Chapter 2 Measurement and Problem Solving
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Chapter 2Measurement andProblem Solving
Book Ch 03Scientific measurement
2009, Prentice Hall
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What Is a Measurement?
• Quantitative observation.
• Comparison to an agreed upon standard.
• Every measurement has a number and a unit.
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A Measurement
• The unit tells you to what standard you are comparing your object.
• The number tells you:1.What multiple of the standard the object
measures.
2. The uncertainty in the measurement.
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Scientists have measured the average global temperature rise over the past
century to be 0.6 °C
• °C tells you that the temperature is being compared to the Celsius temperature scale.
• 0.6 tells you that:1. The average temperature rise is 0.6
times the standard unit of 1 degree Celsius.
2. The confidence in the measurement is such that we are certain the measurement is between 0.5 and 0.7 °C.
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Scientific Notation
A way of writing
large and small numbers.
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Big and Small Numbers
• We commonly measure objects that are many times larger or smaller than our standard of comparison.
• Writing large numbers of zeros is tricky and confusing.Not to mention there’s the 8-
digit limit of your calculator!
The sun’sdiameter is
1,392,000,000 m.
An atom’s average diameter is0.000 000 000 3 m.
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Scientific Notation
• Each decimal place in our number system represents a different power of 10.
• Scientific notation writes the numbers so they are easily comparable by looking at the power of 10. An atom’s
average diameter is3 x 10-10 m.
The sun’sdiameter is
1,392,000,000 m.
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Exponents
• When the exponent on 10 is positive, it means the number is that many powers of 10 larger. Sun’s diameter = 1.392 x 109 m = 1,392,000,000 m.
• When the exponent on 10 is negative, it means the number is that many powers of 10 smaller.Average atom’s diameter = 3 x 10-10m = 0.0000000003m
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Scientific Notation• To compare numbers written in scientific
notation:First compare exponents on 10.If exponents are equal, then compare decimal
numbers
1.23 x 10-8
Decimal part Exponent part
Exponent1.23 x 105 > 4.56 x 102
4.56 x 10-2 > 7.89 x 10-5
7.89 x 1010 > 1.23 x 1010
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123401. Locate the decimal point.
12340.2. Move the decimal point to obtain a number between 1 and 10.
1.2343. Multiply the new number by 10n .
Where n is the number of places you moved the decimal point.
1.234 x 104
4. If you moved the decimal point to the left, then n is +; if you moved it to the right, then n is − .
1.234 x 104
Writing a Number in Scientific Notation, Continued
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Writing a Number in Scientific Notation, Continued0.00012340
1. Locate the decimal point.0.00012340
2. Move the decimal point to obtain a number between 1 and 10.1.2340
3. Multiply the new number by 10n .Where n is the number of places you moved the decimal
point.1.2340 x 104
4. If you moved the decimal point to the left, then n is +; if you moved it to the right, then n is − .
1.2340 x 10-4
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Writing a Number in Standard Form1.234 x 10-6
• Since exponent is -6, make the number smaller by moving the decimal point to the left 6 places.When you run out of digits to move around,
add zeros.Add a zero in front of the decimal point for
decimal numbers.
000 001.2340.000 001 234
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Practice—Write the Following in Scientific Notation, Continued
123.4 = 1.234 x 102
145000 = 1.45 x 105
25.25 = 2.525 x 101
1.45 = 1.45 x 100
8.0012 = 8.0012 x 100
0.00234 = 2.34 x 10-3
0.0123 = 1.23 x 10-2
0.000 008706 = 8.706 x 10-6
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Practice—Write the Following in Standard Form, Continued
2.1 x 103 = 2100
9.66 x 10-4 = 0.000966
6.04 x 10-2 = 0.0604
4.02 x 100 = 4.02
3.3 x 101 = 33
1.2 x 100 = 1.2
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Inputting Scientific Notation into a Calculator
• Input the decimal part of the number.If negative press +/-
key.• (–) on some.
• Press EXP.EE on some.
• Input exponent on 10.Press +/- key to change
exponent to negative.
-1.23 x 10-3
-1.23 -03
Press +/-
Input 1.23 1.23
Press EXP -1.23 00
Input 3 -1.23 03
-1.23 Press +/-
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Inputting Scientific Notation into a TI Graphics Calculator
• Use ( ) liberally!!• Type in the decimal part of
the number. If negative, first press the (-).
• Press the multiplication key.
• Type “10”.• Press the exponent key, ^ .• Type the exponent.
If negative, first press the (-).
-1.23 x 10-3
–Press (-)
Input 1.23 –1.23
Press ^ -1.23*10^
Input 3 -1.23*10^-3
-1.23* Press ×
Input 10 -1.23*10
-1.23*10^-Press (-)
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Numbers
1. Exact2. Measured: Significant Figures
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Exact Numbers vs. Measurements• Exact: Sometimes you can determine an exact
value for a quality of an object.A. Often by counting.
• Pennies in a pile.
B. Sometimes by definition• 1 ounce is exactly 1/16th of 1 pound.
• Measured: Whenever you use an instrument to compare a quality of an object to a standard, there is uncertainty in the comparison.
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Name the 4 measuring instruments
1. Length
2. Mass
3. Time
4. Temperature
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How do we make a Measurement
• Measurements are written to indicate the uncertainty in the measurement.
• The system of writing measurements we use is called significant figures.
• When writing measurements, all the digits written are known with certainty except the last one, which is an estimate.
45.872
CertainEstimated
45.872
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Reading a Measuring Instrument/Device
For any Digital Device record ALL the digits
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Reading a Measuring Instrument/Device
1. Record all the numbers you can see
2. Make ONE Guess!
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Skillbuilder 2.3—Reporting the Right Number of Digits
• A thermometer used to measure the temperature of a backyard hot tub is shown to the right. What is the temperature reading to the correct number of digits?
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Skillbuilder 2.3—Reporting the Right Number of Digits
• A thermometer used to measure the temperature of a backyard hot tub is shown to the right. What is the temperature reading to the correct number of digits?
103.4 °F
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What is the Length?
• We can see the markings between 1.6-1.7cm
• We can’t see the markings between the .6-.7
• We must guess between .6 & .7
• We record 1.67 cm as our measurement
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What is the length of the wooden stick?1) 4.5 cm 2) 4.54 cm 3) 4.547 cm
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8.00 cm or 3 (2.2/8)?
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Counting Significant Figures
• All non-zero digits are significant.1.5 has 2 significant figures.
• Interior zeros are significant.1.05 has 3 significant figures.
• Trailing zeros after a decimal point are significant.1.050 has 4 significant figures.
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Counting Significant Figures, Continued
• Leading zeros are NOT significant. 0.001050 has 4 significant figures.
• 1.050 x 10-3
• Zeros at the end of a number without a written decimal point are NOT significant
If 150 has 2 significant figures, then 1.5 x 102,
but if 150 has 3 significant figures, then 1.50 x 102.
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Exact Numbers• Exact numbers have an unlimited number of
significant figures. • A number whose value is known with
complete certainty is exact.From counting individual objects.From definitions.
• 1 cm is exactly equal to 0.01 m.• 20@ $.05 = $1.0000000000000• 12 inches = 1.000000000000000000000000 ft
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Example 2.4—Determining the Number of Significant Figures in a Number, Continued
• How many significant figures are in each of the following numbers?
0.0035 2 significant figures—leading zeros are not significant.
1.080 4 significant figures—trailing and interior zeros are significant.
2371 4 significant figures—All digits are significant.
2.97 × 105 3 significant figures—Only decimal parts count as significant.
1 dozen = 12 Unlimited significant figures—Definition
100,000 1, no decimal
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Determine the Number of Significant Figures,
• 12000
• 120.
• 12.00
• 1.20 x 103
• 0.0012
• 0.00120
• 1201
• 1201000
2
3
4
3
2
3
4
4
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How many sig figs?
45.8736
.000239
.00023900
48000.
48000
3.982106
1.00040
6
3
5
5
2
4
6
•All digits count
•Leading 0’s don’t
•Trailing 0’s do
•0’s count in decimal form
•0’s don’t count w/o decimal
•All digits count
•0’s between digits count as well as trailing in decimal form
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Rounding• When rounding to the correct number of significant
figures, if the number after the place of the last significant figure is:
1. 0 to 4, round down. Drop all digits after the last significant figure and leave
the last significant figure alone. Add insignificant zeros to keep the value, if necessary.
2. 5 to 9, round up. Drop all digits after the last significat figure and
increase the last significant figure by one. Add insignificant zeros to keep the value, if necessary.
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Examples of RoundingFor example you want a 4 Sig Fig number
4965.03
780,582
1999.5
0 is dropped, it is <5
8 is dropped, it is >5; Note you must include the 0’s
5 is dropped it is = 5; note you need a 4 Sig Fig
4965
780,600
2000.
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Multiplication and Division with Significant Figures
• When multiplying or dividing measurements with significant figures, the result has the same number of significant figures as the measurement with the fewest number of significant figures.
5.02 × 89,665 × 0.10 = 45.0118 = 45 3 sig. figs. 5 sig. figs. 2 sig. figs. 2 sig.
figs.
5.892 ÷ 6.10 = 0.96590 = 0.966 4 sig. figs. 3 sig. figs. 3 sig. figs.
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Determine the Correct Number of Significant Figures for Each Calculation and
1. 1.01 × 0.12 × 53.51 ÷ 96 = 0.067556 = 0.068
2. 56.55 × 0.920 ÷ 34.2585 = 1.51863 = 1.52
3 sf 2 sf 4 sf 2 sf Result should have 2 sf.
7 is in place of last sig. fig., number after
is 5 or greater, so round up.
4 sf 3 sf 6 sf Result should have 3 sf.
1 is in place of last sig. fig., number after
is 5 or greater, so round up.
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Addition/Subtraction
25.5 32.72 320
+34.270 ‑ 0.0049 + 12.5
59.770 32.7151 332.5
59.8 32.72 330
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Addition and Subtraction with Significant Figures
• When adding or subtracting measurements with significant figures, the result has the same number of decimal places as the measurement with the fewest number of decimal places.
5.74 + 0.823 +2.651 = 9.214 = 9.21 2 dec. pl. 3 dec. pl. 3 dec. pl. 2 dec. pl.
4.8 - 3.965 = 0.835 = 0.8 1 dec. pl 3 dec. pl. 1 dec. pl.
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Determine the Correct Number of Significant Figures for Each Calculation and
Round and Report the Result, Continued
1. 0.987 + 125.1 – 1.22 = 124.867 = 124.9
2. 0.764 – 3.449 – 5.98 = -8.664 = -8.66
3 dp 1 dp 2 dp Result should have 1 dp.
8 is in place of last sig. fig., number after
is 5 or greater, so round up.
3 dp 3 dp 2 dp Result should have 2 dp.
6 is in place of last sig. fig., number after is 4 or less,
so round down.
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__ ___ __
Addition and Subtraction
.56 + .153 = .713
82000 + 5.32 = 82005.32
10.0 - 9.8742 = .12580
10 – 9.8742 = .12580
.71
82000
.1
0
Look for the last important digit
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Both Multiplication/Division and Addition/Subtraction with
Significant Figures• When doing different kinds of operations with
measurements with significant figures, evaluate the significant figures in the intermediate answer, then do the remaining steps.
• Follow the standard order of operations.Please Excuse My Dear Aunt Sally.
3.489 × (5.67 – 2.3) = 2 dp 1 dp
3.489 × 3.37 = 12 4 sf 1 dp & 2 sf 2 sf
- n
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Example 1.6—Perform the Following Calculations to the Correct Number of Significant Figures,
Continued652.065219.04555.30015.45120.010.1 a)
4.9 8730.4
5820.100
1.105
355.0
5379904.5233.4526755.45299870.3562.4 c)
1.0142.002.855.084.14 d)
b)
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Basic Units of Measure
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Units• Units tell the standard quantity to which we are
comparing the measured property.Without an associated unit, a measurement is without
meaning.
• Scientists use a set of standard units for comparing all our measurements.So we can easily compare our results.
• Each of the units is defined as precisely as possible.
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The Standard Units• Scientists generally report results in an
agreed upon International System.• The SI System
Aka Système International
Quantity Unit Symbol
Length meter m
Mass kilogram kg
Volume liter L
Time second s
Temperature kelvin K
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Related Units in the SI System
• All units in the SI system are related to the standard unit by a power of 10.
• The power of 10 is indicated by a prefix.
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Common Prefixes in the SI System
Prefix SymbolDecimal
EquivalentPower of 10
mega- M 1,000,000 Base x 106
kilo- k 1,000 Base x 103
deci- d 0.1 Base x 10-1
centi- c 0.01 Base x 10-2
milli- m 0.001 Base x 10-3
micro- or mc 0.000 001 Base x 10-6
nano- n 0.000 000 001 Base x 10-9
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Measurements and SI
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M 1,000,000
k 1,000
d 0.1
c 0.01
m 0.001
or mc 0.000 001
n 0.000 000 001
gram g
meter m
liter L
seconds s
Kelvin K
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Measurements and SI
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m 0.001
liter L
(m = .001)L
mL = .001L or 1000 mL = L
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Prefixes Used to Modify Standard Unit• kilo = 1000 times base unit = 103
1 kg = 1000 g = 103 g
• deci = 0.1 times the base unit = 10-1
1 dL = 0.1 L = 10-1 L; 1 L = 10 dL
• centi = 0.01 times the base unit = 10-2
1 cm = 0.01 m = 10-2 m; 1 m = 100 cm
• milli = 0.001 times the base unit = 10-3
1 mg = 0.001 g = 10-3 g; 1 g = 1000 mg
• micro = 10-6 times the base unit1 m = 10-6 m; 106 m = 1 m
• nano = 10-9 times the base unit1 nL = 10-9L; 109 nL = 1 L
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Volume
1 mL = 1 cm3
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Common Units and Their Equivalents
Length
1 kilometer (km) = 0.6214 mile (mi)
1 meter (m) = 39.37 inches (in.)
1 meter (m) = 1.094 yards (yd)
1 foot (ft) = 30.48 centimeters (cm)
1 inch (in.) = 2.54 centimeters (cm) exactly
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Units• Always write every number with its
associated unit.• Always include units in your calculations.
You can do the same kind of operations on units as you can with numbers.
• cm × cm = cm2
• cm + cm = 2cm• cm ÷ cm = 1
Using units as a guide to problem solving is called dimensional analysis.
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Problem Solving and Dimensional Analysis, Continued
• Arrange conversion factors so the starting unit cancels.Arrange conversion factor so the starting unit is on the
bottom of the conversion factor.
• May string conversion factors.So we do not need to know every relationship, as long as
we can find something else the starting and desired units are related to :
unit desired unit related
unit desired
unitstart
unit relatedunitstart
unit desired unitstart
unit desiredunitstart
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Problem Solving and Dimensional Analysis
• Many problems in chemistry involve using relationships 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 have the same number of
significant figures.
• Conversion factors generated from equivalence statements.e.g., 1 inch = 2.54 cm can give or
in1
cm54.2cm54.2
in1
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Systematic Approach
1. Write down the given amount and unit.
2. Write down what you want to find and unit.
3. Write down needed conversion factors or equations.
a. Write down equivalence statements for each relationship.
b. Change equivalence statements to conversion factors with starting unit on the bottom.
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Systematic Approach, Continued4. Design a solution map for the problem.
Order conversions to cancel previous units orarrange equation so the find amount is isolated.
5. Apply the steps in the solution map. Check that units cancel properly. Multiply terms across the top and divide by each
bottom term.6. Determine the number of significant figures
to report and round.7. Check the answer to see if it is reasonable.
Correct size and unit.
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Solution Maps and Conversion Factors
• Convert inches into centimeters.1. Find relationship equivalence: 1 in = 2.54 cm
2. Write solution map.
inin cmcm
3. Change equivalence into conversion factors with starting units on the bottom.
in 1
cm 2.54
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Example 2.8:• Convert 7.8 km to miles
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Example:Convert 7.8 km to miles.
• Write down the given quantity and its units.
Given: 7.8 km
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• Write down the quantity to find and/or its units.
Find: ? miles
Information
Given: 7.8 kmExample:Convert 7.8 km to miles.
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• Collect needed conversion factors:
1 mi = 0.6214 km
Information
Given: 7.8 km
Find: ? mi
Example:Convert 7.8 km to miles.
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• Write a solution map for converting the units:
InformationGiven: 7.8 kmFind: ? miConversion Factor:
1 mi = 0.6214 km
Example:Convert 7.8 km to miles.
km mi
km 1
mi .62140
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• Apply the solution map:
InformationGiven: 7.8 kmFind: ? miConversion Factor:1 mi = 0.6214 kmSolution Map: km mi
Example:Convert 7.8 km to miles.
km 1
mi .62140
mi km 1
mi 0.6214km 7.8
= 4.84692 mi
= 4.8 mi
• Significant figures and round:
2 significant figures
2 significant figures
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• Check the solution:
InformationGiven: 7.8 kmFind: ? miConversion Factor: 1 mi = 0.6214 kmSolution Map: km mi
Example:Convert 7.8 km to miles.
km 1
mi .62140
7.8 km = 4.8 mi
The units of the answer, mi, are correct.The magnitude of the answer makes sense
since kilometers are shorter than miles.
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Practice—Convert 30.0 g to Ounces(1 oz. = 28.32 g)
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Convert 30.0 g to Ounces
Units and magnitude are correct.
Check:• Check.
Round:• Significant figures and round.
Solution:• Follow the solution map to Solve the problem.
Solution Map:
• Write a Solution Map.
1 oz = 28.35 gConversion Factor:
• Write down the appropriate Conversion Factors.
oz.Find:• Write down the quantity you want to Find and unit.
30.0 gGiven:• Write down the Given quantity and its unit.
g oz
g 28.35
oz 1
oz 1.05820 g 28.35
oz 1g 30.0
3 sig figs
3 sig figs= 1.06 oz
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Example 2.10:• An Italian recipe for making creamy pasta sauce calls for
0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use?
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An Italian recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use?
• Write down the given quantity and its units.
Given: 0.75 L
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• Write down the quantity to find and/or its units.
Find: ? cups
Information
Given: 0.75 L
An Italian recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use?
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• Collect needed conversion factors:
4 cu = 1 qt
1.057 qt = 1 L
Information
Given: 0.75 L
Find: ? cu
An Italian recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use?
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• Write a solution map for converting the units:
InformationGiven: 0.75 LFind: ? cuConversion Factors:
4 cu = 1 qt; 1.057 qt = 1 L
L qt
An Italian recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use?
cu
L 1
qt .0571qt 1
cu 4
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qt 1
cu 4
L 1
qt .0571L 750 .
• Apply the solution map:
= 3.171 cu
= 3.2 cu
• Significant figures and round:
InformationGiven: 0.75 LFind: ? cuConversion Factors: 4 cu = 1 qt; 1.057 qt = 1 LSolution Map: L qt cu
An Italian recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use?
L 1
qt .0571
qt 1
cu 4
2 significant figures
2 significant figures
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• Check the solution:
0.75 L = 3.2 cu
The units of the answer, cu, are correct.The magnitude of the answer makes sense
since cups are smaller than liters.
InformationGiven: 0.75 LFind: ? cuConversion Factors:4 cu = 1 qt; 1.057 qt = 1 LSolution Map: L qt cu
An Italian recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use?
L 1
qt .0571
qt 1
cu 4
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Example 2.12:• A circle has an area of 2,659 cm2. What is the area in
square meters?
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Example:A circle has an area of 2,659 cm2. What is the area in square meters?
• Write down the given quantity and its units.
Given: 2,659 cm2
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• Write down the quantity to find and/or its units.
Find: ? m2
Information
Given: 2,659 cm2
Example:A circle has an area of 2,659 cm2. What is the area in square meters?
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• Collect needed conversion factors:
1 cm = 0.01m
Information
Given: 2,659 cm2
Find: ? m2
Example:A circle has an area of 2,659 cm2. What is the area in square meters?
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• Write a solution map for converting the units:
cm2 m2
2
cm 1
m .010
InformationGiven: 2,659 cm2
Find: ? m2
Conversion Factor:1 cm = 0.01 m
Example:A circle has an area of 2,659 cm2. What is the area in square meters?
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• Apply the solution map:2
2
2-42 m
cm 1
m 101cm 2,659
= 0.265900 m2
= 0.2659 m2
• Significant figures and round:
InformationGiven: 2,659 cm2
Find: ? m2
Conversion Factor:1 cm = 0.01 mSolution Map: cm2 m2
Example:A circle has an area of 2,659 cm2. What is the area in square meters? 2
cm 1
m .010
4 significant figures
4 significant figures
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Practice—Convert 30.0 cm3 to ft3
(1 cm = 1 x 10-2 m) (in class)
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Convert 30.0 cm3 to m3
Units and magnitude are correct.
Check:7. Check.
30.0 cm3 = 3.00 x 10−5 m3Round:6. Significant figures and round.
Solution:5. Follow the solution map to Solve the problem.
Solution Map:
4. Write a Solution Map.
(1 cm = 0.01 m)3Conversion Factor:
3. Write down the appropriate Conversion Factors.
? m3Find:2. Write down the quantity you want to Find and unit.
30.0 cm3Given:1. Write down the Given quantity and its unit.
cm3 m3
3
cm 1
m .010
35-3
3631 m 103
cm 1
m 101cm 103.00
3 sig figs
3 sig figs
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Density
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Density• Ratio of mass:volume.• Its value depends on the kind of material, not the
amount.• Solids = g/cm3
1 cm3 = 1 mL
• Liquids = g/mL• Gases = g/L• Volume of a solid can be determined by water
displacement—Archimedes Principle.• Density : solids > liquids > gases
Except ice is less dense than liquid water!
Volume
MassDensity
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Platinum has become a popular metal for fine jewelry. A man gives a woman an engagement ring and tells her that it is made of platinum.
Noting that the ring felt a little light, the woman decides to perform a test to determine the ring’s
density before giving him an answer about marriage. She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds that the ring displaces 0.556 cm3 of water. Is the ring made of platinum? (Density Pt = 21.4 g/cm3)
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She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds that the ring displaces 0.556 cm3 of water. Is the ring made of
platinum? (Density Pt = 21.4 g/cm3)
Given: Mass = 5.84 grams
Volume = 0.556 cm3
Find: Density in grams/cm3
Equation:
Solution Map:
m and V d
dV
m
m, V dd
V
m
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She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds that the ring displaces 0.556 cm3 of water. Is the ring made of
platinum? (Density Pt = 21.4 g/cm3)
Apply the Solution Map:
33 cmg 10.5
cm 0.556
g 5.84
dV
m
Since 10.5 g/cm3 21.4 g/cm3, the ring cannot be platinum.
m, V dd
V
m
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Practice—What Is the Density of Metal if a 100.0 g Sample Added to a Cylinder of Water Causes the Water Level to Rise from 25.0 mL to 37.8 mL?
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Find Density of Metal if 100.0 g Displaces Water from 25.0 to 37.8 mL
Units and magnitude are correct.
Check:7. Check.
7.8125 g/cm3 = 7.81 g/cm3Round:6. Significant figures and round.
Solution:
V = 37.8-25.0 = 12.8 mL
5. Follow the solution map to Solve the problem.
Solution Map:
4. Write a Solution Map.
CF & Equation:
d, g/cm3Find:2. Write down the quantity you want to Find and unit.
m =100.0 gdisplaces 25.0 to 37.8 mL
Given:1. Write down the Given quantity and its unit.
3 sig figs
3 significant figures
m, V d
V
m d
V
m d 1 mL = 1 cm3
33
g/cm 2518.7cm 12.8
g 0.100d
33
cm .821mL 1
cm 1mL 12.8
3. Write down the appropriate Conv. Factor and Equation.
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Density as a Conversion Factor
• Can use density as a conversion factor between mass and volume!Density of H2O = 1 g/mL 1 g H2O = 1 mL H2O
Density of Pb = 11.3 g/cm3 11.3 g Pb = 1 cm3 Pb
• How much does 4.0 cm3 of lead weigh?
=4.0 cm3 Pb 11.3 g Pb 1 cm3 Pb
45 g Pbx
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Measurement and Problem Solving:Density as a Conversion Factor
• The gasoline in an automobile gas tank has a mass of 60.0 kg and a density of 0.752 g/cm3. What is the volume?
• Given: 60.0 kg
• Find: Volume in cm3
• Conversion factors: 0.752 g/cm3
1000 grams = 1 kg
Solution Map:
kg g cm3
kg 1
g 1000
g .7520
cm 1 3
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Measurement and Problem Solving:Density as a Conversion Factor,
Continued
343
cm107.98 g 0.752
cm 1
kg 1
g 1000kg 60.0
Solution Map:
kg g cm3
kg 1
g 1000
g .7520
cm 1 3
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Practice—What Volume Does 100.0 g of Marble Occupy? (d = 4.00 g/cm3)
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What Volume Does 100.0 g of Marble Occupy?
Units and magnitude are correct.
Check:7. Check.
25 cm3 = 25.0 cm3Round:6. Significant figures and round.
Solution:5. Follow the solution map to Solve the problem.
Solution Map:
4. Write a Solution Map.
CF & Equation:
3. Write down the appropriate Conv. Factor and Equation.
V, cm3Find:2. Write down the quantity you want to Find and unit.
m =100.0 gGiven:1. Write down the Given quantity and its unit.
4 sig figs
3 significant figures
m V
g 4.00
cm 1 3
33
cm 25g 4.00
cm 1g 100.0
4.00 g = 1 cm33 sig figs
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Example 2.17:• A 55.9 kg person displaces 57.2 L of water when
submerged in a water tank. What is the density of the person in g/cm3?