Transcript significant figures

Measurements
and
Calculations
Chapter 2
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Quantitative Observation
Comparison Based on an Accepted Scale
◦ e.g. Meter Stick
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Has 2 Parts – the Number and the Unit
◦ Number Tells Comparison
◦ Unit Tells Scale
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Technique Used to Express Very Large or Very
Small Numbers
Based on Powers of 10
1. Move the decimal point so there is only
one non-zero number to the left of it.
The new number is now between 1 and 9
2. Multiply the new number by 10n
◦ where n is the number of places you moved
the decimal point
3. 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
1
Determine the sign of n of 10n
2
Determine the value of the exponent of 10
3
◦ If n is + the decimal point will move to the right
◦ If n is – the decimal point will move to the left
◦ Tells the number of places to move the decimal
point
Move the decimal point and rewrite the
number
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75,000,000
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0.0000000011
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0.0005710
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8,031,000,000
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2.75 x 10-7
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7.10 x 10-5
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5.22 x 104
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9.38 x 1012
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Change to scientific
notation
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Change to standard
notation
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41080.642
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0.00065 x 106
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1.8732
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391 x 10-2
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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 or basic unit
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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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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 (cm x cm x cm))
◦ 1 m3 = 106 cm3
◦ 1 cm3 = 10-6 m3 = 0.000001 m3
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Commonly measure liquid or gas volume in
milliliters (mL)
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1
1
1
1
L is slightly larger than 1 quart
L = 1 dL3 = 1000 mL = 103 mL
mL = 0.001 L = 10-3 L
mL = 1 cm3
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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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250 mL to Liters
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1.75 kg to grams
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88 daL to mL
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475 cg to mg
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328 hm to Mm
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0.00075 nL to cL
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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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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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Nonzero integers are always significant
◦ How many significant figures are in the
following examples:
 2753
 89.659
 0.281
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Zeros
◦ Captive zeros are always significant
◦ How many significant figures are in the following
examples:
 1001.4
 55.0702
 0.4900008
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Zeros
◦ Leading zeros never count as significant figures
◦ How many significant figures are in the following
examples:
 0.00048
 0.0037009
 0.0000000802
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Zeros
◦ Trailing zeros are significant if the number has a
decimal point
◦ How many significant figures are in the following
examples:
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22,000
63,850.
0.00630100
2.70900
100,000
Scientific Notation
◦ All numbers before the “x” are significant. Don’t
worry about any other rules.
◦ 7.0 x 10-4 g has 2 significant figures
◦ 2.010 x 108 m has 4 significant figures
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If the digit to be removed
• is less than 5, the preceding digit stays the
same
 Round 87.482 to 4 sig figs.
• is equal to or greater than 5, the preceding
digit is increased by 1
 Round 0.00649710 to 3 sig figs.
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In a series of calculations, carry the extra
digits to the final result and then round off
◦ Ex: Convert 80,150,000 seconds to years
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Don’t forget to add place-holding zeros if
necessary to keep value the same!!
◦ Round 80,150,000 to 3 sig figs.
Count the number of significant figures in each
measurement
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Round the result so it has the same number
of significant figures as the measurement
with the smallest number of significant
figures
14.593 cm x 0.200 cm =
3.7 x 103 x 0.00340 =
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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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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
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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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Many problems in chemistry involve using
equivalence statements to convert one unit
of measurement to another
Conversion factors are relationships between
two units
Conversion factors generated from
equivalence statements
◦ e.g. 1 inch = 2.54 cm can give
or
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Arrange conversion factor so starting unit is
on the bottom of the conversion factor
You may string conversion factors together
for problems that involve more than one
conversion factor.
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Find the relationship(s) between the starting
and final units.
Write an equivalence statement and a
conversion factor for each relationship.
Arrange the conversion factor(s) to cancel
starting unit and result in goal unit.
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Check that the units cancel properly
Multiply all the numbers across the top and
divide by each number on the bottom to
give the answer with the proper unit.
Round your answer to the correct number of
significant figures.
Check that your answer makes sense!
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28.5 inches to feet
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4.0 gallons to
quarts
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48.39 minutes to
hours
155.0 pounds to
grams
2.00 x 108 seconds
to hours
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682 mg to pounds
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0.091 ft2 to inches2
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3.5 x 10-4 L to cm3
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47.1 mm3 to kL
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25 miles per hour to feet per second
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4.70 gallons per minute to mL per year
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5.6 x 10-6 centiliters per square meter (cL/m2) to
cubic meters per square foot (m3/ft2)
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Fahrenheit Scale, °F
◦ Water’s freezing point = 32°F, boiling point = 212°F
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Celsius Scale, °C
◦ Temperature unit larger than the Fahrenheit
◦ Water’s freezing point = 0°C, boiling point = 100°C
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Kelvin Scale, K
◦ Temperature unit same size as Celsius
◦ Water’s freezing point = 273 K, boiling point = 373 K
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Fahrenheit to Celsius
oC
= 5/9(oF -32)
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Celsius to Fahrenheit
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Celsius to Kelvin
K = oC + 273
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Kelvin to Celsius
oC
oF
= 1.8(oC) +32
= K – 273
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86oF to oC
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-5.0oC to oF
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352 K to oC
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12oC to K
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248 K to oF
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98.6oF to K
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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
Mass
◦ 1 cm3 = 1 mL
Density
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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
Mass
Density 
Volume
Mass
Volume 
Density
Mass  Density  Volume
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What is the density of a metal with a
mass of 11.76 g whose volume occupies
6.30 cm3?
What volume, in mL, of ethanol (density
= 0.785 g/mL) has a mass of 2.04 lbs?
What is the mass (in mg) of a gas that
has a density of 0.0125 g/L in a 500. mL
container?
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To determine the volume to insert into the
density equation, you must find out the
difference between the initial volume and
the final volume.
A student attempting to find the density of
copper records a mass of 75.2 g. When the
copper is inserted into a graduated cylinder,
the volume of the cylinder increases from
50.0 mL to 58.5 mL. What is the density of
the copper in g/mL?
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Percent error – absolute value of the error
divided by the accepted value, multiplied by
100%.
% error = measured value – accepted value x 100%
accepted value
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Accepted value – correct value based on
reliable sources.
Experimental (measured) value – value
physically measured in the lab.
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In the lab, you determined the density of
ethanol to be 1.04 g/mL. The accepted
density of ethanol is 0.785 g/mL. What is the
percent error?
The accepted value for the density of lead is
11.34 g/cm3. When you experimentally
determined the density of a sample of lead,
you found that a 85.2 gram sample of lead
displaced 7.35 mL of water. What is the
percent error in this experiment?