Reading Chapter Three: 3.4

Connecting to Your World Have you ever wondered why some objects float in water, while others sink? If you think that these lily pads float because they are lightweight, you are only partially correct. The ratio of the mass of an object to its volume can be used to determine whether an object floats or sinks in water. For pure water at 4°C, this ratio is 1.000 g/cm3. If an object has a mass-to-volume ratio less than 1.000 g/cm3, it will float in water. If an object has a mass-to-volume ratio greater than this value, it will sink in water.



Key Concepts

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What determines the density of a substance?
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How does a change in temperature affect density?

Vocabulary

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density

Reading Strategy

Identifying Main Idea As you read, write the main idea of the text that follows each heading.

Determining Density

Perhaps someone has tricked you with this question: “Which is heavier, a pound of lead or a pound of feathers?” Most people would not give the question much thought and would incorrectly answer “lead.” Of course, a pound of lead has the same mass as a pound of feathers. What concept, instead of mass, are people really thinking of when they answer this question?

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Simulation 1 Rank materials according to their densities.

Most people are incorrectly applying a perfectly correct idea: namely, that if a piece of lead and a feather of the same volume are weighed, the lead would have a greater mass than the feather. It would take a much larger volume of feathers to equal the mass of a given volume of lead.

The important relationship in this case is between the object’s mass and its volume. This relationship is called density. Density is the ratio of the mass of an object to its volume.

A 10.0-cm3 piece of lead, for example, has a mass of 114 g. What, then, is the density of lead? You can calculate it by substituting the mass and volume into the equation above.

Note that when mass is measured in grams, and volume in cubic centimeters, density has units of grams per cubic centimeter (g/cm3).

Figure 3.13 compares the density of three substances. Why does each10-g sample have a different volume? The volumes vary because the substances have different densities. Density is an intensive property that depends only on the composition of a substance, not on the size of the sample. With a mixture, density can vary because the composition of a mixture can vary.

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Figure 3.13

What do you think will happen if corn oil is poured into a glass containing corn syrup? Using Table 3.6, you can see that the density of corn oil is less than the density of corn syrup. For that reason, the oil floats on top of the syrup, as shown in Figure 3.14.

Figure 3.14 Because of differences in density, corn oil floats on top of corn syrup.

You have probably seen a helium-filled balloon rapidly rise to the ceiling when it is released. Whether a gas-filled balloon will sink or rise when released depends on how the density of the gas compares with the density of air. Helium is less dense than air, so a helium-filled balloon rises. The densities of various gases are listed in Table 3.6.

Determining Density

Perhaps someone has tricked you with this question: “Which is heavier, a pound of lead or a pound of feathers?” Most people would not give the question much thought and would incorrectly answer “lead.” Of course, a pound of lead has the same mass as a pound of feathers. What concept, instead of mass, are people really thinking of when they answer this question?

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Simulation 1 Rank materials according to their densities.

Most people are incorrectly applying a perfectly correct idea: namely, that if a piece of lead and a feather of the same volume are weighed, the lead would have a greater mass than the feather. It would take a much larger volume of feathers to equal the mass of a given volume of lead.

The important relationship in this case is between the object’s mass and its volume. This relationship is called density. Density is the ratio of the mass of an object to its volume.

A 10.0-cm3 piece of lead, for example, has a mass of 114 g. What, then, is the density of lead? You can calculate it by substituting the mass and volume into the equation above.

Note that when mass is measured in grams, and volume in cubic centimeters, density has units of grams per cubic centimeter (g/cm3).

Figure 3.13 compares the density of three substances. Why does each10-g sample have a different volume? The volumes vary because the substances have different densities. Density is an intensive property that depends only on the composition of a substance, not on the size of the sample. With a mixture, density can vary because the composition of a mixture can vary.

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Figure 3.13

What do you think will happen if corn oil is poured into a glass containing corn syrup? Using Table 3.6, you can see that the density of corn oil is less than the density of corn syrup. For that reason, the oil floats on top of the syrup, as shown in Figure 3.14.

Figure 3.14 Because of differences in density, corn oil floats on top of corn syrup.

You have probably seen a helium-filled balloon rapidly rise to the ceiling when it is released. Whether a gas-filled balloon will sink or rise when released depends on how the density of the gas compares with the density of air. Helium is less dense than air, so a helium-filled balloon rises. The densities of various gases are listed in Table 3.6.

Key Concepts

3.1 Measurements and Their Uncertainty

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Measurements are fundamental to the experimental sciences.Hint
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To evaluate accuracy, the measured value must be compared to the correct value. To evaluate precision, you must compare the values of repeated measurements.Hint
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Calculated answers often depend on the number of significant figures in the values used in the calculation.Hint
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In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated.Hint

3.2 The International System of Units

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Five commonly used SI base units are the meter, kilogram, kelvin, second, and mole.Hint
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Common metric units of length: cm, m, km. Common metric units of volume: μL, mL, L, cm3. Common metric units of mass: mg, g, kg. Common units of temperature: °C and K. Common units of energy: J and cal.Hint

3.3 Conversion Problems

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Multiplying by a conversion factor does not change the actual size of a measurement.Hint
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Dimensional analysis provides an alternative approach to problem solving.Hint
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Conversion problems are easily solved using dimensional analysis.Hint

3.4 Density

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Density is an intensive property that depends only on the composition of a substance.Hint
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The density of a substance generally decreases as its temperature increases.Hint

Vocabulary

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Vocabulary Review

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absolute zero
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accepted value
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accuracy
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calorie (cal)
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Celsius scale
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conversion factor
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density
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dimensional analysis
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energy
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error
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experimental value
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gram (g)
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International System of Units (SI)
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joule (J)
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Kelvin scale
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kilogram (kg)
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liter (L)
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measurement
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meter (m)
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percent error
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precision
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scientific notation
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significant figures
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temperature
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weight

Key Equations

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Error = experimental value − accepted value Hint
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Hint

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K = °C + 273and °C = K − 273Hint
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1 J = 0.2390 cal and 1 cal = 4.184 JHint
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Hint

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.

Reading Chapter Three: 3.2

Connecting to Your World “Are we there yet?” You may have asked this question during a long road trip with family or friends. To find out how much farther you have to go, you can read the roadside signs that list destinations and their distances. In the signs shown here, however, the distances are listed as numbers with no units attached. Is Carrieton 44 kilometers or 44 miles away? Without the units, you can’t be sure. When you make a measurement, you must assign the correct units to the numerical value. Without the units, it is impossible to communicate the measurement clearly to others.



Key Concepts

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Which five SI base units do chemists commonly use?
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What metric units are commonly used to measure length, volume, mass, temperature, and energy?

Vocabulary

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International System of Units (SI)
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meter (m)
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liter (L)
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kilogram (kg)
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gram (g)
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weight
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temperature
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Celsius scale
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Kelvin scale
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absolute zero
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energy
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joule (J)
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calorie (cal)

Reading Strategy

Summarizing As you read about SI units, summarize the main ideas in the text that follows the red and blue headings.

Connecting to Your World “Are we there yet?” You may have asked this question during a long road trip with family or friends. To find out how much farther you have to go, you can read the roadside signs that list destinations and their distances. In the signs shown here, however, the distances are listed as numbers with no units attached. Is Carrieton 44 kilometers or 44 miles away? Without the units, you can’t be sure. When you make a measurement, you must assign the correct units to the numerical value. Without the units, it is impossible to communicate the measurement clearly to others.



Key Concepts

*

Which five SI base units do chemists commonly use?
*

What metric units are commonly used to measure length, volume, mass, temperature, and energy?

Vocabulary

*

International System of Units (SI)
*

meter (m)
*

liter (L)
*

kilogram (kg)
*

gram (g)
*

weight
*

temperature
*

Celsius scale
*

Kelvin scale
*

absolute zero
*

energy
*

joule (J)
*

calorie (cal)

Reading Strategy

Summarizing As you read about SI units, summarize the main ideas in the text that follows the red and blue headings.

Units and Quantities

As you already know, you don’t measure length in kilograms or mass in centimeters. Different quantities require different units. Before you make a measurement, you must be familiar with the units corresponding to the quantity that you are trying to measure.
Units of Length

Size is an important property of matter. In SI, the basic unit of length, or linear measure, is the meter (m). All measurements of length can be expressed in meters. (The length of a page in this book is about one-fourth of a meter.) For very large and very small lengths, however, it may be more convenient to use a unit of length that has a prefix. Table 3.2 lists the prefixes in common use. For example, the prefix milli- means 1/1000 (one-thousandth), so a millimeter (mm) is 1/1000 of a meter, or 0.001 m. A hyphen (-) measures about 1 mm.

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Table 3.2: Commonly Used Metric Prefixes

For large distances, it is usually most appropriate to express measurements in kilometers (km). The prefix kilo- means 1000, so 1 km equals 1000 m. A standard marathon distance race of about 42,000 m is more conveniently expressed as 42 km (42 × 1000 m). Common metric units of length include the centimeter, meter, and kilometer. Table 3.3 summarizes the relationships among metric units of length.

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Table 3.3: Metric Units of Length

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Length of 5 city blocks ≈ 1 km
Units of Volume

The space occupied by any sample of matter is called its volume. You calculate the volume of any cubic or rectangular solid by multiplying its length by its width by its height. The unit for volume is thus derived from units of length. The SI unit of volume is the amount of space occupied by a cube that is 1 m along each edge. This volume is a cubic meter (m3). An automatic dishwasher has a volume of about 1 m3.

A more convenient unit of volume for everyday use is the liter, anon-SI unit. A liter (L) is the volume of a cube that is 10 centimeters (10 cm) along each edge (10 cm × 10 cm × 10 cm = 1000 cm3 = 1 L). A decimeter (dm) is equal to 10 cm, so 1 L is also equal to 1 cubic decimeter (dm3). A smaller non-SI unit of volume is the milliliter (mL); 1 mL is 1/1000 of a liter. Thus there are 1000 mL in 1 L. Because 1 L is defined as 1000 cm3, 1 mL and 1 cm3 are the same volume. The units milliliter and cubic centimeter are thus used interchangeably. Common metric units of volume include the liter, milliliter, cubic centimeter, and microliter. Table 3.4 summarizes the relationships among these units of volume.

Figure 3.6 These photographs above give you some idea of the relative sizes of some different units of volume. a The volume of 20 drops of liquid from a medicine dropper is approximately 1 mL. b A sugar cube is 1 cm on each edge and has a volume of 1 cm3. Note that 1 mL is the same as 1 cm3. c A gallon of milk has about twice the volume of a 2-L bottle of soda. Calculating How many cubic centimeters are in 2 liters?

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Table 3.4: Metric Units of Volume

There're many devices for measuring liquid volumes, including graduated cylinders, pipets, burets, volumetric flasks, and syringes. Note that the volume of any solid, liquid, or gas will change with temperature (although the change is much more dramatic for gases). Consequently, accurate -volume-measuring devices are calibrated at a given temperature—usually 20 degrees Celsius (20°C), which is about normal room temperature.

Reading Checkpoint
Units of Mass

The mass of an object is measured in comparison to a standard mass of 1 kilogram (kg), which is the basic SI unit of mass. A kilogram was originally defined as the mass of 1 L of liquid water at 4°C. A cube of water at 4°C measuring 10 cm on each edge would have a volume of 1 L and a mass of 1000 grams (g), or 1 kg. A gram (g) is 1/1000 of a kilogram; the mass of 1 cm3 of water at 4°C is 1 g. Common metric units of mass include the kilogram, gram, milligram, and microgram. The relationships among units of mass are shown in Table 3.5.

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Table 3.5: Metric Units of Mass

You can use a platform balance to measure the mass of an object. The object is placed on one side of the balance, and standard masses are added to the other side until the balance beam is level. The unknown mass is equal to the sum of the standard masses. Laboratory balances range from very sensitive instruments with a maximum capacity of only a few milligrams to devices for measuring quantities in kilograms. An analytical balance is used to measure objects of less than 100 g and can determine mass to the nearest 0.0001 g (0.1 mg).

The astronaut shown on the surface of the moon in Figure 3.7 weighs one sixth of what he weighs on Earth. The reason for this difference is that the force of gravity on Earth is about six times what it is on the moon. Weight is a force that measures the pull on a given mass by gravity. Weight, a measure of force, is different from mass, which is a measure of the quantity of matter. Although the weight of an object can change with its location, its mass remains constant regardless of its location. Objects can thus become weightless, but they can never become massless.

Figure 3.7 An astronaut’s weight on the moon is one sixth as much as it is on Earth. Earth exerts six times the force of gravity as the moon. Inferring How does the astronaut’s mass on the moon compare to his mass on Earth?

Reading Checkpoint
Units of Temperature

When you hold a glass of hot water, the glass feels hot because heat transfers from the glass to your hand. When you hold an ice cube, it feels cold because heat transfers from your hand to the ice cube. Temperature is a measure of how hot or cold an object is. An object’s temperature determines the direction of heat transfer. When two objects at different temperatures are in contact, heat moves from the object at the higher temperature to the object at the lower temperature.

Almost all substances expand with an increase in temperature and contract as the temperature decreases. (A very important exception is water.) These properties are the basis for the common liquid-in-glass thermometer. The liquid in the thermometer expands and contracts more than the volume of the glass, producing changes in the column height of liquid. Figure 3.8 shows a few different types of thermometers.

Figure 3.8 Thermometers are used to measure temperature. a A liquid-in-glass thermometer contains alcohol or mineral spirits. b A dial thermometer contains a coiled bimetallic strip. c A Galileo thermometer contains several glass bulbs that are calibrated to sink or float depending on the temperature. The Galileo thermometer shown uses the Fahrenheit scale, which sets the freezing point of water at 32°F and the boiling point of water at 212°F.

Several temperature scales with different units have been devised. Scientists commonly use two equivalent units of temperature, the degree Celsius and the kelvin. The Celsius scale of the metric system is named after the Swedish astronomer Anders Celsius (1701–1744). It uses two readily determined temperatures as reference temperature values: the freezing point and the boiling point of water. The Celsius scale sets the freezing point of water at 0°C and the boiling point of water at 100°C. The distance between these two fixed points is divided into 100 equal intervals, or degrees Celsius (°C).

Another temperature scale used in the physical sciences is the Kelvin, or absolute, scale. This scale is named for Lord Kelvin (1824–1907), a Scottish physicist and mathematician. On the Kelvin scale, the freezing point of water is 273.15 kelvins (K), and the boiling point is 373.15 K. Notice that with the Kelvin scale, the degree sign is not used. Figure 3.9 on the next page compares the Celsius and Kelvin scales. A change of one degree on the Celsius scale is equivalent to one kelvin on the Kelvin scale. The zero point on the Kelvinscale, 0 K, or absolute zero, is equal to −273.15°C. For problems in this text, you can round −273.15°C to −273°C. Because one degree on the Celsius scale is equivalent to one kelvin on the Kelvin scale, converting from one temperature to another is easy. You simply add or subtract 273, as shown in the following equations.

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Figure 3.9

K = °C + 273

°C = K − 273

Go Online
For: Links on Temperature Scales
Visit: www.SciLinks.org
Web Code: cdn-1032

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3.4 Converting Between Temperature Scales

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Problem-Solving 3.17 Solve Problem 17 with the help of an interactive guided tutorial.
Units of Energy

Figure 3.10 shows a house equipped with solar panels. The solar panels convert the radiant energy from the sun into electrical energy that can be used to heat water and power appliances. Energy is the capacity to do work or to produce heat.

Figure 3.10 Photoelectric panels convert solar energy into electricity.

Like any other quantity, energy can be measured.The joule and the calorie are common units of energy. The joule (J) is the SI unit of energy. It is named after the English physicist James Prescott Joule (1818–1889). One calorie (cal) is the quantity of heat that raises the temperature of 1 g of pure water by 1°C. Conversions between joules and calories can be carried out using the following relationships.

1J = 0.2390 cal 1 cal = 4.184 J

Reading Chapter Three: 3.1

Connecting to Your World On January 4, 2004, the Mars Exploration Rover Spirit landed on Mars. Equipped with five scientific instruments and a rock abrasion tool (shown at left), Spirit was sent to examine the Martian surface around Gusev Crater, a wide basin that may have once held a lake. Each day of its mission, Spirit recorded measurements for analysis. This data helped scientists learn about the geology and climate on Mars. All measurements have some uncertainty. In the chemistry laboratory, you must strive for accuracy and precision in your measurements.



Key Concepts

*

How do measurements relate to science?
*

How do you evaluate accuracy and precision?
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Why must measurements be reported to the correct number of significant figures?
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How does the precision of a calculated answer compare to the precision of the measurements used to obtain it?

Vocabulary

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measurement
*

scientific notation
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accuracy
*

precision
*

accepted value
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experimental value
*

error
*

percent error
*

significant figures

Reading Strategy

Building Vocabulary As you read, write a definition of each key term in your own words.

Using and Expressing Measurements

Your height (67 inches), your weight (134 pounds), and the speed you drive on the highway (65 miles/hour) are some familiar examples of measurements. A measurement is a quantity that has both a number and a unit. Everyone makes and uses measurements. For instance, you decide how to dress in the morning based on the temperature outside. If you were baking cookies, you would measure the volumes of the ingredients as indicated in the recipe.

Such everyday situations are similar to those faced by scientists. Measurements are fundamental to the experimental sciences. For that reason, it isimportant to be able to make measurements and to decide whether a measurement is correct. The units typically used in the sciences are those of the International System of Measurements(SI).

In chemistry, you will often encounter very large or very small numbers. A single gram of hydrogen, for example, contains approximately 602,000,000,000,000,000,000,000 hydrogenatoms. The mass of an atom of gold is 0.000 000 000 000 000 000 000 327 gram. Writing and using such large and small numbers is very cumbersome. You can work more easily with these numbers by writing them in scientific, or exponential, notation.

In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power. For example, the number 602,000,000,000,000,000,000,000written in scientific notation is 6.02 × 1023. The coefficient in this number is 6.02. In scientific notation, the coefficient is always a number equal to or greater than one and less than ten. The power of 10, or exponent, in this example is 23. Figure 3.1 illustrate show to express the number of stars in a galaxy by using scientific notation. For more practice on writing numbers in scientific notation, refer to page R56 of Appendix C.

Accuracy, Precision, and Error

Your success in the chemistry lab and in many of your daily activities depends on your ability to make reliable measurements. Ideally, measurements should be both correct and reproducible.
Accuracy and Precision

Correctness and reproducibility relate to the concepts of accuracy and precision, two words that mean the same thing to many people. In chemistry, however, their meanings are quite different. Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured. Precision is a measure of how close a series of measurements are to one another.To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.

Darts on a dartboard illustrate accuracy and precision in measurement. Let the bull’s-eye of the dartboard represent the true, or correct, value of what you are measuring. The closeness of a dart to the bull’s-eye corresponds to the degree of accuracy. The closer it comes to the bull’s-eye,the more accurately the dart was thrown. The closeness of several darts to one another corresponds to the degree of precision. The closer together the darts are, the greater the precision and the reproducibility.

Look at Figure 3.2 as you consider the following outcomes.

1.

All of the darts land close to the bull’s-eye and to one another. Closeness to the bull’s-eye means that the degree of accuracy is great. Each dart in the bull’s-eye corresponds to an accurate measurement of a value. Closeness of the darts to one another indicates high precision.
2.

All of the darts land close to one another but far from the bull’s-eye. The precision is high because of the closeness of grouping and thus the high level of reproducibility. The results are inaccurate, however, because of the distance of the darts from the bull’s-eye.
3.

The darts land far from one another and from the bull’s-eye.The results are both inaccurate and imprecise.

Figure 3.2 The distribution of darts illustrates the difference between accuracy and precision. a Good accuracy and good precision: The darts are close to the bull’s-eye and to one another. b Poor accuracy and good precision: The darts are far from the bull’s-eye but close to one another. c Poor accuracy and poor precision: The darts are far from the bull’s-eye and from one another.

Reading Checkpoint
Determining Error

Note that an individual measurement may be accurate or inaccurate. Suppose you use a thermometer to measure the boiling point of pure water at standard pressure. The thermometer reads 99.1°C. You probably know that the true or accepted value of the boiling point of pure water under these conditions is actually 100.0°C. There is a difference between the accepted value, which is the correct value based on reliable references, and the experimental value, the value measured in the lab. The difference between the experimental value and the accepted value is called the error.

Error = experimental value − accepted value

Error can be positive or negative depending on whether the experimental value is greater than or less than the accepted value.

For the boiling-point measurement, the error is 99.1°C − 100.0°C,or − 0.9° C. The magnitude of the error shows the amount by which the experimental value differs from the accepted value. Often, it is useful to calculate the relative error, or percent error. The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%.

Word Origins

Percent comes from the Latin words per, meaning “by” or “through,” and centum, meaning “100.” What do you think the phrase per annum means?

Using the absolute value of the error means that the percent error will always be a positive value. For the boiling-point measurement, the percent error is calculated as follows.

= 0.009 × 100%

= 0.9%

Just because a measuring device works doesn’t necessarily mean that it is accurate. As Figure 3.3 shows, a weighing scale that does not read zero when nothing is on it is bound to yield error. In order to weigh yourself accurately, you must first make sure that the scale is zeroed.

Figure 3.3 The scale below has not been properly zeroed so the reading obtained for the person's weight is inaccurate. There is a difference between the person's correct weight and the measured value. Calculating What is the percent error of a measured value of 114 lb if the person's actual weight is 107 lb?

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Significant Figures in Measurements

Supermarkets often provide scales like the one in Figure 3.4. Customers use these scales to measure the weight of produce that is priced per pound. If you use a scale that is calibrated in 0.1-lb intervals, you can easily read the scale to the nearest tenth of a pound. With such a scale, however, you can also estimate the weight to the nearest hundredth of a pound by noting the position of the pointer between calibration marks.

Figure 3.4 The precision of a weighing scale depends on how finely it is calibrated.

Suppose you estimate a weight that lies between 2.4 lb and 2.5 lb to be 2.46 lb. The number in this estimated measurement has three digits. The first two digits in the measurement (2 and 4) are known with certainty. But the rightmost digit (6) has been estimated and involves some uncertainty. These three reported digits all convey useful information, however, and are called significant figures. The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated. Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significant figures in the values used in the calculation.

Significant Figures in Calculations

Suppose you use a calculator to find the area of a floor that measures 7.7 meters by 5.4 meters. The calculator would give an answer of 41.58 square meters. The calculated area is expressed to four significant figures. However, each of the measurements used in the calculation is expressed to only two significant figures. So the answer must also be reported to two significant figures (42 m2). In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated.
Rounding

To round a number, you must first decide how many significant figures the answer should have. This decision depends on the given measurements and on the mathematical process used to arrive at the answer. Once you know the number of significant figures your answer should have, round to that many digits, counting from the left. If the digit immediately to the right of the last significant digit is less than 5, it is simply dropped and the value of the last significant digit stays the same. If the digit in question is 5 or greater, the value of the digit in the last significant place is increased by 1.

Reading Checkpoint

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3.1 Rounding Measurements

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Problem-Solving 3.3 Solve Problem 3 with the help of an interactive guided tutorial.
Addition and Subtraction

The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. Work through Sample Problem 3.2 below which provides an example of rounding in an addition calculation.

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3.2 Significant Figures in Addition

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Problem Solving 3.6 Solve Problem 6 with the help of an interactive guided tutorial.
Multiplication and Division

In calculations involving multiplication and division, you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements. The position of the decimal point is important only in rounding the answers of addition or subtraction problems.