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Introductory Chemistry , 2 nd Edition Nivaldo Tro. Chapter 2 Part 2: Problem Solving & Dimensional Analysis. 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

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Introductory chemistry 2 nd edition nivaldo tro l.jpg
Introductory Chemistry, 2nd EditionNivaldo Tro

Chapter 2

Part 2: Problem Solving &

Dimensional Analysis


Units l.jpg
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

Tro's Introductory Chemistry, Chapter 2


Problem solving and dimensional analysis l.jpg
Problem Solving and Dimensional Analysis

  • Many problems in Chemistry use relationships to convert one unit to another

  • Conversion Factors are relationships between two units which

  • Conversion factors are generated from equivalence statements:

    • e.g. 1 inch = 2.54 cm can give or

Tro's Introductory Chemistry, Chapter 2


Problem solving and dimensional analysis4 l.jpg

unit 2

unit 1

x

=

unit 2

unit 1

Problem Solving and Dimensional Analysis

  • Arrange conversion factors so starting unit cancels

    • Arrange conversion factor so starting unit is on the bottom of the conversion factor

  • May string conversion factors

    • 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

Tro's Introductory Chemistry, Chapter 2


Solution maps l.jpg
Solution Maps

  • Solution map = a visual outline showing 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

Tro's Introductory Chemistry, Chapter 2


Systematic approach l.jpg
Systematic Approach

  • Write down given amount and unit

  • Write down what you want to find and unit

  • Write down needed conversion factors or equations

    • Write down equivalence statements for each relationship

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

Tro's Introductory Chemistry, Chapter 2


Systematic approach7 l.jpg
Systematic Approach

  • Design a solution map for the problem

    • order conversions to cancel previous units or

    • arrange Equation so Find amount is isolated

  • Apply the steps in the solution map

    • check that units cancel properly

    • multiply terms across the top and divide by each bottom term

  • Check the answer to see if its reasonable

    • correct size and unit

Tro's Introductory Chemistry, Chapter 2


Solution maps and conversion factors l.jpg
Solution Maps and Conversion Factors

  • Convert Inches into Centimeters

    • Find Relationship Equivalence: 1 in = 2.54cm

    • Write Solution Map

in

cm

  • Change Equivalence into Conversion Factors with Starting Units on the Bottom

Tro's Introductory Chemistry, Chapter 2


Convert 7 8 km to miles l.jpg

km

mi

Convert 7.8 km to miles


Solution maps and conversion factors10 l.jpg

c

qt

L

Solution Maps and Conversion Factors

  • Convert Cups into Liters

    • Find Relationship Equivalence: 1 L = 1.057 qt,

      1 qt = 4 c

      2) Write Solution Map

  • Change Equivalence into Conversion Factors with Starting Units on the Bottom

Tro's Introductory Chemistry, Chapter 2




Slide13 l.jpg

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?

Tro's Introductory Chemistry, Chapter 2


Slide14 l.jpg

Write down the given quantity and its units.

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?

Tro's Introductory Chemistry, Chapter 2


Slide15 l.jpg

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?

Tro's Introductory Chemistry, Chapter 2


Slide16 l.jpg

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?

Tro's Introductory Chemistry, Chapter 2


Slide17 l.jpg

Write a Solution Map for converting the units :

Information

Given: 0.75 L

Find: ? cu

Conv. Fact. 4 cu = 1 qt;

1.057 qt = 1 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?

L

qt

cu

Tro's Introductory Chemistry, Chapter 2


Slide18 l.jpg

Apply the Solution Map:

Information

Given: 0.75 L

Find: ? cu

Conv. Fact. 4 cu = 1 qt;

1.057 qt = 1 L

Sol’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?

= 3.171 cu

  • Sig. Figs. & Round:

= 3.2 cu

Tro's Introductory Chemistry, Chapter 2


Slide19 l.jpg

Check the Solution:

Information

Given: 0.75 L

Find: ? cu

Conv. Fact. 4 cu = 1 qt;

1.057 qt = 1 L

Sol’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?

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.

Tro's Introductory Chemistry, Chapter 2


Solution maps and conversion factors20 l.jpg

in3

cm3

Solution Maps and Conversion Factors

  • Convert Cubic Inches into Cubic Centimeters

    • Find Relationship Equivalence: 1 in = 2.54 cm

    • Write Solution Map

  • Change Equivalence into Conversion Factors with Starting Units on the Bottom

Tro's Introductory Chemistry, Chapter 2


Convert 2 659 cm 2 into square meters l.jpg
Convert 2,659 cm2 into square meters

cm2

m2


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Example 2.12: Converting Quantities Involving Units Raised to a Power


Slide23 l.jpg

Example:

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

Tro's Introductory Chemistry, Chapter 2


Example a circle has an area of 2 659 cm 2 what is the area in square meters l.jpg

Write down the given quantity and its units.

Given: 2,659 cm2

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

Tro's Introductory Chemistry, Chapter 2


Example a circle has an area of 2 659 cm 2 what is the area in square meters25 l.jpg

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?

Tro's Introductory Chemistry, Chapter 2


Example a circle has an area of 2 659 cm 2 what is the area in square meters26 l.jpg

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?

Tro's Introductory Chemistry, Chapter 2


Example a circle has an area of 2 659 cm 2 what is the area in square meters27 l.jpg

Write a Solution Map for converting the units :

Information

Given: 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?

cm2

m2

Tro's Introductory Chemistry, Chapter 2


Example a circle has an area of 2 659 cm 2 what is the area in square meters28 l.jpg

Apply the Solution Map:

Information

Given: 2,659 cm2

Find: ? m2

Conv. Fact. 1 cm = 0.01 m

Sol’n Map: cm2 m2

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

= 0.265900 m2

  • Sig. Figs. & Round:

= 0.2659 m2

Tro's Introductory Chemistry, Chapter 2


Example a circle has an area of 2 659 cm 2 what is the area in square meters29 l.jpg

Check the Solution:

Information

Given: 2,659 cm2

Find: ? m2

Conv. Fact. 1 cm = 0.01 m

Sol’n Map: cm2 m2

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

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.

Tro's Introductory Chemistry, Chapter 2



Mass volume l.jpg
Mass & Volume

  • Two main characteristics of matter

  • Cannot be used to identify what type of matter something is

    • if 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!

Tro's Introductory Chemistry, Chapter 2


Mass vs volume of brass l.jpg
Mass vs Volume of Brass

Tro's Introductory Chemistry, Chapter 2


Slide33 l.jpg

Volume vs Mass of Brass

y = 8.38x

160

140

120

100

Mass, g

80

60

40

20

0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

Volume, cm3

Tro's Introductory Chemistry, Chapter 2


Density34 l.jpg
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!

Tro's Introductory Chemistry, Chapter 2


Density35 l.jpg
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 affect their mass!!

    • How would heating an object affect its density?

  • In a heterogeneous mixture, the denser object sinks

    • Why do hot air balloons rise?

Tro's Introductory Chemistry, Chapter 2


Using density in calculations l.jpg

m, V

D

m, D

V

V, D

m

Using Density in Calculations

Solution Maps:

Tro's Introductory Chemistry, Chapter 2


Applying density in problem solving l.jpg
Applying Density in Problem Solving

  • 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)

Tro's Introductory Chemistry, Chapter 2


Slide38 l.jpg

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

Tro's Introductory Chemistry, Chapter 2


Slide39 l.jpg

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:

Since 10.5 g/cm3 21.4 g/cm3 the ring cannot be platinum. Should she marry him? ☺

Tro's Introductory Chemistry, Chapter 2


Density as a conversion factor l.jpg

11.3 g Pb

1 cm3 Pb

x

=

4.0 cm3 Pb

45 g Pb

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?

Tro's Introductory Chemistry, Chapter 2


Measurement and problem solving density as a conversion factor l.jpg
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

Tro's Introductory Chemistry, Chapter 2


Measurement and problem solving density as a conversion factor42 l.jpg
Measurement and Problem SolvingDensity as a Conversion Factor

  • Solution Map: kg  g  cm3

Tro's Introductory Chemistry, Chapter 2



Slide44 l.jpg

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?

Tro's Introductory Chemistry, Chapter 2


Slide45 l.jpg

Write down the given quantity and its units.

Given: m = 55.9 kg

V = 57.2 L

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?

Tro's Introductory Chemistry, Chapter 2


Slide46 l.jpg

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

Find: density, g/cm3

Information

Given: m = 55.9 kg

V = 57.2 L

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?

Tro's Introductory Chemistry, Chapter 2


Slide47 l.jpg

Design a Solution Map:

Information:

Given: m = 55.9 kg

V = 57.2 L

Find: density, g/cm3

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?

m, V D

Tro's Introductory Chemistry, Chapter 2


Slide48 l.jpg

Collect Needed Conversion Factors:

Mass: 1 kg = 1000 g

Volume: 1 mL = 0.001 L; 1 mL = 1 cm3

Information:

Given: m = 55.9 kg

V = 57.2 L

Find: density, g/cm3

Equation:

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?

Tro's Introductory Chemistry, Chapter 2


Slide49 l.jpg

Write a Solution Map for converting the Mass units

Write a Solution Map for converting the Volume units

Information:

Given: m = 55.9 kg

V = 57.2 L

Find: density, g/cm3

Solution Map: m,VD

Equation:

Conversion Factors: 1 kg = 1000 g

1 mL = 0.001 L

1 mL = 1 cm3

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?

kg

g

L

mL

cm3


Slide50 l.jpg

Apply the Solution Maps

Information:

Given: m = 55.9 kg

V = 57.2 L

Find: density, g/cm3

Solution Map: m,VD

Equation:

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?

= 5.59 x 104 g

Tro's Introductory Chemistry, Chapter 2


Slide51 l.jpg

Apply the Solution Maps

Information:

Given: m = 5.59 x 104 g

V = 57.2 L

Find: density, g/cm3

Solution Map: m,VD

Equation:

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?

= 5.72 x 104 cm3

Tro's Introductory Chemistry, Chapter 2


Slide52 l.jpg

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,VD

Equation:

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?

= 0.9772727 g/cm3

= 0.977 g/cm3


Slide53 l.jpg

Check the Solution

Information:

Given: m = 5.59 x 104 g

V = 5.72 x 104 cm3

Find: density, g/cm3

Solution Map: m,VD

Equation:

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?

D = 0.977 g/cm3

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.


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