Significant Figures

Significant Figures - 22 Chapter 1 in this situation,...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 22 Chapter 1 in this situation, students will sometimes mentaiiy divide the space between 1 and 2 into quarters or thirds, and make an estimate of 1.25 or 1.33. However, such an estimate is incorrect, because it really involves dividing the space into hundredths (25i100 or 33/100). Estimates shouid involve V only the firstdecimal place beyond the last certain digit. Matter, Measurement, and Problem Solving 1.7 The Reliability of 3 Measurement Recall from our opening example (Section 1.1) that carbon monoxide is a colorless gas emitted by motor vehicles and found in polluted air. The table below shows carbon monoxide concentrations in Los Angeles County as reported by the U.S. Environmental Protection Agency (EPA) over the period 1997—2002: Carbon Monoxide Year Concentration (ppm)* 1997 15.0 1998 11.5 1999 11.1 2000 9.9 2001 7.2 2002 6.5 ’'Second maximum, 8 hour average; ppm 2 parts per million, defined as mL pollutant per million mL of air. The first thing you should notice about these values is that they are decreasing over time. For this decrease, we can thank the Clean Air Act and its amendments, which have resulted in more efficient engines, in specially blended fuels, and consequently in cleaner air in all major U.S. cities over the last 30 years. The second thing you might notice is the number of digits to which the measurements are reported. The number of digits in a re- ported measurement indicates the certainty associated with that measurement. For exam— ple, a less certain measurement of carbon monoxide levels might be reported as follows: Carbon Monoxide Year Concentration (ppm) 1997 15 1998 12 1999 1 1 2000 10 2001 7 2002 7 Notice that the first set of data is reported to the nearest 0.1 ppm while the second set is reported to the nearest 1 ppm. Scientists agree to a standard way of reporting mea- sured quantities in which the number of reported digits reflects the certainty in the mea- surement: more digits, more certainty; fewer digits, less certainty. Numbers are usually written so that the uncertainty is in the last reported digit. (That uncertainty is assumed to be 3:1 in the last digit unless otherwise indicated.) For example, by reporting the 1997 carbon monoxide concentration as 15.0 ppm, the scientists mean 15.0:l:0.1 ppm. The carbon monoxide concentration is between 14.9 and 15.1 ppm—it might be 15.1 ppm, for example, but it could not be 16.0 ppm. In contrast, if the reported value was 15 ppm (without the .0), this would mean 1511 ppm, or between 14 and 16 ppm. In general, Scientific measurements are reported so that every digit is certain except the last, which is estimated. For example, consider the following reported number: 5.213 7‘ ’\ , certain estimated The first three digits are certain; the last digit is estimated. The number of digits reported in a measurement depends on the measuring device. For example, consider weighing a pistachio nut on two different balances (Figure 1.14 on page 23). The balance on the left has marks every 1 gram, while the balance on the right has marks every 0.1 gram. For the balance on the left, we mentally divide the space between the 1— and 2—gram marks into ten equal spaces and estimate that the pointer is at about 1.2 grams. We then write the measurement as 1.2 grams indicating that we are sure of the “1” but have estimated the “.2.” The balance on the right, with marks every tenth of a gram. 1.7 The Reliability of 8 Measurement 23 Estimation in Weighing in. ll! 2 a“ Ilillli n...— iiiiiiliiil lb) I Markings every 1 g l Markings every 0.1 g l Estimated reading 1.2 g | Estimated reading 1.27 g requires us to write the result with more digits. The pointer is between the 1.2-gram mark and the 1.3-gram mark. We again divide the space between the two marks into ten equal spaces and estimate the third digit. For the figure shown, we report 1.27 g. EHAMPLE 1.4 Reporting the Correct Number of Digits The graduated cylinder shown at right has markings every 0.1 mL. Report the volume (which is read at the bottom of the meniscus) to the correct number of digits. (Note: The meniscus is the crescent—shaped surface at the top of a column of liquid} Solution Since the bottom of the meniscus is between the 4.5 and 4.6 mL markings, mentally divide the space between the markings into ten equal spaces and estimate the next digit. In this case, you should report the result as 4.57 mL. What if you estimated a little differently and wrote 4.56 mL? in general, one unit difference in the last digit is acceptable because the last digit is estimated and different people might estimate it slightly differently. However. if you wrote 4.63 mL, you would have misreported the measurement. for Practice1fi Record the temperature on the thermometer shown at right to the correct number of significant digits. Counting Significant Figures The precision otha measurement—which depends on the instrument used to make the mea- surement—must be preserved. not only when recording the measurement, but also when performing calculations that use the measurement. The preservation of this precision is conveniently accomplished by using significant figures. In any reported measurement, the non-place-holding digits—those that are not simply marking the decimal place—are called significant figures (or significant digits). The greater the number of significant figures, the greater is the certainty of the measurement. For example, the number 23.5 has three signifi— cant figures while the number 23.56 has four. To determine the number of significant figures in a number containing zeroes, we must distinguish between zeroes that are signifi~ cant and those that simply mark the decimal place. For example. in the number 0.0008. the leading zeroes mark the decimal place but do not add to the certainty of the measurement and are therefore not significant; this number has only one significant figure. In contrast. the trailing zeroes in the number 0.000800 do add to the certainty of the measurement and a? therefore counted as significant; this number has three significant figures. 4 FIGURE 1.14 Estimation in Weigh- ing (a) This scale has markings every 1 3. so we estimate to the tenths place by men- tally dividing the space into ten equal spaces to estimate the last digit. This read— ing is 1.2 g. (b) Because this balance has markings every 0.1 3. we estimate to the hundredths place. This reading is 1.27 g. Meniscus 5 Hill” ll .Tlllllllll m Chapter 1 Matter, Measurement, and Problem Solving To determine the number of significant figures in a number, follow these rules (with examples shown on the right). Significant Figure Rules Examples . All nonzero digits are significant. 28.03 0.0540 . Interior zeroes (zeroes between two numbers) 408 7.0301 are significant. . Leading zeroes (zeroes to the left of the first 2 0.0000 nonzero number) are not significant. \ f They only serve to locate the decrmal point. not significant . Trailing zeroes (zeroes at the end of a number) are categorized as follows: - Trailing zeroes after a decimal point are 45.000 3.5600 always significant - Trailing zeroes before an implied decimal 1200 ambiguous point are ambiguous and should be avoided 1.2 X 103 2 Significant figures by usmg sc1ent1fic notation. 1.20 X 103 “ignifiwm figures 1.200 X 103 4significant figures - Some textbooks put a decimal point after one 1200. 4 significant figures or more trailing zeroes if the zeroes are to be (“mm i“ 3"“ tex‘bmks) considered significant. We avoid that practice in this book, but you should be aware ofit. Exact Numbers Exact numbers have no uncertainty, and thus do not limit the number of significant figures in any calculation. In other words, we can regard an exact number as having an unlimited number of significant figures. Exact numbers originate from three sources: ~ From the accurate counting of discrete objects. For example, 3 atoms means 3.00000. . . atoms. - From defined quantities, such as the number of centimeters in 1 m. Because 100 cm is defined as 1 m, 100 cm = 1 m means 100.00000 . . . cm = 1.0000000. . .m - From integral numbers that are part of an equation. For example, in the equation, diameter radius = —E—-, the number 2 is exact and therefore has an unlimited number of significant figures. 1 EXAMPLE 1.5 Determining the Number of Significant Figures in a Number Solution 1 (a) 0.04450m ‘ (b) 5.0003km How many significant figures are in each of the following? i (a) 0.04450 m l (c) 10 dm = 1 m l (6) 0.00002 mm (b) 5.0003km (d) 1.000 x 105s (f) 10,000m , Four sigmficant figures. The two 4’s and the 5 are significant (rule 1). The trailing zero is after a decimal point and is therefore significant (rule 4). The leading zeroes only mark the decimal 3 place and are therefore not significant (rule 3). | Five sigmficant figures. The 5 and 3 are significant (rule 1) as are the three interior zeroes (rule 2). 1(c) 10dm=1m ‘ 1.7 The Reliability of a Measurement 25 Unlimited significantfigures. Defined quantities have an unlimited number of significant figures. y (d) 1.000 x 1055 ' (e) 0.00002 mm and therefore significant (rule 4). and are therefore not significant (rule 3). Four significant figures. The 1 is significant (rule 1). The trailing zeroes are after a decimal point One significant figure. The 2 is significant (rule 1). The leading zeroes only mark the decimal place How many significant figures are in each of the following numbers? ‘ (a) 554 km ‘ (c) 1.01 x 105 m ' (e) 1.4500 km (b) 7pennies ((1) 0.000993 (f) 21,000m Significant Figures in Calculations When you use measured quantities in calculations, the results of the calculation must reflect the precision of the measured quantities. You should not lose or gain precision during mathematical operations. Follow these rules when carrying significant figures through calculations. Rules for Calculations Examples . In multiplication or division, the 1.052 X 12.054 X 0.53 = (5.331113. = 6.7 result carries the same number (“is-figures) (his-figures) (his-figures) (lsisfigures) ofsignificantfiguresasthefactor 20035 _ 320 = {yJfiEELWR = (1626 with the fewest significant figures. (5 Sig. figures) (3 Sig, figures) (3 sig. figures) In addition or subtraction the result carries the same number of decimal places as the quantity with the fewest decimal places. In addition and subtraction, it is helpful to draw a line next to the number with the fewest decimal places. This line determines the number of decimal places in the answer. When rounding to the correct number of significant figures, round down if the last (or leftmost) digit dropped is four or less; round up if the last (or leftmost) digit dropped is five or more. To two significant figures: 5.37 rounds to 5.4 5.34 rounds to 5.3 5.35 rounds to 5.4 5.349 rounds to 5.3 Notice in the last example that only the last (or leftmost) digit being dropped determines in which direction to round—ignore all digits to the right of it. To avoid rounding errors in multistep calculations round only the final answer—do not round intermediate steps. If you write down intermediate anSWers, keep track of significant figures by underlining the least significant digit. 6.78 x 5.903 x (5.489 — 5.01) = 6.78 x 5.903 x 0.429 9.1707 \ = 19 underline least significant digit l (f) 10,000 m Ambiguous. The 1 is significant (rule 1) but the trailing zeroes occur before an implied decimal I point and are therefore ambiguous (rule 4). Without more information, we would assume 1 ‘ significant figure. It is better to write this as 1 X 105 to indicate one significant figure or as __ 1.0000 X 105 to indicate five (rule 4). For Practice 1.5 A few books recommend a slightly different rounding procedure for cases ' where the last digit is 5. However. the procedure presented here is consistent with electronic calculators and will be used throughout this book. Chapter 1 Matter, Measurement, and Problem Solving Notice that for multiplication or division, the quantity with the fewest significant fig- ures determines the number of significant figures in the answer, but for addition and sub— traction, the quantity with the fewest decimal places determines the number of decimal places in the answer. In multiplication and division, we focus on significant figures, but in addition and subtraction we focus on decimal places. When a problem involves addition or subtraction, the answer may have a different number of significant figures than the ini- tial quantities. Keep this in mind in problems that involve both addition or subtraction and multiplication or division. For example, 1.002 — 0.999 0.003 3.754 3.754 = 7.99 x 10“4 = 8 x 10‘4 The answer has only one significant figure, even though the initial numbers had three or four. gig» l EXAMPLE 1.5 Significant Figures in Calculations ‘ Perform the following calculations to the correct number of significant figures. (a) 1.10 X 0.5120 X 4.0015 + 3.4555 , (b) 0.355 1 + 105.1 1 — 100.5820 i I i (c) 4.562 X 3.99870 + (452.6755 — 452.33) , (d) (14.84 X 0.55) — 8.02 j Solution (a) Round the intermediate result (in blue) to three significant fig- ; 1.10 X 0.5120 X 4.0015 + 3.4555 ; ures to reflect the three significant figures in the least precisely ‘ = 0.65219 ' known quantity (1.10). i = 0652 (b) Round the intermediate answer (in blue) to one decimal place to reflect the quantity with the fewest decimal places (105.1). . , Notice that 105.1 is not the quantity with the fewest significant l figures, but it has the fewest decimal places and therefore 1 determines the number of decimal places in the answer. 2 places of the decimal =53 l ‘ (c) Mark the intermediate result to two decimal places to reflect ‘ 4.562 X 3.99870 + (452.6755 — 452.33) : the number of decimal places in the quantity_within the l = 4562 X 3.99870 + 0.3455 1 1: parentheses havmg the fewest number of dec1ma1 places ( _ a L 904 — (452.33). Round the final answer to two significant figures to _ 32'7) \ l reflect the two significant figures in the least precisely known quantity (0.3455). i ((1) Mark the intermediate result to two significant figures to reflect ‘ (14.84 X 0.55) — 8.02 the number of significant figures in the quantity within the ‘ = 3.162 _ 802 parentheses having the fewest number of significant figures ‘ = 0342 (0.55). Round the final answer to one decimal place to reflect 1 0 1 the one decimal place in the least precisely known quantity (8.162). 28 Chapter 1 Matter, Measurement, and Problem Solving ° The results of student B are precise (close to one another in value) but inaccurate. Th inaccuracy is the result of systematic error, error that tends toward being either too hig or too low. Systematic error does not average out with repeated trials. For example, if balance is not properly calibrated, it may systematically read too high or too low. ' The results of student C display little systematic error or random error—they at both accurate and precise. Chemistry in Your Day Integrity in Data Gathering Most scientists spend many hours collecting data in the labora— tory. Often, the data do not turn out exactly as the scientist had expected (or hoped). A scientist may then be tempted to “fudge” his or her results. For example, suppose you are expect— ing a particular set of measurements to follow a certain pattern. After working hard over several days or weeks to make the mea- surements, you notice that a few of them do not quite fit the pattern that you anticipated. You might find yourself wishing that you could simply change or omit the “faulty” measure- ments to better fit your expectations. Altering data in this way is considered highly unethical in the scientific community and, when discovered, is usually punished severely. In 2004, Dr. Hwang Woo Suk, a stem cell researcher at the Seoul National University in Korea, published a research paper in Science (a highly respected research journal) claiming that he and his colleagues had cloned human embryonic stem cells. As part of his evidence, he showed photographs of the cells. The paper was hailed as an incredible breakthrough, and Dr. Hwan traveled the world lecturing on his work. Time magazine eve named him among their “people that matter” for 2004. Sever: months later, however, one of his co-workers revealed that th photographs were fraudulent. According to the co-worker, th photographs came from a computer data bank of stem cell phc tographs, not from a cloning experiment. A university panel in vestigated the results and confirmed that the photographs an other data had indeed been faked. Dr. Hwang was forced to re sign his prestigious post at the university. Although not common, incidents like this do occur fror time to time. They are damaging to a community that is large] built on trust. Research papers are reviewed by peers (other re searchers in similar fields), but usually reviewers are judgin whether the data support the conclusion—they assume that th experimental measurements are authentic. The pressure to suc ceed sometimes leads researchers to betray that trust. Howeve over time, the tendency of scientists to reproduce and build upo one another’s work results in the discovery of the frauduler data. When that happens, the researchers at fault are usually bar. ished from the community and their careers are ruined. I 1.8 Solving Chemical Problems Learning to solve problems is one of the most important skills you will acquire in thi course. No one succeeds in chemistry—or in life, really—without the ability to solv problems. Although no simple formula applies to every problem, you can learn problem solving strategies and begin to develop some chemical intuition. Many of the problem you will solve in this course can be thought of as unit conversion problems, where you at given one or more quantities and asked to convert them into different units. Other prol: lems require the use of specific equations to get to the information you are trying to fin( In the sections that follow, you will find strategies to help you solve both of these types c problems. Of course, many problems contain both conversions and equations, requirin the combination of these strategies. Converting from One Unit to Another In Section 1.6, we learned the SI unit system, the prefix multipliers, and a few other unit.- Knowing how to work with and manipulate these units in calculations is central to $011: ing chemical problems. In calculations, units help to determine correctness. Using unit as a guide to solving problems is often called dimensional analysis. Units should alway be included in calculations; they are multiplied, divided, and canceled like any other alge braic quantity. Consider converting 12.5 inches (in) to centimeters (cm). We know from Table 1. that 1 in = 2.54 cm (exact), so we can use this quantity in the calculation as follows: 12.5111 X = 31.8cm ...
View Full Document

Page1 / 6

Significant Figures - 22 Chapter 1 in this situation,...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online