Chapter 2
Measurement and
Problem Solving
Homework
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Exercises (optional)
 1 through 27 (odd)
Problems
 29-65 (odd)
 67-91 (odd)
 93-99 (odd)
Cumulative Problems
 101-117 (odd)
Highlight Problems (optional)
 119, 121
2.2 Scientific Notation:
Writing Large and Small Numbers
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In scientific (chemistry) work, it is not unusual to
come across very large and very small numbers
Using large and small numbers in
measurements and calculations is time
consuming and difficult
Recording these numbers is also very prone to
errors due to the addition or omission of zeros
A method exists for the expression of awkward,
multi-digit numbers in a compact form: scientific
notation
2.2 Scientific Notation:
Writing Large and Small Numbers
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Scientific Notation
 A system in which an ordinary decimal
number (m) is expressed as a product of a
number between 1 and 10, multiplied by 10
raised to a power (n)
 Used to write very large or very small
numbers
 Based on powers of 10
m  10
n
2.2 Scientific Notation:
Writing Large and Small Numbers
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Scientific notation uses exponents (i.e. powers of
numbers) which are numbers that are written as
superscripts (following another number) which indicate
how many times the number is multiplied by itself
(e.g., 62 = 6 × 6 = 36, 35 = 3 × 3 × 3 × 3 × 3 = 243
Scientific notation exclusively uses powers of 10
When ten is raised to a power, its decimal equivalent is
the number 1 followed by as many zeros as the power
itself
For example:
102 = 100 (two zeros and power of 2)
104 = 10,000 (four zeros and power of 4)
106 = 1,000,000 (six zeros and power of 6)
2.2 Scientific Notation:
Writing Large and Small Numbers
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A negative sign in front of an exponent indicates
that the number and the power to which it is
raised are in the denominator of a fraction in
which 1 is in the numerator.
The number of zeros between the decimal point
and the one is always one less than the absolute
power of the exponent
For example:
10-1 = 1/101 = 1/10 = 0.1
10-2 = 1/102 = 1/10 × 10 = 1/100 = 0.01
10-3 = 1/103 = 1/10 × 10 × 10 = 1/1000 = 0.001
2.2 Scientific Notation:
Writing Large and Small Numbers
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Numbers written in scientific notation
consist of a number (coefficient) followed
by a power of 10 (x 10n)
2
7.03  10
coefficient or
decimal part
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exponent
exponential term
or part
Negative exponent: number is less than 1
Positive exponent: number is greater than 1
2.2 Scientific Notation:
Writing Large and Small Numbers
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In an ordinary cup of water there are:
7,910,000,000,000,000,000,000,000 molecules
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Each molecule has a mass of:
0.0000000000000000000000299 gram
In scientific notation:
7.91 х 1024 molecules
2.99 х 10-23 gram
To Express a Number in Scientific Notation:
For small numbers (<1):
1) Locate the decimal point
2) Move the decimal point to the right to give a number
(coefficient) between 1 and 10
3) Write the new number multiplied by 10 raised to the
“nth power”
 where “n” is the number of places you moved the
decimal point so there is one non-zero digit to the
left of the decimal.
 If the decimal point is moved to the right, from its
initial position, then the exponent is a negative
number (× 10-n)
To Express a Number in Scientific Notation:
For large numbers (>1):
1) Locate the decimal point
2) Move the decimal point to the left to give a number
(coefficient) between 1 and 10
3) Write the new number multiplied by 10 raised to the
“nth power”
 where “n” is the number of places you moved the
decimal point so there is one non-zero digit to the
left of the decimal.
 If the decimal point is moved to the left, from its
initial position, then the exponent is a positive
number (× 10n)
2.2 Scientific Notation:
Writing Large and Small Numbers
 Write
each of the following in
scientific notation
 12,500
 0.0202
 37,400,000
 0.0000104
Examples
12,500
Decimal place is at the far right
 Move the decimal place to a position
between the 1 and 2 (one non-zero digit to
the left of the decimal)
 Coefficient (1.25): only significant digits
become part of the coefficient
 The decimal place was moved 4 places to
the left (large number) so exponent is
positive
 1.25x104
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Examples
 0.0202
 Move the decimal place to a position
between the 2 and 0 (one non-zero digit
to the left of the decimal)
 Coefficient (2.02): only significant digits
become part of the coefficient
 The decimal place was moved 2 places
to the right (small number) so exponent
is negative
 2.02x10-2
Examples
 37,400,000
 Decimal
place is at the far right
 Move the decimal place to a position
between the 3 and 7
 Coefficient (3.74): only significant digits
become part of the coefficient
 The decimal place was moved 7 places
to the left (large number) so exponent is
positive
 3.74x107
Examples
 0.0000104
 Move
the decimal place to a position
between the 1 and 0
 Coefficient (1.04): only significant
digits become part of the coefficient
 The decimal place was moved 5
places to the right (small number) so
exponent is negative
 1.04x10-5
Using Scientific Notation on a
Calculator
1) Enter the coefficient (number)
2) Push the key: EE or EXP
Then enter only the power of 10
3) If the exponent is negative, use the
key: (+/-)
4) DO NOT use the multiplication key:
X
to express a number in sci. notation
Converting Back to Decimal Notation
1) Determine the sign of the exponent, n
 If n is positive (×10n), the decimal point will move
to the right (this gives a number greater than one)
 If n is negative(×10-n), the decimal point will move
to the left (this gives a number less than one)
2) Determine the value of the exponent of 10
 The “power of ten” determines the number of
places to move the decimal point
 Zeros may have to be added to the number as the
decimal point is moved
Using Scientific Notation
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To compare numbers written in scientific notation, with
the same coefficient, compare the exponents of each
number
The number with the larger power of ten (the
exponent) is the larger number
3.4 х 104 < 3.4 х 107
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If the powers of ten (exponents) are the same, then
compare coefficients directly
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Which number is larger?
21.8 х 103 or 2.05 х 104
2.18 х 104 > 2.05 х 104
2.3 Significant Figures:
Writing Numbers to Reflect Precision
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Two kinds of numbers exist:
 Numbers that are exact (defined)
 Numbers that are measured
It is possible to know the exact value of a
counted number
The exact value of a measured number is
never known
Counting objects does not entail reading
the scale of a measuring device
2.3 Exact Numbers
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Exact numbers occur in definitions or in
counting
These numbers have no uncertainty
Counting numbers
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You can count the number of peaches in a bushel of
peaches with absolute certainty
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You can count the number of chairs in a room with absolute
certainty
Defined numbers (one exact value)
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There are exactly twelve inches in one foot (1 ft = 12 in)
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There are exactly four quarts in one gallon (1 gal = 4 quarts)
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There are exactly sixty seconds in one minute (1 min = 60
sec)
Measured Numbers
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Counting objects does not involve a measuring
device and it is not subject to uncertainties
Unlike counted (or defined) numbers, measured
numbers always contain a degree of uncertainty
(or error)
A measurement:
involves reading the scale of a measuring device
 always has some amount of uncertainty which
comes from the tool used for comparison
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A measuring device with a smaller unit will give
a more precise measurement, but some degree
of uncertainty will always be present
Measured Numbers
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This ruler has divisions
every one millimeter
Whenever a
measurement is made,
an estimate is required,
i.e., the value between
the two smallest divisions
on a measuring device
Every person will
estimate it slightly
differently, so there is
some uncertainty present
as to the true value
Mentally divide the space into
10 equal spaces to
estimate the last digit
2.8 cm
2.8 to 2.9 cm
2.85 cm
2.9 cm
Measured Numbers
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This balance has
divisions every one gram
Whenever a
measurement is made,
an estimate is required,
i.e., the value between
the two smallest divisions
on a measuring device
The estimate will be in
the tenths place
Mentally divide the space
into 10 equal spaces to
estimate the last digit
1g
1g
1.2 g
2g
2g
3g
2.3 Significant Figures: Writing Numbers
to Reflect Precision
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Scientific numbers are reported so that all
digits are certain except the last digit
which is estimated
To indicate the uncertainty of a single
measurement, scientists use a system
called significant figures
Significant Figures: All digits known with
certainty plus one digit that is uncertain
2.3 Counting Significant Figures
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The last digit written in a measurement is the
number that is considered to be uncertain
(estimated)
 Unless stated otherwise, the uncertainty in the
last significant digit is ±1 (plus or minus one unit)
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The precision of a measured quantity is
determined by number of sig. figures
A set of guidelines is used to interpret the
significance of
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a reported measurement
values calculated from measurements
2.3 Counting Significant Figures
Four rules (the guidelines):
1. Nonzero integers are always significant
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Zeros (may or may not be significant)
 significant zeros
 place-holding zeros (not significant)
 It is determined by its position in a sequence
of digits in a measurement
2. Leading zeros never count as significant
figures
3. Captive (interior) zeros are always significant
4. Trailing zeros are significant if the number has
a decimal point
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2.4 Significant Figures in Calculations
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Calculations cannot improve the precision of
experimental measurements
The number of significant figures in any
mathematical calculation is limited by the least
precise measurement used in the calculation
Two operational rules to ensure no increase in
measurement precision:
 addition and subtraction
 multiplication and division
2.4 Significant Figures in Calculations:
Multiplication and Division
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Product or quotient has the same number of
significant figures as the factor with the fewest
significant figures
Count the number of significant figures in each
number. The least precise factor (number) has
the fewest significant figures
Rounding
 Round the result so it has the same number
of significant figures as the number with the
fewest significant figures
2.4 Significant Figures in Calculations:
Rounding
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To round the result to the correct number of
significant figures
If the last (leftmost) digit to be removed:
 is less than 5, the preceding digit stays the
same (rounding down)
 is equal to or greater than 5, the preceding
digit is rounded up
 In multiple step calculations, carry the extra
digits to the final result and then round off
2.4 Multiplication/Division Example:
4 SF
5 SF
3 SF
3 SF
0.1021 × 0.082103 × 273 = 2.288481
2.29
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The number with the fewest significant
figures is 273 (the limiting term) so the
answer has 3 significant figures
2.4 Multiplication/Division Example:
5 SF
4 SF
3 SF
0.10210.082103273
1.1
2.1
 2.080438
2 SF
2 SF
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The number with the fewest
significant figures is 1.1 so the
answer has 2 significant figures
2.4 Significant Figures in Calculations:
Addition and Subtraction
 Sum
or difference is limited by the
quantity with the smallest number
of decimal places
 Find quantity with the fewest
decimal places
 Round answer to the same
decimal place
2.4 Addition/Subtraction Example:
1 d.p.
3 d.p.
2 d.p.
171.5 72.9158.23  236.185
236.2
1 d.p.
 The
number with the fewest
decimal places is 171.5 so the
answer should have 1 decimal
place
2.5 The Basic Units of Measurement
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The most used tool of the chemist
Most of the basic concepts of chemistry
were obtained through data compiled by
taking measurements
How much…?
How long…?
How many...?
These questions cannot be answered
without taking measurements
The concepts of chemistry were
discovered as data was collected and
subjected to the scientific method
2.5 The Basic Units of Measurement
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A measurement is the process or the result of
determining the magnitude of a quantity (e.g.,
length or mass) relative to a unit of measurement
Involves a measuring device:
 meter stick, scale, thermometer
The device is calibrated to compare the object to
some standard (inch/centimeter, pound/kilogram)
Quantitative observation with two parts:
A number and a unit
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Number tells the total of the quantity measured
Unit tells the scale (dimensions)
2.5 The Basic Units of Measurement
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A unit is a standard (accepted) quantity
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Describes what is being added up
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Units are essential to a measurement
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For example, you need “six of sugar”
 teaspoons?
 ounces?
 cups?
 pounds?
2.5 The Standard Units (of Measurement)
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The unit tells the magnitude of the standard
Two most commonly used systems of units of
measurement
 U.S. (English) system: Used in everyday
commerce (USA and Britain*)
 Metric system: Used in everyday commerce
and science (The rest of the world)
SI Units (1960): A modern, revised form of the
metric system set up to create uniformity of
units used worldwide (world’s most widely
used)
2.5 The Standard Units (of Measurement):
The Metric/SI System
The metric system is a decimal system of
measurement based on the meter and the
gram
 It has a single base unit per physical
quantity
 All other units are multiples of 10 of the
base unit
 The power (multiple) of 10 is indicated by
a prefix
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2.5 The Standard Units:
The Metric System
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In the metric system there is one base unit for each
type of measurement
 length
 volume
 mass
 (also, time, temperature)
The base units multiplied by the appropriate power of
10 form smaller or larger units
The prefixes are always the same, regardless of the
base unit
 milligrams and milliliters both mean 1/1000 (10-3) of
the base unit
2.5 The Standard Units: Length
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Meter
Base unit of length in metric and SI system
About 3 ½ inches longer than a yard
 1 m = 1.094 yd
2.5 The Standard Units: Length
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Other units of length
are derived from the
meter
Commonly use
centimeters (cm)
 1 m = 100 cm
 1 inch = 2.54 cm
(exactly)
2.5 The Standard Units: Volume
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Volume: Measure of the amount
of three-dimensional space
occupied by a object
Derived from length
volume = side × side × side
Since it is a three-dimensional
measure, its units have been
cubed
SI base unit = cubic meter (m3)
Volume = side × side × side
Metric base unit = liter (L) or
10 cm3
Commonly measure smaller
volumes in cubic centimeters
(cm3)
2.5 The Standard Units: Volume
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SI base unit = 1m3
The volume equal to
that occupied by a
perfect cube that is
one meter on each
side
This unit is too large
for practical use in
chemistry
Take a volume 1000
times smaller than the
cubic meter, 1dm3
2.5 The Standard Units: Volume
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Metric base unit = 1dm3
(one liter, L)
The volume equal to that
occupied by a perfect cube
that is ten centimeters on
each side
1L = 1.057 qt
Commonly measure smaller
volumes in cubic centimeters
(cm3)
Take a volume 1000 times
smaller than the cubic
decimeter, 1cm3
3
10 cm = 1 dm
V =10 cm × 10 cm × 10 cm
2.5 The Standard Units: Volume
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The most commonly used
unit of volume in the
laboratory: milliliter (mL)
The volume equal to that
occupied by a perfect
cube that is one
centimeter on each side
1 mL = 1 cm3
1 L= 1 dm3 = 1000 mL
1 m3 = 1000 dm3 =
1,000,000 cm3
Use a graduated cylinder
or a pipette to measure
liquids in the lab
2.5 The Standard Units: Mass
Measure of the total quantity of matter present
in an object
 SI unit (base) = kilogram (kg)
 Metric unit (base) = gram (g)
 Since the gram is such a relatively small unit,
the kilogram is a very commonly used unit
 1 kg = 1000 g
 1 g = 1000 mg
 1 kg = 2.205 pounds
 1 lb = 453.6 g
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2.5 Prefixes Multipliers
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One base unit for each type of measurement
Length (meter), volume (liter), and mass (gram*)
The base units are then multiplied by the appropriate
power of 10 to form larger or smaller units
base unit = meter, liter, or gram
2.5 Prefixes Multipliers (memorize)
× base unit
mega
 kilo
 base
 deci
 centi
 milli
 micro
 nano
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(M) 1,000,000
(k)
1,000
1 meter liter gram
(d)
0.1
(c)
0.01
(m)
0.001
(µ)
0.000001
(n)
0.000000001
106
103
100
10-1
10-2
10-3
10-6
10-9
2.5 Prefix Multipliers
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For a particular measurement:
 Choose
the prefix which is similar in size
to the quantity being measured
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Keep in mind which unit is larger
A
kilogram is larger than a gram, so
there must be a certain number of
grams in one kilogram
 Choose the prefix most convenient for a
particular measurement
n < µ < m < c < base < k < M
2.6 Converting from One Unit to Another:
Dimensional Analysis
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Many problems in chemistry involve converting the
units of a quantity or measurement to different units
The new units may be in the same measurement
system or a different system, i.e., U.S. System to
metric and the converse
Dimensional Analysis is the method of problem solving
used to achieve this unit conversion
Unit conversion is accomplished by multiplication of a
given quantity (or measurement) by one or more
conversion factors to obtain the desired quantity or
measurement
2.6 Converting from One Unit to Another:
Equalities
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An equality is a fixed relationship between
two quantities
It shows the relationship between two units
that measure the same quantity
These relationships are exact, not measured
 1 min = 60 s
 12 inches = 1 ft
 1 dozen = 12 items (units)
 1L = 1000 mL
 16 oz = 1 lb
 4 quarts = 1 gallon
2.6 Converting from One Unit to Another:
Dimensional Analysis
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Conversion factor: An equality expressed
as a fraction
 It is used as a multiplier to convert a
quantity in one unit to its equivalent in
another unit
 May be exact or measured
 Both parts of the conversion factor
should have the same number of
significant figures
2.7 Solving Multistep Conversion Problems:
Dimensional Analysis Example
(Conversion Factors Stated within a Problem)
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The average person in the U.S.
consumes one-half pound of sugar per
day. How many pounds of sugar
would be consumed in one year?
1) State the initial quantity given (and the unit):
One year
State the final quantity to find (and the unit):
Pounds
2) Write a sequence of units (map) which
begins with the initial unit and ends with the
desired unit: year
day
pounds
1 cal 4.184 J
2.7 Solving Multistep Conversion Problems:
Dimensional Analysis Example
3) For each unit change,
State the equalities:
 Every equality will have two conversion
factors
365 days = 1 year
0.5 lb sugar =1day
year
day
pounds
2.7 Solving Multistep Conversion Problems:
Dimensional Analysis Example
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State the conversion factors:
0.5 lb. sugar and
1 day
1day
0.5 lb. sugar
4) Set Up the problem:
1 year 365 day(s) 0.5 lb sugar  183 lbs. sugar
1 day
1 year
Guide to Problem Solving when
Working Dimensional Analysis Problems
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Identify the known or given quantity and the
units of the new quantity to be determined
Write out a sequence of units which starts with
your initial units and ends with the desired units
(“solution map”)
Write out the necessary equalities and
conversion factors
Perform the mathematical operations that
connect the units
Check that the units cancel properly to obtain
the desired unit
Does the answer make sense?
2.9 Density
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The ratio of the mass of an object to the volume
occupied by that object
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Density tells how tightly the matter within an object is packed
together
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Units for solids and liquids = g/cm3
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1 cm3 = 1 mL so can also use g/mL
Unit for gases = g/L
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Density of three states of matter: solids > liquids >>> gases
mass
Density 
volume
m
d
v
2.9 Density
Can use density as a conversion factor
between mass and volume
 Density of some common substances given
in Table 2.4, page 33
 You will be given any densities on tests
EXCEPT water
 Density of water is 1.0 g/cm3 at room
temperature
 1.0 mL of water weighs how much?
 How many mL of water weigh 15 g?
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2.9 Density
 To
determine the density of an object
 Use a balance to determine the mass
 Determine the volume of the object
 Calculate
it if possible (cube shaped)
 Can also calculate volume by
determining what volume of water is
displaced by an object
Volume of Water Displaced = Volume of Object
Density Problem

Iron has a density of 7.87 g/cm3. If 52.4 g
of iron is added to 75.0 mL of water in a
graduated cylinder, to what volume
reading will the water level in the cylinder
rise?
Vf  ?
m  52.4 g
Vi  75.0 mL
d  7.87 g cm3



Density Problem
Solve for volume of iron
density  mass
volume
volume  mass
density
3
1 cm = 1 mL
52.4 g iron 1 mL iron
 6.658 mL iron
7.87 g iron

6.658 mL iron + 75.0 mL water = 81.7 mL total
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End