Transcript English measurement conversion

Chapter 5
Measurements and
Calculations
Goals of Chapter 5:
Measurement & Calculations
• Express numbers in scientific notation
• Learn English, metric, & SI system of measurement
• Use metric system to measure length, volume, and
mass
• Significant digits
• Dimensional Analysis
• Temperature Scales
• Density/Specific Gravity
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Scientific Notation
• Used to express very large or very small
numbers
• A number is expressed in scientific notation
when it is in the form
a x 10n
– where a is between 1 and 10
– and n is an integer
– use positive power for large numbers
– use negative number for small numbers (decimals)
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Scientific Notation: Large Numbers
210,000,000,000,000,000,000,000
Where is the decimal point?
After the last zero.
Where would you put the decimal to
make this number be between 1 and
10?
Between the 2 and the 1
2.10,000,000,000,000,000,000,000.
How many decimal places did you move the
decimal?
23
When the original number is more than 1,
the exponent is positive.
The answer in scientific notation is
2.1 x 1023
Scientific Notation: Small Numbers
0.0000000902
Where would the decimal go to make the
number be between 1 and 10?
9.02
The decimal was moved how many places?
8
When the original number is less than 1, the
exponent is negative.
9.02 x 10-8
Learning Check
• Convert the following to scientific notation
a. 100,000
c. 45,000,000
e. 9006
b. 0.0056
d. 0.0000101000
f. 0.00000002
• Convert to standard notation
g. 8.9 x 108
i. 7.003 x 106
h. 7.7 x 10-4
j. 1.001 x 10-8
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Units
• Tell us what scale or standard is being used to
represent measurement
• Scientists need common units to represent quantities
like mass, length, time, and temperature
• If everyone had own set of units – chaos would result
• US uses English system, most of world (& scientists)
use metric system, also SI system
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Units of Measurement
Figure 5.1: Comparison of English and
metric units.
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Figure 5.2:
Cube representations.
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Figure 5.3:
A 100 mL
graduated cylinder.
1 mL = 1 cm3
1 milliliter = 1 cubic centimeter
100 mL = 100 cm3
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Move decimal point left
Move decimal point right
Power of 10 between each increment
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Chapter 2
SI Conversions
Learning Check
1000 meters = ___ km (decimal moved ___ places left)
1 meter = ___ cm (decimal moved ___ places right)
10 mm = ___ cm (decimal moved ___ place left)
1 Liter = ___ mL (decimal moved ___ places to right)
600 grams = ___ kg (decimal moved ___ places left)
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Derived Units
Uncertainty of Measurement
• When using an instrument to measure
(such as a ruler or graduated cylinder),
we visualize divisions between markings
and estimate
• When making measurement, record all
certain numbers and first uncertain
number
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Figure 5.5: Measuring a pin.
Reading is between
2.8 cm & 2.9 cm
These divisions
were visualized
2.85 cm is
measurement
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“5” is uncertain
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Significant Figures
• Includes all numbers recorded in a
measurement
– For pin, length = 2.85 cm: 3 significant figures
– All certain numbers plus first uncertain
• Assume to be accurate to ± 1 in last #
– Pin length is 2.85 ± 0.01 cm
– Pin is somewhere between 2.84 & 2.86 cm
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Rules for Counting Significant Figures
1. All non-zero digits ARE significant
- 125.55 contains FIVE significant figures
2. Leading Zeros ARE NOT significant
- 0.00055 contains TWO significant figures
3. Middle Zeros ARE significant
- 505 contains THREE significant figures
4. Ending Zeros ARE significant IF there is a decimal
present
- 500.00 contains FIVE significant figures
5. Ending Zeros ARE NOT significant IF there is no
decimal present
- 500 contains ONE significant figure
Rules for counting significant figures
(continued)
• Significant Figures only apply to measurements and
calculations involving those measurements
• Any number written in scientific notation is
considered significant
• An ending zero can be expressed as significant by
placing a line above it
Remember! Just because a number is not significant,
does not mean it’s not important! A non-significant
zero acts as a placeholder to show the magnitude of
the number!
Learning Check
• How many significant figures are in the
following measurements?
– 0.0108
– 0.0050060
– 110
– 5.030 x 103
– 0.00100
– 480
– 500
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To give answer with correct number
of significant figures – round off
• Look at number to right of last s.f.
– If number is 4 or below, let it go!
– If number is 5 or above, give it a shove!
• Do not round off until end of calculations
Rules for Significant Figures in
Calculations
• Multiplication & Division
– Answer should have same number of significant
figures as measurement with smallest number of
significant figures
– Example: 4.56 x 1.4 = 6.384 → 6.4
(3 s.f.)
(2 s.f.)*
(2 s.f.)*
Rules for Significant Figures in
Calculations
• Addition & Subtraction
– Limited by smallest number of decimal places
– Example:
12.11 (2 decimal places)
18.0
(1 decimal place)*
+
1.013 (3 decimal places)
31.123 → 31.1 (1 decimal place)*
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Learning Check
• Perform the following operations and round to
the correct number of significant figures
a. 5.19 +1.9 + 0.842
b. 1081 – 7.25
c. 2.3 x 3.14
d. 25.36/2.5
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Dimensional Analysis
Dimensional Analysis is a method of problem
solving that allows for the conversion of one unit to
another by way of a conversion factor.
A unit conversion factor is a fraction whose
numerator and denominator are equivalent
measures. Some common unit conversion factors
are given below. You can also use the reciprocal of
these.
Choose a unit conversion factor that…
• Introduces the unit you want in the
answer
• Cancels out the original unit so that the
one you want is all that is left.
“Canceling” out Words
Which conversion factor will convert feet to
inches?
? Feet
12 inches
1 foot
1 foot
12 inches
You can cancel a unit if it appears in both the
numerator and the denominator
Feet x
12 inches
1 foot
Learning Check: Choose the
appropriate conversion factor.
Inches to feet
Minutes to hours
Meters to centimeters
inches
1
1 ft
12 in.
minutes
1
60 min
1 hr
meters
1
1m
100 cm
12 in.
1 ft
1 hr
60 min
100 cm
1m
Single-Step Problems
• Choose a conversion factor that will get you
from the starting unit to the ending unit
• Multiply the given number by the conversion
factor that will cancel out the original unit
• Solve the problem by multiplying all number
on top of the fraction and divide by the
number on the bottom of the fraction
• Be sure to include a UNIT in your answer!
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Example 1: Convert 8 yards to feet
Make a decision: What conversion factor will
you use?
1 yard
3 feet
3 feet
1 yard
Set up the problem: Multiply the measurement
by the conversion factor.
8 yards x 3 feet = 24 feet
1 yard
Solve the problem: Perform the multiplication
Multiply all numbers on the top
Divide by all numbers on the bottom
Example 2: Converts 16 quarts to gallons
What are the two conversion
factors comparing quarts and
gallons?
4 qt
1 gal
Which one will “cancel”
quarts?
16 quarts x
1 gallon =
4 quarts
4 gallons
1 gal
4 qt
Learning Check
• Water is often bottled in 0.750 L containers.
Convert 0.750 L into quarts.
• 1 L = 1.06 quarts
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Multiple-Step Problems
• Problem that requires the use of multiple
conversion factors
1. Determine which conversion factors are needed
to get from starting to ending unit
2. Develop plan to put the required conversion
factors in order to cancel them out
3. Solve the problem
• Multiply all numbers on top of fractions and divide by
all numbers on the bottom of fractions
• Include a UNIT in your answer!
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Example 3 : Convert 26.2 miles to kilometers
• Determine conversion factors
– 1 mile = 1760 yards
– 1 meter = 1.094 yards
– 1000 meter = 1 kilometer
• Outline plan
– Miles  yards  meters  kilometers
• Solve the Problem
26.2 miles x 1760 yards x 1 meter x 1 kilometer
1 mile
1.094 yards 1000 meters
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Learning Check
• If a race car travels around a track at the
speed of 225 miles/hour, what is the speed of
the car in kilometers/hour?
• Use conversion factors from Example 3
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Temperature Conversions
Temperture Scale
Fahrenheit
Celsius
Kelvin
Abbreviation
°F
°C
K
Boiling point H2 0 Freezing Point H 2O
212°F
32°F
100°C
0°C
373 K
273 K
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Figure 5.6: The three major temperature scales.
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Converting between Kelvin & Celsius
To convert from Kelvin to Celsius:
T°C = TK – 273
Liquid Nitrogen boils at 77K, what is this in Celsius?
T°C = 77 – 273 = -196 °C
To convert from Celsius to Kelvin:
TK = T°C + 273
The bp of water on top of Mt. Everest is 70 °C. Convert to K.
TK = 70 + 273 = 343 K
Fahrenheit/Celsius Conversions
To convert from Celsius to Fahrenheit:
T°F = 1.80(T°C) + 32
If the temperature is 28°C, what is this in °F?
T°F = 1.80(28) + 32 = 50.4 + 32 = 82°F (2 s.f.)
To convert from Fahrenheit to Celsius:
T°C = (T°F – 32)/1.80
If you have a temperature of 101°F, what is this in °C?
T°C = (101 – 32)/1.80 = 69/1.8 = 38°C
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Learning Check
a. Which temperature is colder, 172 K or -75 oC?
b. Hot tubs are often maintained at 41 oC, what is the
temperature in Fahrenheit?
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Density
•Defined as the amount of matter present in a
given volume of a substance
Density = mass / volume
•Units are in g/cm3 or g/mL
•Volume is dependent on temperature, so
density is as well
Densities of Common Substances @ 20oC
Substance
Density (g/cm3)
Substance
Density (g/cm3)
Oxygen
0.00133
Aluminum
2.70
Hydrogen
0.000084
Iron
7.87
Ethanol
0.785
Copper
8.96
Benzene
0.880
Silver
10.5
Water
1.000
Lead
11.34
Magnesium
1.74
Mercury
13.6
Salt (sodium
chloride)
2.16
Gold
19.32
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Determining volume by water
displacement
• Place water in graduated cylinder & record
level
• Add object
• Record volume after addition of object
• Volume is difference between second
volume and first volume
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Figure 5.9: Tank of water.
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Figure 5.9: Person submerged in the tank.
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Learning Check
• A student wants to identify the main
component in a liquid cleaner. He finds that
35.8 mL of the liquid weighs 28.1 g. Which is
the main component?
a. chloroform
1.483 g/mL
b. diethyl ether
0.714 g/mL
c. isopropyl alcohol
0.785 g/mL
d. toluene
0.867 g/mL
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