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Chapter 1 Chemical Foundations

1.6Dimensional Analysis

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

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Equalities• use two different units to describe the same measured

amount. • are written for relationships between units of the metric

system, U.S. units, or between metric and U.S. units.

For example, 1 m = 1000 mm1 lb = 16 oz2.205 lb = 1 kg

Equalities

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Exact and Measured Numbers in Equalities

Equalities between units of• the same system are definitions and use exact

numbers.• different systems (metric and U.S.) use measured

numbers and count as significant figures.

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Some Common Equalities

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Equalities on Food Labels

The contents of packaged foods • in the U.S. are listed as both metric and

U.S. units.

• indicate the same amount of a substance in two different units.

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

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A conversion factor• is a fraction obtained from an equality.

Equality: 1 in. = 2.54 cm

• is written as a ratio with a numerator and denominator.• can be inverted to give two conversion factors for

every equality.1 in. and 2.54 cm 2.54 cm 1 in.

Conversion Factors

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Write conversion factors for each pair of units.

A. liters and mL

B. hours and minutes

C. meters and kilometers

Learning Check

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Write conversion factors for each pair of units.

A. liters and mL Equality: 1 L = 1000 mL

1 L and 1000 mL1000 mL 1 L

B. hours and minutes Equality: 1 hr = 60 min

1 hr and 60 min60 min 1 hr

C. meters and kilometers Equality: 1 km = 1000 m

1 km and 1000 m 1000 m 1 km

Solution

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A conversion factor• may be obtained from information in a word problem.• is written for that problem only.

Example 1: The price of one pound (1 lb) of red peppers is $2.39.

1 lb red peppers and $2.39$2.39 1 lb red peppers

Example 2:The cost of one gallon (1 gal) of gas is $2.34.

1 gallon of gas and $2.34$2.34 1 gallon of gas

Conversion Factors in a Problem

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A percent factor• gives the ratio of the parts to the whole.

% = Parts x 100Whole

• uses the same unit to express the percent.• uses the value 100 and a unit for the whole.• can be written as two factors.

Example: A food contains 30% (by mass) fat.

30 g fat and 100 g food100 g food 30 g fat

Percent as a Conversion Factor

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Percent Factor in a Problem

The thickness of the skin fold atthe waist indicates 11% bodyfat. What percent factors can bewritten for body fat in kg?

Percent factors using kg:

11 kg fat and 100 kg mass100 kg mass 11 kg fat

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

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Learning Check

Write the equality and conversion factors for each of thefollowing.

A. square meters and square centimeters

B. jewelry that contains 18% gold

C. One gallon of gas is $2.27

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Solution

A. square meters and square centimeters(1m)2 and (100 cm)2

(100 cm)2 (1m)2

B. jewelry that contains 18% gold 18 g gold and 100 g jewelry

100 g jewelry 18 g gold

C. One gallon of gas is $2.271 gal and $2.27 $2.27 1 gal

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To solve a problem• Identify the given unit• Identify the needed unit.

Example: A person has a height of 2.0 meters. What is that height in inches?

The given unit is the initial unit of height. given unit = meters (m)

The needed unit is the unit for the answer.needed unit = inches (in.)

Given and Needed Units

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Learning Check

An injured person loses 0.30 pints of blood. Howmany milliliters of blood would that be?

Identify the given and needed units given in thisproblem.

Given unit = _______

Needed unit = _______

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Solution

An injured person loses 0.30 pints of blood. How many milliliters of blood would that be?

Identify the given and needed units given in thisproblem.

Given unit = pints

Needed unit = milliliters

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• Write the given and needed units.• Write a unit plan to convert the given unit to the needed

unit.• Write equalities and conversion factors that connect the

units.• Use conversion factors to cancel the given unit and

provide the needed unit.

Unit 1 x Unit 2 = Unit 2Unit 1

Given x Conversion = Neededunit factor unit

Problem Setup

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Guide to Problem Solving (GPS)

The steps in the Guide to Problem Solving are useful in setting up a problem with conversion factors.

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How many minutes are 2.5 hours?Given unit = 2.5 hrNeeded unit = minUnit Plan = hr min

Setup problem to cancel hours (hr). Given Conversion Neededunit factor unit2.5 hr x 60 min = 150 min (2 SF)

1 hr

Setting up a Problem

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

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A rattlesnake is 2.44 m long. How many centimeters long is the snake?

1) 2440 cm2) 244 cm3) 24.4 cm

Learning Check

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A rattlesnake is 2.44 m long. How many centimeters long is the snake?

2) 244 cm

Given Conversion Neededunit factor unit2.44 m x 100 cm = 244 cm

1 m

Solution

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• Often, two or more conversion factors are required to obtain the unit needed for the answer.Unit 1 Unit 2 Unit 3

• Additional conversion factors are placed in the setup to cancel each preceding unit

Given unit x factor 1 x factor 2 = needed unitUnit 1 x Unit 2 x Unit 3 = Unit 3

Unit 1 Unit 2

Using Two or More Factors

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How many minutes are in 1.4 days?Given unit: 1.4 days

Factor 1 Factor 2Plan: days hr min

Set up problem:1.4 days x 24 hr x 60 min = 2.0 x 103 min

1 day 1 hr2 SF Exact Exact = 2 SF

Example: Problem Solving

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• Be sure to check your unit cancellation in the setup.• The units in the conversion factors must cancel to

give the correct unit for the answer.

What is wrong with the following setup?

1.4 day x 1 day x 1 hr24 hr 60 min

Units = day2/min is not the unit neededUnits don’t cancel properly.

Check the Unit Cancellation

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Using the GPS

What is 165 lb in kg?STEP 1 Given 165 lb Need kgSTEP 2 PlanSTEP 3 Equalities/Factors

1 kg = 2.20 lb2.20 lb and 1 kg 1 kg 2.20 lb

STEP 4 Set Up Problem165 lb x 1 kg = 75.0 kg

2.20 lb

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A bucket contains 4.65 L of water. How manygallons of water is that?

Unit plan: L qt gallon

Equalities: 1.06 qt = 1 L 1 gal = 4 qt

Set up Problem:

Learning Check

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Given: 4.65 L Needed: gallonsPlan: L qt gallonEqualities: 1.06 qt = 1 L; 1 gal = 4 qt

Set Up Problem: 4.65 L x x 1.06 qt x 1 gal = 1.23 gal

1 L 4 qt 3 SF 3 SF exact 3 SF

Solution

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If a ski pole is 3.0 feet in length, how long is the ski pole in mm?

Learning Check

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3.0 ft x 12 in x 2.54 cm x 10 mm =1 ft 1 in. 1 cm

Calculator answer: 914.4 mm Needed answer: 910 mm (2 SF rounded)

Check factor setup: Units cancel properlyCheck needed unit: mm

Solution

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If your pace on a treadmill is 65 meters per minute,how many minutes will it take for you to walk adistance of 7500 feet?

Learning Check

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Solution

Given: 7500 ft 65 m/min Need: minPlan: ft in. cm m minEqualities: 1 ft = 12 in. 1 in. = 2.54 cm 1 m = 100 cm

1 min = 65 m (walking pace)Set Up Problem:

7500 ft x 12 in. x 2.54 cm x 1m x 1 min1 ft 1 in. 100 cm 65 m

= 35 min final answer (2 SF)

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Percent Factor in a Problem

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

If the thickness of the skin fold at the waist indicates an 11% body fat, how much fat is in a person with a mass of 86 kg?

percent factor86 kg mass x 11 kg fat

100 kg mass

= 9.5 kg fat

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How many lb of sugar are in 120 g of candy if the candy is 25% (by mass) sugar?

Learning Check

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How many lb of sugar are in 120 g of candy if the candy is 25%(by mass) sugar?

percent factor120 g candy x 1 lb candy x 25 lb sugar

454 g candy 100 lb candy

= 0.066 lb sugar

Solution

Ref: Timberlake, “Chemistry”, Pearson/Benjamin Cummings, 2006, 9th Ed.