When we express a measured quantity with a number, though, we must always include the appropriate unit; otherwise , the measurement is meaningless.. We use the fol-lowing equation to co
Trang 1-What Do Molecules Look Like?
Molecules are far too small for us to observe them directly An
effective means of visualizing them is by the use of molecular
mod-els Throughout this book, we will represent matter at the molecular
level using molecular art, the two-dimensional equivalent of
molec-ular models In these pictures, atoms are represented as spheres and
atoms of particular elements are represented using specific colors
Table 1.1 lists some of the element s that you will encounter most
often and the colors used to represent them in this book
atom s The ball-and-stick model doe s a good job of illustrating the arrangement of atoms, but exaggerates the distances between atoms, relative to their sizes The space-filling model gives a more accurate picture of these interatomic distances but can obscure
the details of the three-dimensional arrangement
Molecular art can be of ball-and-stick models, in which the bonds connecting atoms appear as sticks [Figure 1.2(b)], or of
space-filling models, in which the atoms appear to overlap one
another [Figure 1.2(c)] Ball-and- s tick and space-filling
mod-els illustrate the specific, three-dimensional arrangement of the
Fluorine Iodine Figure 1.2 Water represented with a (a) molecular fonnula, (b)
ball-and-stick model, and (c) space-filling model
The reactions in Figure 1.1 are all things that you can observe at the ma croscopic level In
other words, these processes and their results are visible to the human eye In studying chemistry,
you will learn to visualize and understand these same proce sses at the molecular level
Although it can take many different forms , all matter consists of various combinations of atoms
of only a relatively small number of simple substances called elements The properties of matter
depend on which of these elements it contains, and on how the atoms of tho se elements are arranged
The Scientific Method
Experiments are the key to advancing our understanding of chenustry or any scie nce Although
all scientists will not nece ss arily take the same approach to experimentation, they follow a set of
guidelines known as the scientific method in order to add their result s to the larger body of
knowl-edge within a given field The flowchart in Figure 1.3 illustrates thi s basic proce ss The method
begins with the gathering of data via observations and experiments Scientists st udy these data and
try to identify patterns or trends When they find a pattern or trend, they may summarize their
find-ings with a law, a concise verbal or mathematical s tatement of a reliable relationship between
phe-nomena Scientists may then formulate a hypothesis, a tentative explanation for their observations
Further experiments are designed to test the hypothesis If experiments indicate that the hypothesis
is incorrect, the scientists go back to the drawing board, try to come up with a different
interpre-tation of their data, and formulate a new hypothe sis The new hypothe s is will then be tested by
experiment When a hypothesis stands the test of extensive experimentation, it may evolve into a
theory A theory is a unifying principle that explains a body of experimental observations and the
laws that are based on them Theorie s can also be used to predict related phenomena, so theories
are constantly being tested If a theory is disproved by experiment, then it mu s t be di scarde d or
modified so that it becomes consistent with experimental observations
Trang 26 CHAPTER 1 Chemistry: The Central Science
Set of conceptual
that e xplain s ob se rvation s ; accumulated experiments;
Having contracted
c owpo x, milkmaid s hav e a natural immunity
to smallpox
Int entionally expose
a healthy child to cowpox and later to s mallpox
Because child did not contract smallpox, immunity seemed to have re s ulted from cowpox exposure
Further Experiment:
Many more human s inoculated with cowpox virus, confirming
the model
Figure 1.3 Flowchart of the scientific method
S ome books refer to su b s ta nces as pure
su b s tances Th ese two ter m s generall y mean
the s ame thing alt ho ugh the adj ectiv e pu r e is
unnecessary in th is co ntext because a substan ce
is, by defi n i io n, pure
So ld s and li quids so meti mes are referred to
collecti vely as the co nde ns ed p h ase s Li qu i ds
and g a s es sometim es are r efe rred to co ll ectivel y
as f lu i d s
- - ,
_-
-Multimedia
Matter-three states of matter (Go
to www.mhhe.com/AR I S to view the
further categorized as either an element or a compound A substance is a form of matter that has a
definite (constant) composition and distinct properties Examples are salt (sodium chloride), iron, water , mercury, carbon dioxide, and oxygen Substances can be either elements (such as iron, mer- cury, and oxygen) or compounds (such as sa l t, water, and carbon dioxide) They differ from one another in composition and can be identified by appearance, smell, taste, a n d other properties
States of Matter
All substances can, in principle, exist as a solid, a liquid, and a gas, the three physical states depicted in Figure lA In a solid, particles are held close together in an orderly fashion with little freedom of motion A s a result, a solid does not conform to the shape of its container Particles in a liquid are close together but are not held rigidly in position; they are free to move past one another Thus, a liquid conforms to the shape of the part of the container it fills In a gas, the particles are
s eparated by distances that are very large compared to the size of the particles A sample of gas assumes bot h the shape and the volume of its container
The three states of matter can be interconverted without changing t h e chemical composition
of the substance Upon heating, a solid (e.g., ice) will melt to form a liquid (water) Further heating will vaporize the liquid, converting it to a gas (water vapor) Conversely, cooling a gas will cause
it to condense into a liquid When the liquid is cooled further , it will freeze into the solid form Figure 1.5 shows the three physical states of water
Elements
An element is a substance that cannot be separated into simpler substances by chemical means Iron, mercury, oxygen, and hydrogen are just four of the 117 elements t h a t have been identified Most of the known elements occur naturally o n Earth The others have been produced by scientists via nuclear processes, which are discussed in C h apter 20
For convenience, chemists use symbols of one or two letters to represent the elements Only the first letter of an element's chemical symbol is capitalized A list of the elements and their symbols appears on the in s ide front cover of this book The symbols of some e l ements are derived from their Latin names for example, Ag from argentum (silver), Pb from p l umbum (lead), and
Na from natrium (sodium) while most of them come from their English names for example, H for hydrogen, Co for cobalt, and Br for bromine
Compounds
Most elements can combine with other elements to form compounds Hydrogen gas, for example, bums in the presence of oxygen gas to form water, which has properties that are distinctly differ-
Trang 3SECTION 1.2 Classification of Matter
Figure 1 4 Molecular-level illu strat ions of a solid, liquid , and gas
ent from tho se of either hydrogen or oxygen Thus, water i s a compound, a s ub s tance composed
of atoms of two or more elements chemically united in fixed proportions The elements that make
up a compound are called the compound's constituent elements For example, the constituent
ele-ments of water are hydrogen and oxygen
o
A compound ' caniielt ' be ' sep ' arated Intel s i'mjJier ' substa ' nce s by ' an y ' physbil praces ;,- CA physi'~ A compound may consist of molecules or ions,
which we will discus s in Chapte r 2
cal process is one that doe s not change the identit y of the matter Examples of physical proce sses
include boiling , freezing, and filtering.) In stead, the separat i o n of a compound into its constituent
ele ments require s a chemical r eact ion
Mixtures
A mixture i s a combination of two or more s ub s tances in which the s ub sta nce s retain their distinct
identities Like s ub s tances, mixtures can be so lid s, liquid s, or gases Some familiar examples are
mixed nut s, 14 carat gold, apple juice, milk , and air Mixtures do not hav e a uni ve r sa l co n s tant
co mposition Therefore, samples of air collected in different location s will differ in compo s ition
because of differences in altitude, pollution , and other factor s Various brand s of apple juice may
differ in compo s ition becau se of the use of different var ietie s of apples, or there may be
differ-e nce s in proce ss in g and packaging , and so on
Mixtures are either homogeneous or heterogeneous When we di sso l ve a teaspoon of s ugar
in a glass of water , we get a homogeneous mixture becau se the composition of the mixture is
uniform throughout If we mix s ugar with iron filings, however , the sugar crystals and the iron
fil-ings remain di s tinct and di sce rnible from each o th e r (F igure 1.6 ) This type of mixture is ca lled a
heterogeneous mixture becau se the composition is not unif o rm
Mixtures, whether homogeneou s or h e tero geneo u s, can be se parated b y ph ysic al m ea n s into
pure component s without changing the identitie s of the comp o nent s Thu s, s ugar can be r ecovere d
from a water solution by evaporating the so lution to dryness Condensing the va por will give u s
back the water component To se parate the s ugar-iron mixture , we can u se a magnet to rem ove
the iron filing s from the sugar, becau se s u gar i s not attracted to the magnet [see Figure 1.6 (b)]
•
- ,
Fig u re 1.5 Wat e r as a so lid (ice),
liquid , and gas (We ca n ' t actually see
water vapor, any more than we can
see th e nitrogen and oxyge n that make
up most of the air we breathe When
we see s team or cloud s, what we are
actually seei ng is water vapo r that ha s
conden s ed upon e n co untering cold air.)
7
Trang 48 CHAPTER 1 Ch emistry : Th e Ce n tra l Sc i ence
Atoms of an element
Molecules of an element
Molecules of a compound
Mixture of e lement s and a compound
A ccord in g t o the U.S Me t r i c Associ a tion
(U SMA ) , t h e Unit ed S t ates is " t he only
sig n ific an t h old o u t " w it h r eg a rd to a dopt i on
of th e m e tr i c s yst e m T he othe r count r i e s that
contin ue to use trad i t ion a l uni ts ar e My a nm a r
(f or merl y Burm a) and Li be ri a
•
Separation by
- chemical methods
-Figure 1 7 Flo w chart for the cl a s s ification of matter
After separation , the components of the mixture will have the same composition and properties
a s they did prior to being mixed The relationships among substances, elements, compou n ds, and mixtures are summarized in Figure 1.7
Scientific Measurement
Scientists use a v ariety of devices to meas u re the properties of matter A meterstick is used to
mea-s ure length ; a buret, pipet , graduated cylinder , and volumetric flask are used to measure vo lu me (Figure 1.8); a balance is used to measure mass ; and a thermometer is used to measure tempera- ture Propertie s that can be measured are called quantitative properties because they are expressed using number s When we express a measured quantity with a number, though, we must always include the appropriate unit; otherwise , the measurement is meaningless For example, to say that the depth of a swimming poo l i s 3 is insufficient to distinguish between one that is 3 feet
(0.9 meters) and one that is 3 meters (9.8 feet) deep Uni ts are essential to reporting measurements correctly
The two system s of units with which you are probably most fami li ar are the English system
(foot, gallon, pound, etc.) and the metric system (meter, liter, kilogram, etc.) Although there has been an increase in the use of metric units in the United States in recent years, English units still are ' u sed C · 6iiiiri.6iiiY : For ' m.'ciiii ) i ea ] : s ' sc' i'eiiiii t s ' rec orded ' measurements in metric units, but in 1960,
the General Conference on Weight s and Measure s, the international a ut h ority on units, proposed
a revi s ed metric sy s tem for universal use by scientists We will use both metric and revised metric (SI) units in thi s book
Trang 5SECTION 1.3 Scientific Measurement
51 Base Units
The revi s ed metric s ystem i s called the International System of U nits ( abbre v iated S1 , f r o m th e
French S y stem e Int e rnati o nal e d' Unit es) Tabl e 1.2 li s t s th e s even S1 ba s e unit s All oth e r unit s of
mea s urement can be deriv e d from th ese base unit s The SI unit for vo lu m e , f o r in s tanc e, i s d e ri v ed
by cubing the S1 ba s e unit f o r l e ngth The pr e fi x e s li s ted in T able 1.3 are u se d t o den o t e d e cimal
fraction s and multiples of SI unit s This enabl es s cienti s t s to tailor the magnitude of a unit to a
particular application For e xampl e, th e meter ( m ) i s appropriate f o r de s cribin g the dim e n s ion s
of a clas s room , but the kil o meter ( kIn ), 1000 m , i s mor e a ppropri ate f or de sc ribing th e di s tance
between two citie s Unit s that you will encounter frequently in the s tudy o f c hemi s tr y include
those for mas s , t e mperature , volume , and den s it y
Mass
Although the term s ma s s a nd we i g ht of ten ar e u s ed int e r c h a ngeabl y, the y d o n o t mean t h e s am e
thing Strictly s p e aking , w e i g ht is the f o rce e xe rted by an o bj e ct or sa mple du e t o g ra v it y M ass i s
a mea s ure of the amount of matter in an object or s ample B e cau s e g ra v it y v arie s from l o c a tion to
TABLE 1.2 Base SI Units
Base Quantity Name of Unit
Amount of sub s tance mole
Luminous inten s ity candela
Only one of the seven S I base
units, the kilogram, itself contains
Trang 610 CHAPTER 1 Chemistry: The Central Science
Depending on the precision required, the
conversion from degrees Celsius to kelvin often is
done simply by adding 273, rather than 273.15
Think About It Check your math
and remember that converting a
temperature from degrees Celsius
to kelvin is different from
convert-ing a difference in temperature from
degrees Celsius to kelvin
Prefix
Giga- Mega- Kilo- Deci- Centi- Milli- Micro- Nano- Pico-
kel-point (100 ° C) of pure water at sea level As Table 1.2 s how s, the SI base unit of temperature i s the
kelvin Kelvin i s known as the absolute temperature sca le , meaning that the lowest temperature
po ss ible i s 0 K, a temperature referred to as " absolute zero." No degree sign CO) is u se d to
repre-se nt a temperature on the Kelvin sca le The theoretical ba s is of the Kelvin scale has to do with the
behavior of gases and is discussed in Chapter 11
Units of the Celsius and Kelvin sc ales are equal in magnitude, so a degree Celsius is
equiva-lent to a kelvin Thus, if the temperature of an object increases by S oC , it also increases by S K
Absolute zero on the Kelvin sca le i s equivalent to -273.1S o C on the Celsius scale We use the
fol-lowing equation to convert a temperature from units of degree s Celsius to kelvin:
using the Kelvin scale
Strategy Use Equation l.1 to convert temperatures from the Celsius scale to the Kelvin scale Then convert the range of temperatures from degrees Celsius to kelvin, keeping in mind that 1°C is equiva-
lent to 1 K
Setup Equation l.1 is already set up to convert the two temperatures from degrees Celsius to kelvin
No further manipUlation of the equation is needed The range in kelvin will be the same as the range
in degrees Celsius
Solution 360C + 273 = 309 K, 37°C + 273 = 310 K, and the range of 1 °C is equal to a range of 1 K
Trang 7SECTION 1.3 Scientific Measurement 11
,
I Practice Problem A Express the freezing point of water (O°C), the boiling point of water (100°C),
and the range spanned by the two temperatures using the Kelvin scale
Practice Problem B According to the website of the National Aeronautics and Space
Administra-tion (NASA), the average temperature of the universe is 2.7 K Convert this temperature to degrees
Celsius
Bringing Chemistry to life
Fahrenheit Temperature Scale
Outside of scientific circles, the Fahrenheit temperature s cale is the one most used in the United
States Before the work of Daniel Gabriel Fahrenheit (1686-1736), there were numerous different,
somewhat arbitrarily defined temperature scales, none of which gave consistent measurements
In 1724, Fahrenheit devised a scale based on the lowest artificially attainable temperature at the
time (a mixture of ice, water, and salt), which he labeled 0 ° ; the freezing point of water, which he
labeled 32 ° ; and the temperature of a healthy human body, which he labeled 96 ° The odd numbers
reportedly arose from Fahrenheit's initial use of a traditional scale with 12 degrees, each of which
he divided into 8 smaller degrees to give his thermometers better resolution Thus, water froze at
the fourth degree and body temperature occurred at the twelfth degree, but when each degree was
divided into eight smaller degrees, this put the freezing point of water at 32 ° and body temperature
at 96 ° Today we consider normal body temperature to be somewhat higher than 96 degrees
Fahr-enheit ( O F)
The boiling point of water on the Fahrenheit scale is 212 ° , meaning that there are 180 ° ( 212 - 32) between the freezing and boiling points This is considerably more than the 100 °
between the freezing point and boiling point of water on the Celsius scale Thus , the size of a
degree on the Fahrenheit scale is only 100/180 or five-ninths of a degree on the Celsius scale
Con-version between the Fahrenheit and Celsius scales is done using the following two equations:
= (temperature III degrees FahrenheIt - 32 ° F) X 9
and
~ : ~ X (temperature in degrees Celsius) + 32 ° F
temperature
A body temperature above 39 ° C constitutes a high fever Convert this temperature to the Fahrenheit
Practice Problem In Ray Bradbury's 1953 novel Fahrenheit 451, 451 OF is said to be the temperature
at which books, which have been banned in the story, ignite Convert this temperature to the Celsius
scale
"norma]" body temperature on the Fahrenheit scale is approximately
99°F (98.6°F is the number most
often cited), 102°F seems like a reasonable answer
Trang 812 CHAPTER 1 Chemistry: The Central Science
Oil floating on water is a familiar
demonstration of density differences
1 dm
Derived Units: Volume and Density
There are many quantitie s, s uch a s volume and density , that require units not included in the base
SI unit s In these ca s e s, we must combine base units to derive appropriate units for the quantity
The derived SI unit for volume , the meter cubed ( m3) is a larger volume than is
practi-cal in mo s t laboratory s ettings The more commonly used metric unit, the liter (L), is derived
by cubing the decimet e r (one-tenth of a meter) and is therefore also referred to as the cubic decimeter (dm\ Another commonly used metric unit of volume is the milliliter (mL), which
is derived by cubing the centimeter (11100 of a meter ) The milliliter is also referred to as the cubic centimeter ( cm \ Figure 1.9 illustrates the relationship between the liter (or dm3) and the milliliter (or cm3)
Density is the ratio of mass to volume Oil floats on water, for example, because, in addition
to not mixing with water , oil has a lower density than water That is, given equal volumes of the two liquids , the oil will have a smaller mass than the water Density i s calculated using the follow- ing equation:
Equation 1.4
/
1 dm _
m d=-
Figure 1 9 The larger cube has I-dm (10 em) sides and a volume of 1 L The next smaller cube
has l-cm (10 mm) sides and a volume of 1 em3 or 1 mL The smallest cube has I-mm sides and a ~cm v (j) 1 mm3
£
Trang 9SECTION 1.3 Scientific Measurement 13
where d, m, and V denote den s ity, mass, and volume, respectivel y The SI-derived unit for den s ity is
the kilogram per cubic meter (kg/ m \ This unit i s too large for m ost common u ses, howe ve r , so grams
per cubic centimeter (g/cm 3) and it s equivalent, grams per milliliter (g/ mL ), are used to express the
densities of mo s t solids and liquid s Water, for example, ha s a density of 1.00 g/cm3 at 4 ° C Because
gas densities generally are very low , we typically express them in unit s of grams per liter (gIL) :
1 g/cm3 = 1 g/mL = 1000 kg/m 3
1 gIL = 0.001 g/mL Sample Problem 1.3 illustrate s den s ity calculations
Sample Problem 1.3
Ice cube s float in a glass of water because solid water is less dense than liquid water (a) Calculate
the density of ice given that , at O ° C, a cube that is 2.0 cm on each s i de has a mass of 7 36 g, and
(b) determine t he volume occupied by 23 g of ice at O ° e
Strategy (a) Determine den s ity by di v iding mass by vo lume (Equat ion 1.4 ), and (b) u se the
calculated density to determine the volume occupied by the g i ven mass
Setup (a) We are given the mass of the ice cube , but we must calcu l ate its volume from the
dimension s given The volume of the ice cube is (2 0 cm) 3 , or 8 0 cm3 (b) Rearr anging Equation 1.4
to solve for volume gives V = mid
Practice Problem A Given that 25 0 rnL of mercury has a mass of 340 g, calculate (a ) the density of
mercury and ( b) the ma ss of 120 rnL of mercury
Practice Problem B Calculate (a) the densit y of a so lid substance if a cube mea sur in g 2.33 cm on
one side ha s a mass of 117 g and (b) the mass of a cube of the s ame s ubstance measuring 7.41 cm on
one side
The box on page 14 illu s trate s the importance of u s ing unit s carefully in scie ntific work
Checkpoint 1.3 Scientific Measurement
1.3 1
1 3 2
The coldest temperature ever recorded
on Earth was -128.6 ° F ( recorded at
Vo s tok Station, Antarctica, on Jul y
21, 1983) Express this temperature in
degree s Celsius and kelvin
a) - 89 2 ° C, -89 2 K
b) - 289.1 ° C, -15.9K
c) - 89.2 ° C, 183.9 K
d ) - 173.9 ° C, 99.3 K e) -7.0 ° C, 266.2 K
What i s the density of an object that
ha s a volume of 34.2 cm3 and a ma ss of 19.6 g?
Given that the density of gold i s
19.3 glcm 3 , calculate the volume (in cm3)
of a gold nugget with a mass of 5.98 g
a ) 3 23 cm3
, b) 5.98 cm '
Think About It For a sam ple
with a density less than 1 g/cm 3, the number of cubic centimeters
s hould be greater than the number
of grams In this case, 25 (cm3) >
23 (g)
,
Trang 10How Important Are Units?
On December 11, 1998, NASA launched the 125-million-dollar
Mars Climate Orbiter, which was intended to be the Red Planet's
first weather satellite After a 416-million-mile (mi) journey, the
23, 1999 Instead, it entered Mars's atmosphere about 100 krn
(62 mi) lower than planned and was destroyed by heat Mission
controllers later determined that the spacecraft was lost because
English measurement units were not converted to metric units in
the navigation software
Engineers at Lockheed Martin Corporation, who built the
of force Scientists at NASA's Jet Propulsion Laboratory, on the
that the thrust data they were given were expressed in newtons, a
metric unit To carry out the conversion between pound and
ton, we would start with 1 Ib = 0.4536 kg and, from Newton's
= 4.45 kg m/s2 = 4.45 N
because 1 newton (N) = 1 kg m/s2 Therefore, instead of
convert-ing lIb ofjorce to 4.45 N, the scientists treated it as a force of 1 N
in a lower orbit and the ultimate destruction of the spacecraft
into introduction to the metric system in elementary school, high
Substances are identified by their properties as well as by their composition Properties of a
explicit measurement)
Physical Properties
prop-erty is one that can be observed and measured without changing the identity of a substance For
Trang 11SECTION 1.5 Uncertainty in Measurement 15
but not in composition; both liquid water and ice are H20 Melting is a physical change; one in
which the state of matter changes, but the identity of the matter does not change We can recover
the original ice by cooling the water until it freezes Therefore, the melting point of a substance is
a physical property Similarly, when we say that nitrogen dioxide gas is brown, we are referring to
the physical property of color
Chemical Properties
The statement "Hydrogen gas bums in oxygen gas to form water" describes a chemical property of
hydrogen, because to observe this property we must carry out a chemical change burning in oxygen
(combustion), in this case After a chemical change, the original substance (hydrogen gas in this case)
will no longer exist What remains is a different substance (water, in this case) We cannot recover the
hydrogen gas from the water by means of a physical process, such as boiling or freezing
Every time we bake cookies, we bring about a chemical change When heated, the sodium
bicarbonate (baking soda) in cookie dough undergoes a chemical change that produces carbon
dioxide gas The gas forms numerous little bubbles in the dough during the baking process,
caus-ing the cookies to "rise." Once the cookies are baked, we cannot recover the sodium bicarbonate
by cooling the cookies, or by any physical process When we eat the cookies , we cause further
chemical changes that occur during digestion and metabolism
Extensive and Intensive Properties
All properties of matter are either extensive or intensive The measured value of an extensive
property depends on the amount of matter Mass is an extensive property More matter means
more mass Values of the same extensive property can be added together For example, two gold
nuggets will have a combined mass that is the sum of the masses of each nugget, and the length of
two city buses is the sum of their individual lengths The value of an extensive property depends
on the amount of matter
The value of an intensive property does not depend on the amount of matter Density and
temperature are intensive properties Suppose that we have two beakers of water at the same
tem-perature and we combine them to make a single quantity of water in a larger beaker The density
and the temperature of the water in the larger combined quantity will be the same as they were in
the two separate beakers Unlike mass and length, which are additive , temperature , density, and
other intensive properties are not additive
Uncertainty in Measurement
Chemistry makes use of two types of numbers: exact and inexact Exact numbers include numbers
with defined values, such as 2.54 in the definition 1 inch (in) = 2.54 cm , 1000 in the definition
1 kg = 1000 g, and 12 in the definition 1 dozen = 12 objects (The number 1 in each of these
defi-nitions is also an exact number.) Exact numbers also include those that are obtained by counting
Numbers measured by any method other than counting are inexact
Measured numbers are inexact because of the measuring devices that are used, the
indi-v iduals who use them, or both For example, a ruler that is poorly calibrated will result in
mea-s urements that are in error-no matter how carefully it is used Another ruler may be calibrated
properly but have insufficient resolution for the necessary measurement Finally , whether or not an
instrument is properly calibrated or has sufficient resolution, there are unavoidable differences in
how different people see and interpret measurements
Significant Figures
A n inexact number must be reported in such a way as to indicate the uncertainty in its value This
i s done using significant figures Significant figures are the meaningful digits in a reported
num-ber Consider the measurement of the memory can! in Figure 1.10 u s ing the ruler above it The
c ard's width is between 2 and 3 cm We may record the width as 2.5 cm, but because there are
n o gradations between 2 and 3 cm on this ruler, we are estimating the second digit Although we
are certain about the 2 in 2.5 , we are not certain about the 5 The last digit in a measured number
i s referred to as the uncertain digit; and the uncertainty associated with a measured number is
g enerally considered to be + 1 in the place of the last digit Thus , when we report the width of the
memory card to be 2.5 cm, we are implying that its width is 2.5 + 0.1 cm Each of the digits in a
Figure 1 10 The width we report
for the memory card depends on which
ruler we u s e to mea s ure it
Trang 1216 CHAPTER 1 Chemistry: The Central Science
It is importa nt not to imply greater certainty
in a measured number than is realistic For
example, it would be inappropriate to report
the width of the memory card in F igure 1.10
as 2.4500 cm, because this would imply an
uncertainty of :t o.0001
Appendix 1 re v ie ws scien tif ic notation
Think About It Be sure that you
have identified zeros correctly as
either significant or not significant
They are significant in (b), (d), and
(f); they are not significant in (c);
and it is not possible to tell in (e)
mea s ured number, including the uncertain digit, is a significant figure The reported width of the circle, 2.5 cm, contains two s ignificant figures
A ruler with millimeter gradations would enable u s to be certain about the second digit in this measurement and to estimate a third digit Now consider the measurement of the memory card using the ruler below it We may record the width as 2.45 cm Again, we estimate one digit beyond those we can read The reported width of 2.45 cm contains three significant figures Reporting the
width as 2.45 cm implies that the width is 2.45 + 0.01 cm
The number of significant figures in any number can be determined using the following guidelines:
1 Any digit that is not zero i s significant (112.1 has four significant figures)
2 Zeros located between nonzero digits are significant ( 305 ha s three significant figures, and
50.08 has four significant figures)
3 Zeros to the left of the fir s t nonzero digit are not significant (0.0023 has two significant
fig-ures, and 0.000001 ha s one s ignificant figure)
4 Zeros to the right of the la s t nonzero digit are significant if the number contains a decimal
point (1.200 has four s ignificant figures)
5 Zeros to the right of the last nonzero digit in a number that does not contain a decimal point
mayor may not be s ignificant (100 may have one , two, or three significant figures-it is impossible to tell without additional information) To avoid ambiguity in such cases, it is
Solution (a) 3; ( b) 4; (c) 3; ( d ) 4; (e) 1 or 2, an ambiguous case; (f) 4
Practice Problem Determine the number of significant figures in the following measurements:
(a) 1129 m , ( b ) 0.0003 kg, (c) 1.094 em, ( d ) 3.5 X 1012 atom s, (e) 150 mL, (f) 9.550 km
Calculations with Measured Numbers
Because we often use one or more measured numbers to calculate a desired result, a second set of guidelines specifies how to handle significant figures in calculations
1 In addition and subtraction, the answer cannot have more digits to the right of the decimal
point than any of the original numbers For example:
102.50
+ 0.231 102.731
143.29 -20.1 123.19
~ two digit s after the decimal point
~ three digits after the decimal point
~ round to 102.73
~ two digits after the decimal point
~ one digit after the decimal point
~ round to 123.2
Trang 13SECTION 1.5 Uncertainty in Measurement 17
The rounding procedure works as follows Suppose we want to round 102.13 and 54.86 each
dropped If the leftmost digit to be dropped is less than 5, as in 102.13, we round down ( to 102.1), meaning that we simply drop the digit(s) If the leftmost digit to be dropped is equal
to or greater than 5, a s in 54.86, we round up (to 54.9 ), meaning that we add 1 to the
preced-ing digit
2 In multiplication and division , the number of significant figures in the final product or
quo-tient is determined by the Oliginal number that ha s the s mallest number of significant
fig-ures The following examples illustrate this rule:
11.57/305.88 = 0.037825290964
figures)
significa nt figur es)
limit the number of s ignificant figure s in a calculated result For example, a penny minted
after 1982 has a ma ss of 2.5 g If we have three such pennies, the total mass is
,
4 In calculations with multiple steps, rounding the re s ult of each step can re su lt in "rounding
error." Consider the following two- ste p calculation:
round off C prior to u s ing it in the second s tep of the calculation
In general, it is best to retain at least one extra digit until the end of a multistep calculation,
Strategy Apply the rules for significant figures in calculations, and round each answer to the
appropriate number of digits
Setup (a) The answer will contain one digit to the right of the decimal point to match 317.5, which
has the fewest digits to the right of the decimal point (b) The answer will contain two digits to the
right of the decimal point to match 47.80 (c) The answer will contain three significant figures to
match 13.5, which has the fewest number of significant figures in the calculation (d) The answer will
contain three significant figures to match 6.25 ( e) To add numbers expressed in scientific notation,
first write both numbers to the same power of 10 That is, 4.991 X 103 = 49.91 X 102, so the answer
will contain two digits to the right of the decimal point (when multiplied by 102) to match both 5.46
and 49.91
(Continued)
Note that it is the number of pennies ( 3 ) , not the mass, that is an exact number