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Kanya EllingtonOakhurst Community College Center
Oakhurst, CA
Introductory Chemistry, 2nd EditionNivaldo Tro
Chapter 2Measurement andProblem Solving
2006, 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 what standard you are comparing your object to
• the number tells you1.what multiple of the standard the object
measures2.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 that1. the average temperature rise is 0.6
times the standard unit2. the uncertainty in the measurement is
such that we know 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 confusingnot to mention 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
the sun’sdiameter is
1.392 x 109 m
an atom’s average diameter is
3 x 10-10 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 smalleravg. atom’s diameter = 3 x 10-10 m = 0.0000000003 m
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Scientific Notation• To Compare Numbers Written in Scientific
NotationFirst Compare Exponents on 10If Exponents 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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Writing Numbers in Scientific Notation
1 Locate the Decimal Point2 Move the decimal point to the right of the
first non-zero digit from the left3 Multiply the new number by 10n
where n is the number of places you moved the decimal point
4 if the number is 1, n is +; if the number is < 1, n is -
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123401 Locate the Decimal Point
12340.2 Move the decimal point to the right of the first non-zero digit
from the left1.234
3 Multiply the new number by 10n
where n is the number of places you moved the decimal pt.1.234 x 104
4 if the number is 1, n is +; if the number is < 1, n is -1.234 x 104
Writing a Number In Scientific Notation
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1.23401 Locate the Decimal Point
1.23402 Move the decimal point to the right of the first non-zero digit
from the left1.2340
3 Multiply the new number by 10n
where n is the number of places you moved the decimal pt.1.2340 x 100
4 if the number is 1, n is +; if the number is < 1, n is -1.2340 x 100
Writing a Number In Scientific Notation
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Writing a Number In Scientific Notation0.00012340
1 Locate the Decimal Point0.00012340
2 Move the decimal point to the right of the first non-zero digit from the left
1.23403 Multiply the new number by 10n
where n is the number of places you moved the decimal pt.1.2340 x 104
4 if the number is 1, n is +; if the number is < 1, n is -1.2340 x 10-4
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Writing a Number in Standard Form
1.234 x 10-6
• since exponent is -6, make the number smaller by moving the decimal point to the left 6 placesif you run out of digits, add zeros
000 001.234
0.000 001 234
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Example 2.1• The U.S. population in 2004 was estimated
to be 293,168,000 people. Express this number in scientific notation.
• 293,168,000 people = 2.93168 x 108 people
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Inputting Scientific Notation into a Calculator• input decimal part of the
numberif 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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Significant Figures
Writing Numbers to Reflect Precision
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Exact Numbers vs. Measurements• sometimes you can determine an
exact value for a quality of an objectoften by counting
• pennies in a pilesometimes by definition
• 1 ounce is exactly 1/16th of 1 pound
• 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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Reporting Measurements• 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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Estimating the Last Digit• for instruments marked with a
scale, you get the last digit by estimating between the marks
if possible• mentally divide the space into
10 equal spaces, then estimate how many spaces over the indicator is
1.2 grams
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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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Significant Figures• the non-place-holding digits in a
reported measurement are called significant figuressome zero’s in a written number are
only there to help you locate the decimal point
• significant figures tell us the range of values to expect for repeated measurements the more significant figures there are
in a measurement, the smaller the range of values is
12.3 cmhas 3 sig. figs. and its range is12.2 to 12.4 cm
12.30 cmhas 4 sig. figs. and its range is
12.29 to 12.31 cm
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Counting Significant Figures
• All non-zero digits are significant1.5 has 2 sig. figs.
• Interior zeros are significant1.05 has 3 sig. figs.
• Trailing zeros after a decimal point are significant1.050 has 4 sig. figs.
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Counting Significant Figures4. Leading zeros are NOT significant
0.001050 has 4 sig. figs.• 1.050 x 10-3
5. Zeros at the end of a number without a written decimal point are ambiguous and should be avoided by using scientific notation
if 150 has 2 sig. figs. then 1.5 x 102
but if 150 has 3 sig. figs. then 1.50 x 102
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Significant Figures and Exact Numbers• Exact Numbers have an unlimited number of
significant figures • A number whose value is known with
complete certainty is exactfrom counting individual objectsfrom definitions
• 1 cm is exactly equal to 0.01 mfrom integer values in equations
• in the equation for the radius of a circle, the 2 is exact
radius of a circle = diameter of a circle2
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Example 2.4 – Determining the Number of Significant Figures in a Number
• How many significant figures are in each of the following numbers?
0.00351.08023712.97 × 105
1 dozen = 12100,000
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Example 2.4 – Determining the Number of Significant Figures in a Number
• How many significant figures are in each of the following numbers?
0.0035 2 sig. figs. – leading zeros not sig.
1.080 4 sig. figs. – trailing & interior zeros sig.
2371 4 sig. figs. – all digits sig.
2.97 × 105 3 sig. figs. – only decimal parts count sig.
1 dozen = 12 unlimited sig. figs. – definition
100,000 ambiguous
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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 figures5.02 × 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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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 sig. fig. and leave the last sig.
fig. alone add insignificant zeros to keep the value if necessary
2. 5 to 9, round up drop all digits after the last sig. fig. and increase the last
sig. fig. by one add insignificant zeros to keep the value if necessary
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Rounding• rounding to 2 significant figures• 2.34 rounds to 2.3
because the 3 is where the last sig. fig. will be and the number after it is 4 or less
• 2.37 rounds to 2.4because the 3 is where the last sig. fig. will be
and the number after it is 5 or greater• 2.349865 rounds to 2.3
because the 3 is where the last sig. fig. will be and the number after it is 4 or less
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Rounding• rounding to 2 significant figures• 0.0234 rounds to 0.023 or 2.3 × 10-2
because the 3 is where the last sig. fig. will be and the number after it is 4 or less
• 0.0237 rounds to 0.024 or 2.4 × 10-2
because the 3 is where the last sig. fig. will be and the number after it is 5 or greater
• 0.02349865 rounds to 0.023 or 2.3 × 10-2
because the 3 is where the last sig. fig. will be and the number after it is 4 or less
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Rounding• rounding to 2 significant figures• 234 rounds to 230 or 2.3 × 102
because the 3 is where the last sig. fig. will be and the number after it is 4 or less
• 237 rounds to 240 or 2.4 × 102
because the 3 is where the last sig. fig. will be and the number after it is 5 or greater
• 234.9865 rounds to 230 or 2.3 × 102
because the 3 is where the last sig. fig. will be and the number after it is 4 or less
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Determine the Correct Number of Significant Figures for each Calculation and
Round and Report the Result1. 1.01 × 0.12 × 53.51 ÷ 96 = 0.067556
2. 56.55 × 0.920 ÷ 34.2585 = 1.51863
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Determine the Correct Number of Significant Figures for each Calculation and
Round and Report the Result1. 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 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 Result1. 0.987 + 125.1 – 1.22 = 124.867
2. 0.764 – 3.449 – 5.98 = -8.664
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Determine the Correct Number of Significant Figures for each Calculation and
Round and Report the Result1. 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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Both Multiplication/Division and Addition/Subtraction with
Significant Figures• when doing different kinds of operations with
measurements with significant figures, do whatever is in parentheses first, find the number of significant figures in the intermediate answer, then do the remaining steps
3.489 × (5.67 – 2.3) = 2 dp 1 dp
3.489 × 3.37 = 12 4 sf 1 dp & 2 sf 2 sf
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Basic Units of Measure
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The Standard Units• Scientists have agreed on a set of
international standard units for comparing all our measurements called the SI unitsSystème International = International System
Quantity Unit Symbollength meter mmass kilogram kgtime second stemperature kelvin K
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Some Standard Units in the Metric System
Quantity Measured
Name of Unit Abbreviation
Mass gram gLength meter mVolume liter LTime seconds sTemperature Kelvin K
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Length• Measure of the two-dimensional distance an object covers• SI unit = meter
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 cm1 cm = 0.01 m = 10 mm1 inch = 2.54 cm (exactly)
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Mass• Measure of the amount of matter present
in an object• SI unit = kilogram (kg)
about 2 lbs. 3 oz.• Commonly measure mass in grams (g) or
milligrams (mg)1 kg = 2.2046 pounds, 1 lbs. = 453.59 g1 kg = 1000 g = 103 g, 1 g = 1000 mg = 103 mg1 g = 0.001 kg = 10-3 kg, 1 mg = 0.001 g = 10-3 g
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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• The prefixes are always the same,
regardless of the standard unit
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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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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 unit
1 m = 10-6 m; 106 m = 1 m• nano = 10-9 times the base unit
1 nL = 10-9L; 109 nL = 1 L
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Volume• Measure of the amount of three-dimensional space occupied• SI unit = cubic meter (m3)
a Derived Unit
• 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 quart1 L = 1 dL3 = 1000 mL = 103 mL 1 mL = 0.001 L = 10-3 L1 mL = 1 cm3
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Common Units and Their Equivalents
Length1 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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Common Units and Their Equivalents
Volume1 liter (L) = 1000 milliliters (mL)1 liter (L) = 1000 cubic centimeters (cm3)1 liter (L) = 1.057 quarts (qt)
1 U.S. gallon (gal) = 3.785 liters (L)
Mass1 kilogram (km) = 2.205 pounds (lb)
1 pound (lb) = 453.59 grams (g)1 ounce (oz) = 28.35 (g)
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Which is Larger?• 1 yard or 1 meter?• 1 mile or 1 km?• 1 cm or 1 inch?• 1 kg or 1 lb?• 1 mg or 1 g?• 1 qt or 1 L?• 1 L or 1 gal?• 1 gal or 1000 cm3?
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Which is Larger?• 1 yard or 1 meter?• 1 mile of 1 km?• 1 cm or 1 inch?• 1 kg or 1 lb?• 1 mg or 1 g?• 1 qt or 1 L?• 1 L or 1 gal?• 1 gal or 1000 cm3?
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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 = cm• cm ÷ cm = 1
using units as a guide to problem solving is called dimensional analysis
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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 unitsMay be exact or measuredBoth parts of the conversion factor have the same number of
significant figures• Conversion factors generated from equivalence
statementse.g. 1 inch = 2.54 cm can give or
in1cm54.2
cm54.2in1
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Problem Solving and Dimensional Analysis
• Arrange conversion factors so starting unit cancelsArrange conversion factor so 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 beginning and ending units are related to
unit 1 unit 2unit 1 unit 2x =
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Solution Maps
• a solution map is a visual outline that shows the strategic route required to solve a problem
• for unit conversion, the solution map focuses on units and how to convert one to another
• for problems that require equations, the solution map focuses on solving the equation to find an unknown value
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Systematic Approach
1) Write down Given Amount and Unit2) Write down what you want to Find and Unit3) Write down needed Conversion Factors or
Equationsa) Write down equivalence statements for each
relationshipb) Change equivalence statements to Conversion Factors
with starting unit on the bottom
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Systematic Approach
4) Design a Solution Map for the Problem order conversions to cancel previous units or arrange Equation so 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 term6) Check the Answer to see if its Reasonable
correct size and unit
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Solution Maps and Conversion Factors
• Convert Inches into Centimeters1) Find Relationship Equivalence: 1 in = 2.54 cm2) Write Solution Map
in cm
3) Change Equivalence into Conversion Factors with Starting Units on the Bottom
in 1cm 2.54
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Convert 7.8 km to miles1. Write down the Given
quantity and its unitGiven: 7.8 km
2. Write down the quantity you want to Find and unit
Find: ? miles
3. Write down the appropriate Conversion Factors
Conversion Factors:
1 km = 0.6214 mi
4. Write a Solution Map Solution Map:
5. Follow the Solution Map to Solve the problem
Solution:
6. Sig. Figs. and Round Round: 4.84692 mi = 4.8 mi
7. Check Check: Units & Magnitude are correct
km mi
km 1mi 62140.
mi 4.84692km 1
mi 62140km 7.8 .
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Example 2.8:Unit Conversion
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Example:• 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
InformationGiven: 7.8 km
Example:Convert 7.8 km to miles
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• Collect Needed Conversion Factors:
1 mi = 0.6214 km
InformationGiven: 7.8 kmFind: ? mi
Example:Convert 7.8 km to miles
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• Write a Solution Map for converting the units :
InformationGiven: 7.8 kmFind: ? miConv. Fact. 1 mi = 0.6214
km
Example:Convert 7.8 km to miles
km mi
km 1mi .62140
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• Apply the Solution Map:
InformationGiven: 7.8 kmFind: ? miConv. Fact. 1 mi = 0.6214 kmSoln. Map: km mi
Example:Convert 7.8 km to miles
km 1mi .62140
mi km 1
mi 0.6214km 7.8
= 4.84692 mi
= 4.8 mi • Sig. Figs. & Round:
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• Check the Solution:
InformationGiven: 7.8 kmFind: ? miConv. Fact. 1 mi = 0.6214 kmSoln. Map: km mi
Example:Convert 7.8 km to miles
km 1mi .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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Solution Maps and Conversion Factors
• Convert Cups into Liters1) Find Relationship Equivalence: 1 L = 1.057 qt, 1 qt = 4 c2) Write Solution Map
Lqt
3) Change Equivalence into Conversion Factors with Starting Units on the Bottom
qt 1.0571L
c
c 4qt 1
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How many cups of cream is 0.75 L?1. Write down the Given
quantity and its unitGiven: 0.75 L
2. Write down the quantity you want to Find and unit
Find: ? cu
3. Write down the appropriate Conversion Factors
Conversion Factors:
1 L = 1.057 qt1 qt = 4 cu
4. Write a Solution Map Solution Map:
5. Follow the Solution Map to Solve the problem
Solution:
6. Sig. Figs. and Round Round: 3.171 cu = 3.2 cu
7. Check Check: Units & Magnitude are correct
L
L 1qt .0571
cu 1713qt 1cu 4
L 1qt .0571L 750 ..
qt cu
qt 1cu 4
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Example 2.10:Solving Multistep Unit Conversion Problems
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Example:• 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
InformationGiven: 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 qt1.057 qt = 1 L
InformationGiven: 0.75 LFind: ? 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: ? cuConv. Fact. 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 1qt .0571
qt 1cu 4
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qt 1cu 4
L 1qt .0571L 750 .
• Apply the Solution Map:
= 3.171 cu
= 3.2 cu
• Sig. Figs. & Round:
InformationGiven: 0.75 LFind: ? cuConv. Fact. 4 cu = 1 qt;
1.057 qt = 1 LSol’n 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 1qt .0571
qt 1cu 4
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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: ? cuConv. Fact. 4 cu = 1 qt;
1.057 qt = 1 LSol’n 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 1qt .0571
qt 1cu 4
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Solution Maps and Conversion Factors
• Convert Cubic Inches into Cubic Centimeters1) Find Relationship Equivalence: 1 in = 2.54 cm2) Write Solution Map
in3 cm3
3) Change Equivalence into Conversion Factors with Starting Units on the Bottom
3
3
33
333
in 1cm 16.4
in 1cm 2.54
in 1cm 2.54
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Convert 2,659 cm2 into square meters1. Write down the Given
quantity and its unitGiven: 2,659 cm2
2. Write down the quantity you want to Find and unit
Find: ? m2
3. Write down the appropriate Conversion Factors
Conversion Factors:
1 cm = 0.01 m
4. Write a Solution Map Solution Map:
5. Follow the Solution Map to Solve the problem
Solution:
6. Sig. Figs. and Round Round: 0.2659 m2
7. Check Check: Units & Magnitude are correct
cm2 m22
cm 1m .010
22
242 m 26590
cm 1m 101cm 2,659 .
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Example 2.12:Converting Quantities Involving Units Raised
to a Power
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Example:• 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
InformationGiven: 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
InformationGiven: 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 1m .010
InformationGiven: 2,659 cm2
Find: ? m2
Conv. Fact.: 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 1m 101cm 2,659
= 0.265900 m2
= 0.2659 m2 • Sig. Figs. & Round:
InformationGiven: 2,659 cm2
Find: ? m2
Conv. Fact. 1 cm = 0.01 mSol’n Map: cm2 m2
Example:A circle has an area of 2,659 cm2. What is the area in square meters? 2
cm 1m .010
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• Check the Solution:
2,659 cm2 = 0.2659 m2
The units of the answer, m2, are correct.The magnitude of the answer makes sense
since square centimeters are smaller than square meters.
InformationGiven: 2,659 cm2
Find: ? m2
Conv. Fact. 1 cm = 0.01 mSol’n Map: cm2 m2
Example:A circle has an area of 2,659 cm2. What is the area in square meters? 2
cm 1m .010
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Density
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Mass & Volume• two main characteristics of matter• cannot be used to identify what type of
matter something isif you are given a large glass containing 100 g
of a clear, colorless liquid and a small glass containing 25 g of a clear, colorless liquid - are both liquids the same stuff?
• even though mass and volume are individual properties - for a given type of matter they are related to each other!
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Mass vs Volume of BrassMassgrams
Volumecm3
20 2.4
32 3.8
40 4.8
50 6.0
100 11.9
150 17.9
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Volume vs Mass of Brass y = 8.38x
0
20
40
60
80
100
120
140
160
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0
Volume, cm3
Mas
s, g
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Density• Ratio of mass:volume• 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!
VolumeMassDensity
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Density• For equal volumes, denser object has larger mass• For equal masses, denser object has smaller
volume• Heating objects causes objects to expand
does not effect their mass!!How would heating an object effect its density?
• In a heterogeneous mixture, the denser object sinksWhy do hot air balloons rise?
VolumeMassDensity
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Using Density in Calculations
VolumeMassDensity
DensityMass Volume
Volume Density Mass
Solution Maps:
m, V D
m, D V
V, D m
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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
DVm
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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 0.556g 5.84
DVm
Since 10.5 g/cm3 21.4 g/cm3 the ring cannot be platinum
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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 H2Odensity 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 SolvingDensity 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 L• Conversion Factors:
0.752 grams/cm3
1000 grams = 1 kg
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Measurement and Problem SolvingDensity as a Conversion Factor
• Solution Map: kg g cm3
343
cm107.98 g 0.752
cm 1kg 1
g 1000kg 60.0
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Example 2.16: Density as a Conversion Factor
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Example:• 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?
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Example: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?• Write down the given quantity and its units.
Given: m = 55.9 kg V = 57.2 L
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Example: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?• Write down the quantity to find and/or its units.
Find: density, g/cm3
InformationGiven: m = 55.9 kg
V = 57.2 L
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Example: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?• Design a Solution Map:
Information:Given: m = 55.9 kg
V = 57.2 LFind: density, g/cm3
m, V D
Vm D
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Example: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?• Collect Needed Conversion Factors:
Mass: 1 kg = 1000 gVolume: 1 mL = 0.001 L; 1 mL = 1 cm3
Information:Given: m = 55.9 kg
V = 57.2 LFind: density, g/cm3
Equation:Vm D
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108
Example: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?
• Write a Solution Map for converting the Mass units
• Write a Solution Map for converting the Volume units
kg g
Information:Given: m = 55.9 kg
V = 57.2 L Find: density, g/cm3
Solution Map: m,VDEquation:Conversion Factors: 1 kg = 1000 g
1 mL = 0.001 L1 mL = 1 cm3
Vm D
kg 1g 1000
L mL
L 0.001mL 1
cm3
mL 1cm 1 3
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g kg 1
g 1000kg 55.9
Example: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?
• Apply the Solution Maps
= 5.59 x 104 g
Information:Given: m = 55.9 kg
V = 57.2 L Find: density, g/cm3
Solution Map: m,VDEquation:
Vm D
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33
cm mL 1cm 1
L 0.001mL 1L 57.2
Example: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?
• Apply the Solution Maps
= 5.72 x 104 cm3
Information:Given: m = 5.59 x 104 g
V = 57.2 L Find: density, g/cm3
Solution Map: m,VDEquation:
Vm D
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111
Example: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?
• Apply the Solution Maps - Equation
Information:Given: m = 5.59 x 104 g
V = 5.72 x 104 cm3 Find: density, g/cm3
Solution Map: m,VDEquation:
Vm D
34
4
cm 10 x 72.5g 10 x 59.5
Vm D
= 0.9772727 g/cm3
= 0.977 g/cm3
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112
Example: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?
• Check the Solution
Information:Given: m = 5.59 x 104 g
V = 5.72 x 104 cm3 Find: density, g/cm3
Solution Map: m,VDEquation:
Vm D
The units of the answer, g/cm3, are correct.The magnitude of the answer makes sense.Since the mass in kg and volume in L are
very close in magnitude, the answer’s magnitude should be close to 1.
D = 0.977 g/cm3