1 introductory chemistry, 2 nd edition nivaldo tro chapter 2 part 2: problem solving &...

Tro's Introductory Chemistry, Chapter 2Tro's Introductory Chemistry, Chapter 2 11

Introductory ChemistryIntroductory Chemistry, 2, 2ndnd Edition EditionNivaldo TroNivaldo Tro

Chapter 2

Part 2: Problem Solving &

Dimensional Analysis


UnitsUnits

AlwaysAlways write every write every numbernumber with its with its associated associated unitunit

Always include unitsAlways include units in your calculations in your calculations– you can do the same kind of operations on you can do the same kind of operations on

units as you can with numbersunits as you can with numberscm cm × cm = cm× cm = cm22

cm + cm = cmcm + cm = cm

cm ÷ cm = 1cm ÷ cm = 1

– using units as a guide to problem solving is using units as a guide to problem solving is called called dimensional analysisdimensional analysis


Problem Solving and Dimensional Problem Solving and Dimensional AnalysisAnalysis

Many problems in Chemistry use Many problems in Chemistry use relationships to convert one unit to relationships to convert one unit to anotheranother

Conversion FactorsConversion Factors are relationships are relationships between two units whichbetween two units which

Conversion factors are generated from Conversion factors are generated from equivalence statements:equivalence statements:– e.g. 1 inch = 2.54 cm can givee.g. 1 inch = 2.54 cm can give or or

in1

cm54.2cm54.2

in1


Problem Solving and Problem Solving and Dimensional AnalysisDimensional Analysis

Arrange conversion factors so starting unit Arrange conversion factors so starting unit cancelscancels– Arrange conversion factor so starting unit is on Arrange conversion factor so starting unit is on

the bottom of the conversion factorthe bottom of the conversion factor

May string conversion factorsMay string conversion factors– So we do not need to know every relationship, So we do not need to know every relationship,

as long as we can find something else the as long as we can find something else the beginning and ending units are related to beginning and ending units are related to

unit 1unit 2unit 1

unit 2x =


Solution MapsSolution Maps

Solution map = a visual outline Solution map = a visual outline showing strategic route required to showing strategic route required to solve a problemsolve a problem

For unit conversion, the solution map For unit conversion, the solution map focuses on units and how to convert focuses on units and how to convert one to anotherone to another

For problems that require equations, For problems that require equations, the solution map focuses on solving the solution map focuses on solving the equation to find an unknown valuethe equation to find an unknown value


Systematic ApproachSystematic Approach

1)1) Write down given amount and unitWrite down given amount and unit

2)2) Write down what you want to find and unitWrite down what you want to find and unit

3)3) Write down needed conversion factors or Write down needed conversion factors or equationsequations

a)a) Write down equivalence statements for each Write down equivalence statements for each relationshiprelationship

b)b) Change equivalence statements to Change equivalence statements to conversion factors with starting unit on the conversion factors with starting unit on the bottom bottom


Systematic ApproachSystematic Approach

4)4) Design a solution map for the problemDesign a solution map for the problem– order conversions to cancel previous units order conversions to cancel previous units

oror– arrange Equation so Find amount is isolatedarrange Equation so Find amount is isolated

5)5) Apply the steps in the solution mapApply the steps in the solution map – check that units cancel properlycheck that units cancel properly– multiply terms across the top and divide by multiply terms across the top and divide by

each bottom termeach bottom term

6)6) Check the answer to see if its reasonableCheck the answer to see if its reasonable– correct size and unitcorrect size and unit


Solution Maps and Conversion Solution Maps and Conversion FactorsFactors

Convert Inches into CentimetersConvert Inches into Centimeters

1)1) Find Relationship Equivalence: Find Relationship Equivalence: 1 in = 2.541 in = 2.54

cmcm

2)2) Write Solution MapWrite Solution Mapinininin cmcmcmcm

3)3) Change Equivalence into Conversion Change Equivalence into Conversion Factors with Starting Units on the BottomFactors with Starting Units on the Bottom

2.54

1

cm

in

99

Convert 7.8 km to milesConvert 7.8 km to miles1.1. Write down the Write down the GivenGiven

quantity and its unitquantity and its unitGiven:Given: 7.8 km7.8 km

2.2. Write down the quantity Write down the quantity you want to you want to FindFind and unit and unit

Find:Find: ? miles? miles

3.3. Write down the appropriate Write down the appropriate Conversion FactorsConversion Factors

Conversion Conversion Factors:Factors:

1 km = 0.6214 mi1 km = 0.6214 mi

4.4. Write a Write a Solution MapSolution Map Solution Solution Map:Map:

5.5. Follow the Solution Map to Follow the Solution Map to SolveSolve the problem the problem

Solution:Solution:

6.6. Sig. Figs. and RoundSig. Figs. and Round Round:Round: 4.4.884692 mi = 4.8 mi4692 mi = 4.8 mi

7.7. CheckCheck Check:Check: Units & Magnitude are Units & Magnitude are correctcorrect

km mi

km 1

mi 62140.

mi 4.84692km 1

mi 62140km 7.8

.


Solution Maps and Solution Maps and Conversion FactorsConversion Factors

Convert Cups into LitersConvert Cups into Liters1)1) Find Relationship Equivalence: Find Relationship Equivalence: 1 L = 1.057 qt, 1 L = 1.057 qt,

1 qt = 4 c1 qt = 4 c

2) Write Solution Map2) Write Solution Map

3)3) Change Equivalence into Conversion Factors Change Equivalence into Conversion Factors with Starting Units on the Bottomwith Starting Units on the Bottom

qt 1.057

1L

LLLLqtqtqtqtcccc

c 4

qt 1

How many cups of cream is 0.75 L?How many cups of cream is 0.75 L?1.1. Write down the Write down the GivenGiven

quantity and its unitquantity and its unitGiven:Given: 0.75 L0.75 L


Find:Find: ? cu? cu

3.3. Write down the appropriate Write down the appropriate Conversion FactorsConversion Factors


1 L = 1.057 qt1 L = 1.057 qt1 qt = 4 cu1 qt = 4 cu


5.5. Follow the Solution Map to Follow the Solution Map to SolveSolve the problem the problem

Solution:Solution:

6.6. Sig. Figs. and RoundSig. Figs. and Round Round:Round: 3.3.1171 cu = 3.2 cu71 cu = 3.2 cu

7.7. CheckCheck Check:Check: Units & Magnitude are Units & Magnitude are correctcorrect

L

L 1

qt .0571

cu 1713qt 1

cu 4

L 1

qt .0571L 750 ..

qt cu

qt 1

cu 4

Solving Multistep Unit Solving Multistep Unit Conversion ProblemsConversion Problems


Example:Example:

An Italian recipe for making creamy pasta sauce An Italian recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups should measures only in cups. How many cups should you use?you use?


An Italian recipe for making An Italian recipe for making creamy pasta sauce calls for creamy pasta sauce calls for 0.75 L of cream. Your 0.75 L of cream. Your measuring cup measures only measuring cup measures only in cups. How many cups in cups. How many cups should you use?should you use?

Write down the given quantity and its units. Write down the given quantity and its units.

Given:Given: 0.75 L 0.75 L


Write down the quantity to find and/or its units.Write down the quantity to find and/or its units.

Find: Find: ? cups? cups

InformationInformation

Given:Given: 0.75 L0.75 L



Collect Needed Conversion Factors:Collect Needed Conversion Factors:

4 cu = 1 qt4 cu = 1 qt

1.057 qt = 1 L1.057 qt = 1 L


Given:Given: 0.75 L0.75 L

Find:Find: ? cu? cu



Write a Solution Map for converting the units :Write a Solution Map for converting the units :

InformationInformationGiven:Given: 0.75 L0.75 LFind:Find: ? cu? cuConv. Fact.Conv. Fact. 4 cu = 1 qt; 4 cu = 1 qt;

1.057 qt = 1 L1.057 qt = 1 L

LL qtqt


cucu

L 1

qt .0571qt 1

cu 4


qt 1

cu 4

L 1

qt .0571L 750 .

Apply the Solution Map:Apply the Solution Map:

= 3.171 cu

= 3.2 cu

• Sig. Figs. & Round:


1.057 qt = 1 L1.057 qt = 1 LSol’n Map:Sol’n Map: L L qt qt cu cu

An Italian recipe for An Italian recipe for making creamy pasta making creamy pasta sauce calls for 0.75 L of sauce calls for 0.75 L of cream. Your measuring cream. Your measuring cup measures only in cup measures only in cups. How many cups cups. How many cups should you use?should you use? L 1

qt .0571

qt 1

cu 4


Check the Solution: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.


1.057 qt = 1 L1.057 qt = 1 LSol’n Map:Sol’n Map: L L qt qt cu cu

An Italian recipe for An Italian recipe for making creamy pasta making creamy pasta sauce calls for 0.75 L of sauce calls for 0.75 L of cream. Your measuring cream. Your measuring cup measures only in cup measures only in cups. How many cups cups. How many cups should you use?should you use? L 1

qt .0571

qt 1

cu 4


Solution Maps and Conversion Solution Maps and Conversion FactorsFactors

Convert Cubic Inches into Cubic CentimetersConvert Cubic Inches into Cubic Centimeters

1)1) Find Relationship Equivalence: Find Relationship Equivalence: 1 in = 2.54 cm1 in = 2.54 cm

2)2) Write Solution MapWrite Solution Map

inin33inin33cmcm33cmcm33

3)3) Change Equivalence into Conversion Factors Change Equivalence into Conversion Factors with Starting Units on the Bottomwith Starting Units on the Bottom

3

3

33

333

in 1

cm 16.4

in 1

cm 2.54

in 1

cm 2.54

Convert 2,659 cmConvert 2,659 cm22 into square meters into square meters1.1. Write down the Write down the GivenGiven

quantity and its unitquantity and its unitGiven:Given: 2,659 cm2,659 cm22


Find:Find: ? m? m22

3.3. Write down the Write down the appropriate appropriate Conversion Conversion FactorsFactors


1 cm = 0.01 m1 cm = 0.01 m


5.5. Follow the Solution Map Follow the Solution Map to to SolveSolve the problem the problem

Solution:Solution:

6.6. Sig. Figs. and RoundSig. Figs. and Round Round:Round: 0.2659 m0.2659 m22

7.7. CheckCheck Check:Check: Units & Magnitude Units & Magnitude are correctare correct

cm2 m2

2

cm 1

m .010

22

242 m 26590

cm 1

m 101cm 2,659 .

Example 2.12:Example 2.12:Converting Quantities Involving Converting Quantities Involving

Units Raised Units Raised to a Powerto a Power


Example:Example:

A circle has an area of 2,659 cmA circle has an area of 2,659 cm22. What is the . What is the area in square meters?area in square meters?


Example:Example:A circle has an area of A circle has an area of 2,659 cm2,659 cm22. What is the . What is the area in square meters?area in square meters?


Given:Given: 2,659 cm2,659 cm22



Find: Find: ? m? m22






1 cm = 0.01m1 cm = 0.01m



Find:Find: ? m? m22



Write a Solution Map for converting the units :Write a Solution Map for converting the units :

cmcm22 mm22

2

cm 1

m .010



Find:Find: ? m? m22

Conv. Fact.:Conv. Fact.: 1 cm = 0.01 m1 cm = 0.01 m



Apply the Solution Map:Apply the Solution Map:

22

2-42 m

cm 1

m 101cm 2,659

= 0.265900 m2

= 0.2659 m2

• Sig. Figs. & Round:

InformationInformationGiven:Given: 2,659 cm2,659 cm22

Find:Find: ? m? m22

Conv. Fact.Conv. Fact. 1 cm = 0.01 m1 cm = 0.01 mSol’n Map:Sol’n Map: cmcm22 m m22


2

cm 1

m .010


Check the Solution: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.

InformationInformationGiven:Given: 2,659 cm2,659 cm22

Find:Find: ? m? m22

Conv. Fact.Conv. Fact. 1 cm = 0.01 m1 cm = 0.01 mSol’n Map:Sol’n Map: cmcm22 m m22


2

cm 1

m .010

DensityDensity


Mass & VolumeMass & Volume

Two main characteristics of matterTwo main characteristics of matter

Cannot be used to identify what Cannot be used to identify what typetype of of matter something ismatter something is– if you are given a large glass containing 100 g if you are given a large glass containing 100 g

of a clear, colorless liquid and a small glass of a clear, colorless liquid and a small glass containing 25 g of a clear, colorless liquid - are containing 25 g of a clear, colorless liquid - are both liquids the same stuff?both liquids the same stuff?

even though mass and volume are even though mass and volume are individual properties - for a given type of individual properties - for a given type of matter they are related to each other!matter they are related to each other!


Mass vs Volume of Mass vs Volume of BrassBrass

Massgrams

Volumecm3

20 2.4

32 3.8

40 4.8

50 6.0

100 11.9

150 17.9


Volume vs Mass of Brassy = 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


DensityDensity

Ratio of mass:volumeRatio of mass:volume

Solids = g/cmSolids = g/cm33

– 1 cm1 cm33 = 1 mL = 1 mL

Liquids = g/mLLiquids = g/mL

Gases = g/LGases = g/L

Volume of a solid can be determined by Volume of a solid can be determined by water displacement – Archimedes Principlewater displacement – Archimedes Principle

Density : solids > liquids >>> gasesDensity : solids > liquids >>> gases– except ice is less dense than liquid water!except ice is less dense than liquid water!

Volume

MassDensity


DensityDensity

For equal volumes, denser object has larger For equal volumes, denser object has larger massmass

For equal masses, denser object has smaller For equal masses, denser object has smaller volumevolume

Heating objects causes objects to expandHeating objects causes objects to expand– does not affect their mass!!does not affect their mass!!– How would heating an object affect its How would heating an object affect its

density? density?

In a heterogeneous mixture, the denser object In a heterogeneous mixture, the denser object sinkssinks– Why do hot air balloons rise?Why do hot air balloons rise?

Volume

MassDensity


Using Density in CalculationsUsing Density in Calculations

VolumeMass

Density

DensityMass

Volume

Volume Density Mass

Solution Maps:

m, Vm, V DD

m, Dm, D VV

V, DV, D mm


Applying Density in Problem SolvingApplying Density in Problem Solving

Platinum has become a popular metal for fine Platinum has become a popular metal for fine jewelry. A man gives a woman an engagement jewelry. A man gives a woman an engagement ring and tells her that it is made of platinum. ring and tells her that it is made of platinum. Noting that the ring felt a little light, the woman Noting that the ring felt a little light, the woman decides to perform a test to determine the ring’s decides to perform a test to determine the ring’s density before giving him an answer about density before giving him an answer about marriage. She places the ring on a balance and marriage. She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds finds it has a mass of 5.84 grams. She then finds that the ring displaces 0.556 cmthat the ring displaces 0.556 cm33 of water. Is the of water. Is the ring made of platinum? (Density Pt = 21.4 g/cmring made of platinum? (Density Pt = 21.4 g/cm33))


She places the ring on a balance and finds it has a She places the ring on a balance and finds it has a mass of 5.84 gramsmass of 5.84 grams. She then finds that the ring . She then finds that the ring displaces 0.556 cmdisplaces 0.556 cm33 of water of water. Is the ring made of . Is the ring made of platinum? (Density Pt = 21.4 g/cmplatinum? (Density Pt = 21.4 g/cm33))

Given: Mass = 5.84 grams

Volume = 0.556 cm3

Find: Density in grams/cm3

Equation:

Solution Map:

m and V d

DV

m


She places the ring on a balance and finds it has a She places the ring on a balance and finds it has a mass of mass of 5.84 grams5.84 grams. She then finds that the ring . She then finds that the ring displaces 0.556 cmdisplaces 0.556 cm33 of waterof water. Is the ring made of platinum? (Density Pt = 21.4 . Is the ring made of platinum? (Density Pt = 21.4 g/cmg/cm33))

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 cannotcannot be platinum. Should she marry him? ☺


Density as a Conversion FactorDensity as a Conversion Factor

• Can use density as a conversion factor between Can use density as a conversion factor between

mass and volume!!mass and volume!!

– density of Hdensity of H22O = 1 g/mL O = 1 g/mL 1 g H 1 g H22O = 1 mL HO = 1 mL H22OO

– density of Pb = 11.3 g/cmdensity of Pb = 11.3 g/cm33 11.3 g Pb = 1 cm 11.3 g Pb = 1 cm33 Pb Pb

• How much does 4.0 cmHow much does 4.0 cm33 of Lead weigh? of Lead weigh?

=4.0 cm3 Pb 11.3 g Pb 1 cm3 Pb

45 g Pbx


Measurement and Problem SolvingMeasurement and Problem SolvingDensity as a Conversion FactorDensity as a Conversion Factor

The gasoline in an automobile gas tank has a mass The gasoline in an automobile gas tank has a mass of 60.0 kg and a density of 0.752 g/cmof 60.0 kg and a density of 0.752 g/cm33. What is the . What is the volume?volume?

Given: 60.0 kgGiven: 60.0 kg

Find: Volume in LFind: Volume in L

Conversion Factors: Conversion Factors: – 0.752 grams/cm0.752 grams/cm33

– 1000 grams = 1 kg1000 grams = 1 kg


Measurement and Problem SolvingMeasurement and Problem SolvingDensity as a Conversion FactorDensity as a Conversion Factor

Solution Map: Solution Map: kg kg g g cm cm33

343

cm107.98 g 0.752

cm 1

kg 1

g 1000kg 60.0

Density as a Conversion Density as a Conversion Factor: Another ExampleFactor: Another Example


Example:Example:

A 55.9 kg person displaces 57.2 L of water when A 55.9 kg person displaces 57.2 L of water when submerged in a water tank. What is the density submerged in a water tank. What is the density of the person in g/cmof the person in g/cm33??


Example:Example:A 55.9 kg person displaces A 55.9 kg person displaces 57.2 L of water when 57.2 L of water when submerged in a water tank. submerged in a water tank. What is the density of the What is the density of the person in g/cmperson in g/cm33??


Given:Given: m = 55.9 kg m = 55.9 kg

V = 57.2 LV = 57.2 L




Find: density, g/cmFind: density, g/cm33


Given: m = 55.9 kgGiven: m = 55.9 kg

V = 57.2 LV = 57.2 L



Design a Solution Map:Design a Solution Map:

Information:Information:


V = 57.2 LV = 57.2 L

Find: Find: density, g/cmdensity, g/cm33

m, V D

V

m D




Mass:Mass: 1 kg = 1000 g1 kg = 1000 g Volume:Volume: 1 mL = 0.001 L; 1 mL = 1 cm1 mL = 0.001 L; 1 mL = 1 cm33

Information:Information:Given: m = 55.9 kgGiven: m = 55.9 kg

V = 57.2 LV = 57.2 LFind: Find: density, g/cmdensity, g/cm33

Equation:Equation:V

m D

4949


Write a Solution Map for converting the Write a Solution Map for converting the MassMass units units

Write a Solution Map for converting the Write a Solution Map for converting the VolumeVolume units units

kgkg gg

Information:Information:Given:Given: m = 55.9 kgm = 55.9 kg

V = 57.2 L V = 57.2 L Find: Find: density, g/cmdensity, g/cm33

Solution Map:Solution Map: m,Vm,VDDEquation:Equation:Conversion Factors: Conversion Factors: 1 kg = 1000 g1 kg = 1000 g

1 mL = 0.001 L1 mL = 0.001 L1 mL = 1 cm1 mL = 1 cm33

V

m D

kg 1

g 1000

LL mLmL

L 0.001

mL 1cmcm33

mL 1

cm 1 3


g kg 1

g 1000kg 55.9


Apply the Solution MapsApply the Solution Maps

= 5.59 x 104 g



V = 57.2 L V = 57.2 L


Solution Map:Solution Map: m,Vm,VDD

Equation:Equation:V

m D


33

cm mL 1

cm 1

L 0.001

mL 1L 57.2


Apply the Solution MapsApply the Solution Maps

= 5.72 x 104 cm3


Given:Given: m = 5.59 x 10m = 5.59 x 1044 g g

V = 57.2 L V = 57.2 L



Equation:Equation:V

m D

5252


Apply the Solution Maps - EquationApply the Solution Maps - Equation



V = 5.72 x 10V = 5.72 x 1044 cm cm33



Equation:Equation:V

m D

34

4

cm 10 x 72.5

g 10 x 59.5

V

m D

= 0.9772727 g/cm3

= 0.977 g/cm3

5353


Check the SolutionCheck the Solution



V = 5.72 x 10V = 5.72 x 1044 cm cm33



Equation:Equation:V

m 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

1 introductory chemistry, 2 nd edition nivaldo tro chapter 2 part 2: problem solving &...

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