2.5 Significant Figures and Mathematical Operations

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2.1 Measurement Systems
Measurement is the determination
of the dimensions, capacity,
quantity, or extent of something.
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2–1
2.1 Measurements in Chemistry
always consists of two parts
NUMBER
EXACT or INEXACT
SIGNIFICANT FIGURES
SCIENTIFIC NOTATION
UNIT
METRIC SYSTEM
DIMENSIONAL ANALYSIS
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2–2
2.1 Metric System Units
System Internationale (SI)
Mass
Length
Volume
kilogram (kg), gram (g)
meter (m), centimeter (cm)
cubic meter (m3),
cubic centimeter (cm3)
liter (L) = 1000 cm3 (exact)
milliliter (mL) = 1 cm3 (exact)
Time
second (s)
Temperature kelvin (k)
Celsius (ºC)
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Common Metric System Prefixes with Their
Symbols and Mathematical Meanings.
Numerical prefixes for larger or smaller units:
mega
kilo
deci
centi
milli
micro
(M)
(k)
(d)
(c)
(m)
(µ)
1000000 times unit
1000 times unit
0.01 times unit
0.01 times unit
0.001 times unit
0.000001 times unit
(106)
(103)
(10-1)
(10-2)
(10-3)
(10-6)
My King Died Chewing M & M’s
Memorize these!
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2–4
Figure 2.2 Comparisons of the base metric system
units of length (meter), mass (gram), and
volume (liter) with common objects.
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2–5
Derived Units in the Metric System.
Frequency (cycles/s, hertz)
Density (mass/volume, g/cm3)
Speed (distance/time, m/s)
Acceleration (distance/(time)2, m/s2)
Force (mass x acceleration, kg•m/s2, newton)
Pressure (force/area, kg/(m•s2), pascal)
Energy (force x distance, kg•m2/s2, joule)
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2.3 Exact and Inexact Numbers
Exact Numbers
Have no uncertainty associated with them
From measurement of indivisible objects
10 tennis balls
29 students enrolled
Some conversion factors
Exactly 2.54 centimeters = one inch
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2–7
2.3 Exact and Inexact Numbers
Inexact Numbers
Have some uncertainty associated with them
From scalar measurements
“Stuff” rather than “Things”
Values are limited by the instrument
14.3 gallons ( 6 oz)
14.325 gallons ( 1 tsp)
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2.4 Uncertainty in Measurement
and Significant Figures
Significant figures (sig figs or sig digs) are
the digits in an inexact number.
The last digit in an inexact number is
an estimated value.
The number of sig figs depends upon
the instrument used.
14.3 gallons
14.325 gallons
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2–9
2.4 Uncertainty in Measurement
and Significant Figures
Exact numbers have an infinite number of
significant figures. They do not contain
an estimated value.
10 tennis balls
2.54 cm = 1 inch
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2–10
2.5 Significant Figures and
Mathematical Operations
When is zero a significant figure?
Trailing zeros are always sig. figs.
11.0
100
____ sig. figs.
____ sig. figs.
Trailing zeros are significant because they
are the best estimate of the value.
10.9 11.0 (best estimate) 11.1
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2.5 Significant Figures and
Mathematical Operations
When is zero a significant figure?
Embedded zeros are always sig. figs.
101
____ sig. figs.
10.11 ____ sig. figs.
Leading zeros are never sig. figs.
0.00145
____ sig. figs.
0.0000000000234 ____ sig. figs.
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2–12
Chemistry at a
Glance:
Significant
Figures
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2.5 Significant Figures and
Mathematical Operations
Multiplication and Division
The result can have no more sig. figs.
than the fewest number of sig. figs.
used to obtain the result.
4.242 x 1.23 = 5.21766
12.24 / 2.0 = 6.12
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5.22
6.1
2–14
2.5 Significant Figures and
Mathematical Operations
Addition and Subtraction
Result will have a digit as far to the right as
all the numbers have a digit in common
2.02
8.7397
1.234
-2.123
+ 3.6923
6.6167
6.9463
6.95
6.617
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2–15
2.5 Significant Figures and
Mathematical Operations
Rounding Off
When the last sig. fig. is followed by a
number equal to or greater than 5,
round the last sig. fig. up.
When the last sig. fig. is followed by a
number less than 5, leave the last sig.
fig. as is.
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2.6 Scientific Notation
Used for very large or very small numbers
Distance from earth to sun
93,000,000 miles
Diameter of a carbon atom
0.000000000015 meters
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2.6 Scientific Notation
An ordinary decimal number is expressed
as the product of a coefficient between
1 and 10, and an exponential term.
Express in scientific notation:
93,000,000 miles
0.000000000015 meters
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2.6 Scientific Notation
Scientific notation is useful for handling
significant figures.
Give the result to the proper number of
significant figures:
(25.456 – 25.423) x 64.58 x 430 =
0.01553  45.0 =
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2.7 Conversion Factors and
Dimensional Analysis
A conversion factor is a ratio that
specifies how one unit of measurement
is related to another:
1 inch = 2.54 cm
_1 in__ 2.54 cm
2.54 cm
1 in
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2.7 Conversion Factors and
Dimensional Analysis
Dimensional analysis is a problem-solving
method in which the units of the measurement are used to set up the calculation.
Units behave like numbers, they can be
multiplied, divided, or cancelled.
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2.7 Conversion Factors and
Dimensional Analysis
What is the volume of a cube if its
edges are 2.5 cm long?
How long is a foot in centimeters?
Convert 0.0256 mm to m.
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2.7 Conversion Factors and
Dimensional Analysis
How many miles will I travel in 3.00
hours at 65.0 miles per hour?
How long will it take to travel 500 miles at
65 miles per hour?
What is my speed, in miles per hour, if I
ride my bike 25.5 miles in 1.20 hours?
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2.8 Density
Matter has mass and volume.
Density is the ratio of mass to volume for
a specific type of matter.
Density = _mass_
volume
_grams_
milliliters
Density is a characteristic property of a
pure substance.
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Left:
Shortening floats on water.
Right: Copper floats on mercury
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Liquids that do not dissolve in one another and
that have different densities float on one another,
forming layers.
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Table 2.3
Densities of Selected Substances
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2.8 Density
Water has a density of 1.00 g/mL at 20°C.
What is the mass of 20.0 mL of water?
20.0 mL of gasoline has a mass of 11.2 g.
What is its density?
What volume of ethyl alcohol will have a
mass of 20.0 g? Its density is 0.79 g/mL.
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2–28
2.9 Temperature Scale
and Heat Energy
Heat is a form of energy. Temperature
can be used to measure heat.
1 calorie = amount of heat required to
raise temperature of 1 gram of
water by 1 Celcius
1 Calorie = 1000 calories = 1 kcal
1 Calorie = 4.184 Joule
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The relationships among the Celsius, Kelvin, and
Fahrenheit temperature scales are determined by the
degree sizes and the reference point values.
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2–30
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