Reading Chapter Three: 3.3

Connecting to Your World Perhaps you have traveled abroad or are planning to do so. If so, you know—or will soon discover—that different countries have different currencies. As a tourist, exchanging money is essential to the enjoyment of your trip. After all, you must pay for your meals, hotel, transportation, gift purchases, and tickets to exhibits and events. Because each country’scurrency compares differently with the U.S. dollar, knowing how to convert currency units correctly is very important. Conversion problems are readily solved by a problem-solving approach called dimensional analysis.



Key Concepts

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What happens when a measurement is multiplied by a conversion factor?
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Why is dimensional analysis useful?
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What types of problems are easily solved by using dimensional analysis?

Vocabulary

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conversion factor
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dimensional analysis

Reading Strategy

Monitoring Your Understanding Preview the Key Concepts, the section heads, and boldfaced terms. List three things you expect to learn. After reading, state what you learned about each item listed.


Conversion Factors

If you think about any number of everyday situations, you will realize that a quantity can usually be expressed in several different ways. For example, consider the monetary amount $1.

1 dollar = 4 quarters = 10 dimes = 20 nickels = 100 pennies

These are all expressions, or measurements, of the same amount of money. The same thing is true of scientific quantities. For example, consider a distance that measures exactly 1 meter.

1 meter = 10 decimeters = 100 centimeters = 1000 millimeters

These are different ways to express the same length.

Whenever two measurements are equivalent, a ratio of the two measurements will equal 1, or unity. For example, you can divide both sides of the equation 1 m = 100 cm by 1 m or by 100 cm.

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Animation 3 Learn how to select the proper conversion factor and how to use it.

A conversion factor is a ratio of equivalent measurements. The ratios 100 cm/1 m and 1 m/100 cm are examples of conversion factors. In a conversion factor, the measurement in the numerator (on the top) is equivalent to the measurement in the denominator (on the bottom). The conversion factors above are read “one hundred centimeters per meter” and “one meter per hundred centimeters.” Figure 3.11 illustrates another way to look at the relationships in a conversion factor. Notice that the smaller number is part of the measurement with the larger unit. That is, a meter is physically larger than a centimeter. The larger number is part of the measurement with the smaller unit.

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Figure 3.11

Conversion factors are useful in solving problems in which a given measurement must be expressed in some other unit of measure.When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same. For example, even though the numbers in the measurements 1 g and 10 dg (decigrams) differ, both measurements represent the same mass. In addition, conversion factors within a system of measurement are defined quantities or exact quantities. Therefore, they have an unlimited number of significant figures, and do not affect the rounding of a calculated answer.

Here are some additional examples of pairs of conversion factors written from equivalent measurements. The relationship between grams and kilograms is 1000 g = 1 kg. The conversion factors are:

The scale of the micrograph in Figure 3.12 is in nanometers. Using the relationship 109 nm = 1 m, you can write the following conversion factors.

Figure 3.12 In this computer image of atoms, distance is marked off in nanometers (nm). Inferring What conversion factor would you use to convert nanometers to meters?

Common volumetric units used in chemistry include the liter and the microliter. The relationship 1 L = 106 μL yields the following conversion factors

Based on what you know about metric prefixes, you should be able to easily write conversion factors that relate equivalent metric quantities.

Dimensional Analysis

No single method is best for solving every type of problem. Several good approaches are available, and generally one of the best is dimensional analysis. Dimensional analysis is a way to analyze and solve problems using the units, or dimensions, of the measurements. The best way to explain this problem-solving technique is to use it to solve an everyday situation.

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3.5 Using Dimensional Analysis

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Problem-Solving 3.29 Solve Problem 29 with the help of an interactive guided tutorial.

There is usually more than one way to solve a problem. When you first read Sample Problem 3.5, you may have thought about different and equally correct ways to approach and solve the problem. Some problems are easily worked with simple algebra.Dimensional analysis provides you with an alternative approach to problem solving. In either case, you should choose the problem-solving method that works best.

Converting Between Units

In chemistry, as in many other subjects, you often need to express a measurement in a unit different from the one given or measured initially. Problems in which a measurement with one unit is converted to an equivalent measurement with another unit are easily solved using dimensional analysis.

Suppose that a laboratory experiment requires 7.5 dg of magnesium metal, and 100 students will do the experiment. How many grams of magnesium should your teacher have on hand? Multiplying 100 students by 7.5 dg/student gives you 750 dg. But then you must convert dg to grams. Sample Problem 3.7 shows you how to do the conversion.

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3.7 Converting Between Metric Units

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Problem-Solving 3.33 Solve Problem 33 with the help of an interactive guided tutorial.
Multistep Problems

Many complex tasks in your everyday life are best handled by breaking them down into manageable parts. For example, if you were cleaning a car, you might first vacuum the inside, then wash the exterior, then dry the exterior, and finally put on a fresh coat of wax. Similarly, many complex word problems are more easily solved by breaking the solution down into steps.

When converting between units, it is often necessary to use more than one conversion factor. Sample Problem 3.8 illustrates the use of multiple conversion factors.

Reading Checkpoint

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3.8 Converting Between Metric Units

Chemath

Scientific Notation

A It is often convenient to express very large or very small numbers in scientific notation. The distance between the sun and Earth is 150,000,000 km, which can be written as 1.5 × 108 km. The diameter of a gold atom is 0.000 000 000 274 m, or 2.74 × 10−10 m. When multiplying numbers written in scientific notation, add the exponents. When dividing numbers written in scientific notation, subtract the exponent in the denominator from the exponent in the numerator.

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Problem-Solving 3.35 Solve Problem 35 with the help of an interactive guided tutorial.
Converting Complex Units

Many common measurements are expressed as a ratio of two units. For example, the results of international car races often give average lap speeds in kilometers per hour. You measure the densities of solids and liquids in grams per cubic centimeter. You measure the gas mileage in a car in miles per gallon of gasoline. If you use dimensional analysis, converting these complex units is just as easy as converting single units. It will just take multiple steps to arrive at an answer.