Transcript Number of Significant Figures

Introductory Chemistry:
A Foundation
FIFTH EDITION
by Steven S. Zumdahl
University of Illinois
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1
Measurements
and
Calculations
Chapter 2
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2
Measurement is important
•
•
•
•
•
Correct I.V. solution concentration
Correct lawn fertilizer application
Correct amount of salt on food
Correct amount of oil in engine
Correct force acting on bridge
girders
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3
Automobile as an example
•
•
•
•
•
•
•
Gallons of gas, quarts of oil
Viscosity (thickness) of oil
Antifreeze density (freezing temperature)
Temperature
Air pressure
Voltage of battery
Oxygen sensors on exhaust, intake
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4
A gas
pump
measures
the amount
of
gasoline
delivered.
What is measurement?
• Defined as a quantitative comparison of an unknown
quantity to a known Standard. The measurement
instrument must be standardized against the known
standard
• Every measurement has a value (number) and a unit
Standard kilogram of mass,
officially known as the
“International prototype of
the kilogram” composed of
platinum-iridium alloy,
stored under glass in a
vacuum since 1889
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6
Standards of Measurement
When we measure, we use a measuring tool to
compare some dimension of an object to a
standard.
Historical standard, platinum iridium meter bar
The meter now is defined
as the length of the path
traveled by light in
vacuum during a time
interval of 1/299,792,458
of a second. The speed
of light is
c = 299,792,458 m/s
For example, at one time the
standard for length was the king’s
foot. What are some problems
with this standard?
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7
Measurement In Action
On 9/23/99, $125,000,000 Mars Climate Orbiter entered Mar’s
atmosphere 100 km lower than planned and was destroyed by
heat.
1 lb = 1 N
1 lb = 4.45 N
“This is going to be the
cautionary tale that will be
embedded into introduction
to the metric system in
elementary school, high
school, and college science
courses till the end of time.”
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2.1 Scientific Notation
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Scientific Notation
• Technique Used to Express Very
Large or Very Small Numbers
• Based on Powers of 10
• To Compare Numbers Written in
Scientific Notation
– First Compare Exponents of 10
– Then Compare Numbers
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Writing Numbers in Scientific Notation
1 Locate the Decimal Point
2 Move the decimal point to
the right of the non-zero
digit in the largest place
–The new number is now
between 1 and 10
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Writing Numbers in Scientific
Notation
3 Multiply the new number by 10n
– where n is the number of places you
moved the decimal point
4 Determine the sign on the exponent n
– If the decimal point was moved left, n is +
– If the decimal point was moved right, n is –
– If the decimal point was not moved, n is 0
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Writing Numbers in Standard Form
1 Determine the sign of n of 10n
– If n is + the decimal point will move to
the right
– If n is – the decimal point will move to
the left
2 Determine the value of the exponent of 10
– Tells the number of places to move the
decimal point
3 Move the decimal point and rewrite the
number
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When describing very small distances, such
as the diameter of a swine flu virus, it is
convenient to use scientific notation.
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14
2.2 Units of Measurement
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15
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The meter
• 1795 Royal jeweler made a bar of
platinum 4 mm thick, 25 mm wide,
and 1 m long. This was known as an
“end measure”
• 1872 “line measure” on a platinum
iridium alloy bar
• Now, the distance light travels in
1/299, 792, 458th of a second
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Length
• SI unit = meter (m)
–About 3½ inches longer than a yard
• Commonly use centimeters (cm)
for length
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Length
–1 m = 100 cm
–1 cm = 0.01 m = 10 mm
–1 inch = 2.54 cm (exactly)
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Figure 2.1: Comparison of English
and metric
units for length on a ruler.
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Volume
• Measure of the amount of threedimensional space occupied by a
substance
• SI unit = cubic meter (m3)
• Commonly measure solid volume in
cubic centimeters (cm3)
– 1 m3 = 106 cm3
– 1 cm3 = 10-6 m3 = 0.000001 m3
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Volume
• Commonly measure liquid or gas
volume in milliliters (mL)
– 1 L is slightly larger than 1 quart
– 1 L = 1 dm3 = 1000 mL = 103 mL
– 1 mL = 0.001 L = 10-3 L
– 1 mL = 1 cm3
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Figure 2.2:
The largest
drawing
represents a
cube that has
sides 1 m in
length and a
volume of 1
m3.
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Figure 2.3:
A 100-mL
graduated
cylinder.
Mass
• Measure of the amount of matter
present in an object
• SI unit = kilogram (kg)
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Mass
• Commonly measure mass in grams (g)
or milligrams (mg)
– 1 kg = 2.20 pounds
– 1 kg = 1000 g = 103 g
– 1 g = 1000 mg = 103 mg
– 1 g = 0.001 kg = 10-3 kg
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Figure 2.4:
An electronic
analytical
balance used
in chemistry
labs.
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31
Related Units in the Metric System
• All units in the metric system are
related to the fundamental unit by a
power of 10
• The power of 10 is indicated by a
prefix
• The prefixes are always the same,
regardless of the fundamental unit
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Uncertainty in Measured Numbers
• A measurement always has some
amount of uncertainty
• Uncertainty comes from limitations
of the techniques used for
comparison
• To understand how reliable a
measurement is, we need to
understand the limitations of the
measurement
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Exact Numbers
• Exact Numbers are numbers known with
certainty
• Unlimited number of significant figures
• They are either
– counting numbers
• number of sides on a square
– or defined
• 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm
• 1 kg = 1000 g, 1 LB = 16 oz
• 1000 mL = 1 L; 1 gal = 4 qts.
• 1 minute = 60 seconds
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Figure 2.5: Measuring a pin.
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Precision
• Refers to the reproducibility of a
measurement.
• Measurements with more “decimal places”
are more precise.
• Devices with finer markings are more
precise.
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Estimation in measurement
1 cm
2 cm
A
3 cm
4 cm
B
5 cm
C
• When we measure, the quantity rarely falls exactly on the
calibration marks of the scale we are using.
• Because of this we are estimating the last digit of the
measurement.
• For instance, we could measure “A” above as about 2.3
cm. We are certain of the digit “2”, but the “.3” part is a
guess - an estimate.
• What is your estimate for B and C?
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Higher Precision
1 cm
2 cm
A
B
3 cm
C
D
• A measuring device with more marks on the scale is more
precise. I.e., we are estimating less, and get a more accurate
reading. Here we are estimating the hundredths place instead of
the tenths.
• Here, we can measure A as 1.25 cm. Only the last digit is
uncertain.
• Usually we assume the last digit is accurate ± 1.
• How would you read B, C, and D?
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Accuracy
• Refers to how close a measurement comes
to the true or accepted value.
• This depends on both the measuring device
and the skill of the person using the
measuring device.
• This can be determined by comparing the
measured value to the known or accepted
value.
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Accurate or Precise?
What is the temperature
at which water boils?
•Measurements: 95.0°C,
95.1°C, 95.3°C
•True value: 100°C
Precise!
(but not accurate)
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Accurate or Precise?
What is the temperature
at which water freezes?
•Measurements: 1.0°C,
1.2°C, -5.0°C
•True value: 0.0°C
Accurate!
(it’s hard to be accurate without being
precise)
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Accurate or Precise?
What is the atmospheric
pressure at sea level?
•Measurements: 10.01
atm, 0.25 atm, 234.5 atm
•True value: 1.00 atm
Not Accurate & Not Precise
(don’t quit your day job)
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Accurate or Precise?
What is the mass of one
Liter of water?
•Measurements: 1.000
kg, 0.999 kg, 1.002 kg
•True value: 1.000 kg
Accurate & Precise
(it’s time to go pro)
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Reporting Measurements
• To indicate the uncertainty of a single
measurement scientists use a system
called significant figures
• The last digit written in a measurement
is the number that is considered to be
uncertain
• Unless stated otherwise, the
uncertainty in the last digit is ±1
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Rules for Counting Significant Figures
• Nonzero integers are always significant
• Zeros
– Leading zeros never count as significant
figures
– Captive zeros are always significant
– Trailing zeros are significant if the
number has a decimal point
• Exact numbers have an unlimited
number of significant figures
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Counting Significant Figures
RULE 1. All non-zero digits in a measured number
are significant. Only a zero could indicate that
rounding occurred.
Number of Significant Figures
38.15 cm
5.6 ft
65.6 lb
122.55 m
4
2
___
___
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Leading Zeros
RULE 2. Leading zeros in decimal numbers are
NOT significant.
Number of Significant Figures
0.008 mm
1
0.0156 oz
3
0.0042 lb
____
0.000262 mL
____
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Sandwiched Zeros
RULE 3. Zeros between nonzero numbers are
significant. (They can not be rounded unless they
are on an end of a number.)
Number of Significant Figures
50.8 mm
3
2001 min
4
0.702 lb
____
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Trailing Zeros
RULE 4. Trailing zeros in numbers without decimals
are NOT significant. They are only serving as place
holders.
Number of Significant Figures
25,000 in.
2
200. yr
3
48,600 gal
____
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Examples
EXAMPLES
# OF SIG. DIG.
COMMENT
453
3
All non-zero digits are always significant.
5057
4.06
4
3
Zeros between two significant digits are significant.
5.00
106.00
114.050
3
5
6
Additional zeros to the right of decimal and a
significant digit are significant.
0.007
1
Placeholders are not significant
12000
2
Trailing zeros in numbers with no decimal point are
not significant (= placeholder)
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Learning Check
A. Which answers contain 3 significant figures?
1) 0.4760
2) 0.00476
3) 4760
B. All the zeros are significant in
1) 0.00307
3) 2.050 x 103
2) 25.300
C. 534,675 rounded to 3 significant figures is
1) 535
2) 535,000
3) 5.35 x 105
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Learning Check
In which set(s) do both numbers contain
the
same number of significant figures?
1) 22.0 and 22.00
2) 400.0 and 40
3) 0.000015 and 150,000
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Learning Check
State the number of significant figures in each
of the following:
A. 0.030 m
1
2
3
B. 4.050 L
2
3
4
C. 0.0008 g
1
2
4
D. 3.00 m
1
2
3
E. 2,080,000 bees
3
5
7
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Practice
How many significant digits in the following?
Number
1.4682
# Significant Digits
5
110256.002
0.000000003
114.00000006
110
9
1
11
2
120600
4
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The Problem
Area of a rectangle = length x width
We measure:
Length = 14.26 cm Width = 11.70 cm
Punch this into a calculator and we find the area as:
14.26 cm x 11.70 cm = 166.842 cm2
• But there is a problem here!
• This answer makes it seem like our measurements were more
accurate than they really were.
• By expressing the answer this way we imply
that we estimated the thousandths position,
when in fact we were less precise than that!
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Calculations with Significant Figures
• Calculators/computers do not know
about significant figures!!!
• Exact numbers do not affect the
number of significant figures in an
answer
• Answers to calculations must be
rounded to the proper number of
significant figures
– round at the end of the calculation
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Multiplication/Division with
Significant Figures
• Result has the same number of
significant figures as the
measurement with the smallest
number of significant figures
• Count the number of significant
figures in each measurement
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Adding/Subtracting Numbers
with Significant Figures
• Result is limited by the number with
the smallest number of significant
decimal places
• Find last significant figure in each
measurement
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Adding/Subtracting Numbers with
Significant Figures
• Find which one is “left-most”
• Round answer to the same decimal
place
450 mL + 27.5 mL = 480 mL
precise to 10’s place
precise to 0.1’s place
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precise to 10’s place
61
Rules for Rounding Off
• If the digit to be removed
• is less than 5, the preceding digit stays
the same
• is equal to or greater than 5, the
preceding digit is increased by 1
• In a series of calculations, carry the
extra digits to the final result and then
round off
• Don’t forget to add place-holding zeros
if necessary to keep value the same!!
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A student
performing a
titration in the
laboratory.
Problem Solving and Dimensional Analysis
• Many problems in chemistry involve
using equivalence statements to convert
one unit of measurement to another
• Conversion factors are relationships
between two units
– May be exact or measured
– Both parts of the conversion factor should have
the same number of significant figures
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Problem Solving and Dimensional
Analysis
• Conversion factors generated from
equivalence statements
– e.g. 1 inch = 2.54 cm can give
2.54cm
1in
or
1in
2.54cm
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Problem Solving and Dimensional Analysis
• Arrange conversion factors so
starting unit cancels
– Arrange conversion factor so starting
unit is on the bottom of the conversion
factor
• May string conversion factors
together
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Converting One Unit to Another
•
•
•
Find the relationship(s) between the
starting and goal units. Write an
equivalence statement for each
relationship.
Write a conversion factor for each
equivalence statement.
Arrange the conversion factor(s) to
cancel starting unit and result in goal
unit.
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Converting One Unit to Another
• Check that the units cancel properly
• Multiply and Divide the numbers to give
the answer with the proper unit.
• Check your significant figures
• Check that your answer makes sense!
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Temperature Scales
• Fahrenheit Scale, °F
–Water’s freezing point = 32°F,
boiling point = 212°F
• Celsius Scale, °C
– Temperature unit larger than the
Fahrenheit
– Water’s freezing point = 0°C, boiling
point = 100°C
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Temperature Scales
• Kelvin Scale, K
– Temperature unit same size as
Celsius
– Water’s freezing point = 273 K,
boiling point = 373 K
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Figure 2.7: The three major
temperature scales.
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Figure 2.9: Comparison of the
Celsius and Fahrenheit scales.
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Figure 2.6: Thermometers based on
the three temperature scales in (a) ice
water and (b) boiling water.
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Figure 2.8: Converting 70. 8C to
units measured on the Kelvin
scale.
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Liquid gallium
expands within
a carbon
nanotube as the
temperature
increases
(left to right).
Source: Glenn Izett/U.S. Geological Survey
Density
• Density is a property of matter
representing the mass per unit volume
• For equal volumes, denser object has
larger mass
• For equal masses, denser object has
small volume
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Density
• Solids = g/cm3
Mass
– 1 cm3 = 1 mL
Density 
• Liquids = g/mL
Volume
• Gases = g/L
• Volume of a solid can be determined by water
displacement
• Density : solids > liquids >>> gases
• In a heterogeneous mixture, denser object
sinks
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Spherical droplets of mercury,
a very dense liquid.
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Figure 2.10: (a) Tank of water. (b)
Person submerged in the tank,
raising the level of the water.
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Figure 2.11:
A hydrometer
being used to
determine the
density of the
antifreeze
solution in a
car’s radiator.
Using Density in Calculations
Mass
Density 
Volume
Mass
Volume 
Density
Mass  Density  Volume
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