# Matter and Its Classification

CHAPTER 1Matter and Energy

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1.1Matter and Its Classification

1.2Physical and Chemical Changes and Properties of Matter

1.3Energy and Energy Changes

1.4Scientific Inquiry

Math Toolbox 1.1 Scientific Notation

Math Toolbox 1.2 Significant Figures

Math Toolbox 1.3 Units and Conversions

Chapter Review

Questions and Problems

Page 2Anna and Bill are college students enrolled in an introductory chemistry course. For their first assignment, the professor has asked them to walk around campus, locate objects that have something to do with chemistry, and classify the things they find according to characteristics of structure and form.

Anna and Bill begin their trek at the bookstore. They spot a fountain, a large metallic sculpture, a building construction site, and festive balloons decorating the front of the store. They notice water splashing in the fountain and coins that have collected at the bottom. The metallic sculpture has a unique color and texture. At the building construction site they notice murals painted on the wooden safety barricade. Through a hole in the fence, they see a construction worker doing some welding.

Bill and Anna make a list of the things that attracted their attention and start trying to classify them. Inspecting the fountain, they notice that it appears to be composed of pebbles embedded in cement. As water circulates in the fountain, it travels in waves on the water’s surface. The coins in the fountain, mostly pennies, vary in their shininess. Some look new, with their copper color gleaming in the bright sunshine. Others look dingy, brown, and old. The metal sculpture has a unique, modern design, but it’s showing signs of age. A layer of rust covers its entire surface. Anna and Bill decide to classify the sculpture as a metal, like the coins in the fountain. They also conclude that the water, pebbles, and concrete in the fountain are not metals.

As they approach the construction site, Anna and Bill examine the painted mural on a safety fence that surrounds the site. Through the peephole in the mural, they see gravel, cinder blocks, metallic tubes for ductwork, steel beams, and copper pipe. They add more nonmetals and metals to their list. A welder is joining two pieces of metal. Sparks are flying everywhere. Anna and Bill wonder what is in the sparks. Since the sparks are so small and vanish so rapidly, they don’t know how to classify them.

Electric carts are an ideal way to get across a large campus. They run using an electric battery instead of fuel.

As they continue their walk, they pass the intramural fields and the gym where they see students using tennis rackets, baseball bats, bicycles, and weight belts. They wonder how they will classify these items. For lunch, Bill and Anna buy pizza. They sip soft drinks from aluminum cans. They settle on a bench to enjoy their lunch in the sunshine and watch students playing volleyball in a sandpit. As they put on their sunscreen, they wonder how they might classify sunlight. After lunch, they hurry off to an afternoon class. On the way, they notice a variety of vehicles on campus. Some are gasoline-powered cars and buses, but others have signs on them saying they operate on alternative fuels. Trucks lumber by, exhaust fumes spewing from their tailpipes. Bill and Anna feel the hoods of parked cars. Some are still warm from their engine’s heat.

How are Bill’s and Anna’s observations related to chemistry? What characteristics have they identified that they can use for classification purposes? They have started their classification with metals and nonmetals. What other categories should they devise?

Now it’s your turn. Make a list of things relevant to chemistry in the location where you are reading this. How will you classify the things on your list? What characteristics will you use to organize the items into categories? Most important, why bother to classify things at all?

In this chapter we will explore some answers to these questions. As you learn what chemistry is, you’ll begin to develop explanations for how substances look, change, and behave.

Questions for Consideration

1.1 What characteristics distinguish different types of matter?

1.2What are some properties of matter?

1.3What is energy and how does it differ from matter?

1.4What approaches do scientists use to answer these and other questions?

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Math Tools Used in This Chapter

Scientific Notation (Math Toolbox 1.1)

Significant Figures (Math Toolbox 1.2)

Units and Conversions (Math Toolbox 1.3)

This icon refers to a Math Toolbox that provides more detail and practice.

1.1 Matter and Its Classification

All the things that Anna and Bill observed on campus are examples of matter. The fountain, the metal sculpture, the construction site, the balloons outside the bookstore, the exhaust fumes from buses, the pizza they had for lunch, even Bill and Anna themselves—all are matter. Matter is anything that occupies space and has mass. Mass is a measure of the quantity of matter. The interaction of mass with gravity creates weight, which can be measured on a scale or balance.

Some of Bill’s and Anna’s observations, however, were not of matter. Sunlight, the light from welding, and the heat of automobile engines are not matter. They do not occupy space, and they have no mass. They are forms of energy. Energy is the capacity to move an object or to transfer heat. We’ll discuss energy in Section 1.3, but for now, let’s focus on matter.

All of Anna’s and Bill’s observations are relevant to chemistry, because chemistry is the study of matter and energy. Since the entire physical world is matter and energy, chemistry would be an overwhelming subject of study if we did not classify phenomena in manageable ways. Anna and Bill used characteristics like shininess and hardness when they decided some materials were metals and others were not. Let’s explore some other characteristics that can be used to classify matter.

Composition of Matter

One way to classify matter is by its chemical composition. Some types of matter always have the same chemical composition, no matter what their origin. Such matter is called a pure substance or more briefly, a substance. A pure substance has the same composition throughout and from sample to sample. It cannot be separated into components by physical means.

Some pure substances can be observed. For example, the aluminum in Anna’s soda can is pure. It is not combined with any other substances, although it is coated with plastic and paint. Consider also the sandpit where Bill and Anna watched the volleyball game. The sand is not a pure substance, but if we removed all the dirt, minerals, and other contaminants, it would be the pure substance, silica, which is one kind of sand (Figure 1.1). Grains of silica differ in size, but they all have the same chemical composition, which can be determined in the laboratory.

FIGURE 1.1 Sand is composed of a mineral, silica. It contains the elements silicon and oxygen in specific proportions.

In contrast to pure substances, other materials are mixtures. A mixture consists of two or more pure substances and may vary in composition. The fountain, for example, is made from a mixture of gravel, concrete, and pebbles. Even the water in the fountain is not a pure substance since small amounts of gases and minerals are dissolved in it. Like sand, however, it could be made pure if all the other substances were removed.

There are two types of pure substances: elements and compounds. We will discuss these first, and then we’ll describe types of mixtures.

Are there any things where you are now that might be pure substances? Actually, pure substances are rare in our world. Most things are mixtures of some kind. Pure substances are found most often in laboratories where they are used to determine the properties and behavior of matter under controlled conditions.

Elements All matter consists of pure substances or mixtures of substances. Pure substances, in turn, are of two types: elements and compounds. An element is a substance that cannot be broken down into simpler substances even by a chemical reaction. For example, suppose we first purified the water in a fountain to remove contaminants. Then we used a chemical process called electrolysis to separate it into its component elements. Water can be broken down by chemical means into hydrogen and oxygen, as shown in Figure 1.2, so water is not an element. The hydrogen and oxygen, however, are elements. We cannot break them down into any simpler

 potassium kalium K silver argentum Ag sodium natrium Na tin stannum Sn tungsten wolfram W

To become familiar with the periodic table, you should learn the names and symbols for the first 36 elements, as well as the symbols for silver, tin, gold, mercury, and lead. Your instructor may ask you to learn others.

EXAMPLE 1.2Element Symbols

Potassium is a soft, silver-colored metal that reacts vigorously with water. Write the symbol for the element potassium.

Solution:

The symbol for potassium is K. In the periodic table, potassium is element 19 in group (column) IA (1) of the periodic table.

Consider This 1.2

What if you instinctively identified the element symbol as P or Po? Why are these symbols incorrect for potassium?

Practice Problem 1.2

(a) Lead is a soft, dull, silver-colored metal. Write the symbol for the element lead.

(b) The symbol for a common element used to make jewelry is Ag. What is the name of this element?

Further Practice: Questions 1.39 and 1.40 at the end of the chapter

Compounds A compound, sometimes called a chemical compound, is a pure substance composed of two or more elements combined in definite proportions. A compound has properties different from those of its component elements. For example, iron pyrite can be broken down into its component elements, iron and sulfur, but its characteristics are different from both (Figure 1.5). Anna and Bill Page 7saw many compounds during their trek around campus. These, and all compounds, can be chemically separated into their component elements. Sand is a compound of silicon and oxygen. Water, as discussed earlier, is composed of hydrogen and oxygen. The cheese on their pizza contains many complex compounds, but each of the compounds contains carbon, hydrogen, oxygen, nitrogen, and a few other elements.

FIGURE 1.5 Iron pyrite is composed of the elements iron and sulfur. Iron is magnetic and can be separated from sulfur when the two exist as elements mixed together. Iron pyrite, a compound of iron and sulfur, is not magnetic.

Chemists represent compounds with formulas based on the symbols for the elements that are combined in the compound. (Chemical formulas are not the same as the mathematical formulas that may be familiar to you, such as A = πr2 for the area of a circle.) A chemical formula describes the composition of a compound, using the symbols for the elements that make up the compound. Subscript numbers show the relative proportions of the elements in the compound. If no subscript number is given for an element in a formula, then you may assume that the element has a relative proportion of one. For example, water is known to consist of one unit of oxygen and two units of hydrogen. This compound is represented by the formula H2O. Sodium chloride, the chemical compound commonly called table salt, contains equal portions of the elements sodium and chlorine. Its formula is therefore NaCl. We will discuss formulas in detail in Chapter 3.

Graphite leaves a mark similar to that made by dragging a rod of lead along a surface, so it was called lead. A hardness number indicates the relative amounts of graphite and clay in a pencil lead. A number 2 pencil is fairly soft, while a number 6 pencil is quite hard. Which has more graphite?

Mixtures Some forms of matter, such as pencil lead, do not have the same composition in every sample. (Pencil lead isn’t the element lead. It is a mixture of graphite and clay.) A mixture consists of two or more elements or compounds. It is possible to separate mixtures into their component pure substances. The separation can be done physically, using procedures such as grinding, dissolving, or filtering. Chemical processes are not needed to separate mixtures.

We can illustrate the difference between pure substances and mixtures by looking at salt water. Water that has been purified is a pure substance that is composed of hydrogen and oxygen, always in the same proportions. Salt water, on the other hand, is water mixed with salt and many other substances in varying proportions. For example, the Great Salt Lake in Utah is approximately 10% salt, while the Dead Sea is about 30% salt. In either case, we can readily separate salt from water by evaporating the water (Figure 1.6).

FIGURE 1.6 To collect salt, water is diverted into large ponds. The water evaporates, leaving solid salt behind.

Mixtures differ in uniformity of composition. A homogeneous mixture has a uniform composition throughout and is often called a solution. Most solutions that we commonly encounter are composed of compounds dissolved in water. They are often clear. For example, a well-mixed sample of salt water prepared in a kitchen is uniform in appearance. The salt dissolved in it is invisible. Furthermore, any microscopically small portion of the sample would have the same composition as any other. The particles in the mixture might not be arranged in exactly the same pattern, but each sample, regardless of size, would have the same components in the same proportions.

A mixture that is not uniform throughout—a mixture of salt and pepper, for instance—is a heterogeneous mixture. Different samples have their components present in different proportions. Which of the things that Bill and Anna had for lunch is a homogeneous mixture? Which is heterogeneous? How about your own lunch? How can you tell?

We have considered a number of classes and subclasses of matter: mixtures, homogeneous mixtures, heterogeneous mixtures, pure substances, compounds, elements, metals, and nonmetals. A method for classifying matter into these categories is outlined in Figure 1.7. Note in the figure that yes or no answers to several questions distinguish one type of matter from another. First, we ask if the material can be separated physically. If so, then it is a mixture. If not, it must be a pure substance. If this substance can be decomposed (broken down into simpler substances) by chemical reactions, it is a compound. If it cannot, it is an element.

FIGURE 1.7 We can classify matter by answering the short series of questions in this flowchart.

Not all solutions are liquids. For example, consider air that has been filtered to remove suspended solid particles. Filtered air is a gaseous solution containing a mixture of primarily oxygen and nitrogen. Solid solutions also exist and are called alloys. For example, brass is a solution of zinc and copper.

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EXAMPLE 1.3Elements, Compounds, and Mixtures

Which of the following pictures represent pure substances?

Solution:

The copper on the outside of the coin and the helium inside the balloons are pure substances. (However, the helium and balloons considered together provide an example of a mixture.)

Consider This 1.3

Why isn’t the water in the fountain considered a pure substance?

Practice Problem 1.3

Which of the pictures represent mixtures? Which are heterogeneous? Which are homogeneous?

Further Practice: Questions 1.45 and 1.46 at the end of the chapter

Although chemists generally use color coding to distinguish between atoms of different elements in representations, the atoms themselves do not have colors. Macroscopic samples of matter may have color, but these colors do not usually match those used to represent atoms. In accurate representations, the sizes of the spheres change to reflect the relative differences in the sizes of atoms of different elements.

Representations of Matter

Chemists and other scientists view the world on several different levels. So far we have considered matter on a macroscopic scale. That is, we’ve discussed matter and phenomena we can see with our eyes. But simple observation is limited. Sometimes we cannot classify things merely by looking at them as Anna and Bill did. What do we do then? Chemists try to make sense of the structure of matter and its behavior on a scale that is much, much smaller than what we can see with our eyes.

Consider the copper pipe at the construction site, for example. If we could enlarge the tiniest unit that makes up the pipe, what would we see? Experimental evidence tells us copper is made up of discrete, spherical entities that all appear to be identical (Figure 1.8). Chemists identify these entities as atoms. An atom is the smallest unit of an element that has the chemical properties of that element. For example, we can imagine the helium inside a balloon as many, many atoms of helium, which we represent symbolically as He. In Figure 1.9, each sphere represents a single helium atom. Similarly, if we could magnify the structure of water, we would find two small hydrogen atoms bound separately to a single larger oxygen atom (Figure 1.10). Such a combination of elemental units is a molecule. Molecules are made up of two or more atoms bound together in a discrete arrangement. Several molecules of water, H2O, are shown in Figure 1.10, where the central red sphere represents an oxygen atom and the two smaller, white spheres stand for hydrogen atoms. (Some compounds do not exist as molecules. We will discuss them in Chapter 3.)

FIGURE 1.8 A copper pipe consists of a regular array of copper atoms.
©Thinkstock/Getty ImagesFIGURE 1.9 Helium atoms are present inside the balloon.
©Jules Frazier/Getty ImagesPage 10FIGURE 1.10 Molecules containing hydrogen atoms and oxygen atoms make up the water in the fountain. Note: The molecular-level image does not include dissolved matter that is present in the fountain water.

In addition to molecules of compounds, molecules can also be formed by the combination of atoms of only one element. For example, as shown in Figure 1.11, the oxygen we breathe consists of molecules of two oxygen atoms joined together. We represent oxygen molecules symbolically as O2.

FIGURE 1.11 Oxygen molecules are made up of two interconnected oxygen atoms and are represented symbolically as O2.

Chemists use many different ways to represent matter. Some for water are shown in Figure 1.12. Element symbols with subscripts represent a ratio of elements in a compound. One example is Figure 1.12B. To describe how the atoms are attached to one another, chemists often use lines and element symbols as shown in Figure 1.12C. In Figure 1.12D, spheres represent the atoms, and sticks show how they are connected. Figure 1.12E represents how the atoms fit together and their relative sizes. Macroscopic, molecular-level, and symbolic representations like these all have their advantages, and sometimes one is more convenient than another. You’ll use them all as you progress through this course.

FIGURE 1.12 Different ways of representing water: (A) macroscopic, (B and C) symbolic, and (D and E) molecular.

EXAMPLE 1.4Representations of Matter

(a) Which of these images best represents a mixture of elements?

(b) If image A represents nitrogen, write its formula.

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Solution:

(a) There are two mixtures represented in the images. Since the spheres (representing atoms) in image C have different colors and sizes, we can conclude that image C is a mixture of two elements. Image D is also a mixture, but it is a mixture of an element and a compound.

(b) The formula of the substance represented in image A is N2. Note that two atoms are connected in the molecule.

Consider This 1.4

Which combination of entities shown in the images would represent a mixture of compounds?

Practice Problem 1.4

(a) Which of the images represents an element that exists as a molecule?

(b) If image E represents a compound of oxygen (red) and sulfur (yellow), what is its formula? (Write the symbol for sulfur first.)

Further Practice: Questions 1.51 and 1.52 at the end of the chapter

States of Matter

Earlier we considered the classification of matter based on composition. Let’s look at a different way to classify matter: by its physical state. A physical state is a form that matter can take. The three most familiar to us are solid, liquid, and gas. Some substances, including some of those Anna and Bill observed, can be found in all three states under more or less ordinary conditions. Water, for example, can be a solid (ice), a liquid (flowing water), or a gas (water vapor) at environmental temperatures.

Other substances require extreme conditions to change from one state to another. For example, while carbon dioxide is a gas under normal conditions, it becomes a solid, called dry ice, at very low temperatures (Figure 1.13).

FIGURE 1.13 Dry ice is the solid state of carbon dioxide. It converts from a gas to a solid at a very low temperature.

ANIMATION: Three States of Matter

How do we know if a substance is in the solid, liquid, or gaseous state? Each state has characteristics that we can observe with our eyes and characteristics that are detectable or measurable at the molecular level. These characteristics are summarized in Table 1.2.

TABLE 1.2 Characteristics of the Physical States of Matter

 Solid Liquid Gas fixed shape shape of container (may or may not fill it) shape of container (fills it) its own volume its own volume volume of container no volume change under pressure slight volume change under pressure large volume change under pressure particles are fixed in place and tend to be in a regular (crystalline) array particles are randomly arranged and free to move about until they bump into one another particles are widely separated and move independently of one another

Some solids, called amorphous solids, do not have the high order that most crystalline solids have.

solid has a fixed shape that is not related to the shape of the container holding it. When you place an iron pipe in a box, the pipe does not change shape. Some solids can be made to change shape if enough force is applied. However, if you try to squeeze a solid to make it smaller, you’ll fail. A solid cannot be compressed because its particles are arranged in a tightly packed, highly ordered structure that does not include much free space into which they might be squeezed. Note the closely packed particles in the solid state of iron shown in Figure 1.14.

FIGURE 1.14 The liquid and solid states of iron. Notice that the liquid atoms are randomly arranged and are free to move around each other. In the solid, the atoms are fixed in a regular array.

liquid is different from a solid in that it has no fixed shape. It takes the shape of the filled portion of its container, and it can be poured. Although they touch, the particles in a liquid are not arranged in ordered structures like those in a solid; they are free to move past one another. A liquid can be compressed slightly because its particles have a little free space between them. Note the differences between the liquid and solid states of iron shown in Figure 1.14.

gas has no fixed shape; it adopts the shape of its container, expanding to fill the available space completely. A gas is easily compressed. When squeezed, gases can undergo large changes in volume. The particles of a gas are widely separated with much empty space between them. When a gas is compressed, the amount of space between the particles is reduced. This happens when pressure is applied, such as when a bicycle tire is filled with air, as shown in Figure 1.15. Another characteristic of gases is that they move through space quickly. When Bill and Anna smelled the pizza they had for lunch, they were detecting particles that migrated as gases from the source of the food to their noses. When gases cool sufficiently, they become liquids or even solids. This occurs, for example, when water vapor in the air liquefies on the surface of a cold glass. Note the differences between the liquid and gaseous states of water shown in Figure 1.16.

FIGURE 1.15 At the same temperature, a gas under high pressure has particles closer together than at low pressure. Notice that the composition (1 O2:4 N2) does not change with an increase in pressure.FIGURE 1.16 Water condenses from a gas to a liquid on a cold surface. Air molecules (e.g., oxygen and nitrogen) are not shown.

Does NaCl(aq) represent a pure substance or a mixture?

It is often convenient to show the physical state of a substance when representing it symbolically. For example, solid, liquid, and gaseous water can be represented as H2O(s), H2O(l), and H2O(g), respectively. The symbol (aq) represents an aqueous solution, a solution in which a substance is dissolved in water. A salt and water solution, for instance, can be written as NaCl(aq). These symbols for physical state are listed in Table 1.3.

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TABLE 1.3 Symbols for Physical State

 Physical State Symbol Example (bromine) solid (s) Br2(s) liquid (l) Br2(l) gas (g) Br2(g) aqueous (dissolved in water) (aq) Br2(aq)

1.2 Physical and Chemical Changes and Properties of Matter

MATH
TOOLBOX
1.1

Bill and Anna observed some of the properties of matter, including changes in matter. Observations can be either qualitative, based on some quality of the matter; or quantitative, based on a numerical value. When making qualitative observations, Bill and Anna described color, shape, texture, shininess, and physical state. Quantitative observations are different. They are numbers or measurements, and they must be carefully made and carefully reported.

MATH
TOOLBOX
1.2

Since quantitative data used to describe matter can involve both very large and very small numbers, it is often useful to express such numbers in scientific or exponential notation. Math Toolbox 1.1 (located at the end of this chapter) provides a review of this notation. In addition, it is necessary to express numbers in such a way as to indicate how accurately the value is known and how precisely it has been measured. The use of significant figures to properly express numerical values is presented in Math Toolbox 1.2.

Physical Properties

When reporting qualitative data, we can classify properties as either physical or chemical. When Bill and Anna observed the color, shape, texture, shininess, and physical state of things around them, they were noting their physical properties. A physical property is a characteristic that we can observe or measure without changing the composition of a substance. Other examples of physical properties are odor, taste, hardness, mass, volume, density, magnetism, conductivity, and the temperatures at which a substance changes from one physical state to another. (Later in this section we’ll discuss chemical properties, which include reactivity and flammability.) Let’s take a close look at mass, volume, density, and temperature. These four properties are quantitative; they involve numerical values.Page 14

MATH
TOOLBOX
1.3

Mass Recall that mass is a measure of the quantity of matter. We usually measure the mass of an object by weighing it on a balance. In chemistry, masses are often reported in units of grams (g). Large masses, like people or elephants, may be reported in units of kilograms (kg); and small masses, such as salt crystals or impurities in water, may be reported in units of milligrams (mg) or micrograms (μg), as shown in Figure 1.17. (Math Toolbox 1.3 summarizes the relationships among units such as these.) Sometimes the mass of something is reported in grams, but we might want to know the mass in another mass unit such as milligrams or kilograms. We can easily convert a measurement from one unit to another if we know the relationship between the units. Tables 1.4 and 1.5 summarize common relationships between metric and English units. Example 1.5 shows how to convert between mass units. (See Math Toolbox 1.3 for more information on unit conversions.)

FIGURE 1.17 A salt crystal has a mass of about 50 mg, while a person has a mass of about 70 kg.

EXAMPLE 1.5Units of Mass

MATH
TOOLBOX
1.3

Anna and Bill notice that there are 50.0 mg of sodium in the soda they bought to go with their lunch. How many grams of sodium are present in the can of soda? How many pounds?

Solution:

One way to solve this problem uses the dimensional-analysis approach. Consult Math Toolbox 1.3 for details. The general approach to solving the first part of the problem can be summarized by the following diagram:

The mass in milligrams has to be converted to the mass in grams. We need to find a relationship between these two quantities: 10−3 g = 1 mg (obtained from Table 1.4). Another way of expressing this relationship is 1 g = 1000 mg. Using this relationship, we get the following conversion:

TABLE 1.4 Metric Conversions

 Prefix Factor Symbol giga 109 G mega 106 M kilo 103 k deci 10−1 d centi 10−2 c milli 10−3 m micro 10−6 μ nano 10−9 n pico 10−12 p

MATH
TOOLBOX
1.2

We use the equivalence to set up possible conversion ratios:

To convert milligrams to grams, we can multiply 50.0 mg by the ratio (conversion factor) that will allow like units to cancel:

MATH
TOOLBOX
1.1

Note that the milligram units cancel to leave the appropriate unit of grams. Also notice that the answer is reported to three significant figures because the measured quantity (50.0 mg) is reported to three significant figures and the other numbers in the calculation (1 g and 1000 mg) are exact quantities. (Consult Math Toolbox 1.2 for details about significant figures.) For convenience, we could report the answer in scientific notation: 5.00 × 10−2. (See Math Toolbox 1.1 for a discussion of scientific notation.)

The second part of the question asks you to convert milligrams to pounds:

There isn’t a direct relationship between milligrams and pounds listed in Tables 1.4 and 1.5. However, Table 1.5 lists a relationship between pounds and grams: 1 lb = 453.6 g. We can convert the grams we found in the first part of this example to pounds using the relationship summarized in the following diagram:

TABLE 1.5 Some English-Metric Conversions

 English Unit Metric Unit 1 lb = 16 oz 453.6 g 1 in 2.54 cm (exactly) 1 yd 0.9144 m 1 mi 1.609 km 1 fluid oz 29.57 mL 1 qt 0.9464 L 1 gal 3.785 L 1 ft3 28.32 L

The ratios for converting between grams and pounds are

To convert grams to pounds, we multiply 0.0500 g by the ratio (conversion factor) that will allow like units to cancel:

Does this answer make sense? Yes, it does. There are a lot of grams (453.6) in a pound, so we would expect the answer to be very small.

Page 16Without a single conversion from milligrams to pounds, the problem we just solved involves multiple steps:

The sequence of steps can be summarized as:

Consider This 1.5

Why wouldn’t an answer of 22.7 lb for the second part of the question make sense?

Practice Problem 1.5

Anna and Bill see an aluminum recycling truck pass by on their way to class. If there are 765 lb of aluminum in the truck, how many grams are there? How many kilograms?

Further Practice: Questions 1.67 and 1.68 at the end of the chapter

Student Hot Spot

Student data indicate you may struggle with unit conversions. Access the SmartBook to view additional Learning Resources on this topic.

Volume Volume is the amount of space a substance occupies. We can determine the volume of a rectangular solid such as a cube by measuring its length, width, and height and then multiplying them. For example, the volume of a cube that is 2.0 centimeters (cm) on each side is 8.0 cubic centimeters (cm3):

Volume of a cube = length × width × height Volume = 2.0 cm × 2.0 cm × 2.0 cm = 8.0 cm3

Notice that the units are cm3, consistent with a three-dimensional quantity. One cubic centimeter is equal to 1 mL (1 cm3 = 1 mL), so the volume of 8.0 cm3 could also be reported as 8.0 mL.

If you need to determine the volume of a sphere, the relationship between volume and radius is .

The volumes of liquids are usually measured in units of liters (L) or milliliters (mL), as shown in Figure 1.18. Larger volumes, such as big bottles of soda, are usually reported in liters. A 1-L bottle of soda contains 1000 mL. Example 1.6 shows how to convert between volume units.

FIGURE 1.18 Some 500-mL, 1-L, and 250-mL containers.

EXAMPLE 1.6Units of Volume

For lunch, Anna and Bill had 12-ounce (oz) cans of soda. What is the volume of a 12.0-oz can of soda in units of milliliters? What is its volume in units of liters?

Solution:

To solve this problem using the dimensional-analysis approach (see Math Toolbox 1.3), we determine if there is a relationship between fluid ounces and milliliters:

MATH
TOOLBOX
1.3

To convert fluid ounces to milliliters, we use the following relationship from Table 1.5: 1 oz = 29.57 mL.

We use the equivalence to set up possible conversion ratios:

To convert ounces to milliliters, we can multiply 12.0 oz by the ratio (conversion factor) that will allow like units to cancel:

MATH
TOOLBOX
1.2

The answer is reported to three significant figures, because the quantity we’re given (12.0 oz) has three significant figures. Consult Math Toolbox 1.2 for details.

The second part of this problem asks you to convert milliliters to liters:

To convert volume in milliliters to volume in liters, we use the following relationship from Table 1.4: 1 mL = 10−3 L or 1000 mL = 1 L.

This is a unit conversion similar to the conversion we just did for mass. The ratios for converting between milliliters and liters are

To convert from milliliters to liters, we can multiply 355 mL by the conversion factor that allows like units to cancel:

Without a single conversion from ounces to liters, the problem we just solved involves multiple steps:

The sequence of steps can be summarized as:

Consider This 1.6

What if Anna and Bill shared a 1-L bottle of water over lunch? How many ounces is this?

Practice Problem 1.6

Anna and Bill saw some balloons outside the bookstore. The volume of gas inside one of the helium balloons was 4.60 L. What is the volume of gas in units of milliliters? In units of cubic centimeters? In units of gallons (4 qt = 1 gal)?

Further Practice: Questions 1.71 and 1.72 at the end of the chapter

Density The density of an object is the ratio of its mass to its volume. While mass and volume both depend on the size of the object or sample, density does not. Density is an unvarying property of a substance no matter how much of it is present, as long as temperature and pressure are constant. For example, the density of water at 4°C is 1.00 g/mL. It doesn’t matter if we have 10 mL or 10 L; the ratio of mass to volume would be the same, 1.00 g/mL. However, if the temperature increases, the water would expand to a larger volume, while the mass stays the same. The density of liquid water would decrease as the temperature increases. The densities of water and a few other substances are listed in Table 1.6.

TABLE 1.6 Densities of Some Common Substances

 Substance Physical State Density (g/mL)* helium gas 0.000178 oxygen gas 0.00143 cooking oil liquid 0.92 water liquid 1.00 mercury liquid 13.6 gold solid 19.3 copper solid 8.92 zinc solid 7.14 ice solid 0.92

*At room temperature and at normal atmospheric pressure, except gases at 0 degrees Celsius (°C) and water at 4°C.

As Anna and Bill noted when they observed the fountain, a copper coin sinks in water. It sinks because copper (and the other metals in a penny) have a greater density than water. Conversely, air bubbles, just like other gases, rise to the top of water because gases are less dense than liquids. Oil floats on water for this same reason.

The density column in Figure 1.19 shows a variety of liquids with different densities. Which liquid has the greatest density? Which is the least dense?

FIGURE 1.19 The densities of antifreeze, corn oil, dish detergent, maple syrup, shampoo, and water in g/mL are 1.13, 0.93, 1.03, 1.32, 1.01, and 1.00, respectively. Which layer is which substance?

If we compare equal volumes of two different substances, such as aluminum and gold, as shown in Figure 1.20, the substance with the greater mass has the greater Page 19density. How, though, can we compare densities if we do not have equal volumes? The mathematical relationship of mass, volume, and density reveals the answer:

FIGURE 1.20 Gold (Au) has a greater density than aluminum (Al) because gold has a greater mass per unit volume.

For example, a 1.0-cm3 sample of copper has a mass of 8.9 g. An 8.0-cm3 sample of copper has a mass of 71 g. A 27-cm3 sample of copper has a mass of 240 g. In all these samples (Figure 1.21), the mass of copper divided by its volume is 8.9 g/cm3. This is the density of copper.

FIGURE 1.21 The density of copper is 8.9 g/cm3. All three samples have the same ratio of mass to volume.

If we know the mass and volume of an object, we can determine its density by substituting directly into the density equation. For example, suppose we have a cube of an unknown metal with a mass of 178 g and an edge length of 2.92 cm. The volume of the cube is 24.9 cm3:

Volume = 2.92 cm × 2.92 cm × 2.92 cm = 24.9 cm3

The density can then be calculated by taking the ratio of the mass to volume:

Consulting Table 1.6, we see that the unknown metal could be zinc.

Additionally, if we know the density of a substance and its mass in our sample, we can determine its volume. For example, suppose we want to know the volume occupied by 100 g of copper. Should the volume be greater than or less than 100 cm3? There are many approaches to this problem. One way is to rearrange the density equation to solve for volume. Another way is to solve for the unknown volume in a set of equivalent ratios because density is a ratio of mass and volume that is constant for a given substance at a particular temperature. Both of these methods are shown in Example 1.7.

EXAMPLE 1.7Density, Volume, and Mass

What is the volume of 100.0 g of copper? The density of copper is 8.9 g/cm3.

Solution:

We need to carry out the following conversion:

The relationship between mass and volume is given by density:

First, we rearrange the density equation to get volume on one side by itself. This manipulation involves cross multiplication, which is described in Math Toolbox 1.3 (Ratio Approach). In the expression for density there is an implied 1:

MATH
TOOLBOX
1.3

Page 20Cross multiplying this density expression, we get:

Since we are trying to find the volume, we want to isolate it on one side of the equation. We can do this by dividing both sides by the density. (We’ll also drop the “× 1” because any quantity times 1 is that quantity.)

Now we have an expression that solves for the volume:

Then, we substitute the known values of mass and density into the equation and solve for the value of volume:

In a second approach to this problem, consider that since the density of copper is always the same, the ratio of mass to volume is the same for both what we know and what we don’t:

Cross multiply to solve for x:

In both approaches, the gram units cancel to give the expected volume unit of cm3.

There is yet another approach to solving this problem that involves using density as a conversion factor:

Does the answer make sense? Yes. The density tells us that 8.9 g of copper occupy a volume of 1 cm3. The mass given, 100.0 g, is over 10 times greater than 8.9 so we would expect it to occupy a volume that is over 10 times greater than 1 cm3.

Consider This 1.7

What if the 100.0-g piece of metal were gold? Would you expect its volume to be greater or less than 11 cm3? How much greater or less?

Practice Problem 1.7

Solve the following problems.

(a) The density of pure gold is 19.3 g/cm3. What is the volume of 1.00 g of pure gold?

(b) 14-Carat gold is a homogeneous mixture of metals containing 58% gold by mass. The other 42% is a mixture of silver and copper. Silver and copper are both less dense than gold. Which of the following could be the mass of 1.00 cm3 of 14-carat gold: 16.0 g, 19.3 g, or 23.0 g?

Further Practice: Questions 1.77 and 1.78 at the end of the chapter

These samples of metals have the same mass. Which has the greater density?

A can of diet cola floats in water, but a can of regular cola sinks. Suggest a reason why. How can you use this information to quickly select your preferred type of soft drink from a cooler filled with ice water at a party?

The density 8.9 g/cm3 can be expressed in fractional form as:

Water is unique among liquids because its solid form (ice) floats on its liquid form. This results from the relatively open structure adopted by water molecules in the solid state. What would happen to fish during the winter if water were like other substances that sink in their solid form?

Why do substances have different densities? Gases, in general, have very low densities because gas particles spread out and occupy large volumes. Metals tend to have high densities because their atoms pack together efficiently. Because ice floats on water, we  Page 21can infer that water in its solid form must have a lesser density than water in its liquid form. Example 1.8 shows how to use molecular pictures to predict relative densities.

ANIMATION: Unique Properties of Water

EXAMPLE 1.8Explanations for Density

How do the molecular diagrams of ice and water help explain why ice is less dense than water?

Solution:

In ice, the H2O molecules have more space between them than in liquid water. The total volume occupied by a given number of molecules is greater in ice. Because density is a ratio of mass to volume, the larger volume accounts for the lower density.

Consider This 1.8

What if hexane, with a density of 0.659 g/cm3, were carefully added to a glass of ice water? Where would the hexane be relative to the ice and liquid water?

Practice Problem 1.8

Helium balloons rise in air, which is a mixture of oxygen and nitrogen molecules, so we know helium is less dense than air. Look at the molecular-level diagrams of helium and carbon dioxide. Predict whether a helium balloon rises or falls in an atmosphere of carbon dioxide.

Further Practice: Questions 1.81 and 1.82 at the end of the chapter

Temperatures are written differently for the different scales. While Celsius and Fahrenheit use the superscript ° to indicate degrees, the Kelvin scale does not. The unit is written as K (the capital letter), but temperatures are measured in kelvins (lowercase).

Temperature Bill and Anna weren’t happy with their lunches. The pizza was cold and their sodas were warm. When we make such comparisons, we are observing relative temperatures. Temperature is a measure of how hot or cold something is relative to some standard. We measure temperature with a thermometer.

In the United States, we often use the Fahrenheit scale to measure body temperature and air temperature. Fahrenheit is rarely used in science. Two other temperature scales are standard: the Celsius scale and the Kelvin scale. The relationships between the three temperature scales, Fahrenheit (°F), Celsius (°C), and Kelvin (K), are shown in Figure 1.22.

FIGURE 1.22 The Fahrenheit, Celsius, and Kelvin temperature scales.

Another property of matter that is independent of sample size is the temperature at which the substance changes from one physical state to another. The boiling point is the temperature at which the liquid form of a substance changes to the gaseous form. Page 22At the melting point, the substance changes from a solid to a liquid. Between these two temperatures, the substance is normally in its liquid state. For example, on the Celsius scale, the boiling point of water is 100°C. Water melts (or freezes, depending on its original state) at 0°C. On the Kelvin scale, these values are 373.15 K and 273.15 K, respectively. On the Fahrenheit scale, they are 212°F and 32°F, respectively.

There are no negative values on the Kelvin scale. It is an absolute temperature scale because its zero point is the lowest possible temperature observable in the universe. This value is absolute zero, which is equivalent to −273.15°C. The temperature increments on the Kelvin scale are the same as those on the Celsius scale. The difference in temperature between the boiling point of water and the freezing point of water is 100 in both the Celsius (100°C − 0°C) and Kelvin (373.15 K − 273.15 K) scales, while the difference is 180 on the Fahrenheit scale (212°F − 32°F). Because the temperature in kelvins is always 273.15 greater than the temperature in degrees Celsius, we can easily convert between them:

TK = T°C + 273.15

When converting between the Fahrenheit and Celsius scales, the calculation is more complicated, because the degree increments are not equal:

T°F = 1.8(T°C) + 32

The equation can be rearranged, solving for degrees Celsius:

EXAMPLE 1.9Units of Temperature

The melting point of copper is 1083°C. Above what temperature, in kelvins and degrees Fahrenheit, is copper a liquid?

Solution:

Copper becomes a liquid above its melting point. In units of kelvin this temperature is

TK = T°C + 273.15

We substitute the value of the Celsius temperature into the expression and solve for the temperature in kelvins:

TK = 1083 + 273.15 = 1356 K

To convert degrees Celsius to degrees Fahrenheit, we use the equation:

T°F = 1.8(T°C) + 32

Substituting the temperature in degrees Celsius, we get:

T°F = 1.8(1083) + 32 = 1981°F

Consider This 1.9

What is the physical state of copper at 1000°C?

Practice Problem 1.9

(a) The boiling point of acetylene is −28.1°C. Below what temperature, in kelvins and degrees Fahrenheit, is acetylene a liquid?

(b) The boiling point of helium is 4 K. Below what temperature, in degrees Celsius, is helium a liquid?

(c) Human body temperature is normally 98.6°F. What is this temperature on the Celsius and Kelvin scales?

Further Practice: Questions 1.85 and 1.86 at the end of the chapter

The most common type of heating we encounter involves things we do in the kitchen. When we heat something up, we’re typically not changing the chemical composition. However, the process of cooking or baking often involves a combination of physical and chemical changes. Of course, if you were to overcook something and burn it, that would be a chemical change.

Physical Changes

A process that changes the physical properties of a substance without changing its chemical composition is a physical change. For example, we can change liquid water to water vapor by heating it. This change from a liquid to a gas, called boiling or vaporization, is a physical change, since both forms involve the same chemical substance, water (H2O).

To represent such changes, we can refine the symbolic representations we developed for elements and for compounds. We write the chemical formula for the initial condition and composition of the matter we are considering, then an arrow, and finally the chemical formula for the final condition and composition. The arrow is used to show that a change has occurred and in which direction. Using this symbolism, the change of water from a liquid to a gas would be represented as

The molecular and symbolic representations in Figure 1.23 show that the water molecules do not themselves change, but their physical state does. All the processes that change water from one physical state into another are summarized in Figure 1.24.

FIGURE 1.23 Molecular-level and symbolic representations of the evaporation of water.FIGURE 1.24 The physical states of solid, liquid, and gas can all change into one another either directly or by going through two changes of state. The names of these processes are shown here next to arrows that designate the direction of the change.

Another example of a physical change is the separation of different substances in a mixture. For example, a magnet divides magnetic materials from nonmagnetic materials without changing their identities, as shown earlier in Figure 1.5. A filter separates solid materials from liquid substances without changing either one chemically.

Page 24Many metals combine with oxygen to form a metal oxide compound at the surface of the metal. When this occurs with iron, we call it rust.

Chemical Changes

Remember the pennies in the fountain that Anna and Bill observed? Some were shiny and others looked dingy and brown. They might describe these less-shiny pennies as “tarnished.” The pennies have undergone a chemical change, a process in which one or more substances are converted into one or more new substances. When pennies tarnish, some of the copper and zinc metal atoms in them combine with oxygen, forming compounds called metal oxides. The compounds are chemically different from either of the elements that formed them.

Suppose we clean a tarnished penny. Is the process a physical or a chemical change? It can be either. If you simply rub off the metal oxide coating with an eraser, the change is physical. Most penny collectors, however, prefer a chemical change that removes less metal. Rubbing ketchup on a penny is a great way to make it shiny. The vinegar in the ketchup reacts chemically with the metal oxides, freeing them from the surface of the penny. When the penny is rinsed, the result of the chemical change is easy to see.

Anna and Bill observed other examples of chemical change during their campus walk. When gasoline-powered cars burn fuel, a chemical change occurs. The gasoline reacts with oxygen to form carbon dioxide and water vapor. This chemical change releases the energy that runs the car. Chemical changes that involve burning are often accompanied by the release of energy. Anna and Bill also observed vehicles that run on alternative fuels. In hydrogen-powered vehicles, the hydrogen fuel combines with oxygen to form water vapor—and to release a lot of energy. The molecular-level and symbolic representations for this chemical change are shown in Figure 1.25. A chemical change is often called a chemical reaction. What are some examples of chemical reactions that you can observe around you?

FIGURE 1.25 A chemical change occurs when the atoms in H2 and O2 rearrange to form H2O.

Chemical Properties

The copper and zinc in a penny, the gasoline in a car, and the hydrogen in an alternative-fuel vehicle all share a common chemical property: They react with oxygen. However, they differ in how they react and what products they form. Only the latter two release sufficient energy rapidly enough to make their use as fuels possible.

Page 25A chemical property of a substance is defined by what it is composed of and what chemical changes it can undergo. For example, let’s compare hydrogen and helium. Although they have similar physical properties (colorless gases, similar densities), their chemical properties are very different. While hydrogen reacts with many other elements and compounds, helium is considered inert (Figure 1.26). It has not yet been shown to react with any other element or compound. Other terms used to describe chemical properties include reactivity and flammability.

FIGURE 1.26 (A) The Hindenburg was a giant, rigid balloon filled with hydrogen gas. In 1937, it was destroyed when its hydrogen caught fire. (B) Today, blimps are filled with helium, an inert gas that will not explode.

EXAMPLE 1.10Physical and Chemical Changes

Which of the following are physical changes and which are chemical changes?

(a) evaporation

(b) burning methane gas to form carbon dioxide and water

(c) using a magnet to separate metal and plastic paperclips

(d) rusting (the conversion of iron to iron oxide)

Solution:

(a) Evaporation is a physical change because it involves only a change of state.

(b) Burning methane gas is a chemical change because new substances form.

(c) Separating components of a mixture is a physical change.

(d) Rusting is a chemical change because a new substance forms.

Consider This 1.10

Is digesting the pizza Anna and Bill ate for a lunch a chemical or a physical change?

Practice Problem 1.10

Which of the following are physical properties and which are chemical properties?

(a) boiling point of ethanol

(b) ability of propane to burn

(c) tendency for silver to tarnish

(d) density of aluminum

Further Practice: Questions 1.91 and 1.92 at the end of the chapter

Water molecules surround solute particles in aqueous solutions. For clarity in seeing the particles, this book shows water molecules faded in the background as in Figure 1.27A.

Sometimes simple observation cannot tell us whether a change is chemical or physical. For example, bubbles appear when baking soda and vinegar mix. Bubbles also appear when water boils, but the changes that produce the bubbles are different in these two cases. Baking soda and vinegar release bubbles because a chemical change takes place. They react to form carbon dioxide gas. However, when we warm water in a pan on the stove, small bubbles rise due to the release of dissolved air (mostly oxygen and nitrogen gas) from the water (before the water starts to boil). This process is only a physical change. If we could look at the nitrogen and oxygen molecules, as shown in Figure 1.27A, we would see that they are the same whether they are dissolved in water or not. When these molecules are dissolved in water, they form a homogeneous mixture with it. Heating the water merely separates the oxygen and nitrogen molecules from the water molecules. If we continue to heat the water to boiling, larger bubbles form and then rise from the bottom, as shown in Figure 1.27B. These bubbles are gaseous water, or water vapor. The result of the physical change can be represented symbolically as

FIGURE 1.27 Water contains small amounts of dissolved nitrogen and oxygen gases. (A) When heated, these molecules go to the gaseous state in bubbles that rise to the surface. (B) When the water begins to boil, it no longer contains dissolved gases, and the bubbles contain gaseous water.

EXAMPLE 1.11Physical and Chemical Changes

Do the following molecular-level images represent a chemical change or a physical change?

Solution:

The substances after the change have a different composition than the substances before the change. Therefore, this is a chemical change.

Consider This 1.11

How do atom attachments differ for each of the elements in the change shown in the example?

Practice Problem 1.11

Do the following molecular-level images represent a chemical change or a physical change?

Further Practice: Questions 1.95 and 1.96 at the end of the chapter

1.3 Energy and Energy Changes

Physical and chemical changes involve energy. Energy is hard to define, but we see and feel evidence of it when something moves or changes temperature. At the construction site, Anna and Bill saw a worker pushing a wheelbarrow up a ramp. If released at the top of the ramp, the wheelbarrow would roll back down, converting energy from one form to another in the process. This release of energy is related to the spontaneous process of rolling down the ramp. (A spontaneous process is one that doesn’t have to be forced to occur after it gets started.) But returning the wheelbarrow to the top of the ramp is not spontaneous. It requires a continuous energy input. Similarly, chemical and physical changes are usually accompanied by energy changes. Some chemical reactions are spontaneous. They happen on their own. Others need a continuous energy input from an external source. Consider the reaction (Figure 1.28) of hydrogen and oxygen gases to form water vapor, represented by this equation:

FIGURE 1.28 In the reaction shown here, a balloon was filled with appropriate amounts of hydrogen and oxygen gas. When a lit candle touched the balloon, the hydrogen and oxygen reacted explosively to form water vapor.

Page 28This reaction is spontaneous and explosive. It releases a tremendous amount of energy. But the opposite reaction—the breakdown of water into hydrogen and oxygen gases—is not spontaneous. It occurs only if sufficient energy is continuously added, such as by passing electricity through liquid water. This process, called electrolysis, was shown in Figure 1.2, and can be described symbolically as follows:

But what is energy? Energy is the capacity to do work or to transfer heat. Work, usually taken to mean mechanical work, occurs when a force acts over a distance. For example, work is done when the construction worker pushes the wheelbarrow up a ramp. Work is done when compressed gases, resulting from the combustion of a fuel, push the piston in the cylinder of an automobile engine. Not all reactions can be made to do work directly, but heat energy can be harnessed to do work. For example, boiling water produces steam, which turns the turbines in power plants. The turbines spin copper coils inside a magnetic field in a generator to produce an electric current.

Energy takes many different forms, and it can be converted from one form to another. Scientists describe two types of energy: kinetic energy and potential energy. Kinetic energy is the energy of motion. The wheelbarrow rolling down a ramp possesses kinetic energy. Potential energy is energy possessed by an object because of its position. The wheelbarrow resting at the top of the ramp has potential energy. If it did not contain this stored energy, it could not release energy when it rolled down the ramp. Any object in a position to be rolled, dropped, or otherwise allowed to move spontaneously has potential energy that will be converted to kinetic energy once the motion starts.

To distinguish kinetic from potential energy, consider the volleyball game Anna and Bill watched while they ate lunch. After they ate, Anna and Bill joined the game. When Anna served the ball, kinetic energy was transferred from her hand to the volleyball (Figure 1.29). As it ascended, the ball transferred some of its kinetic  Page 29energy to the surrounding air molecules, but most of its kinetic energy was converted to potential energy. Nearly all of the kinetic energy was converted to potential energy when the ball reached the top of its ascent. As the volleyball descended, its potential energy was converted to kinetic energy. Most of its kinetic energy was transferred to the ground when Bill missed the ball. The ball’s remaining kinetic energy allowed it to bounce back up, where again its kinetic energy was converted to potential energy during its ascent.

FIGURE 1.29 Kinetic energy is transferred from the server to the volleyball when it is served. As the volleyball rises in the air, its kinetic energy is converted to potential energy. When it descends, potential energy is converted to kinetic energy.

Other forms of energy—chemical, mechanical, electric, and heat energy, for example—are really just forms of kinetic or potential energy. For example, chemical compounds can release chemical energy, the energy associated with a chemical reaction. Chemical energy is potential energy arising from the positions of the atoms and molecules in the compounds. A compound releases its potential energy when it undergoes a spontaneous chemical reaction that forms substances with less potential energy. For example, the explosive material TNT (trinitrotoluene) contains considerable potential energy that is released as kinetic energy when it reacts. Chemical compounds can also have kinetic energy. Molecules move faster as the temperature rises. The motion of molecules or atoms is associated with heat energy, or the kinetic energy that increases with increasing temperature. The fast-moving gases produced by the explosion of TNT have high kinetic energy.

EXAMPLE 1.12Molecular Motion and Kinetic Energy

Which of these two samples of argon gas has more kinetic energy?

Solution:

The atoms that are moving faster have the greater kinetic energy. Thus, the atoms in A have the greater kinetic energy.

Consider This 1.12

How would the trail lengths have to be changed if the kinetic energy of A decreased and B increased?

Practice Problem 1.12

Which of the two samples of argon gas is at a lower temperature?

Further Practice: Questions 1.103 and 1.104 at the end of the chapter

We will use the lengths of trails behind atoms or molecules to depict their relative speeds. Longer trails correspond to higher speeds.

Electric energy is associated with the passage of electricity, generally through metals. The electric energy from a battery arises from a chemical reaction. The chemical energy stored in the compounds that make up the battery is converted to electric energy. Electric current passed through the filament of a lightbulb causes the metal to glow red and increases the motion of the atoms. This is a conversion of electric energy to kinetic energy. A lightbulb also gives off light energy. Nuclear energy involves both light and heat. Energy is released when one element is converted to another, as in a nuclear reactor or in the Sun, where hydrogen atoms fuse to form helium.

Page 30

EXAMPLE 1.13Forms of Energy

Identify examples of potential and kinetic energy in this picture.

Solution:

Anything that might move in the picture has potential energy. Kinetic energy is evident in the moving people and vehicles.

Consider This 1.13

What are some examples of electric energy in the picture?

Practice Problem 1.13

Identify three additional forms of energy in the photograph.

Further Practice: Questions 1.107 and 1.108 at the end of the chapter

All these forms of energy can be converted into one another. For example, the welder at the construction site starts a gasoline engine that runs a generator that makes the electricity the welder uses to join two pieces of metal together. The chemical energy in the gasoline is converted to mechanical energy that turns the generator. The mechanical energy is converted to electric energy by the generator. The electric energy is converted to heat in an arc that is formed between the welding rod and the metal to be welded. This heat melts the metal and creates the weld. Some of the electric energy is also converted to light. What energy conversions can you observe going on around you right now?

Common units of energy are calories, Calories (with a capital C), and joules. The energy input you need per day is about 2000 Calories, 2 million calories, or 8 million joules. We will study ways of measuring energy changes in Chapter 6.

1.4 Scientific Inquiry

We began this chapter by describing some of the things that Anna and Bill saw around their campus. To classify the items, they observed similarities and differences in properties. They were making observations to help them understand nature.

Observation is one of the tools of scientific inquiry, but it is not the only one. The scientific method is an approach to asking questions and seeking answers that employs a variety of tools, techniques, and strategies. Although the scientific method is often explained as a series of steps and procedures, it is more accurately described as a way of looking at the world that differs from nonscience forms of inquiry. Scientists, like all humans, use intuition. They generalize about the world, sometimes with insufficient data. Chemists, especially, make inferences about atoms and molecules from data obtained from instruments that aren’t quite capable of showing these tiny particles.

Scientists differ from professionals in nonscience disciplines in at least three important ways: (1) They test ideas by experimentation, (2) they organize their findings in particular (often mathematical) ways, (3) and they try to explain why things happen. Scientists use what is already known or believed about particular phenomena to gain insight into new observations from their experiments. Careful reasoning and insightful analogy are often employed, but sometimes intuition and luck play a part. Good scientists have an ability to couple objective scientific thinking with creative problem solving. In addition, a scientist must be curious enough to pursue the study of a seemingly trivial observation that can sometimes—albeit rarely—lead to a major advance in understanding.

Practicing scientists employ a variety of approaches to generate new knowledge or solve problems. Scientific inquiry generally includes observations, hypotheses, laws, and theories.

Observations

Scientific inquiry begins with ideas, knowledge, and curiosity in the minds of scientists. To look for answers to their questions, scientists collect data. Data may derive from the observation of a naturally occurring event or from deliberate experimentation. When experimenting, scientists set conditions, allow events to occur, and observe the result. This procedure permits scientists to examine events under controlled conditions not found in nature. In addition, scientists repeat one another’s experiments and compare observations, thus checking the accuracy of their findings. A common experimental design involves the isolation of one factor at a time to determine which of many variables influences the outcome. The results of experiments may be qualitative (descriptive) or quantitative (numeric).

Consider this example. Suppose Anna and Bill want to experiment to find a way to clean the dull, brown coating from the pennies they saw in the fountain. After reading the results of previous studies, they think an acid might be a good agent to use. They try several, always being sure to work under the same conditions of temperature, light, ventilation, degree of tarnish, and so on. The pennies look a little cleaner after being placed in acetic acid, but the dull coating remains unaffected. With nitric acid, a reaction occurs. A blue-green liquid and a red gas form. The tarnish disappears, but so do the pennies (Figure 1.30). Hydrochloric acid appears to work best because the surface tarnish vanishes quickly, leaving the penny intact. However, Anna and Bill notice that one of the pennies appears to react with the hydrochloric acid to form bubbles. After a time, this penny begins to float (Figure 1.31). Upon closer examination of this penny, they notice that bubbles are coming from scratches on its surface. Their descriptions of the behavior of pennies in acids are observations made from deliberate and controlled experimentation.

FIGURE 1.30 Copper reacts vigorously with nitric acid to form copper nitrate in solution and gaseous nitrogen dioxide.
©Charles D. Winters/Science SourceFIGURE 1.31 When some pennies were cleaned with hydrochloric acid, one began to bubble and eventually floated.

Hypotheses

To organize and correlate multiple observations and sets of collected data, scientists propose hypotheses. A hypothesis is a tentative explanation for the properties or behavior of matter that accounts for a set of observations and can be tested. Because hypotheses are usually starting points in the explanation of natural phenomena, they normally lead to further experimentation. In practice, hypotheses are intuitive guesses that may be based on small amounts of data. Often a hypothesis is modified repeatedly in light of the results of additional experimentation. The building of scientific knowledge involves a cyclic interplay between observations and the making and testing of hypotheses (Figure 1.32).

FIGURE 1.32 Observations and hypotheses are linked in a cyclic fashion during the development of scientific knowledge.

Consider the experiments with pennies and acids. Earlier, Bill and Anna observed that one of the pennies reacted with hydrochloric acid to form a colorless gas. That gas formed along scratches on the penny’s surface. Their observation might lead them to form a hypothesis: Pennies react with hydrochloric acid to form bubbles of a gas because they have scratches that make the reaction faster. How might they Page 32test this hypothesis? They could scratch several pennies, dropping them into hydrochloric acid and checking for gas bubble formation. Suppose they performed this experiment with 10 pennies (Figure 1.33) and found that 7 of them formed bubbles at the scratches, but 3 did not. They would have to conclude that their hypothesis was only partially correct. Why? Because some of the pennies did not form bubbles as predicted by their hypothesis. Therefore, they might think further, asking themselves what other characteristics of pennies could influence the outcome.

FIGURE 1.33 Some of the pennies react with hydrochloric acid at the location of the scratches, while others do not.

While all pennies appear the same, close observation reveals detectable differences. For example, mints located in different cities produce pennies. The first letter of the city name appears just below the year in which the penny was minted (Figure 1.34). A D is stamped on pennies minted in Denver; S is used for San Francisco; and pennies minted in Philadelphia have no identifying mark. Given this new information, Anna and Bill might state a new hypothesis: Pennies react with hydrochloric acid to form bubbles if they are scratched and if they come from a specific mint—perhaps because that mint formulates its pennies differently. However, examination of the pennies from the previous experiment indicates that the mint location doesn’t matter. Some scratched pennies from all the mints react and some don’t. The amended hypothesis is incorrect.

FIGURE 1.34 Pennies are marked with the year and city where they were minted.

As Bill and Anna set out to amend their hypothesis again, they realize they must isolate and assess one variable at a time. If they try to test too many ideas at once, they’d end up in a muddle of effects with several possible causes. So, while it’s clear that scratches are necessary for bubbling to occur, some other variable or variables must be involved. As a next step, Anna and Bill reexamine the pennies from their previous experiment. They find that all the reactive pennies were minted in 1984 or more recently, while all the nonreactive pennies were minted in 1982 or before. This observation leads them to another amended hypothesis: Scratched pennies minted in 1982 or before do not react with hydrochloric acid to form bubbles, while scratched pennies minted since 1984 do—perhaps because the metal used to make the pennies was changed between 1982 and 1984.

Page 33How could they test their modified hypothesis? First, they must make sure that all old and new pennies respond in the same way as their original 10. They must then rule out any effects of chance by testing dozens, maybe even hundreds, of pennies. Assuming they get the results their hypothesis predicted, they may go on to cut some pennies open, perhaps verifying with direct observation that the insides of new and old pennies contain different materials.

Such findings would allow them to predict the behavior of pennies in hydrochloric acid, but they would not have a complete explanation for the chemical behavior. They would need to carry the process of scientific inquiry even further.

When nuclear reactions were discovered—such as the fusion reactions that power our Sun—scientists realized that processes that release very large amounts of energy do not conserve mass. To accommodate this new information, the law of conservation of mass had to be modified to become the law of conservation of mass and energy.

Laws

When the behavior of matter is so consistent that it appears to have universal validity, we call this behavior a law. A scientific law describes the way nature operates under a specified set of conditions. For example, in the late eighteenth century, observations of the amounts of materials consumed and produced in chemical reactions led to the formulation of the law of conservation of mass. This law states that the mass of products obtained from a chemical reaction equals the mass of the substances that react. Every known chemical reaction that has been studied follows this law. For example, we could measure the mass of a penny and the mass of the nitric acid before allowing the reaction in Figure 1.30 to proceed. Upon completion of the reaction, we would find that the mass of the blue-green liquid plus the mass of the red gas is equal to the masses of our starting materials.

We will discuss the reactions of metals with acid in Chapter 5.

Theories

Earlier we developed a hypothesis about the behavior of pennies in hydrochloric acid. In the Laws section we described an example of a scientific law. Hypotheses and laws only describe how nature works, not why. Theories explain why observations, hypotheses, or laws apply under many different circumstances. For example, the atomic theory, which we will discuss in Chapter 2, explains many aspects of the behavior of matter, including the law of conservation of mass. Theories often employ mathematical or physical models that, if correct, explain the behavior of matter. Like hypotheses, a theory fits known observations. If new facts become known, theories may have to be modified or amended.

Let’s return to the behavior of pennies in hydrochloric acid. Anna and Bill need to propose a theory that explains the behavior of all the pennies. Their Internet and library research reveals that before 1982 all pennies were minted from a copper alloy containing 5% zinc. After 1984 all pennies were copper-coated disks of zinc. The newer pennies contain approximately 97.5% zinc. If Anna and Bill can find out how pure zinc and pure copper react with hydrochloric acid, they might develop an explanation for the chemical behavior of pennies in the presence of hydrochloric acid. A test with pieces of copper and zinc (Figure 1.35) indicates that zinc does indeed react with hydrochloric acid to release bubbles, while copper does not. They now have an explanation for the behavior of pennies in hydrochloric acid that is consistent with all the relevant observations. They can even explain why some of the pennies float. If the zinc is completely removed by reaction with hydrochloric acid, the copper shell that remains may fill with gas and rise to the surface of the liquid.

FIGURE 1.35 Zinc, but not copper, reacts with hydrochloric acid, producing hydrogen gas.

Scientific Inquiry in Practice

There is a perception that scientists have wild hair, dress funny, wear pocket protectors, lack social graces, and work in isolation in a laboratory. This stereotype is often perpetuated in movies, but is it real? Do you know any scientists who fit this description? Perhaps you do, but the stereotype is best left to the movies. Scientists are people, and they vary as much in their appearance, personalities, and preferences as and preferences as Page 34people in any other vocation (Figure 1.36). Although many scientists do indeed work independently in the lab, research groups who share common interests do most scientific work collaboratively. Sometimes they make discoveries that follow long periods of painstaking work. At other times, new insights arise rapidly through serendipity. These are accidental, fortunate breakthroughs, such as Alexander Fleming’s discovery of penicillin or Henri Becquerel’s discovery of radioactivity. Such serendipitous events happen only occasionally, and they happen to individuals who can recognize their importance. Both Fleming and Becquerel noticed anomalies that less skilled observers might have overlooked. As Louis Pasteur stated, “In the fields of observation, chance favors only the prepared mind.”

FIGURE 1.36 Chemists are a diverse group of people who work in a variety of environments.

While most of scientific inquiry proceeds through hypothesis testing, this is not the only form the scientific method takes. A new approach, especially useful in the search for new drugs, is combinatorial chemistry. A series of related chemical compounds is systematically prepared and tested for effectiveness in disease treatment. Many different combinations are tried, using techniques involving miniaturization, robotics, and computer control. The compounds are screened as possible candidates for drug action. For example, if a drug is needed that binds to a particular enzyme to produce a biological effect, then the various compounds would be added to that enzyme to see if they do bind. If so, additional testing would be carried out.

An alternate approach involves the mixing of many chemicals, producing many different products in one container. These are then tested, either in the mixture or after separation. With current techniques, it is possible for one laboratory to synthesize and test as many as 100,000 new compounds in a month.

To avoid generating a lot of chemical waste from the synthesis of compounds tested by combinatorial techniques or waste generated at the end of testing, scientists run these experiments on a microlevel scale. The machine like the one shown in Figure 1.37 Page 35can dispense as little as 10 to 50 μL in each sample well. An important initiative within the chemistry community is to develop chemical processes that prevent pollution and reduce the amount of natural resources used to manufacture various consumer, research, and industrial products. Green chemistry and sustainability are terms often given to describe these efforts.

FIGURE 1.37 Many chemicals can be tested at once using equipment such as that shown here. Each tip dispenses a very small quantity of chemical into a corresponding well where the chemical’s action as a potential pharmaceutical agent can be tested.

One of the primary principles of green chemistry is to prevent waste. Industrial and research facilities are working diligently to develop processes that do not generate waste that must be treated or disposed of. One way of eliminating waste is to design products that break down to harmless materials after they are used. Another principle of green chemistry is to design processes that do not deplete natural resources. Instead, green-chemistry principles recommend using raw materials that are renewable—that is, raw materials that come from agricultural sources or from the waste products of other processes. Fossil fuels are the primary source of depleting raw materials.

Math Toolbox 1.1 Scientific Notation

Many numbers used in chemistry are either very large, such as 602,200,000,000,000,000,000,000 atoms in 12 g of carbon, or very small, such as 0.0000000001 centimeter per picometer (cm/pm). It is very easy to make mistakes with such numbers, so we express them in a shorthand notation called exponential or scientific notation. The numbers used in the examples are written as (6.022 × 1023 atoms)/(12 g C) and 1 × 10−10 cm/pm in scientific notation. In this form, the numbers are easier to write, and it is easier to keep track of the position of the decimal point when carrying out calculations.

A number in scientific notation is expressed as C × 10n, where C is the coefficient and n is the exponent. The coefficient C is a number equal to or greater than 1 and less than 10 that is obtained by moving the decimal point the appropriate number of places. The exponent n is a positive or negative integer (whole number) equal to the number of places the decimal point must be moved to give C. For numbers greater than 1, the decimal point is moved to the left and the exponent is positive:

For numbers smaller than 1, the exponent is negative, since the decimal point must be moved to the right:

EXAMPLE 1.14 Converting Between Decimal Form and Scientific Notation

(a)Convert 453,000 to scientific notation.

(b)Convert 0.00052 to scientific notation.

(c)Write the value 4.0 × 10−3 in decimal form.

(d)Write the value 8.657 × 104 in decimal form.

(e)Place the following numbers in order from smallest to largest: 103, 1, 0, 10−3, 10−9.

Solution:

(a)The number 453,000 is greater than 1, so our exponent n in 10n will be a positive value. To convert to scientific notation, we move the decimal point between the 4 and 5. The decimal point is moved five decimal places to the left, so the exponent is 5. The value 453,000 in scientific notation is therefore 4.53 × 105.

(b)The number 0.00052 is less than 1, so our exponent n in 10n will be a negative value. To convert to scientific notation, we move the decimal point between the 5 and 2. The decimal point is moved four decimal places to the right, so the exponent is −4. The value 0.00052 in scientific notation is therefore 5.2 × 10−4.

(c)The number 4.0 × 10−3 has a negative exponent, so it is less than 1. To convert to decimal form, we move the decimal point to the left three places: 0.0040.

(d)The number 8.657 × 104 has a positive exponent, so it is greater than 1. To convert to decimal form, we move the decimal point to the right four places: 86,570.

(e)Zero is the lowest value since none of the numbers are negative in value. The values with negative exponents are between 0 and 1. The value with a positive exponent is the only value greater than 1. The order is 0 < 10−9 < 10−3 < 1 < 103.

Practice Problem 1.14

(a)Convert 0.0123 to scientific notation.

(b)Convert 123 to scientific notation.

(c)Write the value 2.55 × 104 in decimal form.

(d)Write the value 1 × 10−3 in decimal form.

(e)Place the following numbers in order from smallest to largest: 10−6, 0, 1, 104, −2.

Further Practice: Questions 1.3 and 1.5 at the end of the chapter

Such operations can be carried out more easily with a scientific calculator. Methods of use vary with different brands and models, so users should always consult their manual for specific instructions. However, the general approach is as follows. First enter the coefficient, including the decimal point. Then press the appropriate button for entering the exponent followed by the value of the exponent. The button is generally labeled EE, EXP, or 10x. If the exponent has a negative value, press the change-sign button before entering the
Page 36value of the exponent. Don’t confuse this button with the subtraction button. The change-sign button is usually labeled +/− or (−).

Generally, it takes fewer key strokes to input a number in exponential form using the EE or EXP buttons. For a number in the format C × 10n, the sequence of calculator buttons is C EE n or C EXP n. For example, the number 3.4 × 10−5 is input as 3.4 EE +/− 5. To test your calculator, try the mathematical operation (8 × 106) ÷ 2. We input this as 8 EE 6 ÷ 2. The answer is 4 × 106, so 4 EE 6 should appear on your calculator.

Rules of Exponents

When you carry out mathematical operations with numbers written in scientific notation, you need to know how to work on the exponential part of the number. You can usually do this with your calculator, but you should always check whether an answer from a calculator makes sense. To do this, verify the calculation by hand or make an estimate in your mind. The following instructions show how to work without a calculator. You should also perform these manipulations with your calculator to assure yourself that you can do them all.

In several of the examples, the coefficient is understood to be 1. That is, 1.0 × 104 is the same as 104. Since 1 multiplied or divided by 1 still equals 1, the coefficient does not need to be written. However, if you want to enter 104 into your calculator using the EE button, you would enter it as 1 EE 4.

To multiply exponential notation numbers, multiply the coefficients and add the exponents:

104 × 107 = 10(4 + 7) = 1011

103 × 10−5 = 10(3 +−5) = 10−2

10−4 × 104 = 10(−4 + 4) = 100 = 1

(2 × 10−4) × (4 × 104) = (2 × 4) × 10(−4 + 4) = 8 × 100 = 8

To divide numbers written in exponential notation, divide the coefficients and subtract the exponents:

To raise an exponential number to a power, raise both the coefficient and the exponent to the power by multiplying:

(104)3 = 10(4 × 3) = 1012

(10−2)−3 = 10(−2 × −3) = 106

(2 × 104)3 = (2)3 × 10(4 × 3) = 8 × 1012

To extract the root of an exponential number, take the root of both the coefficient and the exponent:

If the exponent is not evenly divisible by the root, we rewrite the number so the exponent is evenly divisible:

(107)1/2 = (10 × 106)1/2 = 101/2 × (106)1/2 = 3.16 × 103

To add or subtract numbers in exponential notation, first express both numbers with identical exponents. To do so, shift the decimal point in one of the numbers so its exponential part is the same as that of the other number. Then simply add or subtract the coefficients and carry the exponential part through unchanged. For example, suppose we want to add the following quantities:

5.2 × 104 + 7.0 × 103

We first change one number so they both have the same exponents:

1. × 103+ 7.0 × 103

Then we can add the two coefficients:

(52. + 7.0) × 103

If necessary, we shift the decimal point to return the number to proper scientific notation:

1. × 103= 5.9 × 104

EXAMPLE 1.15Mathematical Operations with Numbers Written in Scientific Notation

(a)(2.0 × 10−2) × (2.5 × 104)

(b)(4.0 × 10−2) ÷ (2.0 × 103)

(c)(3.0 × 104)2

(d)(4.3 × 102) + (6.90 × 103)

Solution:

(a)To multiply numbers in scientific notation, multiply the coefficients and add the exponents:

(2.0 × 2.5) × (10−2 + 4) = 5.0 × 102

(b)To divide numbers in scientific notation, divide the coefficients and subtract the exponents:

(4.0 ÷ 2.0) × (10−2 − 3) = 2.0 × 10−5

(c)To raise a number in scientific notation to a power, raise both the coefficient and the exponent to the power:

(3.0 × 104)2 = (3.0)2 × (104)2

When we raise an exponent to a power, we multiply the exponent by the power:

(3.0)2 × (104)2 = 9.0 × 108

(d)To add two numbers in scientific notation, we must first express both numbers with identical exponents:

(4.3 × 102) + (6.90 × 103) = (0.43 × 103) + (6.90 × 103)

(0.43 + 6.90) × 103 = 7.33 × 103

Now, confirm your answers with a calculator. If you don’t get the answers shown, you’re not inputting numbers in scientific notation correctly.

Practice Problem 1.15

(a)(4.0 × 106) × (1.5 × 10−3)

(b)(3.0 × 108) ÷ (1.0 × 10−7)

(c)(2.0 × 10−2)3

(d)(4.10 × 1010) − (2.0 × 109)

Further Practice: Questions 1.7 and 1.8 at the end of the chapter

Page 37

Math Toolbox 1.2 Significant Figures

Numbers used in chemistry can be placed into two categories. Some numbers are exact. They are established by definition or by counting. Defined numbers are exact because they are assigned specific values: 12 in = 1 ft, 2.54 cm = 1 in, and 10 mm = 1 cm, for example. Numbers established by counting are known exactly because they can be counted with no errors. Since defined and counted numbers are known precisely, there is no uncertainty in their values.

Other numbers are not exact. They are numbers obtained by measurement or from observation. They may also include numbers resulting from a count if the number is very large. There is always some uncertainty in the value of such numbers because they depend on how closely the measuring instrument and the experimenter can measure the values.

Precision and Accuracy

Uncertainty in numbers can be described in terms of either precision or accuracy. The precision of a measured number is the extent of agreement between repeated measurements of its value. Accuracy is the difference between the value of a measured number and its expected or correct value. If repetitive measurements give values close to one another, the number is precise, whether or not it is accurate. The number is accurate only if it is close to the true value.

We usually report the precision of a number by writing an appropriate number of significant figures. Significant figures in a number are all the digits of which we are absolutely certain, plus one additional digit, which is somewhat uncertain. For example, if we measure the height of a line on a graph calibrated with a line every 10 cm, we might obtain a value between 10 and 20 cm, which we estimate to be 18 cm. We are certain of the 1 in 18 cm, but not of the 8. Both of these digits are considered significant; that is, they have meaning. Unless we have other information, we assume that there is an uncertainty of at least one unit in the last digit.

With a graph calibrated more finely, in centimeters, we see that the line height is indeed somewhat more than 18 cm. The line appears to reach about three-tenths of the distance between the 18-cm and 19-cm marks, so we can estimate the height at 18.3 cm. The number of significant figures is three, two of which are certain (1 and 8) and one of which is somewhat uncertain (3).

Determining the Number of Significant Figures

Every number represents a specific quantity with a particular degree of precision that depends on the manner in which it was determined. When we work with numbers, we must be able to  Page 38recognize how many significant figures they contain. We do this by first remembering that nonzero digits are always significant, no matter where they occur. The only problem in counting significant figures, then, is deciding whether a zero is significant. To do so, use the following rules:

• A zero alone in front of a decimal point is not significant; it is used simply to make sure we do not overlook the decimal point (0.2806, 0.002806).
• A zero to the right of the decimal point but before the first nonzero digit is simply a place marker and is not significant (0.002806).
• A zero between nonzero numbers is significant (2806, 0.002806).
• A zero at the end of a number and to the right of the decimal point is significant (0.0028060,2806.0).
• A zero at the end of a number and to the left of the decimal point (28060) may or may not be significant. We cannot tell by looking at the number. It may be precisely known, and thus significant, or it may simply be a placeholder. If we encounter such a number, the best we can assume is that any trailing zeros are not significant.

The following table summarizes the significant figures in the numbers we just considered. The digits that are significant are highlighted.

 Number Count of Significant Figures 0.2806 4 0.002806 4 2806 4 0.0028060 5 2806.0 5 28060 Assume 4

To avoid creating an ambiguous number with zeros at its end like 28060, we write the number in scientific notation (see Math Toolbox 1.1), so that the troublesome zero occurs to the right of the decimal point. In this case, it is simple to show whether the zero is significant (2.8060 × 104) or not (2.806 × 104). The power of ten, 104, is not included in the count of significant figures, since it simply tells us the position of the decimal point.

EXAMPLE 1.16Number of Significant Figures

Determine the number of significant figures in the following:

(a)0.060520

(b)5020.01

Solution:

(a)Zeros before the first nonzero digit are not significant. Zeros at the end of a number and to the right of the decimal point are significant. Zeros between nonzero digits are significant. Thus, all but the first two zeros are significant, giving five significant figures.

(b)Zeros between nonzero digits are significant, so all digits in this number are significant. There are six significant figures in this number.

Practice Problem 1.16

Determine the number of significant figures in the following:

(a)14800.0

(b)0.00076

Further Practice: Questions 1.9 and 1.10 at the end of the chapter

Significant Figures in Calculations

We must also be concerned about the proper expression of numbers calculated from measured numbers. Calculators will not do this for us, so we have to modify their output. The rules for determining the proper number of significant figures in mathematical manipulations of measured numbers are simple. To determine the proper number of significant figures in the answer to a calculation, we consider only measured numbers. We need not consider numbers that are known exactly, such as those in conversions such as 1 foot (ft) = 12 inches (in). In the following examples, we will not use units, so that we can focus on the manipulation of the numbers. Remember, however, that a unit must accompany every measured number.

Multiplication and Division

In a multiplication or division problem, the product or quotient must have the same number of significant figures as the least precise number in the problem. Consider the following example:

2.4 × 1.12 = ?

The first number has two significant figures, and the second has three. Since the answer must match the number with the fewest significant figures, it should have two:

2.4 × 1.12 = 2.7  (not 2.69)

EXAMPLE 1.17 Significant Figures in Multiplication and Division

Express the answers to the following operations with the proper number of significant figures:

(a)5.27 × 3.20

(b)(1.5 × 106) × 317.832

(c)6.0/2.9783

(d)(2.01)3

Solution:

(a)Each number has three significant figures, so the answer should have three:

5.27 × 3.20 = 16.9

(b)The first number has two significant figures, while the second has six. The answer should have two significant figures, the same as the number with fewer significant figures:

(1.5 × 106) × 317.832 = 4.8 × 108

(c)The numerator has two significant figures; the denominator has five. The answer should have two significant figures:

Page 39(d)A number raised to a power is the same as multiplying that number by itself as many times as indicated by the power, so the answer should have the same number of significant figures as the number:

(2.01)3 = 2.01 × 2.01 × 2.01 = 8.12

Practice Problem 1.17

Express the answers to the following operations with the proper number of significant figures:

(a)5.01 × 1.5

(b)0.009/3.00

(c)3.60 × 4.2 × 1010

(d)(1.85)2

Further Practice: Questions 1.11 and 1.12 at the end of the chapter

A sum or difference can only be as precise as the least precise number used in the calculation. Thus, we round off the sum or difference to the first uncertain digit. If we add 10.1 to 1.91314, for instance, we get 12.0, not 12.01314, because there is uncertainty in the tenths’ position of 10.1, and this uncertainty carries over to the answer. In this case, it is not the number of significant figures that is important but rather the place of the last significant digit. Consider another example:

0.0005032 + 1.0102 = ?

The first number has four significant figures and the second number has five. However, this information does not determine the number of significant figures in the answer. If we line these numbers up, we can see which digits can be added to give an answer that has significance:

The digits after the 5 in the first number have no corresponding digits in the second number, so it is not possible to add them and obtain digits that are significant in the sum. The answer is 1.0107, which has five significant figures, dictated by the position of the last significant digit in the number we added with the greatest uncertainty (1.0102).

EXAMPLE 1.18Significant Figures in Addition and Subtraction

Express the answers to the following operations with the proper number of significant figures:

(a)24 + 1.001

(b)24 + 1.001 + 0.0003

(c)428 − 0.01

(d)14.03 − 13.312

Solution:

(a)The first number has no digits after the decimal point, while the second has three. The digits after the decimal point will have no significance:

24 + 1.001 = 25

(b)Only the numbers to the left of the decimal point will have significance in the answer:

24 + 1.001 + 0.0003 = 25

(c)The second number has no digits to the left of the decimal point, while the first number has all its significant digits to the left of the decimal point, so the subtracted number does not affect the answer:

428 − 0.01 = 428

(d)The first number has two digits to the right of the decimal point; the second has three. The answer should have only two digits past the decimal:

14.03 − 13.312 = 0.72

Practice Problem 1.18

Express the answers to the following operations with the proper number of significant figures:

(a)1.003 + 1.02 + 1.0

(b)1.0003 − 0.10

(c)2.5020 − 2.50

(d)7.10 + 13.473

Further Practice: Questions 1.13 and 1.14 at the end of the chapter

Calculations Involving Multiple Steps

When calculations involve multiple steps, the number of significant figures in subsequent steps requires us to know the number of significant figures in the answers from the previous steps. Therefore, we must keep track of the last significant figure in the answer to each step. Let’s consider a two-step calculation involving both subtraction and division:

We always begin with steps set off in parentheses. In this case the first step is the subtraction step:

The significant figure rules for subtraction tell us that this answer should be rounded to one decimal place (2.2 with two significant figures). However, to prevent rounding errors, we do not yet want to round. To keep note that the answer to our first step has only two significant figures, we will underline the significant digit farthest to the right:

12.26 − 10.1 = 2.16

Now we can use this value for the division step. With only two significant figures, the result from the previous step limits the number of significant figures in the second step to two significant figures:

EXAMPLE 1.19Significant Figures in Calculations Involving Multiple Steps

Express the answers to the following calculations with the proper number of significant figures:

Solution:

(a)We begin with the step in parentheses, the one in the denominator. This gives a value to the ones place:

When we divide a number with four significant figures by a number with two significant figures, we get an answer with two significant figures:

(b)Squaring the value in parentheses gives a value with two significant figures with the last significant figure in the tens’ place:

The result of the division step gives a value with two significant figures with the last significant figure in the one-hundredths’ place:

The sum of these values will have two decimal places:

(c)The result of multiplication gives a value with two significant figures:

(8.9 × 10.2) + 4.25 = 90.78 + 4.24

The sum gives a value with the last significant digit in the ones’ place:

(d)In this problem, we need to consider the number of decimal places in each value in the numerator. The numbers in this problem also have units we’ll want to consider. Rewriting the numbers in the numerator in decimal form shows us that the answer must have the last significant figure in the 0.001 place. Also, the unit on the subtraction is grams.

For the denominator, we have to cube both the number and the cm units:

Dividing a value with one significant figure by a value with three significant figures gives a result with just one significant figure:

Practice Problem 1.19

Express the answers to the following calculations with the proper number of significant figures:

Further Practice: Questions 1.15 and 1.16 at the end of the chapter

Rounding Off Numbers

When a number contains more digits than are allowed by the rules of significant figures, we drop the digits after the last significant figure, using the following procedure:

• If the first digit being dropped is less than 5, leave the last significant figure unchanged. (For example, rounding to three significant figures, 6.073 becomes 6.07, and 6073 becomes 6.07 ×103.)

Page 41•If the first digit being dropped is greater than 5 or is 5 followed by digits other than zero, increase the last significant figure by one unit (rounding to three significant figures, 6.077 becomes 6.08, and 60,751 becomes 6.08 × 104).

• If the first digit being dropped is 5 followed only by zeros or by no other digits, then increase the last significant figure by one unit if it is odd (6.075 becomes 6.08) but leave it unchanged if it is even (6.085 becomes 6.08).

EXAMPLE 1.20Rounding Numbers

Round each of the following numbers to three significant figures:

(a)3245

(b)12.263

(c)0.001035

(d)312,486

(e)312,586

Solution:

(a)With four significant figures, the last digit must be dropped to convert to three significant figures. Since the second to the last digit is even, it remains unchanged: 3240 or 3.24 × 103.

(b)The last two digits must be dropped. Since the first one dropped is greater than 5, the previous digit is increased by 1: 12.3.

(c)The last digit must be dropped. Since this digit is a 5 and the previous digit is odd, the previous digit is increased by 1: 0.00104.

(d)The last three digits must be dropped. Since the first one dropped is less than 5, the previous digit is unchanged: 312,000 or 3.12 × 105.

(e)The last three digits must be dropped. Since the first one dropped is 5 followed by nonzero digits, the previous digit must be increased by 1: 313,000 or 3.13 × 105.

Practice Problem 1.20

Round each of the following numbers to two significant figures:

(a)1.502

(b)13.50

(c)0.001056

(d)1200.0

(e)0.2035

Further Practice: Questions 1.17 and 1.18 at the end of the chapter

Math Toolbox 1.3 Units and Conversions

All experimental sciences are based on observation. To be useful to the experimenter and to others, observations often must involve making measurements. Measurement is the determination of the size of a particular quantity—the number of nails, the mass of a brick, the length of a wall. Measurements are always defined by both a quantity and a unit, which tells what it is we are measuring.

But what system of units do we use? Most countries use the metric system, while the United States still uses primarily the English system. The English system is based on various units that are not related to one another by a consistent factor. Common units of length, for example, are inches, feet (12 in), yards (3 ft), and miles [1760 yards (yd)]. The metric system, on the other hand, uses units that are always related by a factor of 10, or by some power of 10. Some units of length are centimeters (10−2 m), decimeters (10 cm or 10−1 m), meters, and kilometers (1000 m). Scientists in all countries use the metric system in their work.

The metric system is convenient to use because its units are related by powers of 10. Converting between related units is simply a matter of shifting a decimal point. The metric system adds a further convenience. It does not use arbitrary names like the English system, but rather, it defines base units of measure, and any multiple or fraction of these base units is defined by a special prefix. Some of the most common prefixes are listed here.

 Prefix Factor Symbol Example giga 109 G 1 Gg = 1,000,000,000 g mega 106 M 1 Mg = 1,000,000 g kilo 103 k 1 kg = 1000 g deci 10−1 d 1 dg = 0.1 g centi 10−2 c 1 cg = 0.01 g milli 10−3 m 1 mg = 0.001 g micro* 10−6 μ 1 μg = 0.000001 g nano 10−9 n 1 ng = 0.000000001 g pico 10−12 p 1 pg = 0.000000000001 g

*The microgram (μg) is sometimes abbreviated mcg in the health professions.

For factors that have a negative exponent, alternative relationships can be derived. For example, 100 cg = 1 g, 1000 mg = 1 g, 1,000,000 μg = 1 g, and so forth.

Many science and health-related disciplines require conversions between different systems. Some of the relationships listed here may be useful in converting between English and metric units.

 English Unit Metric Unit 1 lb = 16 oz 453.6 g 1 in 2.54 cm (exactly) 1 yd 0.9144 m 1 mi 1.609 km 1 fluid oz 29.57 mL 1 qt 0.9464 L 1 gal 3.785 L 1 ft3 28.32 L

Page 42

Metric Base Units and Derived Units

In 1960 an organization of scientists met in France to determine standards for scientific measurements. This group established the SI units (from the French, Système Internationale) listed here.

 Unit Symbol Quantity meter m length kilogram kg mass second s time ampere A electric current kelvin K temperature mole mol amount of substance candela cd luminous intensity

All other units, called derived units, are based on these by multiplication or division and then applying the appropriate prefix or combination of units. For example, if the mass of an object is measured in grams, it could be converted to the SI unit of mass, the kilogram, by dividing the measured mass in grams by 1000 and adding the prefix kilo to the quantity. (We’ll discuss conversions of units in the next section.) An example of a derived unit that involves a combination of units is volume. The SI-derived unit for volume is cubic meters (m3), which for a box would mean measuring length, width, and height in meters, and then multiplying them together. Another derived unit, discussed in Section 1.2, is density. Density is a ratio of mass to volume for a given substance. What is the SI-derived unit for density?

Conversion of Units

Measurements must be interpreted, often by mathematical manipulation of the data. This manipulation often involves converting one set of units into another using relationships between them. Two approaches to such conversions are ratios and dimensional analysis. In both cases, an analysis of the units provides clues to the correct solution of the problem.

Consider a problem that can be solved almost automatically: “Convert 30 min into hours.” You probably gave the answer without thinking: 30 min is one-half hour (0.50 h). But how did you know this answer? Let’s systematize the problem-solving process. First—even if we don’t always consciously think about it—we must decide what the problem is asking for. This problem requires converting a number of minutes into the corresponding number of hours. Next, we must know the number of minutes in 1 h. From this information, write a mathematical expression that shows the equivalence of two quantities having different units:

1 h = 60 min

We can use this expression in two ways to solve this problem. First, we will describe the ratio approach.

Ratio Approach

In this approach, we use a known relationship to compare with our unknown relationship. We know how many minutes make up 1 h, but we want to know how many hours there are in 30 min. From the known relationship, we develop a ratio of equivalent quantities that is equal to 1 and that has different units in the numerator and denominator. To develop a ratio to convert minutes into hours, we use the equivalent quantities and divide 1 h by 60 min.

The ratio for our unknown relationship should equal the known ratio as long as we use the same units, so we can set the unknown ratio equal to the known ratio:

We can cross multiply to solve for x:

x h × 60 min = 1 h × 30 min

We can then solve for x by dividing both sides by 60 min:

We then carry out the mathematical operations:

Notice that the units of minutes cancel out, leaving the units of hours, which is the unit called for in the problem.

Dimensional Analysis

We also use the ratio of equivalent quantities in the dimensional-analysis approach. In this approach, however, we get to the form of the manipulated ratio more quickly. We multiply the known quantity by the ratio, so that units cancel and we get the unknown quantity with the desired units:

Notice how the starting units (minutes) cancel out, leaving the desired units (hours) to accompany the calculated number. In the dimensional-analysis approach, units in the numerator and denominator of a fraction are treated exactly the same as numbers—canceled out, multiplied, divided, squared, or whatever the mathematical operations demand. We set up the conversions so that desired units are introduced and beginning units cancel out.

Now reexamine the conversion just developed. From the same equivalence expression, 1 h = 60 min, we can derive another ratio by dividing 60 min by 1 h:

What happens when the original quantity is multiplied by this conversion factor?

This answer is clearly wrong because the units make no sense. There are always two possible ratios of the known equivalent quantities, but only one gives the proper units in the answer.
Page 43Which ratio is appropriate to use depends on how the problem is stated. But in every case, using the proper ratio cancels out the old units and leaves intact the desired units. Using the improper ratio results in nonsensical units (like min2/h). This is a clear warning that the ratio was upside down or otherwise incorrect.

EXAMPLE 1.21Converting Units

Use the dimensional-analysis approach to convert 75.0 km to meters.

Solution:

We begin by determining if there is a relationship between kilometers and meters:

From the table of metric units we see that 1 km = 1000 m, so there is a relationship between these quantities:

From the equivalence expression, 1 km = 1000 m, we can derive two ratios (conversion factors), one of which we’ll use in the calculation:

Only one of these expressions, when multiplied by the given quantity in kilometers, will correctly cancel the units. What are the units if we multiply 75.0 km by the first ratio?

The units tell us that our first attempt leads to an answer that is nonsense. Applying the other conversion factor, units cancel and lead to an answer that makes sense:

Notice that our given quantity has three significant figures. To show that we know the answer in the conversion to three significant figures, we should express it in scientific notation: 7.50 × 104 m. (See Math Toolboxes 1.1 and 1.2 for details about scientific notation and significant figures.)

Practice Problem 1.21

Convert 0.0276 kg to grams.

Further Practice: Questions 1.19 and 1.20 at the end of the chapter

An Approach to Problem Solving

The processes just discussed can be generalized by the following diagram:

Consider the following problem: A liquid fertilizer tank has a volume of 6255 ft3. How many gallons of liquid fertilizer can fit into this tank?

1.Decide what the problem is asking for.

First, read the problem carefully. If you are not sure what a term means, look it up. If it’s necessary to use an equation to solve the problem, be sure you understand the meaning of each symbol. Look for clues in the problem itself—words or phrases such as determine, calculate, what mass, what volume, or how much. After deciding what quantity the problem is asking for, write the units in which this quantity must be stated. In the fertilizer tank example, we want to find the volume of the tank in units of gallons from units of cubic feet.

2.Decide what relationships exist between the information given in the problem and the desired quantity.

If necessary, recall or look up equivalence relationships (such as 1 h = 60 min) inside the back cover. This information may not be given in the problem itself. In the fertilizer tank example, we need to convert volume in cubic feet to volume in gallons, so we need an equivalence relationship between those two quantities. The expression is 1 ft3 = 7.481 gal.

The additional information that describes the relationship between two quantities won’t always be a direct equivalence. A series of equivalence expressions and their derived conversion factors may be needed. Sometimes a mathematical equation is required to express the equivalence between two quantities [e.g., T°C = (T°F − 32)/1.8]. At other times, some chemical principle may have to be applied.

A word of caution: Just as a problem may contain less information than you need to solve it, it may also contain more information than you need. Never assume that you must use all of the information. Examine all information critically and reject any that is not pertinent.

3.Set up the problem logically, using the relationships decided upon in step 2.

Starting with the relationships you obtained in step 2, develop the ratios needed to arrive at the final answer. Be sure to set up the ratios so that the old units cancel out and the desired units are introduced in the appropriate positions.

In the fertilizer tank example, the ratio is (7.481 gal)/(1 ft3). The conversion is as follows:

Page 44If you need a series of ratios to find the final answer, it is often helpful to “map out” the route you will follow to get there. Suppose you need to know how many centimeters are in 1.00 mile (mi). This conversion can be summarized by a diagram:

Unfortunately you can only find a table that gives the number of feet in a mile, the number of inches in a foot, and the number of centimeters in an inch. The diagram to get from miles to centimeters might look like this:

The problem setup would follow this progression:

4.Check the answer to make sure it makes sense, both in magnitude and in units.

This step is just as important as the others. You must develop an intuitive feeling for the correct magnitude of physical quantities. Suppose, for example, you were calculating the volume of liquid antacid an ulcer patient needed to neutralize excess stomach acid, and the result of your calculation was 32 L. Now, units of liters look reasonable, since the liter is a valid unit of volume; but it should be obvious that something is wrong with the numerical answer. Since a liter is about the same volume as a quart, this patient would need to drink 8 gal of antacid! The source of the problem is probably an arithmetic error, or possibly omission of the metric prefix milli- (10−3) somewhere, since 32 mL would be an appropriate volume to swallow. In any event, a result such as this one should be recognized as unacceptable and the setup and calculations checked for errors.

In the fertilizer tank example, the units cancel properly, so the answer should be correct, if there are no arithmetic errors. Since there are about 7.5 gal in 1 ft3, the answer should be greater than the volume in cubic feet by something less than a factor of 10. The answer is indeed within this expected range.

EXAMPLE 1.22Problem Solving

If a laser beam fired from the moon takes 1.30 s to reach Earth, what is the distance in meters between the moon and Earth? Light travels in a vacuum at a speed of 3.00 × 1010 cm in 1 s or 3.00 × 1010 cm/s.

Solution:

We begin by determining if there is a relationship between seconds and meters:

There isn’t, but the speed given, 3.00 × 1010 cm/s, is a relationship between time and distance in centimeters. We could use this relationship to calculate the distance in centimeters and then convert to meters.

The ratios for converting seconds to centimeters can be expressed in two ways:

>

When multiplied by the time, 1.30 s, the first one cancels units of seconds and gives an answer in centimeters. To convert centimeters to meters, we use the equivalence expression

1 cm = 10−2 m

Rather than working with negative exponents, it is sometimes useful to work in whole numbers. Dividing both sides of the expression by the quantity of the negative exponent provides a whole number ratio that may be more convenient:

Two ratios can be written to describe this relationship:

When multiplied by the distance in centimeters, the second expression will correctly cancel units and give an answer in meters.

Multiplying the given time by the two conversion factors provides a sequence of steps that allows us to cancel the time units (seconds) and distance in centimeters, giving the distance from the moon to Earth in meters:

Does this answer make sense? Yes, because we would expect there to be a great distance between the moon and Earth. If we had inversed the conversion factors, we would have gotten 4.33 × 10−9 and units that do not make sense. The answer, 3.90 × 108 m, has three significant figures, which is the same as the number in the given quantity.

Practice Problem 1.22

A tablet of a typical pain reliever contains 200 mg of active ingredient. How many pounds of active ingredient are in a tablet? (Assume there are three significant figures in the 200-mg quantity.)

Further Practice: Questions 1.21 and 1.22 at the end of the chapter

EXAMPLE 1.23Problem Solving with More Complex Units

Amoxicillin is an antibiotic commonly used to treat bacterial infections. When given to infants, the dosage must be carefully determined. The usual daily dosage for infants under 3 months old is 30.0 milligrams (mg) per kilogram (kg) of body weight. What mass of amoxicillin in milligrams should be given to an 8.00 lb infant in one day?

Solution:

We begin by determining a plan. If we knew the body weight of the infant in kg, then we could use the daily dosage per 1 kg body weight given to determine the mass of the daily dose:

However, we were given the body weight in pounds (lb), not kg, so we must first convert 8.00 lb to units of kg:

Now we can set up the two-step calculation, starting with the given body weight in units of lbs, and then multiplying by the conversion ratios written such that units cancel:

Practice Problem 1.23

The TGV POS high-speed train in France has a clocked speed of 574.8 km/h. Convert this speed to the following units:

(a)miles per hour (mi/h)

(b)meters per second (m/s)

Further Practice: Questions 1.23 and 1.24 at the end of the chapter

CHAPTER REVIEW

KEY CONCEPTS

• Matter can be classified in different ways based on its characteristics. Matter can be classified as an element, a compound, or a mixture of one or both.

°An element is a pure substance that cannot be broken down into substances by a chemical reaction.

°Elements generally fall into two main categories: metals and nonmetals.

°A compound is a pure substance composed of two or more elements combined in definite proportions.

°A mixture consists of two or more pure substances and may vary in composition. A homogeneous mixture (solution) has a uniform composition, while a heterogeneous mixture does not.

°An atom is the smallest unit of an element that has the chemical properties of that element.

°Molecules are made up of two or more atoms bound together in a discrete arrangement. Molecules can be either elements or compounds.

°Elements and compounds are commonly represented using element symbols and chemical formulas.

°Matter can also be classified by its physical state: solid, liquid, or gas.

Page 46•A change in a substance can be classified as either physical or chemical.

°In a physical change, the identity of the substance(s) remains unchanged.

°A physical property is a characteristic that we can observe or measure without changing the composition of a substance. Examples include color, odor, physical state, mass, volume, density, and temperature.

°In a chemical change (chemical reaction), atoms rearrange to form new substances.

°A chemical property is determined by what it is composed of and what chemical changes it can undergo.

• Physical and chemical changes involve energy, the capacity to do work or to transfer heat.

°The energy of an object is a combination of its kinetic and potential energy.

°Energy can change from one form to another, and it can be transferred as heat.

• The scientific method includes a variety of methods of inquiry employed by scientists.

°Observation (or data collection) frequently is a necessary part of the scientific method, and frequently involves the design of carefully controlled experiments.

°Hypotheses are tentative explanations for the results of experiments.

°Laws result when observations appear to have universal validity, and theories are explanations for laws.

KEY RELATIONSHIPS

 Relationship Equation The density of an object is the ratio of its mass and volume. The absolute temperature in kelvins is offset from the Celsius temperature by 273.15. TK = T°C + 273.15 The Fahrenheit temperature has degrees that are 1.8 times as large as Celsius degrees. The freezing point of water is set at 32°F and 0°C. T°F = 1.8(T°c) + 32

KEY TERMS

aqueous solution (1.1)

atom (1.1)

chemical change (1.2)

chemical formula (1.1)

chemical property (1.2)

chemical reaction (1.2)

compound (1.1)

density (1.2)

element (1.1)

element symbol (1.1)

energy (1.3)

gas (1.1)

heterogeneous mixture (1.1)

homogeneous mixture (1.1)

hypothesis (1.4)

kinetic energy (1.3)

liquid (1.1)

mass (1.1)

matter (1.1)

metal (1.1)

mixture (1.1)

molecule (1.1)

nonmetal (1.1)

physical change (1.2)

physical property (1.2)

physical state (1.1)

potential energy (1.3)

pure substance (1.1)

scientific method (1.4)

solid (1.1)

solution (1.1)

temperature (1.2)

volume (1.2)

work (1.3)

Page 47

QUESTIONS AND PROBLEMS

The following questions and problems, except those in Additional Questions and Concept Review Questions, are paired. Questions in a pair focus on the same concept. Answers to selected questions and problems are in Appendix D.

Matching Definitions with Key Terms

1.1Match the key terms with the following descriptions.

 (a) a measure of the quantity of matter (b) a characteristic of a substance involving the possible transformations that the substance can undergo to produce a new substance (c) a combination of two or more substances that can be separated by physical means (d) a pure substance that cannot be broken down into simpler stable substances in a chemical reaction (e) the capacity to do work or to transfer heat (f) a characteristic of a substance that can be observed without changing its composition (g) the physical state in which matter has no characteristic shape but takes the shape of the filled portion of its container (h) the ratio of the mass of a substance to its volume (i) a mixture with uniform composition (j) the physical state of matter characterized by a fixed shape and low compressibility

1.2Match the key terms with the following descriptions.

 (a) the smallest particle of an element that retains the characteristic chemical properties of that element (b) a change in which substances are converted into new substances that have compositions and properties different from those of the original substances (c) anything that occupies space and is perceptible to the senses (d) a substance composed of two or more elements combined in definite proportions (e) a combination of atoms of one or more elements (f) a process characterized by changes only in the physical properties of a substance, not in its composition (g) the physical state in which matter has no fixed shape or volume but expands to fill its container completely (h) the energy possessed by an object because of its position (i) a tentative explanation for the properties or behavior of matter that accounts for a set of observations and can be tested (j) the energy possessed by an object because of its motion

Math Toolbox Questions

1.3Convert each of the following values to scientific notation.

 (a) 29,500 (b) 0.000082 (c) 650,000,000 (d) 0.0100

1.4Convert each of the following values to scientific notation.

 (a) 0.00010 (b) 4500 (c) 90,100,000 (d) 0.0000079

1.5Convert each of the following values from scientific notation to decimal form.

 (a) 1.86 × 10−5 (b) 1 × 107 (c) 4.53 × 105 (d) 6.1 × 10−3

1.6Convert each of the following values from scientific notation to decimal form.

 (a) 8.2 × 103 (b) 2.025 × 10−6 (c) 7 × 10−2 (d) 3.0 × 10−8

1.7For each of the following, carry out the mathematical operation and report answers in scientific notation.

 (a) (3.1 × 105) × (2.0 × 10−2) (b) (7.0 × 109) ÷ (2.0 × 102) (c) (2.8 × 10−4) ÷ (9.6 × 10−2) (d) (5.0 × 10−4)2 (e) (8.50 × 105) − (3.0 × 104) (f) (6.4 × 10−3) ÷ (4.0 × 103)

1.8For each of the following, carry out the mathematical operation and report answers in scientific notation.

 (a) (8.2 × 106) ÷ (4.1 × 10−2) (b) (3.5 × 103)4 (c) (6.6 × 10−5) ÷ (2.2 × 105) (d) (4.23 × 102) × (2.0 × 10−8) (e) (8.44 × 105) + (1.2 × 104) (f) (4.58 × 105) ÷ (3.64 × 10−3)

1.9Determine the number of significant figures in the following.

 (a) 0.0950 (b) 760 (c) 1.005 (d) 0.0052 (e) 3.46 × 1015

1.10Determine the number of significant figures in the following.

 (a) 1.48 × 10−2 (b) 156.0 (c) 0.04350 (d) 70.25 (e) 140

1.11Express the answers to the following operations with the proper number of significant figures.

 (a) 1.2 × 1.216 (b) 3.000/2.0 (c) (1.201 × 103) × (1.2 × 10−2) (d) (1.44)1/2

1.12Express the answers to the following operations with the proper number of significant figures.

 (a) (1.600 × 10−7)(2.1 × 103) (b) (1.33)3 (c) 1.93 × 2.651 (d) 4.4/2.200

1.13Express the answers to the following operations with the proper number of significant figures.

 (a) 1.6 + 1.15 (b) 41.314 − 41.03 (c) 1.8 + 1.022 + 0.001 (d) 0.050 − 0.0012

1.14Express the answers to the following operations with the proper number of significant figures.

 (a) 87.5 + 1.3218 (b) 422 − 410.2 (c) 1.2316 − 1.01 (d) 1.5678 + 0.23 + 0.0006

1.15Express the answers to the following calculations with the proper number of significant figures and with the correct units.

(Assume that the 2 with no units is an exact number.)

 (c) (0.35 m × 0.55 m) + 25.2 m2

1.16Express the answers to the following calculations with the proper number of significant figures and with the correct units.

 (a) (0.25 m/s) × (45.77 s) + 5.0 m (b) (c) (9.0 cm × 15.1 cm × 10.5 cm) + 75.7 cm3

1.17Round each of the following numbers to three significant figures.

 (a) 1.2126 (b) 0.2045 (c) 1.8351 (d) 42.186 (e) 0.007101

1.18Round each of the following numbers to three significant figures.

 (a) 0.020450 (b) 1.3602 × 104 (c) 13.475 (d) 16.225 (e) 1.0001

1.19Carry out the following conversions. Report your answers to the correct number of significant figures.

 (a) 36 mm to m (b) 357 kg to g (c) 76.50 mL to L (d) 0.0084670 m to cm (e) 597 nm to m (f) 36.5 in to cm (g) 168 lb to g (h) 914 qt to L (i) 44.5 cm to in (j) 236.504 g to lb (k) 2.0 L to qt

1.20Carry out the following conversions. Report your answers to the correct number of significant figures.

 (a) 75.5 km to m (b) 25.7 g to mg (c) 0.516 L to dL (d) 5.2 cm to m (e) 0.000000450 m to nm (f) 12 in to cm (g) 25.6 lb to g (h) 4.005 qt to L (i) 934 cm to in (j) 155 g to lb (k) 22.4 L to qt

1.21Carry out the following conversions. Report your answers to the correct number of significant figures.

 (a) 947 m to mi (b) 6.74 kg to lb (c) 250.4 mL to gal (d) 2.30 dL to mL (e) 0.000450 cm to nm (f) 37.5 in to m (g) 689 lb to kg (h) 0.5 qt to mL (i) 125 cm to ft (j) 542 mg to lb (k) 25 nL to gal

1.22Carry out the following conversions. Report your answers to the correct number of significant figures.

 (a) 32 cm to ft (b) 0.579 kg to lb (c) 22.70 μL to qt (d) 9212 mm to km (e) 465 nm to mm (f) 4 ft to cm (g) 2.7 lb to kg (h) 8.320 qt to mL (i) 375 km to ft (j) 62 g to oz (k) 752 mL to gal

1.23Carry out the following conversions. Report your answers to the correct number of significant figures.

 (a) 375 m/s to ft/min (b) 24.5 cm3 to in3 (c) 19.3 g/mL to lb/in3

1.24Carry out the following conversions. Report your answers to the correct number of significant figures.

 (a) 27 ft/s to cm/min (b) 2764 ft3 to m3 (c) 0.927 g/mL to lb/gal

Matter and Its Classification

1.25How would you classify the following items observed by Bill and Anna?

 (a) water in a fountain containing dissolved dye (b) a copper pipe (c) the contents of a balloon after blowing it up by mouth (d) a slice of pizza

1.26How would you classify the following items observed by Bill and Anna?

 (a) sand in a volleyball court (b) a baseball bat made entirely of aluminum (c) the contents of a balloon filled from a helium tank (d) a glass filled with a soft drink

1.27Which of the following are examples of matter?

 (a) sunlight (b) gasoline (c) automobile exhaust (d) oxygen gas (e) iron pipe

1.28Which of the following are not examples of matter?

 (a) light from a fluorescent bulb (b) sand (c) wheelbarrow rolling down a ramp (d) helium balloons (e) heat from a welding torch

1.29How are elements distinguished from compounds?

1.30How are homogeneous mixtures distinguished from heterogeneous mixtures?

1.31List characteristics of metals.

1.32List characteristics of nonmetals.

1.33Name the following elements.

 (a) Ti (b) Ta (c) Th (d) Tc (e) Tl

1.34Name the following elements.

 (a) C (b) Ca (c) Cr (d) Co (e) Cu (f) Cl (g) Cs

1.35Name the following elements.

 (a) B (b) Ba (c) Be (d) Br (e) Bi

1.36Name the following elements.

 (a) S (b) Si (c) Se (d) Sr (e) Sn

1.37Name the following elements.

 (a) N (b) Fe (c) Mn (d) Mg (e) Al (f) Cl

1.38Name the following elements.

 (a) Be (b) Rb (c) Ni (d) Sc (e) Ti (f) Ne

Page 491.39What are the symbols for the following elements?

 (a) iron (b) lead (c) silver (d) gold (e) antimony

1.40What are the symbols for the following elements?

 (a) copper (b) mercury (c) tin (d) sodium (e) tungsten

1.41A chemical novice used the symbol Ir to represent iron. Is this an acceptable symbol for the element? If not, what is the correct symbol for iron?

1.42A chemical novice used the symbol SI to represent silicon. Is this an acceptable symbol for the element? If not, what is the correct symbol?

1.43The symbol NO was used by a student to represent nobelium, an unstable, synthetic element. Is this an acceptable symbol for the element? If not, what is the correct symbol?

1.44A student used the symbol CO to represent cobalt, an element found in vitamin B12. Is this an acceptable symbol for the element? If not, what is the correct symbol?

1.45Classify each of the following as a pure substance, a homogeneous mixture (solution), or a heterogeneous mixture: hamburger, salt, soft drink, and ketchup.

1.46Classify each of the following as a pure substance, a homogeneous mixture (solution), or a heterogeneous mixture: sand, boardwalk, ocean, and roller coaster.

1.47Elemental hydrogen normally exists as two hydrogen atoms bound together. Write the formula for this molecule, and draw a picture to represent how it might look on a molecular level.

1.48Elemental chlorine normally exists as two chlorine atoms bound together. Write the formula for this molecule, and draw a picture to represent how it might look on a molecular level.

1.49This image is a representation for a compound containing nitrogen and oxygen. Write the formula for this compound.

1.50This image represents a compound containing phosphorus and chlorine. Write the formula for this compound.

1.51Which of the images represents a mixture of an element and a compound?

1.52Which of the images in Question 1.51 represents a pure substance that is a compound?

1.53Classify each of the following as an element or a compound.

 (a) O2 (b) Fe2O3 (c) P4 (d) He (e) NaCl (f) H2O

1.54Classify each of the following as an element or a compound.

 (a) hydrogen gas (b) water (c) salt (d) nitrogen dioxide (e) aluminum chloride (f) neon

1.55Under normal conditions, mercury is a liquid. Draw molecular-level pictures of what mercury atoms might look like in the liquid and solid states.

1.56Under normal conditions, bromine is a liquid. Draw molecular-level pictures of what bromine, Br2, might look like in the liquid and gaseous states.

1.57What type of matter expands to fill its container and can be compressed to a smaller volume?

1.58What type of matter is composed of particles that do not move past one another?

1.59Identify the physical state of each of the following elements from their symbols.

 (a) Cl2(g) (b) Hg(l) (c) C(s)

Page 501.60Identify the physical state of each of the following compounds from their symbols.

 (a) NaCl(s) (b) CH3OH(l) (c) CO2(g)

1.61What physical state is represented in this diagram?

1.62Draw a picture of the gaseous state of the substance shown in Question 1.61.

1.63How might you symbolically represent a homogeneous mixture of oxygen, O2, and water?

1.64Why does the symbol H2O(aq) make no sense?

Physical and Chemical Changes and Properties of Matter

1.65At the beginning of the chapter, Anna and Bill made many observations about what they saw. Some of their observations included color, texture, and shininess. Are these physical or chemical properties?

1.66At the beginning of the chapter, you were asked to classify processes like a truck running on gas or welding pieces of metal. After reading the chapter, would you classify these processes as physical or chemical changes?

1.67A slice of Swiss cheese contains 45 mg of sodium.

 (a) What is this mass in units of grams? (b) What is this mass in units of ounces (oz)? (16 oz = 453.6 g) (c) What is this mass in pounds (lb)? (1 lb = 453.6 g)

1.68A package of Swiss cheese has a mass of 0.340 kg.

 (a) What is this mass in grams? (b) What is this mass in ounces (oz)? (16 oz = 453.6 g) (c) What is this mass in pounds (lb)? (1 lb = 453.6 g)

1.69A grain of salt has a mass of about 1.0 × 10−4 g. What is its mass in the following units?

 (a) milligrams (b) micrograms (c) kilograms

1.70If a dog has a mass of 15.2 kg, what is its mass in the following units?

 (a) grams (b) milligrams (c) micrograms

1.71If you drank 1.2 L of a sports drink, what volume did you consume in the following units?

 (a) milliliters (b) cubic centimeters (c) cubic meters

1.72If the volume of helium in a balloon is 145 cm3, what is its volume in the following units?

 (a) milliliters (b) liters (c) cubic meters

1.73If the length, width, and height of a box are 8.0 cm, 5.0 cm, and 4.0 cm, respectively, what is the volume of the box in units of milliliters and liters?

1.74If a cubic box (all sides the same length) has a volume of 1.0 L, what is the length of each side of the box?

1.75A slice of cheese has a mass of 28 g and a volume of 21 cm3. What is the density of the cheese in units of g/cm3 and g/mL?

1.76Two stones resembling diamonds are suspected of being fakes. To determine if the stones might be real, the mass and volume of each were measured. Both stones have the same volume, 0.15 cm3. However, stone A has a mass of 0.52 g and stone B has a mass of 0.42 g. If diamond has a density of 3.5 g/cm3, could the stones be real diamonds? Explain.

1.77If the density of a sugar solution is 1.30 g/mL, what volume of this solution has a mass of 50.0 g?

1.78The density of a certain type of plastic is 0.75 g/cm3. If a sheet of this plastic is 10.0 m long, 1.0 m wide, and 1 cm thick, what is its mass?

1.79Why do liquids have greater densities than gases?

1.80When a balloon filled with air is heated, the balloon increases in volume. Does the density of the air in the balloon increase, decrease, or remain the same?

1.81A piece of plastic sinks in oil but floats in water. Place these three substances in order from lowest density to greatest density.

1.82What special molecular-level feature of ice explains why ice floats in water?

1.83Acetone, a component of some types of fingernail polish remover, has a boiling point of 56°C. What is its boiling point in units of kelvin?

1.84The boiling point of liquid nitrogen is 77 K. What is its boiling point in units of degrees Celsius?

1.85What is the difference in temperature between the boiling point of water and the freezing point of water in each of the following temperature scales?

 (a) Celsius scale (b) Kelvin scale (c) Fahrenheit scale

1.86If the temperature of a cup of coffee decreases from 60.0°C to 25.0°C, what is the decrease in temperature in units of degrees Celsius and kelvin?

1.87Does the boiling point of a substance depend on how much of this substance you have?

1.88Does the melting point of a substance depend on how much of this substance you have?

1.89Identify each of the following as a physical property or a chemical property.

 (a) mass (b) density (c) flammability (d) resistance to corrosion (e) melting point (f) reactivity with water

1.90Identify each of the following as a physical property or a chemical property.

 (a) boiling point (b) reactivity with oxygen (c) resistance to forming compounds with other elements (d) volume

1.91Identify each of the following as a physical change or a chemical change.

 (a) boiling acetone (b) dissolving oxygen gas in water (c) combining hydrogen and oxygen gas to make water (d) burning gasoline (e) screening rocks from sand (f) the conversion of ozone to oxygen, 2O3(g) ⟶ 3O2(g)

1.92Identify each of the following as a physical change or a chemical change.

 (a) condensation of ethanol (b) combining zinc and oxygen to make the compound zinc oxide (c) dissolving sugar in water (d) burning a piece of paper (e) combining sodium metal with water, producing sodium hydroxide and hydrogen gas (f) filtering algae from water

1.93Write a symbolic representation and a molecular-level representation for the change that occurs during the condensation of chlorine, Cl2.

1.94Write a symbolic representation and a molecular-level representation for the process of freezing oxygen, O2.

1.95Do the changes shown in this diagram represent a physical or chemical change?

1.96Do the changes shown in this diagram represent a physical or chemical change?

1.97Draw a picture that shows CH4 (shown in the “After” image in Question 1.95) condensing from a gas to a liquid. Does this picture represent a physical or a chemical change?

1.98Draw a picture that shows water boiling. Does this picture represent a physical or chemical change?

1.99The image shows what happens when iodine, I2, is placed in the bottom of a beaker and heated. Is this a physical or chemical change? (Note that the photo shows an inner flask containing ice.)

1.100The picture shows natural gas (CH4) burning. In the process, carbon dioxide and water form. Is this a physical change or a chemical change?

Energy and Energy Changes

1.101Anna and Bill saw a construction worker welding pipe. Classify the forms of energy they were observing.

1.102Bill and Anna watched students playing volleyball in the sunshine. Classify the forms of energy they were observing.

Page 521.103Which of these two samples of carbon dioxide gas has more kinetic energy? Explain your answer.

1.104Which of these two samples of methane gas is at a higher temperature? Explain your answer.

1.105Give examples of potential energy that you can find in your room.

1.106Give examples of kinetic energy that you can find in your room.

1.107Distinguish between the different types of energy shown in the following picture.

1.108Distinguish between the different types of energy shown in the following picture.

1.109Describe how some common types of energy can change from one type to another.

1.110If energy cannot be created, what is the source of the energy that is released when gasoline is burned?

1.111In terms of kinetic and potential energy, describe what happens when a home run is hit on a softball field.

1.112In terms of kinetic and potential energy, describe what happens when a basketball is dribbled down a basketball court and shot to score three points.

1.113Body mass index (BMI) is a number calculated from a person’s weight and height. BMI provides a reliable indicator of body fatness for most people and is used to screen for weight categories that may lead to health problems. The formula for a person’s BMI is

For an adult, a BMI value between 18.5 and 24.9 is healthy and considered normal.

(a) Calculate the BMI for a 6-ft, 2-in man who weighs 169 lb.

(b) Is this person underweight, at a healthy weight, or overweight?

1.114Consider propane fuel being burned to power a vehicle. How would you classify the transformation of energy in this process in terms of kinetic and potential energy?

1.115Consider water falling into a fountain. How would you classify the transformation of energy in this process in terms of kinetic and potential energy?

Scientific Inquiry

1.116Explain the difference between a hypothesis and a theory.

1.117Explain how a hypothesis is used in scientific research.

1.118Classify each of the following as an observation, hypothesis, law, or theory.

 (a) When wood burns, oxygen is consumed. (b) Heavier-than-air objects always fall toward the center of Earth. (c) Matter is composed of atoms. (d) Crime rates increase when the moon is full.

1.119Classify each of the following as an observation, an hypothesis, law, or theory.

 (a) Bad luck results from walking under a ladder. (b) Oil floats on water. (c) Oil floats on water because it is less dense. (d) Wood burns.

1.120You observe coins in a fountain and propose the hypothesis that fountains are built with coins in them. Suggest experiments to test this hypothesis.

1.121You observe a piece of balsa wood floating on water and propose the hypothesis that all wood floats on water because wood is less dense than water. Suggest experiments to test this hypothesis.

Page 53

1.122Rank the following measurements in order from smallest to largest: 1.0 × 10−4 m, 2.0 × 10−5 m, 3.0 × 10−6 km, 4.0 × 102 mm, 0.0 m, 1.0 m.

1.123The density of air in a balloon is less at high altitudes than at low altitudes. Explain this difference.

1.124If the temperature in a room increases from 20.0°C to 30.0°C, what is the temperature change in units of kelvin?

1.125If you have a sample of zinc and a sample of copper, and both have the same mass, which has the greatest volume?

1.126Give the symbols for potassium and phosphorus.

1.127Give the symbols for the following noble gas elements:

 (a) helium (b) neon (c) argon (d) krypton (e) xenon (f) radon

1.128The red blood cell (RBC) count for a normal female is about 5 million RBCs per μL (1 μL = 1 mm3). How many red blood cells are in a gallon of normal female blood?

1.129Recycling facilities around the world use a variety of techniques to separate different types of recyclable waste. Wet separation is a method that involves processing certain kinds of waste (for example, glass, sand, and metal) based on their densities relative to water. Is this type of separation based on the physical or chemical properties of the recyclable waste?

1.130These samples of metals have the same mass. Which has the greater density?

1.131The typical dose of epinephrine at a particular concentration administered to a patient under cardiac arrest is 0.1 mg per kilogram of body weight. If a patient weighs 180 lb, how much epinephrine should be administered?

1.132About 70 million tons of paper are used per year in the United States. If the U.S. population is about 301.6 million people, how many kilograms of paper are used by the average U.S. citizen?

1.133During a typical physical exam, blood tests to measure cholesterol levels are usually run. A total cholesterol level of 240 mg/dL or above is considered high. What is this threshold level in units of pounds per fluid ounce? If a typical adult has 4 to 6 L of blood, how many pounds of cholesterol are present in the blood of a patient with a total cholesterol level of 260 mg/dL?

1.134The densities of antifreeze, corn oil, dish detergent, maple syrup, shampoo, and water in g/mL are 1.13, 0.93, 1.03, 1.32, 1.01, and 1.00, respectively. Which layer is which substance in the figure?

1.135The lowest possible temperature is the temperature at which a substance would have no kinetic energy (no motion). What is the lowest possible temperature in each of the following temperature units?

 (a) Kelvin scale (b) Celsius scale (c) Fahrenheit scale

1.136Classify the substance in the molecular-level image as:

 (a) an element, compound, or mixture (b) atoms or molecules

1.137Classify each of the following as a symbolic representation for a pure element, pure compound, or mixture.

 (a) NaNO3(s) (b) N2(g) (c) NaCl(aq)

1.138Titanium is a strong metal with a low density that is shiny with a white-metallic color. It is relatively resistant to corrosion by acids and chlorine and resists tarnishing by air by forming a protective oxide layer. It is described as a nontoxic, inert biomaterial, which means that it does not interact unfavorably with human tissues and fluids. Identify all the chemical and physical properties of titanium in this description.

1.139Blood is a water-based liquid in which solids (blood cells and platelets) are suspended. Classify blood as an element, compound, homogeneous mixture, or heterogeneous mixture.

1.140Explain how a fuel such as gasoline has both kinetic and potential energy.

1.141Convert 10.0 m3 to units of cm3 using the conversion factor 1 m = 100 cm.

1.142The average blood volume in the human body is about 4.8 L. Convert this volume to the following units:

 (a) milliliters (mL) (b) cubic centimeters (cm3) (c) cubic meters (m3)

1.143The average density of human blood is 1060 kg/m3. What is the mass of blood in a person who has a blood volume of 0.00500 m3? Report your answer in units of kilograms and in units of pounds.

1.144What is the name for the change in physical state described in each of the following processes?

 (a) (b) (c) (d)

1.145A car traveling at 29.1 m/s drives for 2.5 hours then goes another 75 km. How far did the driver travel in miles?

1.146A large chunk of metal weighing 2.00 lb was found in a mine and appears to be gold. The metal was placed in 1.0 qt of water and the final volume was 38.4 fl oz. What is the density of this metal in g/cm3? Could it be gold?

Concept Review Questions

1.147Which of the following is an example of a mixture?

 A. aluminum B. lake water C. sodium chloride D. iron E. carbon dioxide

For those that are not mixtures, classify them as elements or compounds.

1.148Which of the following is a pure substance that is composed of molecules?

For the correct answers, are they elements or compounds? How would you classify the incorrect answers?

1.149Which of the following statements regarding liquids is correct?

 A. They consist of particles that are relatively far apart. B. They can be easily compressed. C. The symbol for a liquid is (aq). D. Their particles can move around each other. E. They have a fixed shape.

Modify each incorrect statement to make it a true statement.

1.150A rectangular block of an unknown metal with a mass of 456 g is measured to have a length, width, and height of 10.2 cm, 5.08 cm, and 3.26 cm, respectively. What is the likely identity of the metal?

 A. zinc, density = 7.14 g/cm3 B. lead, density = 11.3 g/cm3 C. aluminum, density = 2.70 g/cm3 D. nickel, density = 8.91 g/cm3 E. titanium, density = 4.51 g/cm3

What would be the masses of the same size blocks of the other metals?

1.151Which of the following statements is correct?

 A. Natural gas burning to produce water and carbon dioxide is an example of a chemical change. B. In a physical change, atoms rearrange to form new substances. C. When a substance undergoes a chemical change, its chemical composition does not change. D. Formation of rust on an old car is an example of a physical change. E. The boiling point of nitrogen is −196°C. At −210°C, nitrogen is expected to be a gas.

Modify each incorrect statement to make it a true statement.

1.152Which of the following statements is correct?

 A. Heat coming off a car engine is an example of matter. B. Potential energy is the energy of motion. C. When a compound undergoes a spontaneous chemical reaction, it releases its potential energy. D. Exhaust released from the tailpipe of an old truck is an example of energy. E. A skateboarder rolling down a hill is converting kinetic energy to potential energy.

Modify each incorrect statement to make it a true statement.

1.153Which of the following is an example of a hypothesis?

 A. The mass of products obtained in a chemical reaction always equals the mass of the substances that react. B. Zinc reacts with hydrochloric acid but copper doesn’t because zinc is more reactive than copper. C. Silver tarnishes. D. Oil and water don’t mix. E. All matter is composed of atoms.

How are the incorrect statements classified?

Page 551.154The number 0.00063780 correctly expressed in scientific notation is:

 A. 6.38 × 10−4 B. 6.378 × 10−4 C. 6.3780 × 10−4 D. 6.3780 × 104 E. 6.3780 × 10−5

Write each incorrect answer in decimal form and compare to the value given in the problem.

1.155Which of the following mathematical operations would have two significant figures in the answer?

Carry out each mathematical operation and express your answers to the correct number of significant figures.

1.156Which of the following has the largest mass?

 A. 4.1 × 10−2 kg B. 1.8 × 104 mg C. 25 g D. 6.5 × 106 μg E. 7.6 × 10−1 g

Arrange these quantities in order of increasing mass.

1.157A bicyclist is traveling at 6.7 meters per second. Which of the following mathematical operations would correctly determine the speed in miles per hour?

Explain what is wrong with the incorrect responses.

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