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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 1.2gram mark and the 1.3gram 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
nonplaceholding 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 (“isﬁgures) (hisﬁgures) (hisﬁgures) (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 coworkers revealed that th
photographs were fraudulent. According to the coworker, 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 ...
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 Spring '11
 Yizak
 Chemistry

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