulting answer is reported to the correct number of decimal places.

369.76 meters

+ 8.24 meters349.0 meters

12.52 meters

withChemASAP

MathMath Handbook

Addition and SubtractionAddition and Subtraction The answer to an addition or subtractioncalculation should be rounded to the same number of decimal places (notdigits) as the measurement with the least number of decimal places. Workthrough Sample Problem 3.2 below which provides an example of roundingin an addition calculation.

SAMPLE PROBLEM 3.2

5. Perform each operation. Express your answers to the correct number of significant figures.a. 61.2 meters � 9.35 meters �

8.6 metersb. 9.44 meters � 2.11 metersc. 1.36 meters � 10.17 metersd. 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.Problem-Solving 3.6 Solve

Problem 6 with the help of an interactive guided tutorial.

For help with significant figures, go to page R59.

chem_TE_ch03_FPL.fm Page 70 Wednesday, August 4, 2004 6:32 PM

Scientific Measurement 71

Quick LABQuick LAB

Accuracy and PrecisionObjectives After completing this activ-ity, students will be able to• measure length with accuracy and

precision.• apply rules for rounding answers cal-

culated from measurements.• determine experimental error and

express it as percent error.

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 mea-surement? (Acceptable answers include misreading the ruler or not holding the ruler parallel to the length of the object.)

Skills Focus Measuring, calculating

Prep Time 5 minutesMaterials 3 inch × 5 inch index cards, metric rulersClass Time 15 minutes

Teaching TipsEmphasize 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 Conclude1. Four for length; three for width2. 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.

L2

For EnrichmentFor EnrichmentHave students devise methods of calculating the volume of one card. Point out that measur-ing the thickness of one card with a ruler would be very inaccurate. Ask, How might the mea-surement 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.

L3

Answers to...

Checkpoint The same num-ber of significant figures as the mea-surement with the least number of significant figures.

71

withChemASAP

Practice Problems

Multiplication and DivisionMultiplication and Division In calculations involving multiplicationand division, you need to round the answer to the same number of signifi-cant figures as the measurement with the least number of significant fig-ures. The position of the decimal point has nothing to do with the roundingprocess when multiplying and dividing measurements. The position of thedecimal point is important only in rounding the answers of addition orsubtraction problems.

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 DivisionPerform the following operations. Give the answers to the correctnumber of significant figures.a. 7.55 meters � 0.34 meterb. 2.10 meters � 0.70 meterc. 2.4526 meters � 8.4

Analyze Identify the relevant concepts.Perform the required math operation and then analyze each of theoriginal numbers to determine the correct number of significantfigures required in the answer.

Solve Apply the concepts to this problem.Round the answers to match the measurement with the least numberof 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)

Evaluate Do the results make sense?The mathematical operations have been performed correctly, and theresulting answers are reported to the correct number of places.

7. Solve each problem. Give your answers to the correct num-ber of significant figures and in scientific notation.a. 8.3 meters � 2.22 metersb. 8432 meters � 12.5

c. 35.2 seconds � 1 minute60 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)


MathMath Handbook

For help with using a calculator, go to page R62.

chem_TE_ch03_FPL.fm Page 71 Sunday, August 8, 2004 3:24 PM

72 Chapter 3


ASSESSEvaluate UnderstandingEvaluate UnderstandingWrite the following sets of measure-ments 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 pre-cise? (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)

ReteachReteachUse Figure 3.5 to reteach the method of correctly recording the number of significant figures in a measurement. Then have students convert each mea-surement 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.

with ChemASAP

If your class subscribes to the Interactive Textbook, use it to review key concepts in Section 3.1.

33

L2

L1

Section 3.1 Assessment9. Making correct measurements is funda-

mental to the experimental sciences.10. Accuracy is the measured value compared

to the correct values. Precision is compar-ing 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 pre-cise 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. 315. a. 6.6 × 104 b. 4.0 × 10–7 c. 107

d. 8.65 × 10–1 e. 1.9 × 1014

72 Chapter 3

Quick LABQuick LAB

withChemASAP

Accuracy and Precision

PurposeTo measure the dimensions of an object as accurately and precisely as possible and to apply rules for round-ing answers calculated from the measurements.

Materials

• 3 inch � 5 inch index card

• metric ruler

Procedure1. Use a metric ruler to measure in centi-

meters the length and width of an index card as accurately and precisely as you can. The hundredths place in your measurement should be estimated.

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 Conclude1. 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 perime-ter? Do your rounded answers have as many significant figures as your class-mates’ 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.

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 sig-nificant figures?

12. Key Concept How does the precision of a cal-culated 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.

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

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 � 10�4) � (1.8 � 103) c. 104 � 10�3 � 106

d. (9.12 � 10�1) � (4.7 � 10�2) e. (5.4 � 104) � (3.5 � 109)

Explanatory Paragraph Explain the differences between the accuracy, precision, and error of a measurement.

Assessment 3.1 Test yourself on the concepts in Section 3.1.

chem_TE_ch03_FPL.fm Page 72 Monday, August 2, 2004 1:09 PM

3.1 measurements and their uncertainty 3 - henry...

Documents