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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

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**Introductory Chemistry, 2nd EditionNivaldo 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 • 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**• 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**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**• 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**• 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 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**• 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**km**mi Convert 7.8 km to miles**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**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**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**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**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**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**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**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**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**Example 2.12: Converting Quantities Involving Units Raised**to a Power**Example:**A circle has an area of 2,659 cm2. What is the area in square meters? Tro's Introductory Chemistry, Chapter 2**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**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**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**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**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**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**• 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**Tro's Introductory Chemistry, Chapter 2**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**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**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**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**• 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**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**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**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 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 SolvingDensity as a Conversion**Factor • Solution Map: kg g cm3 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**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**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**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**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**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,VD 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**Apply the Solution Maps**Information: Given: m = 55.9 kg V = 57.2 L Find: density, g/cm3 Solution Map: m,VD 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

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