Transcript No Slide Title

Measurements
and
Calculations
Chapter 2
1
Measurement
• Quantitative Observation
• Comparison Based on an Accepted Scale
– e.g. Meter Stick
• Has 2 Parts – the Number and the Unit
– Number Tells Comparison
– Unit Tells Scale
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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
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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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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Length
• SI unit = meter (m)
– About 3½ inches longer than a yard
• 1 meter = one ten-millionth the distance from the North Pole to
the Equator = distance between marks on standard metal rod in
a Paris vault = distance covered by a certain number of
wavelengths of a special color of light
• Commonly use centimeters (cm)
– 1 m = 100 cm
– 1 cm = 0.01 m = 10 mm
– 1 inch = 2.54 cm (exactly)
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10
Artist's conception of the lost Mars
Climate Orbiter.
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Volume
• Measure of the amount of three-dimensional 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
• Commonly measure liquid or gas volume in milliliters
(mL)
–
–
–
–
1 L is slightly larger than 1 quart
1 L = 1 dL3 = 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
1 m in length and a volume
of 1 m3. The middle-size
cube has sides 1 dm in
length and a volume of 1
dm3. The smalles cube has
sides 1 cm in length and a
volume of 1cm3.
Figure 2.3:
A 100-mL
graduated
cylinder.
Mass
• Measure of the amount of matter present
in an object
• SI unit = kilogram (kg)
• Commonly measure mass in grams (g) or
milligrams (mg)
– 1 kg = 2.2046 pounds, 1 lbs.. = 453.59 g
– 1 kg = 1000 g = 103 g, 1 g = 1000 mg = 103
mg
– 1 g = 0.001 kg = 10-3 kg, 1 mg = 0.001 g =
10-3 g
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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
• Kelvin Scale, K
– Temperature unit same size as Celsius
– Water’s freezing point = 273 K, boiling point = 373 K
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Temperature
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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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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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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 (embedded) 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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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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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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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
Round the result so it has the same number of
significant figures as the measurement with the
smallest number of significant figures
4.5 cm
2 sig figs
x
0.200 cm =
3 sig figs
0.90 cm2
2 sig figs
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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
Find which one is “left-most”
Round answer to the same decimal place
450 mL + 27.5 mL =
precise to 10’s place
precise to 0.1’s place
480 mL
precise to 10’s place
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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
• Conversion factors generated from equivalence
2.54cm
1in
statements
– e.g. 1 inch = 2.54 cm can give1in
2.54cm
or
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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
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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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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
• Solids = g/cm3
– 1 cm3 = 1 mL
Mass
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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Using Density in Calculations
Mass
Density 
Volume
Mass
Volume 
Density
Mass  Density  Volume
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Figure 2.11:
A hydrometer
being used to
determine the
density of the
antifreeze
solution in a
car’s radiator.