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Matter: Its Properties and Measurement CONTENTS 1 1-1 The Scientific Method 1-2 Properties of Matter 1-3 Classification of Matter 1-4 Measurement of Matter: SI (Metric) Units 1-5 Density and Percent Composition: Their Use in Problem Solving 1-6 Uncertainties in Scientific Measurements 1-7 Significant Figures A Hubble Space Telescope image of a cloud of hydrogen gas and dust (lower right half of the image) that is part of the Swan Nebula (M17). The colors correspond to light emitted by hydrogen (green), sulfur (red), and oxygen (blue). The chemical elements discussed in this text are those found on Earth and, presumably, throughout the universe. F rom the clinic that treats chemical dependency to a theatrical perfor- mance with good chemistry to the food label stating no chemicals added, chemistry and chemicals seem an integral part of life, even if everyday references to them are often misleading. A label implying the absence of chemicals in a food makes no sense. All foods consist entirely of chemicals, even if organically grown. In fact, all material objects whether living or inanimate are made up only of chemicals, and we should begin our study with that thought clearly in mind. By manipulating materials in their environment, people have always practiced chemistry. Among the earliest applications were the glazing of pottery, the smelting of ores to produce metals, the tanning of hides, the dyeing of fabrics, and the making of cheese, wine, beer, and soap. With modern knowledge, though, chemists can decompose matter into its smallest components (atoms) and reassemble those components into mate- rials that do not exist naturally and that often exhibit unusual properties. 1 2 Chapter 1 Matter: Its Properties and Measurement Thus, motor fuels and thousands of chemicals used in the manufacture of plastics, synthetic fabrics, pharmaceuticals, and pesticides can all be made from petroleum. Modern chemical knowledge is also needed to understand the processes that sustain life and to understand and control processes that are detrimental to the environment, such as the formation of smog and the destruction of stratospheric ozone. Because it relates to so many areas of human endeavor, chemistry is sometimes called the central science. Early chemical knowledge consisted of the how to of chemistry, discov- ered through trial and error. Modern chemical knowledge answers the why as well as the how to of chemical change. It is grounded in principles and theory, and mastering the principles of chemistry requires a systematic approach to the subject. Scientific progress depends on the way scientists do their work asking the right questions, designing the right experiments to supply the answers, and formulating plausible explanations of their findings. We begin with a closer look into the scientific method. 1-1 The Scientific Method Science differs from other fields of study in the method that scientists use to acquire knowledge and the special significance of this knowledge. Scientific knowledge can be used to explain natural phenomena and, at times, to predict future events. The ancient Greeks developed some powerful methods of acquiring knowl- edge, particularly in mathematics. The Greek approach was to start with cer- tain basic assumptions, or premises. Then, by the method known as deduction, certain conclusions must logically follow. For example, if a = b and b = c, then a = c. Deduction alone is not enough for obtaining scientific knowledge, however. The Greek philosopher Aristotle assumed four fundamental sub- stances: air, earth, water, and fire. All other materials, he believed, were formed by combinations of these four elements. Chemists of several centuries ago (more commonly referred to as alchemists) tried, in vain, to apply the four-element idea to turn lead into gold. They failed for many reasons, one being that the four-element assumption is false. The scientific method originated in the seventeenth century with such people as Galileo, Francis Bacon, Robert Boyle, and Isaac Newton. The key to the method is to make no initial assumptions, but rather to make careful observations of natural phenomena. When enough observations have been made so that a pat- tern begins to emerge, a generalization or natural law can be formulated describ- ing the phenomenon. Natural laws are concise statements, often in mathematical form, about natural phenomena. The form of reasoning in which a general state- ment or natural law is inferred from a set of observations is called induction. For example, early in the sixteenth century, the Polish astronomer Nicolas Copernicus (1473 1543), through careful study of astronomical observations, concluded that Earth revolves around the sun in a circular orbit, although the general teaching of the time, not based on scientific study, was that the sun and other heavenly bodies revolved around Earth. We can think of Copernicus s statement as a natural law. Another example of a natural law is the radioactive decay law, which dictates how long it takes for a radioactive substance to lose its radioactivity. The success of a natural law depends on its ability to explain, or account for, observations and to predict new phenomena. Copernicus s work was a great success because he was able to predict future positions of the planets more accurately than his contemporaries. We should not think of a natural law as an absolute truth, however. Future experiments may require us to modify the law. For example, Copernicus s ideas were refined a half-century later by Johannes Kepler, who showed that planets travel in elliptical, not circular, orbits. To verify a natural law, a scientist designs experiments that show whether the conclusions deduced from the natural law are supported by experimental results.  1-1 The Scientific Method 3 Theory established: Theory (or model): unless later observations Observation: natural Hypothesis: amplifies hypothesis or experiments show or experimental tentative explanation and gives predictions inadequacies of model Experiments designed Experiments: to test to test hypothesis predictions of theory Revise hypothesis: Modify theory: if experiments show if experiments show that it is inadequate that it is inadequate FIGURE 1-1 The scientific method illustrated A hypothesis is a tentative explanation of a natural law. If a hypothesis sur- vives testing by experiments, it is often referred to as a theory. In a broader sense, a theory is a model or way of looking at nature that can be used to explain natural laws and make further predictions about natural phenomena. When differing or conflicting theories are proposed, the one that is most suc- cessful in its predictions is generally chosen. Also, the theory that involves the smallest number of assumptions the simplest theory is preferred. Over time, as new evidence accumulates, most scientific theories undergo modifica- tion, and some are discarded. The scientific method is the combination of observation, experimentation, and the formulation of laws, hypotheses, and theories. The method is illus- trated by the flow diagram in Figure 1-1. Scientists may develop a pattern of thinking about their field, known as a paradigm. Some paradigms may be suc- cessful at first but then become less so. When that happens, a new paradigm may be needed or, as is sometimes said, a paradigm shift occurs. In a way, the method of inquiry that we call the scientific method is itself a paradigm, and some people feel that it, too, is in need of change. That is, the varied activ- ities of modern scientists are more complex than the simplified description of Louis Pasteur (1822 1895). This great practitioner of the the scientific method presented here.* In any case, merely following a set of scientific method was the procedures, rather like using a cookbook, will not guarantee scientific success. developer of the germ theory Another factor in scientific discovery is chance, or serendipity. Many discover- of disease, the sterilization of ies have been made by accident. For example, in 1839, the American inventor milk by pasteurization, and Charles Goodyear was searching for a treatment for natural rubber that would vaccination against rabies. He has been called the greatest make it less brittle when cold and less tacky when warm. In the course of this physician of all time by some. work, he accidentally spilled a rubber sulfur mixture on a hot stove and found He was, in fact, not a physi- that the resulting product had exactly the properties he was seeking. Other chance cian at all, but a chemist by discoveries include X-rays, radioactivity, and penicillin. So scientists and inven- training and by profession. tors always need to be alert to unexpected observations. Perhaps no one was more aware of this than Louis Pasteur, who wrote, Chance favors the prepared mind. 1-1 CONCEPT ASSESSMENT Is the common saying The exception proves the rule a good statement of the Answers to Concept scientific method? Explain. Assessment questions are given in Appendix G. *W. Harwood, JCST, 33, 29 (2004). JCST is an abbreviation for Journal of College Science Teaching. 4 Chapter 1 Matter: Its Properties and Measurement 1-2 Properties of Matter Dictionary definitions of chemistry usually include the terms matter, composition, and properties, as in the statement that chemistry is the science that deals with the composition and properties of matter. In this and the next section, we will consider some basic ideas relating to these three terms in hopes of gaining a better understanding of what chemistry is all about. Matter is anything that occupies space and displays the properties of mass and inertia. Every human being is a collection of matter. We all occupy space, and we describe our mass in terms of weight, a related property. (Mass and weight are described in more detail in Section 1-4. Inertia is described in Appendix B.) All the objects that we see around us consist of matter. The gases of the atmosphere, even though they are invisible, are matter they occupy space and have mass. Sunlight is not matter; rather, it is a form of energy. Energy is discussed in later chapters. Composition refers to the parts or components of a sample of matter and their relative proportions. Ordinary water is made up of two simpler substances hydrogen and oxygen present in certain fixed proportions. A chemist would say that the composition of water is 11.19% hydrogen and 88.81% oxygen by mass. Hydrogen peroxide, a substance used in bleaches and antiseptics, is also made up of hydrogen and oxygen, but it has a different com- position. Hydrogen peroxide is 5.93% hydrogen and 94.07% oxygen by mass. Properties are those qualities or attributes that we can use to distinguish one sample of matter from others; and, as we consider next, the properties of matter are generally grouped into two broad categories: physical and chemical. Physical Properties and Physical Changes A physical property is one that a sample of matter displays without changing its composition. Thus, we can distinguish between the reddish brown solid, copper, and the yellow solid, sulfur, by the physical property of color (Fig. 1-2). Another physical property of copper is that it can be hammered into a thin sheet of foil (see Figure 1-2). Solids having this ability are said to be malleable. Sulfur is not malleable. If we strike a chunk of sulfur with a hammer, it crum- bles into a powder. Sulfur is brittle. Another physical property of copper that sulfur does not share is the ability to be drawn into a fine wire (ductility). Also, sulfur is a far poorer conductor of heat and electricity than is copper. Sometimes a sample of matter undergoes a change in its physical appear- ance. In such a physical change, some of the physical properties of the sample may change, but its composition remains unchanged. When liquid water freezes into solid water (ice), it certainly looks different and, in many ways, it is different. Yet, the water remains 11.19% hydrogen and 88.81% oxygen by mass. FIGURE 1-2 * Physical properties of sulfur and copper A lump of sulfur (left) crumbles into a yellow powder when hammered. Copper (right) can be obtained as large lumps of native copper, formed into pellets, hammered into a thin foil, or drawn into a wire.  1-3 Classification of Matter 5 Chemical Properties and Chemical Changes In a chemical change, or chemical reaction, one or more kinds of matter are converted to new kinds of matter with different compositions. The key to identifying chemical change, then, comes in observing a change in composition. The burning of paper involves a chemical change. Paper is a complex material, but its principal constituents are carbon, hydrogen, and oxygen. The chief products of the combustion are two gases, one consisting of carbon and oxy- gen (carbon dioxide) and the other consisting of hydrogen and oxygen (water, as steam). The ability of paper to burn is an example of a chemical property. A chemical property is the ability (or inability) of a sample of matter to undergo a change in composition under stated conditions. Zinc reacts with hydrochloric acid solution to produce hydrogen gas and a solution of zinc chloride in water (Fig. 1-3). This reaction is one of zinc s distinc- FIGURE 1-3 tive chemical properties, just as the inability of gold to react with hydrochloric A chemical property of acid is one of gold s chemical properties. Sodium reacts not only with hydrochlo- zinc and gold: reaction ric acid but also with water. In some of their physical properties, zinc, gold, and with hydrochloric acid The zinc-plated (galvanized) sodium are similar. For example, each is malleable and a good conductor of heat nail reacts with hydrochloric and electricity. In most of their chemical properties, though, zinc, gold, and acid, producing the bubbles sodium are quite different. Knowing these differences helps us to understand of hydrogen gas seen on its why zinc, which does not react with water, is used in roofing nails, roof flashings, surface. The gold bracelet is and rain gutters, and sodium is not. Also, we can appreciate why gold, because unaffected by hydrochloric of its chemical inertness, is prized for jewelry and coins: It does not tarnish or acid. In this photograph, the rust. In our study of chemistry, we will see why substances differ in properties zinc plating has been and how these differences determine the ways in which we use them. consumed, exposing the underlying iron nail. The reaction of iron with 1-3 Classification of Matter hydrochloric acid imparts some color to the acid Matter is made up of very tiny units called atoms. Each different type of atom is solution. the building block of a different chemical element. Presently, the International Union of Pure and Applied Chemistry (IUPAC) recognizes 112 elements, and all The International Union of matter is made up of just these types! The known elements range from common Pure and Applied Chemistry substances, such as carbon, iron, and silver, to uncommon ones, such as (IUPAC) is recognized as the lutetium and thulium. About 90 of the elements can be obtained from natural world authority on chemical sources. The remainder do not occur naturally and have been created only in nomenclature, terminology, standardized methods for laboratories. On the inside front cover you will find a complete listing of the ele- measurement, atomic ments and also a special tabular arrangement of the elements known as the mass, and more. Along periodic table. The periodic table is the chemist s directory of the elements. We with many other activities, will describe it in Chapter 2 and use it throughout most of the text. IUPAC publishes journals, Chemical compounds are substances comprising atoms of two or more ele- technical reports, and ments joined together. Scientists have identified millions of different chemical chemical databases, most of compounds. In some cases, we can isolate a molecule of a compound. A molecule which are available at www. is the smallest entity having the same proportions of the constituent atoms as iupac.org. does the compound as a whole. A molecule of water consists of three atoms: two The identity of an atom hydrogen atoms joined to a single oxygen atom. A molecule of hydrogen perox- is established by a feature ide has two hydrogen atoms and two oxygen atoms; the two oxygen atoms are called its atomic number (see joined together and one hydrogen atom is attached to each oxygen atom. By con- Section 2-3). Recent report of trast, a molecule of the blood protein gamma globulin is made up of 19,996 other new elements, such as atoms, but they are of just four types: carbon, hydrogen, oxygen, and nitrogen. elements 113 to 116 and 118, await confirmation. Characterizing superheavy O elements is a daunting chal- H H lenge; they are produced only a few atoms at a time and the atoms disintegrate almost instantaneously. H H O O Gamma globulin 6 Chapter 1 Matter: Its Properties and Measurement The composition and properties of an element or a compound are uniform throughout a given sample and from one sample to another. Elements and compounds are called substances. (In the chemical sense, the term substance should be used only for elements and compounds.) A mixture of substances can vary in composition and properties from one sample to another. One that is uniform in composition and properties throughout is said to be a homogeneous mixture or a solution. A given solution of sucrose (cane sugar) in water is uniformly sweet throughout the solution, but the sweetness of Is it homogeneous or another sucrose solution may be rather different if the sugar and water are heterogeneous? When viewed present in different proportions. Ordinary air is a homogeneous mixture of through a microscope, homogenized milk is seen several gases, principally the elements nitrogen and oxygen. Seawater is a solu- to consist of globules of fat tion of the compounds water, sodium chloride (salt), and a host of others. dispersed in a watery Gasoline is a homogeneous mixture or solution of dozens of compounds. medium. Homogenized milk In heterogeneous mixtures sand and water, for example the compo- is a heterogeneous mixture. nents separate into distinct regions. Thus, the composition and physical prop- erties vary from one part of the mixture to another. Salad dressing, a slab of concrete, and the leaf of a plant are all heterogeneous. It is usually easy to dis- tinguish heterogeneous from homogeneous mixtures. A scheme for classifying matter into elements and compounds and homogeneous and heterogeneous mixtures is summarized in Figure 1-4. Separating Mixtures A mixture can be separated into its components by appropriate physical It is composition, particu- means. Consider again the heterogeneous mixture of sand in water. When we larly its variability, that helps pour this mixture into a funnel lined with porous filter paper, the water passes us distinguish the several through and sand is retained on the paper. This process of separating a solid classifications of matter. from the liquid in which it is suspended is called filtration (Fig. 1-5a). You will Solutions can be gaseous probably use this procedure in the laboratory. Conversely, we cannot separate and liquids as described here, a homogeneous mixture (solution) of copper(II) sulfate in water by filtration but they can also be solids. because all components pass through the paper. We can, however, boil the Some alloys are examples of solution of copper(II) sulfate and water. In the process of distillation, a pure solid solutions. liquid is condensed from the vapor given off by a boiling solution. When all All matter No Can it be Yes separated by physical means? Substance Mixture Yes Can it be No Yes Is it No decomposed by uniform chemical process? throughout? Compound Element Homogeneous Heterogeneous FIGURE 1-4 A classification scheme for matter Every sample of matter is either a single substance (an element or compound) or a mixture of substances. At the molecular level, an element consists of atoms of a single type and a compound consists of two or more different types of atoms, usually joined into molecules. In a homogeneous mixture, atoms or molecules are randomly mixed at the molecular level. In heterogeneous mixtures, the components are physically separated, as in a layer of octane molecules (a constituent of gasoline) floating on a layer of water molecules.  1-3 Classification of Matter 7 (a) (b) FIGURE 1-5 Separating mixtures: a physical process (a) Separation of a heterogeneous mixture by filtration: Solid copper(II) sulfate is retained on the filter paper, while liquid hexane passes through. (b) Separation of a homogeneous mixture by distillation: Copper(II) sulfate remains in the flask on the left as water passes to the flask on the right, by first evaporating and then condensing back to a liquid. (c) Separation of the components of ink using chromatography: A dark spot of black ink can be seen just above the water line as water moves up the paper. (d) Water has dissolved the colored components of the ink, and these components are retained in different regions on the paper according (c) (d) to their differing tendencies to adhere to the paper. the water has been removed by boiling a solution of copper(II) sulfate in water, solid copper(II) sulfate remains behind (Fig. 1-5b). Another method of separation available to modern chemists depends on the differing abilities of compounds to adhere to the surfaces of various solid substances, such as paper and starch. The technique of chromatography relies on this principle. The dramatic results that can be obtained with chromatography are illustrated by the separation of ink on a filter paper (Fig. 1-5c d). Decomposing Compounds A chemical compound retains its identity during physical changes, but it can be decomposed into its constituent elements by chemical changes. The decomposition of compounds into their constituent elements is a more difficult matter than the mere physical separation of mixtures. The extraction of iron from iron oxide ores requires a blast furnace. The industrial production of pure magnesium from mag- nesium chloride requires electricity. It is generally easier to convert a compound into other compounds by a chemical reaction than it is to separate a compound into its constituent elements. For example, when heated, ammonium dichromate decomposes into the substances chromium(III) oxide, nitrogen, and water. This reaction, once used in movies to simulate a volcano, is illustrated in Figure 1-6. States of Matter Matter is generally found in one of three states: solid, liquid, or gas. In a solid, atoms or molecules are in close contact, sometimes in a highly organized arrangement called a crystal. A solid has a definite shape. In a liquid, the atoms or molecules are usually separated by somewhat greater distances than in a FIGURE 1-6 solid. Movement of these atoms or molecules gives a liquid its most distinctive A chemical change: property the ability to flow, covering the bottom and assuming the shape of decomposition of ammonium dichromate its container. In a gas, distances between atoms or molecules are much greater 8 Chapter 1 Matter: Its Properties and Measurement FIGURE 1-7 * Macroscopic and microscopic views of (c) matter The picture shows a block of ice on a heated surface and the three states of water. The circular insets show how chemists conceive of these states microscopically, in terms of molecules with two hydrogen atoms joined to (a) one of oxygen. In ice (a), the molecules are arranged in a regular pattern in a rigid framework. In liquid water (b), the molecules are rather closely packed but move freely. In gaseous water (c), the molecules are widely separated. (b) than in a liquid. A gas always expands to fill its container. Depending on con- ditions, a substance may exist in only one state of matter, or it may be present in two or three states. Thus, as the ice in a small pond begins to melt in the spring, water is in two states: solid and liquid (actually, three states if we also consider water vapor in the air above the pond). The three states of water are illustrated at two levels in Figure 1-7. The macroscopic level refers to how we perceive matter with our eyes, through the outward appearance of objects. The microscopic level describes matter as chemists conceive of it in terms of atoms and molecules and their behavior. In this text, we will describe many macroscopic, observable proper- ties of matter, but to explain these properties, we will often shift our view to the atomic or molecular level the microscopic level. 1-4 Measurement of Matter: SI (Metric) Units Nonnumerical information Chemistry is a quantitative science, which means that in many cases we can * is qualitative, such as the color measure a property of a substance and compare it with a standard having a blue. known value of the property. We express the measurement as the product of a number and a unit. The unit indicates the standard against which the measured quantity is being compared. When we say that the length of the playing field in football is 100 yd, we mean that the field is 100 times longer than a standard of length called the yard (yd). In this section, we will introduce some basic units of measurement that are important to chemists. The scientific system of measurement is called the Système Internationale d Unités (International System of Units) and is abbreviated SI. It is a modern version of the metric system, a system based on the unit of length called a The definition of the meter, * formerly based on the atomic meter (m). The meter was originally defined as 1>10,000,000 of the distance from spectrum of 86Kr, was changed the equator to the North Pole and translated into the length of a metal bar kept to the speed of light in 1983. in Paris. Unfortunately, this length is subject to change with temperature, Effectively, the speed of light and it cannot be exactly reproduced. The SI system substitutes for the standard is now defined as meter bar an unchanging, reproducible quantity: 1 meter is the distance traveled 2.99792458 * 108 m>s. by light in a vacuum in 1>299,792,458 of a second. Length is one of the seven  1-4 Measurement of Matter: SI (Metric) Units 9 TABLE 1.1 SI Base Quantities TABLE 1.2 SI Prefixes Physical Quantity Unit Symbol Multiple Prefix Length metera m 1018 exa (E) Mass kilogram kg 1015 peta (P) Time second s 1012 tera (T) Temperature kelvin K 109 giga (G) Amount of substanceb mole mol mega (M) 106 Electric currentc ampere A kilo (k) 103 Luminous intensityd candela cd 102 hecto (h) aThe deka (da) official spelling of this unit is metre, but we will use the American spelling. 101 bThe mole is introduced in Section 2-7. 10-1 deci (d) cElectric current is described in Appendix B and in Chapter 20. dLuminous intensity is not discussed in this text. 10-2 centi (c) 10-3 milli (m) 10-6 micro (m)a fundamental quantities in the SI system (see Table 1.1). All other physical quan- 10-9 nano (n) tities have units that can be derived from these seven. SI is a decimal system. 10-12 pico (p) prefixes. For example, the prefix kilo means one thousand 11032 times the base Quantities differing from the base unit by powers of ten are noted by the use of 10-15 femto (f) 10-18 atto (a) unit; it is abbreviated as k. Thus 1 kilometer = 1000 meters, or 1 km = 1000 m. 10-21 zepto (z) The SI prefixes are listed in Table 1.2. 10-24 yocto (y) Most measurements in chemistry are made in SI units. Sometimes we must aThe Greek letter m convert between SI units, as when converting kilometers to meters. At other (pronounced mew ). times we must convert measurements expressed in non-SI units into SI units, or from SI units into non-SI units. In all of these cases we can use a conversion factor or a series of conversion factors in a scheme called a conversion path- It is a good idea to memorize * way. Later in this chapter, we will apply conversion pathways in a method of the most common SI prefixes problem solving known as dimensional analysis. The method itself is described (such as G, M, k, d, c, m, m, n, in some detail in Appendix A. and p) because you can t sur- vive in a world of science Mass without knowing the SI Mass describes the quantity of matter in an object. In SI the standard of mass prefixes. is 1 kilogram (kg), which is a fairly large unit for most applications in chem- istry. More commonly we use the unit gram (g) (about the mass of three aspirin tablets). Weight is the force of gravity on an object. It is directly proportional to mass, as shown in the following mathematical expressions. W r m and W = g * m (1.1) The symbol r means * proportional to. It can be An object has a fixed mass (m), which is independent of where or how the mass replaced by an equality sign is measured. Its weight (W), however, may vary because the acceleration due to and a proportionality gravity (g) varies slightly from one point on Earth to another. Thus, an object that constant. In expression (1.1), weighs 100.0 kg in St. Petersburg, Russia, weighs only 99.6 kg in Panama (about the constant is the accelera- 0.4% less). The same object would weigh only about 17 kg on the moon. tion due to gravity, g. (See Although the weight of an object varies from place to place, its mass is the same Appendix B.) in all locations. The terms weight and mass are often used interchangeably, but only mass is a measure of the quantity of matter. A common laboratory device for measuring mass is called a balance. A balance is often called, incorrectly, a scale. The principle used in a balance is that of counteracting the force of gravity on an unknown mass with a force of equal magnitude that can be precisely measured. In older two-pan beam balances, the object whose mass is being deter- mined is placed on one pan and counterbalancing is achieved through the force of gravity acting on weights, objects of precisely known mass, placed on the other pan. In the type of balance most commonly seen in laboratories today the electronic balance the counterbalancing force is a magnetic force produced by passing an electric current through an electromagnet. First, an initial balance condition is achieved when no object is present on the balance pan. When the 10 Chapter 1 Matter: Its Properties and Measurement object to be weighed is placed on the pan, the initial balance condition is upset. To restore the balance condition, additional electric current must be passed through the electromagnet. The magnitude of this additional current is proportional to the mass of the object being weighed and is translated into a mass reading that is displayed on the balance. An electronic balance is shown in the margin. 1-2 CONCEPT ASSESSMENT Would either the two-pan beam balance or the electronic balance yield the same result for the mass of an object measured on the moon as that measured for the same object on Earth? Explain. An electronic balance. Time In daily use we measure time in seconds, minutes, hours, and years, depend- ing on whether we are dealing with short intervals (such as the time for a 100 m race) or long ones (such as the time before the next appearance of Halley s comet in 2062). We can use all these units in scientific work also, although in SI the standard of time is the second (s). A time interval of 1 second is not easily established. At one time it was based on the length of a day, but this is not constant because the rate of Earth s rotation undergoes slight variations. In 1956, the second was defined as 1>31,556,925.9747 of the length of the year 1900. With the advent of atomic clocks, a more precise definition became possible. The second is now defined as the duration of Electromagnetic radiation 9,192,631,770 cycles of a particular radiation emitted by certain atoms of the is discussed in Section 8-1. element cesium (cesium-133). Temperature To establish a temperature scale, we arbitrarily set certain fixed points and temperature increments called degrees. Two commonly used fixed points are the temperature at which ice melts and the temperature at which water boils, both at standard atmospheric pressure.* On the Celsius scale, the melting point of ice is 0 °C, the boiling point of water is 100 °C, and the interval between is divided into 100 equal parts called Celsius degrees. On the Fahrenheit temperature scale, the melting point of ice is 32 °F, the boiling point of water is 212 °F, and the interval between is divided into 180 equal parts called Fahrenheit degrees. Figure 1-8 compares the Fahrenheit and Celsius temperature scales. The SI temperature scale, called the Kelvin scale, assigns a value of zero to the lowest possible temperature. The zero on the Kelvin scale is denoted 0 K and it comes at 273.15 °C. We will discuss the Kelvin temperature scale in detail in Chapter 6. For now, it is enough to know the following: The interval on the Kelvin scale, called a kelvin, is the same size as the Celsius degree. When writing a Kelvin temperature, we do not use a degree symbol. That is, we write 0 K or 300 K, not 0 °K or 300 °K. The SI symbol for Kelvin The Kelvin scale is an absolute temperature scale; there are no negative temperature is T and that for Kelvin temperatures. Celsius temperature is t but shown here as t(°C). The In the laboratory, temperature is most commonly measured in Celsius Fahrenheit temperature, degrees; however, these temperatures must often be converted to the Kelvin shown here as t(°F), is not scale (in describing the behavior of gases, for example). Occasionally, particu- recognized in SI. larly in some engineering applications, temperatures must be converted *Standard atmospheric pressure is defined in Section 6-1. The effect of pressure on melting and boiling points is described in Chapter 12.  1-4 Measurement of Matter: SI (Metric) Units 11 bp of water 373 K 100 °C 212 °F 100 °C 212 °F 100 °C 212 °F hot day 303 K 30 °C 86 °F mp of ice 273 K 0 °C 32 °F very cold day 238 K *35 °C *31 °F 0 °C 32 °F 0 °C 32 °F bp of liquid nitrogen 77 K *196 °C *321 °F (a) (b) 0 K *273.15 °C *459.67 °F Absolute zero FIGURE 1-8 A comparison of temperature scales (a) The melting point (mp) of ice. (b) The boiling point (bp) of water. between the Celsius and Fahrenheit scales. Temperature conversions can be made in a straightforward way by using the algebraic equations shown below. Kelvin from Celsius T1K2 = t1°C2 + 273.15 9 Fahrenheit from Celsius t1°F2 = t1°C2 + 32 5 3t1°F2 - 324 5 Celsius from Fahrenheit t1°C2 = 9 The factors 95 and 59 arise because the Celsius scale uses 100 degrees between the two chosen reference points and the Fahrenheit scale uses 180 degrees: 180>100 = 95 and 100>180 = 59. The diagram in Figure 1-8 illustrates the rela- tionship among the three scales for several temperatures. EXAMPLE 1-1 Converting Between Fahrenheit and Celsius Temperatures The predicted high temperature for New Delhi on a given day is 41 °C. Is this temperature higher or lower than the predicted daytime high of 103 °F for the same day in Phoenix, Arizona, reported by a newscaster? Analyze We are given a Celsius temperature and seek a comparison with a Fahrenheit temperature. To convert the given Celsius temperature to a Fahrenheit temperature, we use the equation given previously that expresses t(°F) as a function of t(°C). Solve t1°C2 + 32 = 1412 + 32 = 106 °F 9 9 t1°F2 = 5 5 The predicted temperature for New Delhi, 106 °F, is 3 °F higher than for Phoenix, 103 °F. (continued)  12 Chapter 1 Matter: Its Properties and Measurement Assess For temperatures at which t(°C) 7 - 40 °C, the Fahrenheit temperature is greater than the Celsius tempera- ture. If the Celsius temperature is lower than - 40 °C, then t(°F) is lower than (more negative than) t(°C) (Fig. 1-8). Concept Assessment 1-3 asks you to think further about the relationship between t(°C) and t(°F). PRACTICE EXAMPLE A: A recipe in an American cookbook calls for roasting a cut of meat at 350 °F. What is this temperature on the Celsius scale? PRACTICE EXAMPLE B: A particular automobile engine coolant has antifreeze protection to a temperature of - 22 °C. Will this coolant offer protection at temperatures as low as - 15 °F? Answers to Practice Examples are given on the Mastering Chemistry site: www. masteringchemistry.com. 1-3 CONCEPT ASSESSMENT Can there be a temperature at which °C and °F have the same value? Can there be more than one such temperature? Explain. 1 L * 1 dm3 1 cm3 * 1 mL 1 m3 10 cm 10 cm Derived Units The seven units listed in Table 1.1 are the SI units for the fundamental quantities of length, mass, time, and so on. Many measured properties are expressed as combinations of these fundamental, or base, quantities. We refer to the units of such properties as derived units. For example, velocity is a distance divided by the time required to travel that distance. The unit of velocity is length divided by 10 cm time, such as m>s or m s -1. Some derived units have special names. For example, the combination kg m 1 s 2 is called the pascal (Chapter 6) and the combination kg m2 s 2 is called the joule (Chapter 7). Other examples are given in Appendix C. An important measurement that uses derived units is volume. Volume has FIGURE 1-9 the unit (length)3, and the SI standard unit of volume is the cubic meter (m3). Some metric volume units compared More commonly used volume units are the cubic centimeter (cm3) and the The largest volume, shown in liter (L). One liter is defined as a volume of 1000 cm3, which means that one part, is the SI standard of milliliter (1 mL) is equal to 1 cm3. The liter is also equal to one cubic decimeter 1 cubic meter (m3). A cube (1 dm3). Several volume units are depicted in Figure 1-9. with a length of 10 cm (1 dm) on edge (in blue) has a volume Non-SI Units of 1000 cm3 (1 dm3) and is called 1 liter (1 L). The smallest Although its citizens are growing more accustomed to expressing distances in cube is 1 cm on edge (red) and kilometers and volumes in liters, the United States is one of the few countries has a volume of 1 cm3 = 1 mL. where most units used in everyday life are still non-SI. Masses are given in pounds, room dimensions in feet, and so on. In this book, we will not routinely The official spelling is litre, use these non-SI units, but we will occasionally introduce them in examples and but we will use the American end-of-chapter exercises. In such cases, any necessary relationships between spelling, liter. non-SI and SI units will be given or can be found on the inside back cover. 1-1 ARE YOU WONDERING... Why attaching the units to a number is so important? In 1993, NASA started the Mars Surveyor program to conduct an ongoing series of missions to explore Mars. In 1995, two missions were scheduled that would be launched in late 1998 and early 1999. The missions were the Mars Climate Orbiter (MCO) and the Mars Polar Lander (MPL). The MCO was launched December 11, 1998, and the MPL, January 3, 1999.  1-5 Density and Percent Composition: Their Use in Problem Solving 13 The development of * Nine and a half months after launch, the MCO was to fire its main engine to science requires careful achieve an elliptical orbit around Mars. The MCO engine start occurred on quantitative measurement. September 23, 1999, but the MCO mission was lost when the orbiter entered the Theories have stood or fallen Martian atmosphere on a lower-than-expected trajectory. The MCO entered the based on their agreement or low orbit because the computer on Earth used British Engineering units, whereas otherwise with experiments the MCO computer used SI units. in the fourth significant This error in units brought the MCO 56 km above the surface of Mars instead figure or beyond. Problem of the desired 250 km. At 250 km, the MCO would have successfully entered the solving, the handling of units, desired elliptical orbit, and the $168 million orbiter would probably not have and the use of significant been lost. figures (Section 1-7) are important in all areas of science. 1-5 Density and Percent Composition: Their Use in Problem Solving Throughout this text, we will encounter new concepts about the structure and behavior of matter. One means of firming up our understanding of these new concepts is to work problems that relate concepts that we already know to those we are trying to understand. In this section, we will intro- duce two quantities frequently required in problem solving: density and percent composition. Density Here is an old riddle: What weighs more, a ton of bricks or a ton of cotton? If you answer that they weigh the same, you demonstrate a clear understand- ing of the meaning of weight and, indirectly, of the quantity of matter to which weight is proportional, that is, mass. Anyone who answers that the bricks weigh more than the cotton has confused the concepts of weight and density. Matter in a brick is more concentrated than in cotton that is, the matter in a brick is confined to a smaller volume. Bricks are more dense than cotton. Density is the ratio of mass to volume. mass (m) density (d) = (1.2) volume (V) Mass and volume are both extensive properties. An extensive property is dependent on the quantity of matter observed. However, if we divide the mass of a substance by its volume, we obtain density, an intensive property. An intensive property is independent of the amount of matter observed. Thus, the density of pure water at 25 °C has a unique value, whether the sample fills a small beaker (small mass/small volume) or a swimming pool (large mass/large volume). Intensive properties are especially useful in chemical studies because they can often be used to identify substances. The SI base units of mass and volume are kilograms and cubic meters, respectively, but chemists generally express mass in grams and volume in cubic centimeters or milliliters. Thus, the most commonly encountered den- sity unit is grams per cubic centimeter (g>cm3) or the identical unit grams per milliliter (g>mL). The mass of 1.000 L of water at 4 °C is 1.000 kg. The density of water at 4 °C is 1000 g>1000 mL, or 1.000 g>mL. At 20 °C, the density of water is 0.9982 g>mL. Density is a function of temperature because volume varies with temperature, whereas mass remains constant. One reason that climate change is a concern is because as the average temperature of seawater increases, the seawater will become less dense, its volume will increase, and sea level will rise even if no continental ice melts. Like temperature, the state of matter affects the density of a substance. In general, solids are denser than liquids and both are denser than gases, but 14 Chapter 1 Matter: Its Properties and Measurement there are notable overlaps in densities between solids and liquids. Following are the ranges of values generally observed for densities; this information should prove useful in solving problems. KEEP IN MIND * Solid densities: from about 0.2 g>cm3 to 20 g>cm3 that recognizing the scale of * Liquid densities: from about 0.5 g>mL to 3 4 g>mL things is one important step * Gas densities: mostly in the range of a few grams per liter in avoiding mistakes. If a solid density calculates out as In general, densities of liquids are known more precisely than those of 0.05 g>cm3, or a gas density solids (which may have imperfections in their microscopic structures). Also, as 5.0 g>cm3, review the work densities of elements and compounds are known more precisely than densi- done up to that point! ties of materials with variable compositions (such as wood or rubber). There are several important consequences of the different densities of solids and liquids. A solid that is insoluble and floats on a liquid is less dense than the liquid, and it displaces a mass of liquid equal to its own mass. An insoluble solid that sinks to the bottom of a liquid is more dense than the liquid and dis- places a volume of liquid equal to its own volume. Liquids that are immiscible in each other separate into distinct layers, with the most dense liquid at the bottom and the least dense liquid at the top. 1-4 CONCEPT ASSESSMENT Approximately what fraction of its volume is submerged when a 1.00 kg block of wood (d = 0.68 g>cm3) floats on water? KEEP IN MIND Density in Conversion Pathways that in a conversion pathway, If we measure the mass of an object and its volume, simple division gives us its all units must cancel except density. Conversely, if we know the density of an object, we can use density as a for the desired unit in the final conversion factor to determine the object s mass or volume. For example, a cube result (see Appendix A-5). of osmium 1.000 cm on edge weighs 22.59 g. The density of osmium (the densest Also, note that the quantities of the elements) is 22.59 g>cm3. What would be the mass of a cube of osmium given and sought are typically that is 1.25 in. on edge (1 in. = 2.54 cm)? To solve this problem, we begin by extensive properties and the relating the volume of a cube to its length, that is, V = l3. Then we can map out conversion factor(s) are often intensive properties the conversion pathway: (here, density). in. osmium ¡ cm osmium ¡ cm3 osmium ¡ g osmium (converts in. to cm) (converts cm to cm3) (converts cm3 to g osmium) 3 2.54 cm 22.59 g osmium 723 g osmium 1 in. 1 cm3 At 25 °C the density of mercury, the only metal that is liquid at this tempera- ture, is 13.5 g>mL. Suppose we want to know the volume, in mL, of 1.000 kg of mercury at 25 °C. We proceed by (1) identifying the known information: 1.000 kg of mercury and d = 13.5 g>mL (at 25 °C); (2) noting what we are trying to determine a volume in milliliters (which we designate mL mercury); and (3) looking for the relevant conversion factors. Outlining the conversion pathway will help us find these conversion factors: kg mercury ¡ g mercury ¡ mL mercury We need the factor 1000 g>kg to convert from kilograms to grams. Density provides the factor to convert from mass to volume. But in this instance, we need to use density in the inverted form. That is, 1000 g 1 mL mercury ? mL mercury = 1.000 kg * * = 74.1 mL mercury 1 kg 13.5 g  1-5 Density and Percent Composition: Their Use in Problem Solving 15 Examples 1-2 and 1-3 further illustrate that numerical calculations involv- ing density are generally of two types: determining density from mass and volume measurements and using density as a conversion factor to relate mass and volume. EXAMPLE 1-2 Relating Mass, Volume, and Density The stainless steel in the solid cylindrical rod pictured below has a density of 7.75 g>cm3. If we want a 1.00 kg mass of this rod, how long a section must we cut off? Refer to the inside back cover for the formula to calculate the volume of a cylinder. 1.000 in. Analyze We are given the density, d, and the desired mass, m. Because d = m/V, we can solve for V and then use the formula for the volume of a cylinder, V = pr2h, to calculate h, the length of rod we seek. Two different mass units (g and kg) and two different length units (centimeters and inches) appear in the information given in this problem, so we anticipate having to make at least two unit conversions. To avoid errors, we include units in all steps. Solve Solve equation (1.2) for V. The reciprocal of m 1 density, 1/d, is a conversion factor for con- V = = m * d d verting from mass to volume. Calculate the volume of the rod that will have 1000 g 1 cm3 a mass of 1.00 kg. A conversion from kg to g is V = 1.00 kg * * = 129 cm3 1 kg 7.75 g required in this step. 3.1416 * 10.500 in. * 2.54 cm>1 in.22 Solve V = pr2h for h and then calculate h. We V 129 cm3 must be certain to use the radius of the rod h = = = 25.5 cm pr2 (one-half the diameter) and to express the radius in centimeters. Assess late d = 1.00 * 103 g/33.1416 * 11.27 cm22 * 25.5 cm4 = 7.74 g/cm3, which is very close to the given den- One way to check whether our answer is correct is to work the problem in reverse. For example, we calcu- sity. We are confident that our answer, h = 25.5 cm, is correct. PRACTICE EXAMPLE A: To determine the density of trichloroethylene, a liquid used to degrease electronic components, a flask is first weighed empty (108.6 g). It is then filled with 125 mL of the trichloroethylene to give a total mass of 291.4 g. What is the density of trichloroethylene in grams per milliliter? PRACTICE EXAMPLE B: Suppose that instead of using the cylindrical rod of Example 1-2 to prepare a 1.000 kg mass we were to use a solid spherical ball of copper (d = 8.96 g>cm3). What must be the radius of this ball? EXAMPLE 1-3 Determining the Density of an Irregularly Shaped Solid A chunk of coal is weighed twice while suspended from a spring scale (see Figure 1-10). When the coal is sus- pended in air, the scale registers 156 g; when the coal is suspended underwater at 20 °C, the scale registers 59 g. What is the density of the coal? The density of water at 20 °C is 0.9982 g cm 3. Analyze We need the ratio of mass to volume of the chunk of coal. The mass of the coal is easily obtained; it is what registers on the scale when the coal is suspended in air: 156 g. But what is the volume of this chunk of coal? The key to this calculation is the weight measurement under water. The coal weighs less than 156 g when sub- merged in water because the water exerts a buoyant force on the coal. The buoyant force is the difference (continued) 16 Chapter 1 Matter: Its Properties and Measurement FIGURE 1-10 * Measuring the volume of an irregularly shaped solid When submerged in a liquid, an irregularly shaped solid displaces a volume of liquid equal to its own. The necessary data can be obtained by two mass measurements of the type illustrated here; the required calculations are like those in Example 1-3. between the two weight measurements: 156 g - 59 g = 97 g. Recall the statement on page 14 that a submerged solid displaces a volume of water equal to its own volume. We don t know this volume of water directly, but we can use the mass of displaced water, 97 g, and its density, 0.9982 g/cm3, to calculate the volume of displaced water. The volume of the coal is equal to the volume of displaced water. Solve The mass of the chunk of coal is 156 g. If we use mwater to denote the mass of displaced water, then the volume of the displaced water is calculated as follows: mwater 156 g - 59 g V = = = 97 cm3 d 0.9982 g>cm3 The volume of the chunk of coal is the same as the volume of displaced water. Therefore, the density of the coal is 156 g d = = 1.6 g>cm3 97 cm3 Assess To determine the density of an object, we might think it is necessary to make measurements of both the mass and volume of the object. Example 1-3 shows that a volume measurement is not necessary. The steps in our calculation can be combined to give the following expression: (density of object)>(density of water) = (weight in water)>(weight in air - weight in water). The expression above clearly shows that the density of an object can be deter- mined by making two weight measurements: one in air, and the other in a fluid (such as water) of known density. PRACTICE EXAMPLE A: A graduated cylinder contains 33.8 mL of water. A stone with a mass of 28.4 g is placed in the cylinder and the water level rises to 44.1 mL. What is the density of the stone? PRACTICE EXAMPLE B: In the situation shown in the photograph, when the ice cube melts completely, will the water overflow the container, will the water level in the container drop, or will the water level remain unchanged? Explain.  1-5 Density and Percent Composition: Their Use in Problem Solving 17 Percent Composition as a Conversion Factor In Section 1-2, we described composition as an identifying characteristic of a sample of matter. A common way of referring to composition is through percent- ages. Percent (per centum) is the Latin for per (meaning for each ) and centum (meaning 100 ). Thus, percent is the number of parts of a constituent in 100 parts of the whole. To say that a seawater sample contains 3.5% sodium chloride by mass means that there are 3.5 g of sodium chloride in every 100 g of the sea- water. We make the statement in terms of grams because we are talking about percent by mass. We can express this percent by writing the following ratios: 3.5 g sodium chloride 100 g seawater and (1.3) 100 g seawater 3.5 g sodium chloride In Example 1-4, we will use one of these ratios as a conversion factor. EXAMPLE 1-4 Using Percent Composition as a Conversion Factor A 75 g sample of sodium chloride (table salt) is to be produced by evaporating to dryness a quantity of sea- water containing 3.5% sodium chloride by mass. What volume of seawater, in liters, must be taken for this pur- pose? Assume a density of 1.03 g/mL for seawater. Analyze The conversion pathway is g sodium chloride : g seawater : mL seawater : L seawater. To convert from g sodium chloride to g seawater, we need the conversion factor in expression (1.3), with g seawater in the numerator and g sodium chloride in the denominator. To convert from g seawater to mL of seawater, we use the reciprocal of the density of seawater as the conversion factor. To make the final conversion, from mL sea- water to L of seawater, we use the fact that 1 L = 1000 mL. Solve Following the conversion pathway described above, we obtain 100 g seawater ? L seawater = 75 g sodium chloride * 3.5 g sodium chloride 1 mL seawater 1 L seawater * * 1.03 g seawater 1000 mL seawater = 2.1 L seawater Assess In solving this problem, we set up a conversion pathway, and then we thought about the conversion factors that were required. We will make use of this approach throughout the text. PRACTICE EXAMPLE A: How many kilograms of ethanol are present in 25 L of a gasohol solution that is 90% gasoline to 10% ethanol by mass? The density of gasohol is 0.71 g>mL. PRACTICE EXAMPLE B: Common rubbing alcohol is a solution of 70.0% isopropyl alcohol by mass in water. If a 25.0 mL sample of rubbing alcohol contains 15.0 g of isopropyl alcohol, what is the density of the rubbing alcohol? 1-2 ARE YOU WONDERING... When to multiply and when to divide in doing problems with percentages? A common way of dealing with a percentage is to convert it to decimal form (3.5% becomes 0.035) and then to multiply or divide by this decimal, but students some- times can t decide which to do. Expressing percentage as a conversion factor and (continued) 18 Chapter 1 Matter: Its Properties and Measurement using it to produce a necessary cancellation of units gets around this difficulty. Also, remember that The quantity of a component must always be less than the quantity of the entire mixture. Component (Multiply by percentage.) The quantity of the entire mixture must always be greater than the MIXTURE quantity of any of the components. (Divide by percentage.) KEEP IN MIND If, in Example 1-4, we had not been careful about the cancellation of units and had multiplied by percentage (3.5>100) instead of dividing by it (100>3.5), we would that a numerical answer that have obtained the numerical answer 2.5 * 10-3. This would be a 2.5 mL sample defies common sense is of seawater, weighing about 2.5 g. Clearly, a sample of seawater that contains 75 g probably wrong. of sodium chloride must have a mass greater than 75 g. 1-6 Uncertainties in Scientific Measurements All measurements are subject to error. To some extent, measuring instru- ments have built-in, or inherent, errors, called systematic errors. (For exam- ple, a kitchen scale might consistently yield results that are 25 g too high or a thermometer a reading that is 2°C too low.) Limitations in an experi- menter s skill or ability to read a scientific instrument also lead to errors and give results that may be either too high or too low. Such errors are Random errors are observed called random errors. * by scatter in the data and can Precision refers to the degree of reproducibility of a measured quantity be dealt with effectively by that is, the closeness of agreement when the same quantity is measured taking the average of many several times. The precision of a series of measurements is high (or good) measurements. Systematic if each of a series of measurements deviates by only a small amount errors, conversely, are the bane from the average. Conversely, if there is wide deviation among the mea- of the experimental scientist. They are not readily apparent surements, the precision is poor (or low). Accuracy refers to how close and must be avoided by a measured value is to the accepted, or actual, value. High-precision carefully calibrating a measurements are not always accurate a large systematic error could method against a known be present. (A tight cluster of three darts near the edge of a dart board can sample or result. Systematic be considered precise but not very accurate if the intention was to strike the errors influence the accuracy center of the board.) Still, scientists generally strive for high precision in of a measurement, whereas measurements. random errors are linked To illustrate these ideas, consider measuring the mass of an object to the precision of by using the two balances shown on page 19. One of the balances is a single- measurements. pan balance that gives the mass in grams with only one decimal place. The other balance is a sophisticated analytical balance that gives the mass in grams with four decimal places. The accompanying table gives results obtained when the object is weighed three times on each balance. For the single-pan balance, the average of the measurements is 10.5 g, with measurements ranging from 10.4 g to 10.6 g. For the analytical balance, the average of the measurements is 10.4978 g, with measurements ranging from 10.4977 g to 10.4979 g. The scatter in the data obtained with the single- pan balance ( ; 0.1 g) is greater than that obtained with the analytical balance ( ; 0.0001 g). Thus, the results obtained by using the single-pan bal- ance have lower (or poorer) precision than those obtained by using the ana- lytical balance.  1-7 Significant Figures 19 Pan Balance Analytical Balance Three measurements 10.5, 10.4, 10.6 g 10.4978, 10.4979, 10.4977 g Their average 10.5 g 10.4978 g Reproducibility ; 0.1 g ; 0.0001 g Precision low or poor high or good 1-5 CONCEPT ASSESSMENT Can a set of measurements be precise without being accurate? Can the average of a set of measurements be accurate and the individual measurements be imprecise? Explain. 1-7 Significant Figures Consider these measurements made on a low-precision balance: 10.4, 10.2, and 10.3 g. The reported result is best expressed as their average, that is, 10.3 g. A scientist would interpret these results to mean that the first two digits 10 are known with certainty and the last digit 3 is uncertain because it was estimated. That is, the mass is known only to the nearest 0.1 g, a fact that we could also express by writing 10.3 ; 0.1 g. To a scientist, the measurement 10.3 g is said to have three significant figures. If this mass is reported in kilograms rather than in grams, 10.3 g = 0.0103 kg, the measurement is still expressed to three significant figures even though more than three digits are shown. When measured on an analytical balance, the corresponding reported value might be 10.3107 g a value with six significant figures. The number of significant figures in a measured quantity gives an indication of the capabilities of the measuring device and the precision of the measurements. We will frequently need to determine the number of significant figures in a numerical quantity. The rules for doing this, outlined in Figure 1-11, are as follows: All nonzero digits are significant. Zeros are also significant, but with two important exceptions for quantities less than one. Any zeros (1) preceding the decimal point, or (2) following the decimal point and preceding the first nonzero digit, are not significant. The case of terminal zeros that precede the decimal point in quantities greater than one is ambiguous. The quantity 7500 m is an example of an ambiguous case. 20 Chapter 1 Matter: Its Properties and Measurement Not significant: Not significant: Significant: zero for zeros used only all zeros between cosmetic to locate the nonzero numbers FIGURE 1-11 * purpose decimal point Determining the number of significant figures in a quantity 0. 0 0 4 0 0 4 5 0 0 The quantity shown here, 0.004004500, has seven Significant: Significant: significant figures. All nonzero all nonzero zeros at the end of digits are significant, as are integers a number to the right the indicated zeros. of decimal point Do we mean 7500 m, measured to the nearest meter? Nearest 10 meters? If all the zeros are significant if the value has four significant figures we can write 7500. m. That is, by writing a decimal point that is not otherwise needed, we show that all zeros preceding the decimal point are significant. This tech- nique does not help if only one of the zeros, or if neither zero, is significant. The best approach here is to use exponential notation. (Review Appendix A if necessary.) The coefficient establishes the number of significant figures, and the power of ten locates the decimal point. 2 significant figures 3 significant figures 4 significant figures 3 3 7.5 * 10 m 7.50 * 10 m 7.500 * 103 m Significant Figures in Numerical Calculations Precision must neither be gained nor be lost in calculations involving mea- sured quantities. There are several methods for determining how precisely to express the result of a calculation, but it is usually sufficient just to observe A more exact rule on some simple rules involving significant figures. * multiplication/division is that the result should have about the same relative The result of multiplication or division may contain only as many error for example, significant figures as the least precisely known quantity in the expressed as parts per calculation. hundred (percent) or parts per thousand as the least precisely known quantity. Usually the significant figure In the following chain multiplication to determine the volume of a rectan- rule conforms to this require- gular block of wood, we should round off the result to three significant figures. ment; occasionally, it does not Figure 1-12 may help you to understand this. (see Exercise 67). 14.79 cm * 12.11 cm * 5.05 cm = 904 cm3 (4 sig. fig.) (4 sig. fig.) (3 sig. fig.) (3 sig. fig.) In adding and subtracting numbers, the applicable rule is as follows: In addition and subtraction * The result of addition or subtraction must be expressed with the same the absolute error in the result number of digits beyond the decimal point as the quantity carrying the can be no less than the smallest number of such digits. absolute error in the least pre- cisely known quantity. In the summation at the right, the absolute error in one quantity Consider the following sum of masses. is ;0.1 g; in another, ; 0.01 g; 15.02 g and in the third, ; 0.001 g. The sum must be expressed 9986.0 g with an absolute error of 3.518 g ; 0.1 g. 10,004.5 3 8 g  1-7 Significant Figures 21 FIGURE 1-12 Significant-figure rule in multiplication In forming the product 14.79 cm * 12.11 cm * 5.05 cm, the least precisely known quantity is 5.05 cm. Shown on the calculators are the products of 14.79 and 12.11 with 5.04, 5.05, and 5.06, respectively. In the three results, only the first two digits, 90 Á , are identical; variations begin in the third digit. We are certainly not justified in carrying digits beyond the third. We express the volume as 904 cm3. Usually, instead of a detailed analysis of the type done here, we can use a simpler idea: The result of a multiplication may contain only as many significant figures as does the least precisely known quantity. The sum has the same uncertainty, ; 0.1 g, as does the term with the smallest number of digits beyond the decimal point, 9986.0 g. Note that this calculation Later in the text, we will is not limited by significant figures. In fact, the sum has more significant figures need to apply ideas about (six) than do any of the terms in the addition. significant figures to There are two situations when a quantity appearing in a calculation may be logarithms. This concept is exact, that is, not subject to errors in measurement. This may occur discussed in Appendix A. by definition (such as 1 min = 60 s, or 1 in. = 2.54 cm) as a result of counting (such as six faces on a cube, or two hydrogen atoms in a water molecule) Exact numbers can be considered to have an unlimited number of significant figures. As added practice in 1-6 CONCEPT ASSESSMENT working with significant figures, review the calcula- Which of the following is a more precise statement of the length 1 inch: tions in Section 1-6. You will 1 in. = 2.54 cm or 1 m = 39.37 in.? Explain. note that they conform to the significant figure rules presented here. Rounding Off Numerical Results To three significant figures, we should express 15.453 as 15.5 and 14,775 as Some people prefer the 1.48 * 104. If we need to drop just one digit, that is, to round off a number, the round 5 to even rule. Thus rule that we will follow is to increase the final digit by one unit if the digit 15.55 rounds to 15.6, and dropped is 5, 6, 7, 8, or 9 and to leave the final digit unchanged if the digit 17.65 rounds to 17.6. In dropped is 0, 1, 2, 3, or 4.* To three significant figures, 15.44 rounds off to 15.4, banking and with large data sets, rounding needs to be and 15.45 rounds off to 15.5. unbiased. With a small number of data, this is less *C. J. Guare, J. Chem. Educ., 68, 818 (1991). important. 22 Chapter 1 Matter: Its Properties and Measurement EXAMPLE 1-5 Applying Significant Figure Rules: Multiplication/Division Express the result of the following calculation with the correct number of significant figures. 0.225 * 0.0035 = ? 2.16 * 10-2 Analyze By inspecting the three quantities, we see that the least precisely known quantity, 0.0035, has two significant figures. Our result must also contain only two significant figures. Solve When we carry out the calculation above by using an electronic calculator, the result is displayed as 0.0364583. In our analysis of this problem, we determined that the result must be rounded off to two significant figures, and so the result is properly expressed as 0.036 or as 3.6 * 10 - 2. Assess tion by using exponential numbers. The answer should be 12 * 10 - 1214 * 10 - 32>12 * 10 - 22 L 4 * 10 - 2, and To check for any possible calculation error, we can estimate the correct answer through a quick mental calcula- it is. Expressing numbers in exponential notation can often help us quickly estimate what the result of a calcu- lation should be. PRACTICE EXAMPLE A: Perform the following calculation, and express the result with the appropriate number of significant figures. 62.356 = ? 0.000456 * 6.422 * 103 PRACTICE EXAMPLE B: Perform the following calculation, and express the result with the appropriate number of significant figures. 8.21 * 104 * 1.3 * 10-3 = ? 0.00236 * 4.071 * 10-2 EXAMPLE 1-6 Applying Significant Figure Rules: Addition/Subtraction 12.06 * 1022 + 11.32 * 1042 - 11.26 * 1032 = ? Express the result of the following calculation with the correct number of significant figures. Analyze If the calculation is