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

2. Tro's Introductory Chemistry, Chapter 2
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

3. 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

4. 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
unit 1
unit 2 x =
Tro's Introductory Chemistry, Chapter 2

5. Tro's Introductory Chemistry, Chapter 2
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

6. Tro's Introductory Chemistry, Chapter 2
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

7. Tro's Introductory Chemistry, Chapter 2
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

8. Solution Maps and Conversion Factors
Convert Inches into Centimeters
Find Relationship Equivalence: 1 in = cm
Write Solution Map in cm
Change Equivalence into Conversion Factors with Starting Units on the Bottom
Tro's Introductory Chemistry, Chapter 2

9. Units & Magnitude are correct
Convert 7.8 km to miles
Write down the Given quantity and its unit
Given:
7.8 km
Write down the quantity you want to Find and unit
Find:
? miles
Write down the appropriate Conversion Factors
Conversion Factors:
1 km = mi
Write a Solution Map
Solution
Map:
Follow the Solution Map to Solve the problem
Solution:
Sig. Figs. and Round
Round:
mi = 4.8 mi
Check
Check:
Units & Magnitude are correct km mi

10. Solution Maps and Conversion Factors
Convert Cups into Liters
Find Relationship Equivalence: 1 L = qt,
1 qt = 4 c
2) Write Solution Map L qt c
Change Equivalence into Conversion Factors with Starting Units on the Bottom
Tro's Introductory Chemistry, Chapter 2

11. How many cups of cream is 0.75 L?
Write down the Given quantity and its unit
Given:
0.75 L
Write down the quantity you want to Find and unit
Find:
? cu
Write down the appropriate Conversion Factors
Conversion Factors:
1 L = qt
1 qt = 4 cu
Write a Solution Map
Solution
Map:
Follow the Solution Map to Solve the problem
Solution:
Sig. Figs. and Round
Round:
3.171 cu = 3.2 cu
Check
Check:
Units & Magnitude are correct L qt cu

12. Solving Multistep Unit Conversion Problems

13. Tro's Introductory Chemistry, Chapter 2
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

14. Tro's Introductory Chemistry, Chapter 2
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
Tro's Introductory Chemistry, Chapter 2

15. Tro's Introductory Chemistry, Chapter 2
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?
Information
Given: 0.75 L
Write down the quantity to find and/or its units.
Find: ? cups
Tro's Introductory Chemistry, Chapter 2

16. Tro's Introductory Chemistry, Chapter 2
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?
Information
Given: 0.75 L
Find: ? cu
Collect Needed Conversion Factors:
4 cu = 1 qt
1.057 qt = 1 L
Tro's Introductory Chemistry, Chapter 2

17. Tro's Introductory Chemistry, Chapter 2
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?
Information
Given: 0.75 L
Find: ? cu
Conv. Fact. 4 cu = 1 qt;
1.057 qt = 1 L
Write a Solution Map for converting the units : L qt cu
Tro's Introductory Chemistry, Chapter 2

18. Tro's Introductory Chemistry, Chapter 2
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?
Information
Given: 0.75 L
Find: ? cu
Conv. Fact. 4 cu = 1 qt;
1.057 qt = 1 L
Sol’n Map: L  qt  cu
Apply the Solution Map:
= cu
Sig. Figs. & Round:
= 3.2 cu
Tro's Introductory Chemistry, Chapter 2

19. Check the Solution: 0.75 L = 3.2 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?
Information
Given: 0.75 L
Find: ? cu
Conv. Fact. 4 cu = 1 qt;
1.057 qt = 1 L
Sol’n Map: L  qt  cu
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.
Tro's Introductory Chemistry, Chapter 2

20. Solution Maps and Conversion Factors
Convert Cubic Inches into Cubic Centimeters
Find Relationship Equivalence: 1 in = 2.54 cm
Write Solution Map
in3
cm3
Change Equivalence into Conversion Factors with Starting Units on the Bottom
Tro's Introductory Chemistry, Chapter 2

21. Convert 2,659 cm2 into square meters
Write down the Given quantity and its unit
Given:
2,659 cm2
Write down the quantity you want to Find and unit
Find:
? m2
Write down the appropriate Conversion Factors
Conversion Factors:
1 cm = 0.01 m
Write a Solution Map
Solution
Map:
Follow the Solution Map to Solve the problem
Solution:
Sig. Figs. and Round
Round: m2
Check
Check:
Units & Magnitude are correct
cm2 m2

22. Example 2.12: Converting Quantities Involving Units Raised to a Power

23. Tro's Introductory Chemistry, Chapter 2
Example:
A circle has an area of 2,659 cm2. What is the area in square meters?
Tro's Introductory Chemistry, Chapter 2

24. Tro's Introductory Chemistry, Chapter 2
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
Tro's Introductory Chemistry, Chapter 2

25. Tro's Introductory Chemistry, Chapter 2
Example: A circle has an area of 2,659 cm2. What is the area in square meters?
Information
Given: 2,659 cm2
Write down the quantity to find and/or its units.
Find: ? m2
Tro's Introductory Chemistry, Chapter 2

26. Tro's Introductory Chemistry, Chapter 2
Example: A circle has an area of 2,659 cm2. What is the area in square meters?
Information
Given: 2,659 cm2
Find: ? m2
Collect Needed Conversion Factors:
1 cm = 0.01m
Tro's Introductory Chemistry, Chapter 2

27. Tro's Introductory Chemistry, Chapter 2
Example: A circle has an area of 2,659 cm2. What is the area in square meters?
Information
Given: 2,659 cm2
Find: ? m2
Conv. Fact.: 1 cm = 0.01 m
Write a Solution Map for converting the units :
cm2 m2
Tro's Introductory Chemistry, Chapter 2

28. Tro's Introductory Chemistry, Chapter 2
Example: A circle has an area of 2,659 cm2. What is the area in square meters?
Information
Given: 2,659 cm2
Find: ? m2
Conv. Fact. 1 cm = 0.01 m
Sol’n Map: cm2  m2
Apply the Solution Map:
= m2
Sig. Figs. & Round:
= m2
Tro's Introductory Chemistry, Chapter 2

29. Check the Solution: 2,659 cm2 = 0.2659 m2
Example: A circle has an area of 2,659 cm2. What is the area in square meters?
Information
Given: 2,659 cm2
Find: ? m2
Conv. Fact. 1 cm = 0.01 m
Sol’n Map: cm2  m2
Check the Solution:
2,659 cm2 = 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

30. Density

31. Tro's Introductory Chemistry, Chapter 2
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

32. Tro's Introductory Chemistry, Chapter 2
Mass vs Volume of Brass
Tro's Introductory Chemistry, Chapter 2

33. Tro's Introductory Chemistry, Chapter 2
Volume vs Mass of Brass
y = 8.38x 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
Mass, g
Tro's Introductory Chemistry, Chapter 2

34. Tro's Introductory Chemistry, Chapter 2
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

35. Tro's Introductory Chemistry, Chapter 2
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

36. Using Density in Calculations
Solution Maps:
m, V D
m, D V
V, D m
Tro's Introductory Chemistry, Chapter 2

37. 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 cm3 of water. Is the ring made of platinum? (Density Pt = 21.4 g/cm3)
Tro's Introductory Chemistry, Chapter 2

38. Tro's Introductory Chemistry, Chapter 2
She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds that the ring displaces cm3 of water. Is the ring made of platinum? (Density Pt = 21.4 g/cm3)
Given: Mass = 5.84 grams
Volume = cm3
Find: Density in grams/cm3
Equation:
Solution Map:
m and V  d
Tro's Introductory Chemistry, Chapter 2

39. Tro's Introductory Chemistry, Chapter 2
She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds that the ring displaces 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

40. 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 Pb x
Tro's Introductory Chemistry, Chapter 2

41. 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 g/cm3. What is the volume?
Given: kg
Find: Volume in L
Conversion Factors:
0.752 grams/cm3
1000 grams = 1 kg
Tro's Introductory Chemistry, Chapter 2

42. Measurement and Problem Solving Density as a Conversion Factor
Solution Map: kg  g  cm3
Tro's Introductory Chemistry, Chapter 2

43. Density as a Conversion Factor: Another Example

44. Tro's Introductory Chemistry, Chapter 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?
Tro's Introductory Chemistry, Chapter 2

45. Tro's Introductory Chemistry, Chapter 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?
Write down the given quantity and its units.
Given: m = 55.9 kg
V = 57.2 L
Tro's Introductory Chemistry, Chapter 2

46. Tro's Introductory Chemistry, Chapter 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?
Information
Given: m = 55.9 kg
V = 57.2 L
Write down the quantity to find and/or its units.
Find: density, g/cm3
Tro's Introductory Chemistry, Chapter 2

47. Tro's Introductory Chemistry, Chapter 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?
Information:
Given: m = 55.9 kg
V = 57.2 L
Find: density, g/cm3
Design a Solution Map:
m, V D
Tro's Introductory Chemistry, Chapter 2

48. Tro's Introductory Chemistry, Chapter 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?
Information:
Given: m = 55.9 kg
V = 57.2 L
Find: density, g/cm3
Equation:
Collect Needed Conversion Factors:
Mass: 1 kg = 1000 g
Volume: 1 mL = L; 1 mL = 1 cm3
Tro's Introductory Chemistry, Chapter 2

49. Write a Solution Map for converting the Mass units
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?
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 = L
1 mL = 1 cm3
Write a Solution Map for converting the Mass units
Write a Solution Map for converting the Volume units kg g L mL
cm3

50. Tro's Introductory Chemistry, Chapter 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?
Information:
Given: m = 55.9 kg
V = 57.2 L
Find: density, g/cm3
Solution Map: m,VD
Equation:
Apply the Solution Maps
= 5.59 x 104 g
Tro's Introductory Chemistry, Chapter 2

51. Tro's Introductory Chemistry, Chapter 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?
Information:
Given: m = 5.59 x 104 g
V = 57.2 L
Find: density, g/cm3
Solution Map: m,VD
Equation:
Apply the Solution Maps
= 5.72 x 104 cm3
Tro's Introductory Chemistry, Chapter 2

52. Apply the Solution Maps - 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?
Information:
Given: m = 5.59 x 104 g
V = 5.72 x 104 cm3
Find: density, g/cm3
Solution Map: m,VD
Equation:
Apply the Solution Maps - Equation
= g/cm3
= g/cm3

53. The units of the answer, g/cm3, are correct.
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?
Information:
Given: m = 5.59 x 104 g
V = 5.72 x 104 cm3
Find: density, g/cm3
Solution Map: m,VD
Equation:
Check the Solution
D = 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.